T. Nombebe J.S. Allison L. Santana and I.J.H. Visagie

There are relatively few goodness-of-fit tests specifically developed for the Pareto distribution when compared to other well-known distributions like the normal or exponential distributions. This is the case even though there are a host of practical applications where it would be required to first check the assumption that the data were realised from a Pareto distribution. We propose and investigate new goodness-of-fit tests for the Pareto Type I distribution based on a specific conditional expectation that characterises the Pareto distribution. Currently, the literature contains no other tests for the Pareto distribution based on conditional expectation. We conduct a thorough Monte Carlo power study in order to assess the finite sample performance of the newly developed tests using various estimation methods. The results from the simulation study show that the newly proposed tests are competitive in terms of power performance when compared to some existing tests. It also shows that the majority of tests produce their highest powers when the unknown shape parameter is estimated by the method of moments. A practical example, where we consider the annual salaries of English Premier League football players for two consecutive seasons, is also included to illustrate the use of the newly proposed tests. We find that the salaries in the 2021–2022 season can be adequately modelled with the Pareto distribution, but not the salaries for the 2022–2023 season.

]]>Koushika Dhevi S and Sangeetha S

Functional equations are important and exciting concepts in mathematics. They make it possible to investigate fundamental algebraic operations and create fascinating solutions. The concept of functional equations develops further creative methods and techniques for resolving issues in information theory, finance, geometry, wireless sensor networks, and other domains. These include geometry, algebra, analysis, and so on. In recent decades, several writers in many domains have covered the study of various types of stability. Many authors have studied the stability of various functional equations in great detail, with the traditional case (Archimedean) revealing more fascinating results. Recently, researchers have used NANS to study the equivalent conclusions of stability problems from various functional equations. In this research, we examine the Hyers-Ulam stability of the hexic-quadraticadditive mixed-type functional equation where is fixed such that and in NANS and also provided some suitable counterexamples.

]]>C. Rajapandiyan and V. Visalakshi

This paper explores the homogeneous spaces and induced transformation groups of S-topological transformation group. S-topological transformation group is a structure constructed by concatenating a topological group with a topological space through a semi totally continuous action. It is shown that any map from a topological group to the quotient group of a finite Hausdorff topological group by the isotropy group is surjective, continuous, open and it has been proven that any map from the quotient group of a finite Hausdorff topological group by the isotropy group to the homogenous space is both H-isomorphism and semi totally continuous. Furthermore, an equivariant map has been established between homogeneous spaces and discussed the partial order relation on the family of all Hausdorff homogeneous spaces for a compact Hausdorff topological group. Subsequently, an induced S-topological transformation group is constructed by an induced H-action. For any compact subgroup K of a topological group H, it is verified that any map from the topological spcae Y to the orbit space of K-action is continuous and a K-map. For any H-space, K-map and an induced S-topological transformation group; it is proved that there is a unique semi totally continuous H-map. Additionally, it is shown that for a topological group, a subgroup K of topological group and a K-space, there is a unique H-space and a unique injective K-map and also it is established that for a H-space and a semi totally continuous K-map, there exists a unique semi totally continuous H-map. Finally, it is demonstrated that for a finite Hausdorff topological group, finite Frechet space and a M-space, any map from the orbit space of M-action to is semi totally continuous, for the subgroups M and N of topological group.

]]>Sunil B. Thakare Archana Bhange and H. R. Bhapkar

In graph theory, the notion of graph coloring plays an important role and has several applications in the fields of science and engineering. Since the concept of map coloring was first proposed, many researchers have invented a wide range of graph coloring techniques, among which are vertex coloring, edge coloring, total coloring, perfect coloring, list coloring, acyclic coloring, strong coloring, radio coloring, and rank coloring; these are some of the important graph coloring methods that color the graph's vertices, edges, and regions with certain conditions. One of the coloring method is Incident Vertex PI coloring. This is a function of coloring from the set of pairs of incident vertices of every edge of a graph to the power set of colors. This method ensures that all vertices are properly colored, with an additional condition that ordered pair vertices for all edges of graph receive distinct colors. Many types of graphs are defined in the graph theory. In this paper, we have discussed the Incident Vertex PI Coloring numbers for the class of graph families, Fan graph, Book graph, Gear graph, Windmill graph, Dutch Windmill graph and Crown graph.

]]>Rafi Raza Ahmad Termimi Ab Ghani∗ and Lazim Abdullah

The hesitant fuzzy set (HFS) is an innovative approach to decision-making under uncertainty. This study addresses the aggregated operation of the HFS decision matrix. The introduction of induced VIKOR procedures, various extensions of HFSs aggregation operator, and essential approaches for multi-criteria decision-making (MCDM) are presented. This technique uses the aggregation operator, HFWA operator, to rank alternatives and identify the compromise solution that comes closest to the ideal solution. We developed the hesitant fuzzy weight averaging VIKOR (HFWA-VIKOR) model as a novel technique to achieve this. By combining the hesitant fuzzy elements, the HFWA aggregation operator creates aggregated values that are expressed as a single value. The primary advantage of the HFWA-VIKOR model lies in its initial step of aggregating the hesitant fuzzy element. This results in an initial hesitant fuzzy decision matrix, which provides much more detailed information for decision-making and, through the use of the inducing HFWA operator, represents the complex attitudinal nature of the decision-makers. The multi-criteria location selection problem is then solved using the combined HFWA-VIKOR technique, and the outcomes are presented in an easy-to-understand way owing to aggregation operators. A numerical example is also applied in this new method which gives out the best alternative result. As per the scope of our research work, MCDM under hesitant fuzzy sets with HFWA-VIKOR method have been used and their result revealed the best alternative is to find out. These results indicate good potential for objectives. This technique may also be used for other studies or applications. Further research in this area may provide a more developed technique for this application.

]]>Ahmed J. Sabali Saad A. Manaa and Fadhil H. Easif

Solving nonlinear partial differential equations (PDEs) is crucial in various scientific and engineering domains. The Adomian Decomposition Method (ADM) has emerged as a promising technique for tackling such problems. However, its effectiveness diminishes over extended time intervals due to divergence issues. This limitation hampers its practical applicability in real-world scenarios where stable and accurate numerical solutions are essential. To address the divergence problem associated with ADM, this research explores the combination of the Adomian Decomposition Method (ADM) with the Padé technique – a method known for its accuracy and efficiency. This combination's purpose is to mitigate ADM's shortcomings, particularly when dealing with extended time intervals. Experimental analysis was conducted across varying time intervals to compare the performance of the combined technique with traditional ADM. Mathematica software was used to obtain all calculations, including the creation of tables and figures. Results from the experiments demonstrate the superiority of the combined technique in producing accurate results regardless of the time interval used. Furthermore, the combined method improves accuracy and ensures result stability over long time intervals, creating new possibilities for its use in scientific and engineering fields. This research contributes to the field by offering a solution to the divergence issue associated with ADM, thereby enhancing its applicability in solving nonlinear PDEs. While acknowledging limitations such as reliance on numerical simulations, the study highlights the practical implications of its findings, including more accurate predictions and modeling in complex systems, with potential social implications in decision-making and problem-solving contexts.

]]>Prashant Patel and Rahul Shukla

Fixed points (FP) of infinite families of nonexpansive mappings find diverse applications across various disciplines. In economics, they help to find stable prices and quantities in markets. In game theory, fixed points help to find Nash equilibria. In computer science, fixed points are used to understand program meanings and help in making better algorithms for tasks like data analysis, checking models, and improving compilers. Solutions to equilibrium problems have practical uses in various areas. For instance, in physics, these solutions assist in analyzing systems at rest or in motion. In engineering, they aid in designing structures that can withstand forces without collapsing, ensuring safety and stability in construction projects. The main aim of the article is to present the concept of generalized modification of the system of equilibrium problems (GMSEP) for an infinite family of nonexpansive mappings. In this paper, we study viscosity approximation methods and present a new algorithm to find a common element of the fixed point of an infinite family of nonexpansive mappings and the set of solutions of generalized modification of the system of equilibrium problem in the setting of Hilbert spaces. Under some conditions, we prove that the sequence generated by the algorithm converges strongly to this common solution.

]]>S. Sripriya and A. Anuradha

Energy of the graph G is the sum of absolute values of eigenvalues of its adjacency matrix. Given a simple connected graph G, its first (second) Zagreb matrix is constructed by including the sum (product) of the degrees of each pair of adjacent vertices of G. Computation of sum of absolute eigen values of these matrices yields the corresponding Zagreb energies. In this paper, the first and second Zagreb energies of certain families of graphs have been computed and a criterion to discern the nature of graph G based on their energies is obtained. The paper focuses on the comparative analysis of first and second Zagreb energies in terms of regular graphs such as cycle graphs, bipartite and tripartite graphs. Our findings reveal that the second Zagreb energy is always greater than first Zagreb energy for all complete bipartite graphs of even order greater than or equal to 4. Also we have established that the same is the case for complete tripartite graphs too. Furthermore, we illustrate that the two Zagreb energies coincide exclusively for the complete bipartite graph with equal partite sets if and only if the graph is of order 2. Additionally, we provide a criterion leading to an infinite set of non-isomorphic Zagreb equi-energetic graphs for all r>1 within partite graphs. The computations of two Zagreb energies for graph operations like t-splitting graph and t-shadow graph are also illustrated. The first and second Zagreb energies for some specific graphs along with bounds on Zagreb energies for wheel graphs are also discussed.

]]>S. Priyadharshini V. Sadhasivam and K. K. Viswanathan

The main objective of this study is to investigate some new oscillation criteria for Euler-Bernoulli beam equations with damping term by using the integral average method and Riccati technique. Philo introduces the following new integral operator, which is the main tool in this paper. Our plan of action is to reduce the multidimensional problems to ordinary differential problem by using Jenson's inequality, assuming the assumptions and integration by parts with boundary conditions. With hinged, sliding and hinged-sliding end boundary conditions, several new sufficient conditions are established. The results improve and generalize those given in some previous papers, which can be seen by the examples given at the end of this paper. The majority of engineering constructions, ships, support buildings, airplanes, and rotor blades all use beams as structural elements. It is presumed that these elements are only subjected to static loads; yet, dynamic loads induce vibrations, which affect the stress and strain values. These mechanical phenomena also result in noise, instability, and the potential for resonance, which enhances deflections and failure. We analyze the spatial force load the equations of a damped Euler-Bernoulli beam derived from the equation for the velocity or final time displacement that we measured. Usually, internal damping determines the nature of this term.

]]>Prem Antony J and Edwin Prabakaran T

Spatial econometrics is pivotal in understanding spatial dependencies across diverse fields like urban economics, environmental economics, and disease spread. This study highlights the significance of spatial grouping for data management and pattern detection, particularly in epidemiological analysis and policy planning. The Spatial Autoregressive random effect (SAR-RE) model is a classical model for analysing datasets with repeated observations across units over time, particularly when these units are situated in a spatial context. The mixture effect models account for the presence of different sub-groups within the overall population, each of which has a unique response pattern. In this paper, the proposed methodology integrates the SAR-RE model into a mixture framework, allowing for the consideration of diverse spatial patterns and class-specific coefficients. By incorporating class-specific coefficients, the model accommodates heterogeneous spatial structures within the data, providing a more nuanced understanding of spatial dependencies. The Spatial autoregressive model along with the assumption that the random effect follows a mixture of Gaussian distributions is developed to analyse panel data with spatial dependency and unobserved heterogeneity. The parameters of the model are estimated using the Limited-Memory BFGS (L-BFGS) quasi-Newton method-based EM algorithm for good convergence of the estimated. The classification of subjects into different latent classes is carried out based on their posterior probabilities. The model is applied to state-wise COVID-19 confirmed rates, revealing insightful patterns. The analysis employs the estimated model for the interpretation and comprehensive understanding of spatially dependent panel data with unobserved heterogeneity. The results of the empirical study show that the proposed model outperforms the existing model based on performance metrics criteria.

]]>Hayfa Abdul Jawad Saieed Mhasen Saleh Altalib Safwan Nathem Rashed and Manaf Hazim Ahmed

In multivariate data modeling, the statistical analyst can desire to construct a multivariate distribution with correlated variables. For this reason, there is a need to generalize univariate distributions, but this generalization is not easy. Many methods have been presented for construction of continuous multivariate families with univariate distributions. Some of these methods are based on a single baseline, while others are based on more than one baseline, so that their variables are dependent. Some authors were interested in expanding a univariate transmuted family into multivariate case. Some suggestions were made about extension of univariate quadratic transmuted (QT) family to bivariate ones, and another modification was made to this family by replacing the (c.d.f.) with exponentiated (c.d.f.). Another construction of bivariate family is based on probability distribution of paired order Statistics for a sample size two drawn from quadratic ranked transmuted (QRT) margin, and this bivariate family allows for positive and negative dependence between variables. Another family proposed an extension of univariate mixture of standard continuous uniform, with decreasing densities to a bivariate case. Our proposed (CT_{2}) reduces to a bivariate quadratic transmuted (QT_{2}) family if the cubic transmutation parameters equal to zero. (CT_{2}) family can be used for modeling positive and negative correlated variables. Some statistical properties of (CT_{2}) family have been studied which comprise joint, marginal and conditional (c.d.f., p.d.f), joint, marginal and conditional moments, data generation and dependence coefficients. It is seen that (joint, marginal and conditional) moments depend on raw moments of (baseline variables and largest order statistics of samples sizes 2 and 3). The Egyptian bivariate economic data are fitted by (CT_{2}Fr), (FGMFr), (T_{2}Fr) and (DSASFr). The (CT_{2}Fr) is the fit to which has smallest (AIC) and (BIC) criteria.

Jebisha Esther S and Veninstine Vivik J

In the fields of combinatorics and graph theory, prism graphs are very important. They provide insights into the structural features of many real-world networks and act as a model for them. In graph theory, the study of dominant sets is essential for a variety of applications, including social network research and network design. A dominating set in a graph G is a subset D of vertices V having the property that each vertex w belongs to V − D is neighbouring to at least one vertex D. Determining the minimum cardinal number of dominating sets, locating dominating sets, and local locating dominating sets is of critical importance in such fields as network design and social network analysis. In this paper, we determine these minimum cardinal bounds for families of prism graphs. The study adds to the basic understanding of graph theory by methodically disentangling the intricate relationships between dominating sets in prism graphs. The exploration of lowest cardinal value of locating dominating sets yields solutions to optimisation issues in network design. In this work, we determine the upper bounds of locating domination and local locating domination for prsim, antiprism, crossed prism and circulant ladder prism graph.

]]>Omdutt Sharma Surender Kumar Naveen Kumar and Pratiksha Tiwari

Rolle's Theorem (RT) and Lagrange's Mean-value Theorem (LMVT) are significant for pure and applied mathematics, and they have applications in various other fields such as management, physics etc. RT is significant in finding the projectile trajectory's maximum height and in information theory, and the entropy function (measure) is used to measure the uncertainty of information. RT is used to analyze the graphs of annual performance in any field. Since information is necessary to analyze any performance and in information theory, entropy measure is a significant tool to quantize the uncertainty so by using the concept of RT and LMVT in information theory the uncertainty and vagueness or noise can be minimized or maximized. In this manuscript, the concept of differential calculus, i.e., RT and LMVT are used for validation of the entropy function. In this paper, characteristics of differential calculus in information entropy function have been discussed. It has been shown that the entropy function satisfies RT and LMVT. It also describes the conditions when Rolle's Theorem becomes the necessary and sufficient condition for entropy function. Theorems are proved related to the concept of differential calculus in information theory which shows that by using the existing entropy function some new entropies can be derived.

]]>Maheshwar Pathak Rachna Bhatia Pratibha Joshi and Ramesh Chand Mittal

Newell-Whitehead-Segel (NWS) type equations arise in solid-state physics, optics, dispersion, convection system, mathematical biology, quantum mechanics, plasma physics and oil pollution in ocean environment. Extensive applications of such type of equations draw attention of scientists toward their numerical solutions. In this work, we propose fourth order numerical method based on cubic B-spline functions for the numerical solutions of nonlinear NWS type equations. The Crank Nicolson finite difference scheme is used to discretize the equation and quasi-linearization is use to linearize the nonlinear term. As a result, we get a system of linear equation, which we solve using Gauss elimination method. Stability analysis has been carried out by a thorough Fourier series analysis and stability conditions have been obtained. The scheme has been applied to five numerical problems having quadratic, cubic and forth order nonlinear terms. The effectiveness and robustness of the proposed technique have been demonstrated by comparing the obtained numerical results with the exact solutions and numerical results obtained by other existing methods. A comparison of the numerical results obtained using the proposed technique with exact solutions shows excellent agreement. Graphs of numerical solutions have been drawn at different times and also compared with the graphs of the exact solutions. The comparative analysis shows that the proposed scheme outperformed other methods in terms of accuracy and produced good results.

]]>Patchanok Srisuradetchai and Wararit Panichkitkosolkul

This paper presents four types of confidence intervals (CIs) for parameter estimation of the Juchez distribution, a robust model in the domain of lifetime data analysis. The likelihood-based, Wald-type, bootstrap-t, andbias-corrected and accelerated (BCa) bootstrap confidence intervals are proposed and evaluated through simulation studies and application to real datasets. The effectiveness of these methods is assessed in terms of the empirical coverage probability (CP) and average length (AL) of the confidence intervals, providing an understanding of their performance under various conditions. Additionally, we derive the Wald-type CI formula in explicit form, making it readily calculable. The results show that when the sample size is small, such as 10, 20, or 30, the bootstrap-t and BCa bootstrap methods produce CPs less than 0.95. However, as sample sizes increase, the CPs of all methods tend to converge towards the nominal level of 0.95. The parameter values also affect the CP. At low values of the parameter, the CPs are quite close to the ideal, with both the Wald-type and likelihood-based methods achieving a CP of approximately 0.95. However, at higher parameter values with small sample sizes, the CPs for the bootstrap-t and BCa bootstrap methods tend to have lower coverage.

]]>Gagandeep Kaur and Sarbjit Singh Brar

This research introduces the Partial Product- Exponential Method of Estimation, focusing on utilizing partial auxiliary information for estimating population mean in simple random sampling without replacement. The method proposes novel estimators tailored for situations where only partial auxiliary information is available, particularly when it demonstrates a negative correlation with the study variable within sub-populations. The paper evaluates the performance of the suggested method under two cases: when sub-population weights are known and when they are unknown. Approximate expressions for bias and variance, up to the first order, are derived for the suggested estimators. A comprehensive comparative analysis concludes that the proposed estimators are more efficient than existing estimators, such as mean per unit estimator, partial product estimator, and weighted post-stratified estimator, under specific conditions. Particularly, the proposed estimators outperform the corresponding existing methods when certain conditions are true, demonstrating superiority in both known and unknown weight cases. Furthermore, a simulation study using R software validates the theoretical findings for normal and non-normal populations. The study showcases the practical utility of the proposed estimators, emphasizing their superiority over existing counterparts in real-world applications. Particularly, the proposed estimators are increased accuracy and efficiency in estimating the population mean, enhancing the reliability of sample survey results. In summary, the Partial Product-Exponential Method of Estimation presents a valuable addition to the domain of sample survey methodology, addressing the challenge of partial auxiliary information. The suggested methods demonstrated advantages in efficiency and accuracy, and highlights its potential for practical applications, promising enhanced estimation accuracy in various cases of sample survey.

]]>Nechirvan B. Ibrahim and Alias B. Khalaf

The primary aim of this paper is to establish and analyze certain topological structures linked with a specified graph . In a graph , a vertex u is considered a neighbor of another vertex v if there exists an edge uv in . Furthermore, we define two vertices (or edges) in as coneighbors if they share identical sets of neighboring vertices (or edges). The topology under consideration arises from the collections of vertices that are coneighbor and the collections of edges that are coneighbor within the graph. It is proved that the coneighbor topology of every non-coneighbor graph is homeomorphic to the included point topology while this space is quasi-discrete if and only if the graph contains at least one coneighbor set of vertices and some examples of coneighbor topologies of special graphs are presented to be quasi-discrete spaces such as (a path, a cycle and a bipartite) graphs. Moreover, several topological properties of the coneighbor space are presented. We proved that the coneighbor topological space associated with a graph always has dimension one and satisfies the T_{1/2} axiom. Also, the family of θ-open sets is determined in this spaces and it is proved that this space is almost compact whenever the family of coneighbor sets is finite. Finally, we looked at some graphs in which the coneighbor space fulfills other topological concepts such as connectedness, compactness and countable compactness.

R. Nishanthini and R. Jeyabalan

A vertex N-magic total labeling is a bijective function that maps the vertices and edges of a graph G onto the successive integers from 1 to p + q. The labeling exhibits two distinct properties: First, the count of unique magic constants k_{i} for i belonging to the set {1, 2, ...,N} is equivalent to the cardinality of N; secondly, the magic constants ki must be arranged in a strictly ascending order. In the present context, the constant N is employed to represent different degrees of vertices. The term “magic constant values k_{i}” for i ∈ {1, 2, ...,N} refers to specific numbers that exhibit unique and interesting properties and are employed in the context of this investigation. By adding up the weights of each vertex in V (G), we might receive a magical constant number k_{i} for i ∈ {1, 2, ...,N}. Within the scope of this study, we discuss the sharp bounds of vertex N-magic total labeling graphs. In terms of magic constants k_{i} for i ∈ {1, 2, ...,N}, we also found the requirement for vertex N-magic total labeling of trees. We investigated the potential for vertex N-magic total labeling at vertices in graphs with varying vertex degrees.

Ram Dayal A. Parthiban and P. Selvaraju

Graph labeling is an allocation of labels (mostly integers) to the nodes/lines or both of a graph G_{α} subject to a few conditions. The field of graph theory, specifically graph labeling, plays a vital role in various fields. To name a few, graph labeling is utilized in coding, x−ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management. It can also be applied to network security, network addressing, channel assignment process, and social networks. A graph G_{β} is a prime distance graph (PDG) if its nodes can be assigned with distinct integers such that for any two adjacent nodes, the positive difference of their labels is a prime number. A complete characterization of prime distance graphs is an open problem of high interest. This paper contributes partially towards the same. More specifically, Laison et al. raised the following questions. (1) Is there a family of graphs which are PDGs if and only if Goldbach’s Conjecture is true? (2) What other families of graphs are PDGs? In this paper, these questions are answered partially and also show certain families of graphs that admit prime distance labeling (PDL) if and only if the Twin Prime Conjecture holds, besides establishing PDL of some special graphs.

Hafed. H. Saleh Amirah Azmi and Ali. F. Jameel

In order to obtain sufficient solutions for fuzzy differential equations (FDEs), reliable and efficient approximation methods are necessary. Approximate numerical methods can not directly solve fuzzy HIV models. Meanwhile, the approximate analytical methods can potentially provide more straightforward solutions without the need for extensive numerical computations or linearization and discretization techniques, which may be challenging to apply to fuzzy models. One significant advantage of approximate analytical methods is their ability to provide insights into solution accuracy without requiring an exact solution for comparison, where an exact solution may not be readily available. In this work, the fuzzy nonlinear HIV infection model is analyzed and solved using the new fuzzy form of an approximate analytical method. Fuzzy set theory mixed with standard fuzzy variational iteration method (FVIM) properties is utilized to produce a new formulation denoted by the multistage fuzzy variational iteration method (MFVIM) to process and solve a fuzzy nonlinear HIV infection model. MFVIM offers an effective method for attaining convergence in the series solution presented as a polynomial function. This approach enables efficient solutions to diverse mathematical challenges. The solution methodology is reliant on fuzzy differential equations conversion into systems of ordinary differential equations, utilizing the parametric form regarding its r-level representations, and considering the approximate solution of the system in a sequence of intervals. Subsequently, the equivalent classical systems are resolved by applying FVIM algorithms in each subinterval. Also, the existence and unique solution analysis of the proposed problem have been described, along with a fuzzy optimal control analysis. A tabular and graphical representation of the MFVIM of the proposed models is presented and analyzed in comparison with the numerical method and FVIM. The new method produces better performance in terms of solutions than a numerical method with a simple implementation for solving fuzzy nonlinear HIV infection model associated with FIVPs. The ability to better comprehend the behavior of the system under investigation can enable researchers and scientists to work on models incorporating systems with long memories and ill-defined notions to make more effective design and decision-making.

]]>B. I. Andrew and A. Anuradha

Balanced multi-star graph is a specialized type of graph formed by connecting apex vertices of star graphs to create a cohesive structure known as a clique. These graphs comprise r star graphs, where each star graph has an apex vertex connected to n pendant vertices. Balanced multistar graphs offer benefits in scenarios requiring equal distances between peripheral nodes, such as sensor networks, distributed computing, traffic engineering, telecommunications, supply chain management, and power distribution. The integral graph spectrum derived from the adjacency matrix of balanced multistar graphs holds significance across various domains. It aids in network analysis to understand connectivity patterns, facilitates efficient computation of structural properties through graph algorithms, and enables graph partitioning and community detection. Spectral graph theory assists in identifying connectivity patterns in network visualization, supports modeling biological networks in biomedical research, aids in generating personalized recommendations in recommendation systems and contributes to graph-based segmentation and scene analysis tasks in image processing. This paper aims to characterize the integral graph spectrum of balanced multi-star graphs by focusing on spectral parameters of double-star graphs (r=2), triple-star graphs (r=3), and quadruple-star graphs (r=4). This spectrum serves as an important tool across disciplines, providing insights into graph structure and facilitating tasks ranging from network analysis to computational biology and image processing.

]]>Inessa I. Pavlyuk and Sergey V. Sudoplatov

A series of basic characteristics of structures and of elementary theories reflects their complexity and richness. Among these characteristics, four kinds of degrees of rigidity and the index of rigidity are considered as measures of how far the given structure is situated from rigid one, both with respect to the automorphism group and to the definable closure, for some or any subset of the universe, which has the given finite cardinality. Thus, a natural question arises on a classification of model-theoretic objects with respect to rigidity characteristics. We apply a general approach of studying the rigidity values and related classification to abelian groups and their theories. We describe possibilities of degrees and indexes of rigidity for finite abelian groups and for standard infinite abelian groups. This description is based both on general consideration of rigidity, on its application for finite structures, and on their specificity for abelian groups including Szmielew invariants, combinatorial formulas for cardinalities of orbits, links with dimensions, and on their combinations. It shows how characteristics of infinite abelian groups are related to them with respect to finite ones. Some applications for non-standard abelian groups are discussed.

]]>B. Surender Reddy S. Vijayabalaji N. Thillaigovindan and K. Punniyamoorthy

This new study helps us understand 2-multiplicative or product metric spaces and normed linear spaces (NDLS) better than before, going beyond what we already know. Seeing a gap in existing research, our main aim is to thoroughly explore the natural properties of 2-multiplicative NDLS. Using a careful approach that looks at continuity, compactness, and convergence properties, our research finds results that point out the special features of these spaces and show the connections between their algebraic and topological sides. The importance of our findings goes beyond just theory, affecting practical uses and encouraging collaboration across different fields. Our research builds a strong base in mathematical analysis, giving useful insights for making nuanced decisions. Acknowledging some limitations in our study opens the door for future improvements, creating promising paths for further exploration. In real-world terms, what we learn from this thorough study not only informs but also changes how we make decisions in mathematical analysis. In research community, our work makes people appreciate the connection between algebraic and topological spaces more deeply, sparking curiosity and inspiring future research. In essence, this research acts as a guiding light, showcasing the unique features of 2-multiplicative NDLS and paving the way for a deeper understanding of mathematical structures and their flexible uses in both theory and practice. Furthermore, our exploration motivates future researchers to dive into the details of 2-multiplicative NDLS, expanding their knowledge and looking into broader implications in the field of mathematical analysis.

]]>Zuhair A. Al-Hemyari Alaa Khlaif Jiheel and Iman Jalil Atewi

For the purpose of modelling the Reliability of Burr XII Distribution, a family of shrinkage estimators is proposed for any parameter of any distribution when a prior guess value of is available from the past. In addition, two sub-models of the shrinkage type estimators for estimating the reliability and parameters of the Burr XII Distribution using two types of shrinkage weight functions with the preliminary test of the hypothesis against the alternative have been proposed and studied. The criteria for studying the properties of two sub-models of the reliability estimators which are the Bias, Bias ratio, Mean Squared Error and Relative Efficiency were derived and computed numerically for each sub-model for the purpose of studying the behavior of the estimators for the Burr XII Distribution because they are complicated and contain many complex functions. The numerical results showed the usefulness of the proposed two sub-models of the reliability estimators of Burr XII Distribution relative to the classical estimators for both of the shrinkage functions when the value of the a priori guess value is close to the true value of . In addition, the comparison between the proposed two sub-models of the shrinkag

]]>T. M. Velammal A. Nagarajan and K. Palani

Domination plays a very important role in graph theory. It has a lot of applications in various fields like communication, social science, engineering, etc. Let be a simple graph. A function is said to be a product signed dominating function if each vertex in satisfies the condition where denotes the closed neighborhood of . The weight of a function is defined as . The product signed domination number of a graph is the minimum positive weight of a product signed dominating function and is denoted as . Product signed dominating function assigns 1 or -1 to the nodes of the graph. This variation of dominating function has applications in social networks of people or organizations. Probabilistic Neural Network (PNN) was first proposed by Specht. This is a classifier that maps input patterns in a number of class levels and estimates the probability of a sample being part of learned theory. This paper studies the existence of product signed dominating functions in probabilistic neural networks and calculates the accurate values of product signed domination numbers of three layered and four layered probabilistic neural networks.

]]>Suparman Eviana Hikamudin Hery Suharna Aryanti In Hi Abdullah and Rina Heryani

Polynomial regression (PR) is a stochastic model that has been widely used in forecasting in various fields. Stationary stochastic models play a very important role in forecasting. Generally, PR model parameter estimation methods have been developed for non-stationary PR models. This article aims to develop an algorithm to estimate the parameters of a stationary polynomial regression (SPR) model. The SPR model parameters are estimated using the Bayesian method. The Bayes estimator cannot be determined analytically because the posterior distribution for the SPR model parameters has a complex structure. The complexity of the posterior distribution is caused by the SPR model parameters which have a variable dimensional space. Therefore, this article uses the reversible jump MCMC algorithm which is suitable for estimating the parameters of variable-dimensional models. Applying the reversible jump MCMC algorithm to big data requires many iterations. To reduce the number of iterations, the reversible jump MCMC algorithm is combined with the Bootstrap algorithm via the resampling method. The performance of the Bootstrap-reversible jump MCMC algorithm is validated using 2 simulated data sets. These findings show that the Bootstrap-reversible jump MCMC algorithm can estimate the SPR model parameters well. These findings contribute to the development of SPR models and SPR model parameter estimation methods. In addition, these findings contribute to big data modeling. Further research can be done by replacing Gaussian noise in SPR with non-Gaussian noise.

]]>Sonali Priyadarsini Ajay Vikram Singh and Said Broumi

The neutrosophic soft set is one of the most significant mathematical approaches for uncertainty description, and it has a multitude of practical applications in the realm of decision making. On the other hand, the decision-making process is often made more difficult and complex since these situations contain criteria that are less significant and more redundant. In neutrosophic soft set-based decision-making problems, parameter reduction is an efficient method for cutting down on redundant and superfluous factors, and it does so without damaging the decision-makers' ability to make decisions. In this work, a parametric reduction strategy has been proposed. This approach lessens the difficulties associated with decision making while maintaining the existing order of available options. Because the decision sequence is maintained while the process of reduction is streamlined, utilizing this tactic results in an experience that is both less difficult and more convenient. This article demonstrates the applicability of this method by outlining a decision-making dilemma that was taken from the actual world and providing a solution for it. This article discusses a novel method for dealing with neutrosophic soft graphs by merging graph theory with neutrosophic soft set theory. An illustration of a graphical depiction of a neutrosophic soft set is provided alongside an explanation of neutrosophic graphs and neutrosophic soft set graphs in this article.

]]>D. A. N. Njamen B. Baldagaï G. T. Nguefack and A. Y. Nana

The recursive method known as the stochastic approximation method, can be used among other things, for constructing recursive nonparametric estimators. Its aim is to ease the updating of the estimator when moving from a sample of size n to n + 1. Some authors have used it to estimate the density and distribution functions, as well as univariate regression using Bernstein's polynomials. In this paper, we propose a nonparametric approach to the multidimensional recursive estimators of the distribution function using Bernstein's polynomial by the stochastic approximation method. We determine an asymptotic expression for the first two moments of our estimator of the distribution function, and then give some of its properties, such as first- and second-order moments, the bias, the mean square error (MSE), and the integrated mean square error (IMSE). We also determine the optimal choice of parameters for which the MSE is minimal. Numerical simulations are carried out and show that, under certain conditions, the estimator obtained converges to the usual laws and is faster than other methods in the case of distribution function. However, there is still a lot of work to be done on this issue. These include the studies of the convergence properties of the proposed estimator and also the estimation of the recursive regression function; the developments of a new estimator based on Bernstein polynomials of a regression function using the semi-recursive estimation method; and also a new recursive estimator of the distribution function, density and regression functions; when the variables are dependent.

]]>K Krishna Sowmya and V Srinivas

The concept of onto functions plays a very important role in the theory of Analysis and has got rich applications in many engineering and scientific techniques. Here in this paper, we are proposing a new application in the field of cryptography by using onto functions on the algebraic structures like rings and fields to get a strong encryption technique. A new symmetric cryptographic system based on Hill ciphers is developed using onto functions with two keys- Primary and Secondary, to enhance the security. This is the first algorithm in cryptography developed using onto functions which ensures a strong security for the system while maintaining the simplicity of the existing Hill cipher. The concept of using two keys is also novel in the symmetric key cryptography. The usage of onto functions in the encryption technique eventually gives the highest security to the algorithm which has been discussed through different examples. The original Hill cipher is obsolete in the present-day technology and serves as pedagogical purpose but whereas this newly proposed algorithm can be safely used in the present-day technology. Vulnerability from different types of attacks of the algorithm and the cardinality of key spaces have also been discussed.

]]>D. Bharathi and A. Saraswathi

Fuzzy nonlinear programming plays a vital role in decision-making where uncertainties and nonlinearity significantly impact outcomes. Real-world situations often involve imprecise or vague information. Fuzzy nonlinear programming allows for the representation of uncertainty through fuzzy sets, enabling more accurate modeling of real-world complexities. Many optimization problems exhibit nonlinear relationships among variables. Fuzzy nonlinear programming addresses these complex relationships, providing solutions that linear programming methods cannot accommodate. The objective of this research article proposes Fuzzy Non-Linear Programming Problems (FNLPP) under environment of triangular Fuzzy numbers. This paper proposed a method based on the pivotal operation with aid of Wolfe's technique. Fuzzy non-linear programming is an area of study that deals with optimization problems in which the objective function and constraints involve fuzzy numbers, which represent uncertainty or vagueness in real-world data. We claim that the proposed method is easier to understand and apply compared to existing methods for solving similar problems that arise in real-life situations. To demonstrate the effectiveness of the method, the authors have solved a numerical example and provided illustrations in the paper. This proposed method in the paper aims to address such complexities and find solutions to these problems more efficiently.

]]>Chori Normurodov Akbar Toyirov Shakhnoza Ziyakulova and K. K. Viswanathan

In this study, initial-boundary value problem for the Burgers equation is solved using the theoretical substantiation of the spectral-grid method. Using the theory of Green's functions, an operator equation of the second kind is obtained with the corresponding initial-boundary conditions for a continuous problem. To approximately solve the differential problem, the spectral grid method is used, i.e. a grid is introduced on the integration interval, and approximate solutions of the differential problem on each of the grid elements are presented as a finite series in Chebyshev polynomials of the first kind. At the internal nodes of the grid, the requirement for the continuity of the approximate solution and its first derivative is imposed. The corresponding boundary conditions are satisfied at the boundary nodes. A discrete analogue of the operator equation of the second kind is obtained using the spectral-grid method. The convergence theorems for the spectral-grid method are proven and estimates for the method's convergence rate are obtained. To discretize the Burgers equation in time on the interval [0,T], a grid with a uniform step of is introduced, i.e. , where - given number. Numerical calculations have been carried out at sufficiently low values of viscosity, which cannot be obtained by other numerical methods. The high accuracy and efficiency of the spectral-grid method used in solving the initial-boundary value problem for the Burgers equation is shown.

]]>Bader Alruwaili

In this article, we introduce a new model entitled a mixture of the Ailamujia and size biased Ailamujia distributions. We present and discuss some statistical properties of this mixture of the Ailamujia and size biased Ailamujia distributions, such as moments, skewness, and kurtosis. We also provide some graphical results on the mixture of the Ailamujia and size biased Ailamujia distributions and provide some numerical results to understand the behavior of the proposed mixture and its properties. Also, we provide some reliability analysis results on the proposed mixture. The parameters of the Ailamujia and size biased Ailamujia distributions are estimated by using the maximum likelihood method. The usefulness of the proposed combination is illustrated by using a real-life dataset. We use the Ailamujia distribution and the size biased Ailamujia distribution, in addition to the mixture of the Ailamujia and size biased Ailamujia distributions to fit the real-life dataset. We use different criteria in this comparison; the results show that the proposed mixture fits the dataset better than the use of the Ailamujia distribution and the size biased Ailamujia distribution alone.

]]>Ali Sadig Mohommed Bager

The chaotic nature of the earth's atmosphere and the significant impact of weather on various fields necessitate accurate weather forecasting. Time series analysis plays a crucial role in predicting future values based on past data. The Autoregressive Conditional Heteroscedasticity (ARCH) model is widely used for forecasting, especially in the field of temperature analysis. This study focuses on the ARCH model for analyzing and forecasting temperature changes. The ARCH model is selected based on its ability to capture the regular variations in the predictability of meteorological variables. The methodology section explains the ARCH model and various statistical tests used, such as the heteroscedasticity test (ARCH test), Jarque-Bera test, and Augmented Dickey-Fuller test (ADF). A sample study is conducted on monthly average temperature data from Athenry, Ireland, over a period of four years. The study utilizes the ARCH model to calculate temperature series volatility and assesses the model's performance using goodness-of-fit measures and predictive accuracy. The results show that the ARCH model successfully predicts temperature changes for three years, as indicated by the forecasted temperature series. The statistical performance of the ARCH model is evaluated using in-sample and out-of-sample analyses, demonstrating its effectiveness in capturing temperature variations. The study highlights the importance of time series forecasting and the significant impact of the ARCH model in temperature analysis.

]]>Florian Heiser and E W Knapp

Central moments of statistical samples provide coarse-grained information on width, symmetry and shape of the underlying probability distribution. They need appropriate corrections for fulfilling two conditions: (1) yielding correct limiting values for large samples; (2) yielding these values also, if averaged over many samples of the same size. We provide correct expressions of unbiased central moments up to the fourth and provide an unbiased expression for the kurtosis, which generally is available in a biased form only. We have verified the derived general expressions by applying them to the Gaussian probability distribution (GPD) and we show how unbiased central moments and kurtosis behave for finite samples. For this purpose, we evaluated precise distributions of all four moments for finite samples of the GPD. These distributions are based on up to 3.2*10^{8} randomly generated samples of specific sizes. For large samples, these moment distributions become Gaussians whose second moments decay with the inverse sample size. We parameterized the corresponding decay laws. Based on these moment distributions, we demonstrate how p-values can be computed to compare the values of mean and variance evaluated from a sample with the corresponding expected values. We also show how one can use p-values for the third moment to investigate the symmetry and for the fourth moment to investigate the shape of the underlying probability distribution, certifying or ruling out a Gaussian distribution. This all provides new power for the usage of statistical moments. Finally, we apply the evaluation of p-values for a dataset of the percent of people of age 65 and above in the 50 different states of the USA.

Ismah Ismah Erfiani Aji Hamim Wigena and Bagus Sartono

Functional data has a data structure with large dimensions and is a broad source of information, but it is very possible that there are problems in analyzing functional data. Functional continuum regression is an alternative method that can be used to overcome calibration modeling with functional data. This study aimed to determine the robustness of Functional continuum regression in overcoming multicollinearity problems or the number of independent variables greater than the number of observations, with functional data. The research method used in this study is the analysis of the Functional continuum regression method on the results of the Wavelet transform of blood glucose measurements with noninvasive techniques in the calibration model, and making comparisons with non-functional methods, namely Principal component regression, partial least square regression, least square regression, and functional method namely functional regression. The results of the analysis using the five methods obtained the root mean square error prediction (RMSEP), the correlation between the observed data and the estimated observation data, and the mean absolute error (MAE). The results of the analysis can be said that reduction methods such as Functional continuum regression, Principal component regression, and partial least square regression are superior methods when used when multicollinearity occurs or the number of independent variables is greater than the number of observations. In the case of functional data analysis, the application of Functional continuum regression is better because it does not eliminate data patterns. Thus it can be said that Functional continuum regression is an effective approach in analyzing calibration models which generally have functional data, and there are several problems which include multicollinearity or the number of independent variables is greater than the number of observations.

]]>Defita Intan Muchtadi-Alamsyah and Aleams Barra

An MDS (maximum distance separable) matrix is a square matrix where all its submatrices are non-singular. The MDS matrices are used in some cryptographic systems' encryption and decryption processes. The matrix used in the process of decryption is the inverse matrix used in the encryption. Therefore, choosing a matrix which inverse is easy to find, is more efficient. Orthogonal and involutory matrices are two kinds of matrices in which inverses are easy to find. On the other hand, in terms of storage memory, a circulant matrix is more advantageous than any square matrix. In 2019, Cauchois and Loidreau proved that there is no involutory circulant MDS matrices of order 2m for m≥2 over a field of characteristics prime number p≥2. In 2022, Adhiguna et al. stated that there is no orthogonal circulant MDS matrix of even order and of order divisible by p > 2 over a field with characteristic p. This research concerns about observing the existence of MDS matrices over ring , that is, the ring where and q is the power of p. This paper shows that there is no involutory circulant MDS and no orthogonal circulant MDS matrices of certain order over . Furthermore, we can generalize these results for ring .

]]>Carlos Lopez Garces and Nayeong Kong

This paper investigates three distinct Monte Carlo estimators derived from the research of Sbert et al. These estimators are specifically tailored to the scattering equations using the Ward Bidirectional Reflectance Distribution Function (BRDF) integrated with a designed cosine-weighted environment map. In this paper, we have two goals. First, to bridge the gap between theoretical foundations and practical applicability to understand how these estimators can be seamlessly integrated as extensions to the acclaimed PBRT renderer. And the second is to measure their real-world performance. We aim to validate our methodology by comparing rendered images with varying convergence rates and deviations to the results of Sbert et al. This validation will ensure the robustness and reliability of our approaches. We analyze the analytical structure of these estimators to derive their precise form. We then implement the three estimators as extensions to the PBRT renderer, subjecting them to a numerical evaluation. We further evaluate the results of the estimator set and sampling strategy by utilizing another pair of incident radiance functions and BRDFs. The final step is to generate rendered images from the implementation to verify the results observed by Svart et el. and extend them with this new pair of functions.

]]>Sunil M.P. and J. Suresh Kumar

A graph is a basic representation of relationship between vertices and edges. This can be used when the relationships are normal and straight forward. But most of the real life situations are rather complex and it calls for advance development in graph theory. The concept of fuzzy graph addresses uncertainity to a certain extent. But, situations arise when we have to address complex hesitant situations such as taking major decisions regarding merging of companies. Intuitionistic fuzzy graph (IFG) and Hesitancy fuzzy graph (HFG) were developed to resolve this uncertainity. But it also fell short in resolving problems related to hesitant situations. In this paper, we present the concepts of IFG and HFG, which serve as the foundation for introducing, defining and analysing Intuitionistic hesitancy fuzzy graph (IHFG). We explore the concepts such as λ-strong, δ-strong and ρ-strong IHFGs. Also, we make a detailed comparative study on the cartesian product and composition of HFGs and IHFGs, establishing essential theorems related to the properties of such products. We prove that the cartesian product and composition of two strong HFGs need not be a strong HFG, but the cartesian product and composition of two strong IHFGs is a strong IHFG. Also we prove that if the cartesian product of two IHFGs is strong, then, at least one of the IHFG will be strong and if the composition of two IHFGs is strong, then at least one of the IHFG will be strong. IHFG models provide exact and accurate results for taking apt decisions in problems involving hesitant situations.

]]>V. Kalaiyarasan and K. Muthunagai

Complex number system is an extension of the real number system which came into existence during the attempts to find a solution for cubic equations. A set characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one is called a Fuzzy set. A new development of Fuzzy system is a Complex Fuzzy system in which the membership function is complex- valued and the range of which is represented by the unit disk. The fuzzy similarity measure helps us to find the closeness among the Fuzzy sets. Due to the wide range of applications to various fields, Fuzzy Multi Criteria Decision Making (FMCDM) has gained its importance in Fuzzy set theory. A combination of Complex Fuzzy set, Fuzzy similarity measure and Fuzzy Multi Criteria Decision Making has resulted in this research contribution. In this article, we have introduced and investigated Complex neutrosophic fuzzy set, which involves complex- valued neutrosophic component. We have discussed two real life examples, one on selecting the best variety of a seed that gives the maximum yield and profit in a short period of time and another on choosing the best company to invest. Similarity measure between Complex neutrosophic fuzzy sets has been used to take a decision.

]]>Faris Muslim Al-Athari

The issue of obtaining accurate interval estimates for the median of an exponential population when some of the observations are extreme values is an important issue for researchers in the fields of reliability applications and survival analysis. In this research paper, a new method is proposed for obtaining a robust confidence interval which is a substitute for the known ordinary (classical) confidence interval when there are extreme values in the sample. The proposed method is simply a result of changing the sample mean by a constant multiple of a sample median and adjusting the upper percentile point of the chi-square of the ordinary confidence interval formula. Further, the performance of the proposed method is evaluated and compared with the ordinary one by using Monte Carlo simulation based on 100,000 trials for each sample size with 5% and 10% extreme values showing that the proposed method, under the contaminated exponential distribution, is always performing better than the ordinary method in the sense of having simulated confidence probability quite close to the aimed confidence level with shorter width and smaller standard error. The use and the application of the proposed method to real-life data are presented and compared with the simulation results.

]]>Dhurata Valera Bederiana Shyti and Silvana Paralloj

The Theory of Pseudo-Additive Measures has been studied by analyzing and evaluating significant results. The system of pseudo-arithmetic operations (SPAO) as a system generated by the generator is shown directly by taking results of Rybárik and Pap, but is a further development of . Using the meaning of entropy as a logarithmic measure in information theory. Through examples we present the relation between the and the entropy, realized by the , i.e. a . The paper studies the construction of relationships between entropy and supported by and the connection with Shannon Entropy. For the pseudo-additive probabilistic measure , using as well as in the system generated by , the problem of modification of this measure by is addressed. The modifications of the Pseudo-Additive Probability Measure and the Induced Probability Measure supported by are presented, showing the relationships between the two modifications of the Pseudo-Additive Probability Measure (PAPM) and the Induced Probability Measure (IPM). Further, the Bi-Pseudo-Integral for and the Lebesgue Integral are represented in a relationship.

]]>Alaa. M. F. AL. Jumaili

The concepts of neutrosophic connectedness and compactness between neutrosophic sets find extensive applications in various fields, including sensor networks, physics, mechanical engineering, robotics and data analysis involving numerous variables. Neutrosophic set theory also plays a pivotal role in addressing complex problems in engineering, environment science, economics, and advanced mathematical disciplines. Hence, this paper aims to extend the classical definitions of neutrosophic connectedness and compactness within neutrosophic topological spaces. We introduce new classes of neutrosophic connectedness and compactness, specifically, neutrosophic δ-ß-connectedness and neutrosophic δ-ß-compactness, defined using a generalized neutrosophic open set known as "neutrosophic δ-ß-open sets". We explore several essential properties and characterizations of these spaces and introduce new notions of neutrosophic covers, which lead to the concept of neutrosophic compact spaces. Additionally, we present characterizations related to neutrosophic δ-ß-separated sets. A noteworthy feature of these concepts is their ability to model intricate connectedness networks and facilitate optimal solutions for problems involving a multitude of variables, each with degrees of acceptance, rejection, and indeterminacy. We provide relevant examples to illustrate our main findings.

]]>S. Manicka Vinayagam L. Meenakshi Sundaram and C. Devamanoharan

The main objective of this paper is to introduce a new type of contra continuous function namely based on the concept of set and function in Nano Ideal Topological Spaces. The conceptualisation of contra continuous functions, which is an alteration of continuity that requires inverse images of open sets to be closed rather than open. We compare function with function and establish the independent relation between and functions by providing suitable counter examples. Fundamental properties of with and are investigated. We study the behaviour of with . We define space and describe its relation upon space and space. Characterizations of based on space, space and graph function namely are explored. As like the continuity, the preserves the property that it maps and sets to the same type of sets in co-domain. We defined space and described its nature over . Also we have introduced functions with an example and discussed its relation with and analysed its basic properties. Composition of functions under , and are examined.

]]>V Dhanya M Selvarathi and M Ambika

Neutrosophic fuzzy sets are an extension of fuzzy sets. Fuzzy sets can only handle vague information, and it cannot deal with incomplete and inconsistent information. But neutrosophic fuzzy sets and their combinations are one technique for handling incomplete and inconsistent information. Neutrosophic fuzzy set theory provides the groundwork for a whole group of new mathematical theories and summarizes both the traditional and fuzzy counterparts. Following this, the area of neutrosophic fuzzy sets is being developed intensively, with the goal of strengthening the foundations of the theory, creating new applications, and enhancing its practicality in a range of real-life scenarios. Further, neutrosophic fuzzy sets are characterized by three components. One is truth (), the second is indeterminacy (), and the third is falsity (). In this paper, we have examined the idea of homomorphism of implication-based () neutrosophic fuzzy subgroups over a finite group. Then, neutrosophic fuzzy subgroups over a finite group and neutrosophic fuzzy normal subgroups over a finite group were defined. Finally, we have demonstrated some basic properties of homomorphism of neutrosophic fuzzy subgroups over a finite group in this study.

]]>Abd El-Monem A. Megahed Mohamed R. Zeen El Deen and Asmaa A. Ahmed

The purpose of this paper is to investigate the Nash equilibrium concept for differential games when there is uncertainty in the available information for the players. Our study involves examining the problem of uncertainty in player information during the game using the "rough sets" concept, which is widely used for many such problems. Furthermore, we also explore the possible alliance between continuous differential games and the rough programming approach. Our primary aim is to ascertain the Nash equilibrium for a differential game in situations where the players have uncertain information, so they are exerting rough control, along with the trajectory of the system state being rough as well. We derive the necessary and sufficient conditions for the open-loop Nash equilibrium of the rough differential game. Additionally, we make use of the expected value operator and trust measure of rough interval to convert the rough problem into a crisp problem, allowing us to calculate the expected Nash equilibrium strategies and α-trust Nash equilibrium strategies for the game. Finally, a numerical example that outlines the steps involved in producing the rough interval of the Nash equilibrium and system state trajectory for the rough differential game is given. Moreover, this example demonstrates how to obtain each crisp problem from a rough one and then determines its Nash equilibrium and the corresponding state trajectory.

]]>B.M. Sultanov A. Kurudirek and Sh.Sh. Ismoilov

Currently, the study of the geometry of semi-Euclidean spaces is an urgent task of geometry. In the singular parts of pseudo-Euclidean spaces, a geometry associated with a degenerate metric appears. A special case of this geometry is the geometry of Galileo. The basic concepts of the geometry of Galilean space are given in the monograph by A. Artykbaev. Here the differential geometry "in the small" is studied, the first and second fundamental forms of surfaces and geometric characteristics of surfaces are determined. The derivational equations of surfaces, analogs of the Peterson-Codazzi and Gauss formulas are calculated. This paper studies the development and isometry of surfaces in Galilean space. Moreover, the isometry of surfaces in Galilean space is divided into three types: semi-isometry, isometry and completely isometry. This separation is due to the degeneracy of the Galilean space metric. The existence of a development of a surface projecting uniquely onto a plane in general position is proved, as well as the conditions for isometric and completely isometric surfaces of Galilean space. We present the conditions associated with the analog of the Christoffel symbol, providing isometries of the surfaces of Galilean space. An example of isometric, but not completely isometric surfaces in G3 is given. The concept of surface development for Galilean space is generalized. A development of the surface is obtained, which is uniquely projected onto the plane of the general position. In addition, the Gaussian curvature of the surface has been shown to be completely defined by Christoffel symbols.

]]>Isakjan Khamdamov and Azam Imomov

In this article, we study the functionals of the convex hull generated by independent observations over two-dimensional random points. When the random points are given in the polar coordinate system, their components are independent of each other, the angular coordinate is distributed uniformly, and the tail of the distribution of the radial coordinate is a regularly varying function near the circle of the unit disk – support. Here, with the approximation of the binomial point process by an inhomogeneous Poisson one, it is possible to study the asymptotic properties of the main functionals of the convex hull. Using the independence property of the increment of Poisson processes, we find an asymptotic expression for the mean values and variances for the main functionals of the convex hull. Uniform boundedness of exponential moments is proved for the same functionals, in the case when the convex hull is generated from an inhomogeneous Poisson point process inside the disk. The indicated independence property of the increment of the Poisson process allows us to express the area of the convex hull as a sum of independent identically distributed random variables, with which we prove the central limiting theorem for the number of vertices and the area of the convex hull. From the results obtained, we can conclude that if the tail of the distribution near the boundary is heavier, then there are many elements of the sample near the support boundary, and this means that there are many vertices of the convex hull, but the area bounded by the perimeter of the convex hull and the circle, as well as the difference between the perimeter of the convex hull and the circle, becomes negligible.

]]>R. Nithya Raj R. Sundara Rajan Haewon Byeon CT. Nagaraj and G. Kokila

The metric dimension of a chemical graph is a fundamental parameter in the study of molecular structures and their properties. This metric dimension is a numerical measure of the smallest set of atoms required to uniquely determine the location of all other atoms within the molecule. In this abstract, we explore the concept of metric dimension in chemical graphs, discussing its theoretical foundations and its applications in various fields such as navigation, network theory, drug design, optimization, pattern recognition, and other related fields computational chemistry, and material science. Understanding the metric dimension of chemical graphs enables the identification of crucial atoms or bonds that significantly impact the properties and behavior of molecules, aiding in the design of more effective drugs, catalysts, and materials. Finding the metric dimension of any given graph poses a computational challenge classified within the NP-complete problem category. A group of nodes, denoted as , is regarded as a locating set if, every pair of nodes and within the graph , there is a minimum of one node in such a way that the separation between and is not the same as the separation between and . The metric dimension is represented by and corresponds to the minimum size of a locating set for . The primary objective of this effort is to establish the proof that, for , the metric dimensions of the line network for the Hammer and triangular benzene structures are 2 and 3, respectively. We also established the existence of a constant metric dimension for this class of line graphs, which includes Hammer and triangular benzene structures.

]]>Nahathai Rerkruthairat and Noppadon Wichitsongkram

Sometimes draws or ties occur in sports. Tiebreakers are the forms of competition that break ties and decide the winner when a draw or a tie occurs. Depending on types of tiebreakers, some take shorter and some take longer to end the competition. In this article, we are interested in calculating the expectation and variance of the number of games that will continue after a draw from types of tiebreakers that require players to win by two points. We focus on three types of win by two points that are used in many popular sports, such as tennis, volleyball and racquetball. By calculating the expected number of games, we can compare the number of games in each type of tiebreakers that will approximately be taken to end the game. In these kinds of sports, the rules to gain each point are usually the same. This means that there are the same finite states that the players or teams can reach in each point and each possible state depends only on the previous state. Since we know that a Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event, we can use an application of Markov chains to solve the problems.

]]>Ahmed Al-Adilee and Wasan Al-Shemmari

In this study, the generalized half-logistic distribution (GHLD) was expanded by replacing the shape parameter with a linear model, denoted by the notation . This model involves a vector of explanatory variables denoted by , where with a vector of coefficients of each one of those explanatory variables, denoted by . The linear model represents several explanatory variables with their coefficients that represent effects on some items. Briefly, the proposed distribution is denoted by, LM-GHLD. Afterward, by finding the pdf, and cdf of LM-GHLD, many mathematical and statistical characteristics were investigated, such as the survival function, the hazard function, the moments, the moment generating function, quantiles, the Rényi entropy, and the order statistic function. The unknown parameters of the modern distribution were estimated with the non-Bayesian method, which is known as the Maximum Likelihood Estimate (MLE). An important part of such a study is related to the simulation, which is shown within a generation of different sample sizes. A goodness-of-fit measure has been implemented on real data sets to compare the classical distribution (GHLD) and the proposed distribution (LM-GHLD) enabling us to determine which distribution is better. Eventually, we provide some conclusions and summarize our findings.

]]>V. Padmapriya and M. Kaliyappan

The field of fractional calculus is mainly concerned with the differentiation as well as integration of arbitrary orders. This concept is obviously present in various domains of science and engineering. Most people are familiar with the Caputo and Riemann-Liouville fractional definitions. Recently, Hilfer has related the Caputo and Riemann-Liouville derivatives by a general formula; this connection is referred to as the Hilfer or generalized Riemann-Liouville derivative. The Hilfer fractional derivative serves as an intermediary between the Riemann-Liouville and Caputo fractional derivatives, providing a means of interpolation. Parameters in the Hilfer derivative provide more degrees of freedom. Adomian decomposition method (ADM) is widely regarded as a highly effective mathematical technique for solving both linear and nonlinear differential equations. ADM provides an analytical solution in the form of a series solution. Motivated by the growing number of real-life applications for fractional calculus, the objective of this work is to explore the solutions of Hilfer fractional differential equations in a fuzzy sense using the ADM. The efficiency and accuracy of the proposed method are demonstrated by the solution of numerical examples. Graphical representations are provided to visualize the solutions' behavior. This shows that as the number of terms in the series goes up, the numerical results get closer and closer to the exact solutions.

]]>Ayazul Hasan and Jules Clement Mba

The close association of abelian group theory and the theory of modules have been extensively studied in the literatures. In fact, the theory of abelian groups is one of the principal motives of new research in module theory. As it is well-known, module theory can only be processed by generalizing the theory of abelian groups that provide novel viewpoints of various structures for torsion abelian groups. The theory of torsion abelian groups is significant as it generates the natural problems in QTAG-module theory. The notion of QTAG (torsion abelian group like) module is one of the most important tools in module theory. Its importance lies behind the fact that this module can be applied in order to generalized torsion abelian group accurately. Significant work on QTAG-module was produced by many authors, concentrating on establishing when torsion abelian groups are actually QTAG-modules. There are two rather natural problems which arise in connection with the Σ-uniserial modules. Namely: The QTAG-module M is Σ-uniserial if and only if all N-high submodules of M are Σ-uniserial, for some basic submodules N of M, and M is not a Σ-uniserial module if and only if it contains a proper (ω + 1)-projective submodule. The current work explores these two problems for QTAG-modules. Some related concepts and problems are also considered. Our global aim here is to review the relationship between the aspects of group theory in the form of torsion abelian groups and theory of modules in the form of QTAG-modules.

]]>Iweobodo D. C. Njoseh I. N. and Apanapudor J. S.

In this paper, we developed a new wavelet-based Galerkin method of weighted residual function. In order to achieve this, we considered the wavelet transform as it relates to orthogonal polynomials, developed new wavelets using the Mamadu-Njoseh Polynomials, and formulated a base function with the newly developed wavelets. We considered the method of implementing solutions with the newly developed wavelet-based Galerkin method of weighted residual function, and applied it in obtaining approximate solutions of some one-dimensional differential equations having the Dirichlet boundary conditions. The results obtained from the newly developed method were compared with the results obtained from the exact solution and that from the classical Finite Difference Method (FDM) in literature. It was observed that the newly developed wavelet-based Galerkin method of weighted residual function demonstrated a high efficiency in providing approximate solutions to differential equations. The study revealed that the newly developed wavelet-based Galerkin method of weighted residual function converges at a good pace to the exact solution, and iterated the accuracy and effectiveness of its solutions. We used the MAPLE 18 software in carrying out all computations in this work.

]]>Asep Rusyana Aji Hamim Wigena I Made Sumertajaya and Bagus Sartono

Different machine learning algorithms may produce different orders of the variable importance measures even though they use an identical dataset. The measures raise the difficulty of concluding which predictor variables are the most important. Therefore, there is a requirement to unify those scores into a single order so that the analyst can withdraw a conclusive decision more easily. This research applied the Cuckoo Search algorithm approach to obtain the unification of those orders into a single one. A simulation study was conducted to justify that the approach could work well in several circumstances of data. We implemented the algorithm to identify the importance of the variables where the correlations among them are low, moderate, and high. The result of the paper shows that the proposed variable importance measure is the best if it is applied to predictors independent of each other. Generally, it is more accurate than variable importance measures of machine learning. The algorithm was also applied to identify the proposed important variable measure for recognizing food insecurity in households in Indonesia. The proposed variable importance has good accuracy. The accuracy is higher if the number of variables is greater than ten.

]]>Rasha A. Farghali Hemat M. Abd-Elgaber and Essam A. Ahmed

In this paper, a novel method is presented for simultaneously identifying and estimating Seasonal Moving Average (SMA) models, which are considered a special case of Seasonal Autoregressive Integrated Moving Average (SARIMA) models introduced by Box and Jenkins. To accomplish this, we utilize a mixed-integer nonlinear programming (MINLP) model, which falls within the class of optimization problems involving integer and continuous decision variables, as well as non-linear objective functions and/or constraints. The advantage of employing MINLP lies in its ability to provide a more flexible representation of real-world problems. The aim of employing the MINLP is to identify and estimate the appropriate SMA model, specifically determining whether it is Multiplicative or Non-multiplicative. To evaluate the effectiveness of the proposed MINLP approach, we conducted both a simulation study and real-world applications. In the simulation study, we generate 1000 time series datasets from each of the twelve SMA models, which comprised six multiplicative SMA models and six non-multiplicative SMA models, with different orders. Additionally, we examine the effectiveness of MINLP through two real-world applications: Carbon Dioxide Levels data and College Enrollment data. The results obtained from both the simulation study and real-world applications consistently demonstrate the effectiveness of MINLP in accurately identifying the appropriate SMA model. These findings support the applicability and reliability of the proposed method in practical scenarios. Overall, our research contributes to the field of time series analysis by providing a new approach for identifying and estimating SMA models using MINLP, paving the way for improved forecasting and decision-making in various domains.

]]>Samar A. A. Quota Faten. R. Kara O. H. Fathy and W. M. Mahmoud

A scheme is a type of mathematical construction that extends the concept of algebraic variety in a number of ways, including accounting for multiplicities and being defined over any commutative ring. In this article, we study some properties of the cyclic and dihedral homology theory in schemes. We study the long exact sequence of cyclic homology of scheme and prove some results. So, we introduce and study Morita-equivalence in cyclic homology of schemes and proof the main relation between trace map and inclusion map. Our goal is to explain product structures on cyclic homology groups . Especially, we show of algebra. We give the relations between dihedral homology and cyclic homology of schemes, therefore: . We explain the trace map and inclusion map of cyclic homology for scheme algebra which takes form: and . For the shuffle map , we obtain the long exact sequence of cyclic homology for scheme: . We give the long exact sequence of dihedral homology for scheme: . For any three and algebra, we write the next long exact sequence as a commutative diagram: . For all and schemes, we give the long exact sequence of dihedral homology as: .

]]>Bourakadi Ahssaine Baraka Achraf Chakir and Khalifi Hamid

In this article, we have discussed the properties of the probability law "T" called functional capacity and other closely related functionals "Q and C" pertaining to random closed sets. We are interested in the most widely used functional in random set theory "T". We have established the belonging of "T" to the interval [0,1], and proven that it is increasing in the sense of inclusion, and its sub-additivity property through probability techniques. Moreover, we have explored the various types of convergences of a sequence of random closed sets, such as weak convergence, strong convergence (almost surely in the sense of Hausdorff), convergence in the sense of Painlevé-Kuratowski and Wijsman-Mosco, as well as convergence in probability. In the second part of our work, we have proven a new corollary which states that the strong convergence in the sense of Hausdorff implies the convergence in probability of a sequence of random closed sets at infinity. Our proof involves the definition of mathematical expectation for a discrete variable and the indicator variable, which is a random variable that takes two possible values, 0 or 1.

]]>Adisak Moumeesri and Weenakorn Ieosanurak

We have introduced a novel continuous distribution known as the Klongdee distribution, which is a combination of the exponential distribution with parameter and the gamma distribution with parameters . We thoroughly examined various statistical properties that provide insights into probability distributions. These properties encompass measures such as the cumulative distribution function, moments about the origin, and the moment-generating function. Additionally, we explored other important measures including skewness, kurtosis, C.V., and reliability measures. Furthermore, we explore parameter estimation using nonlinear least squares methods. The numerical results presented compare the unweighted and weighted least squares (UWLS and WLS) methods, maximum likelihood estimation (MLE), and method of moments (MOM). Based on our findings, the MLE demonstrates superior performance compared to other parameter estimation methods. Moreover, we demonstrate the application of this distribution within an actuarial context, specifically in the analysis of collective risk models using a mixed Poisson framework. By incorporating the proposed distribution into the mixed Poisson model and analyzing a real-life dataset, it has been determined that the Poisson-Klongdee model outperforms alternative models in terms of performance. Highlighting its capability to mitigate the problem of overcharges, the Poisson-Klongdee model has been proven to be a valuable tool.

]]>R. Muthuraj K. Nachammal M. Jeyaraman and D. Poovaragavan

In the context of the Neutrosophic Norm, the essay explores the challenge of constructing precise sequence spaces whose elements' convergence is a generalised form of the Cauchy convergence. It has proven to be a crucial tool, opening the door to the theory of functions and the law of large numbers applications. Numerous authors, including those who investigated the Euler totient matrix operator, have studied the strategy for building new sequence spaces that are specified as the domain of matrix operators. Recently, the Jordan totient function generalised the Euler totient function . In the context of neutrosophic Norm spaces, we establish some sequence spaces, specifically , and as a domain of the triangular Jordan totient matrix operator, and investigate the ideal convergence of these sequences. These concepts serve as an introduction to a new sort of convergence that Fast and Steinhaus presented as more general than normal convergence and statistical convergence. According to Kostyrko et al., this form is known as ideal convergence. In order to arrive at a finite limit, the Jordan totient operator, an infinite matrix operator, is used. We also construct a number of inclusion connections between the spaces as we explain various topological and algebraic properties. The Jordan totient operator, an infinite matrix operator, is used to accomplish the task of reaching a finite limit. As we discuss various topological and algebraic features, we also create several inclusion relations between the spaces.

]]>Benhari Mohamed amine and Kaicer Mohammed

This article presents an innovative approach to solving linear systems with interval coefficients efficiently. The use of intervals allows the uncertainty and measurement errors inherent in many practical applications to be considered. We focus on the solution algorithm based on the Cholesky decomposition applied to positive symmetric matrices and illustrate its efficiency by applying it to the Leontief economic model. First, we use Sylvester's criterion to check whether a symmetric matrix is positive, which is an essential condition for the Cholesky decomposition to be applicable. It guarantees the validity of our solution algorithm and avoids undesirable errors. Using theoretical analyses and numerical simulations, we show that our algorithm based on the Cholesky decomposition performs remarkably well in terms of accuracy. To evaluate our method in concrete terms, we apply it to the Leontief economic model. This model is widely used to analyze the economic interdependencies between different sectors of an economy. By considering the uncertainty in the coefficients, our approach offers a more realistic and reliable solution to the Leontief model. The results obtained demonstrate the relevance and effectiveness of our algorithm for solving linear systems with interval coefficients, as well as its successful application to the Leontief model. These advances are crucial for fields such as economics, engineering, and the social sciences, where data uncertainty can greatly affect the results of analyses. In summary, this article highlights the importance of interval arithmetic and Cholesky's method in solving linear systems with interval coefficients. Applying these tools to the Leontief model can help you better understand the impact of uncertainty and make informed decisions in a variety of fields, including economics and engineering.

]]>Sudhanshu Aggarwal Shahida A. T. Ekta Pandey and Aakansha Vyas

Diophantine equations have great importance in research and thus among researchers. Algebraic equations with integer coefficients having integer solutions are Diophantine equations. For tackling the Diophantine equations, there is no universal method available. So, researchers are keenly interested in developing new methods for solving these equations. While handling any such equation, three issue arises, that is whether the problem is solvable or not; if solvable, possible number of solutions and lastly to find the complete solutions. Fermat's equation and Pell's equation are most popularly known as Diophantine equations. Diophantine equations are most often used in the field of algebra, coordinate geometry, group theory, linear algebra, trigonometry, cryptography and apart from them, one can even define the number of rational points on circle. In the present manuscript, the authors demonstrated the problem of existence of a solution of a non-linear (exponential) Diophantine equation , where are non-negative integers and are primes such that has the form of a natural number n. After it, authors also discussed some corollaries as special cases of the equation in detail. Results of the present manuscript depict that the equation of the study is not satisfied by the non-negative integer values of the unknowns and . The present methodology of this paper suggests a new way of solving the Diophantine equation especially for academicians, researchers and people interested in the same field.

]]>Nahed I. Eassa Hegazy M. Zaher and Noura A. T. Abu El-Magd

The purpose of this paper is to present a neutrosophic form of the generalized Pareto distribution (NGPD) which is more flexible than the existing classical distribution and deals with indeterminate, incomplete and imprecise data in a flexible manner. In addition to this, NGPD will be obtained as a generalization of the neutrousophic Pareto distribution. Also, the paper introduces its special cases as neutrosophic Lomax distribution. The mathematical properties of the proposed distributions, such as mean, variance and moment generating function are derived. Additionally, the analysis of reliability properties, including survival and hazard rate functions, is mentioned. Furthermore, neutrosophic random variable for Pareto distribution was presented and recommended using it when data in the interval form follow a Pareto distribution and have some sort of indeterminacy. This research deals the statistical problems that have inaccurate and vague data. The proposed model NGPD is widely used in finance to model low probability events. So, it is applied to a real-world data set to modelling the public debt in Egypt for the purpose of dealing with neutrosophic scale and shape parameters, finally the conclusions are discussed.

]]>S. Sangeetha and Shakeela Sathish

Rough sets are extensions of classical sets characterized by vagueness and imprecision. The main idea of rough set theory is to use incomplete information to approximate the concept of imprecision or uncertainty, or to treat ambiguous phenomena and problems based on observation and measurement. In Pawlak rough set model, equivalence relations are a key concept, and equivalence classes are the foundations for lower and upper approximations. Developing an algebraic structure for rough sets will allow us to study set theoretic properties in detail. Several researchers studied rough sets from an algebraic perspective and a number of structures have been developed in recent years, including rough semigroups, rough groups, rough rings, rough modules, and rough vector spaces. The purpose of this study is to demonstrate the usefulness of rough set theory in group theory. There have been several papers investigating the roughness in algebraic structures by substituting an algebraic structure for the universe set. In this paper, rough groups are defined using upper and lower approximations of rough sets from a finite universe instead of considering the whole universe. Here we have considered a finite universe along with a relation which classifies the universe into equivalence classes. We have identified all rough sets with respect to this relation. The upper and lower approximated sets have been taken separately and these form a rough group equivalence relation () and it partitions the group () into equivalence classes. In this paper, the rough group approximation space () has been defined along with upper and lower approximations and properties of subsets of with respect to rough group equivalence relations have been illustrated.

]]>S. Senthil and R. Perumal

In this paper, we focus on a subclass of duo-seminearrings called as right duo-seminearrings. We also focus on the algebraic properties and peculiarities of mid-units within this class. As a logical extension of the concept of mid-identities in semirings, the concept of mid-units in right pair seminearings is introduced. Mid-units are elements with both left and right invertibility, making them essential for understanding the structure and behaviour of right duo-seminearrings. In particular, we examine the interaction between idempotents and seminearring mid-units. We have also investigated regular right duo-seminearring which is a semilattice of subseminearrings with mid-units. In order to have a mid-unit in duo-seminearrings, we have established the necessary and sufficient conditions. The aim of this work is to carry out an extensive study on algebraic structure of right duo-seminearrings and the major objective is to further enhance the theory of right duo-seminearrings in order to find special structures of right duo-seminearrings. Throughout the research, rigorous proofs are provided to support the theoretical developments and ensure the validity of the findings. Concrete examples are also presented to illustrate the concepts and facilitate a better understanding of the algebraic structures associated with duo seminearings and mid-units. These examples serve as valuable tools for researchers and practitioners interested in the application of right duo-seminearrings and mid-units in their respective fields. Due to their applicability in domains such as computer science, cryptography, and coding theory, the topic of duo seminearrings, which generalise both semirings and duo-rings, have received substantial attention in algebraic research.

]]>Mahesh Puri Goswami and Raj Kumar

In this article, we generalize the Riemann-Liouville fractional differential and integral operators that can be applied to the functions of a bicomplex variable. For this purpose, we consider the bicomplex Cauchy integral formula and some contours in bicomplex space. We elaborate these operators through some examples. Also, we contemplate some significant properties of these operators which include a discussion of bicomplex analytical behavior of generalized bicomplex functions through Pochhammer contours, the law of exponents, generalized Leibniz rule along with a depiction of the region of convergence, and generalized chain rule for Riemann-Liouville fractional operators of bicomplex order. We give an application of our work in the construction of fractional Maxwell's type equations in vacuum and sourcefree domains equipped with the Riemann-Liouville derivative operator. For this, we define bicomplex grad, div, and curl operator with the help of these newly defined operators. The advantage of this fractional construction of Maxwell's equation is that it may be used to build fractional non-local electronics in bicomplex space. By considering bicomplex vector fields for the respective domains, we reduce the number of these fractional Maxwell's type equations by half, which makes it easier to extract electric and magnetic fields from the bicomplex vector fields.

]]>Siti Humaira Pudji Astuti Intan Muchtadi Alamsyah and Edy Tri Baskoro

Some researchers have studied some properties of the Jacobson graph of commutative rings. In this study, we expand these results by examining the Jacobson graph of a non-commutative ring with identity, where we focus on the case of matrix rings. Initially, we update the definition of the Jacobson graph of non-commutative rings as a directed graph. Then we find that the Jacobson graph of the matrix rings case is undirected. We can classify matrices based on rank by viewing the matrix as a linear transformation. The main result is that the order of the matrix rank values will be proportional to the order of the matrix degrees as vertices of the graph. So that one can identify the maximum and minimum degrees in this graph. Sequentially, we describe the graph properties starting from the Jacobson graph of matrices over fields, then expanding to the Jacobson graph of matrices over local commutative rings and the Jacobson graph of matrices over non-local rings. In the end, we also give different results on the Jacobson graph of triangular matrices. The main contribution of this paper is to review the relationship between the aspects of linear algebra in the form of matrix rings and combinatorics in the form of diameter and vertex degree on this graph.

]]>V.Kamalakannan P.Murugadas and M.Kavitha

The generalized inverse is crucial in matrix theory. In many applications, such as control systems, robotics, and signal processing, the generalized inverse of matrices is critical. The generalized inverse of a picture fuzzy matrix is critical to solving a variety of real-world problems. Because of their ability to handle uncertain and imprecise medical data, applications of the generalized inverse of picture fuzzy matrix have gained significant attention in the medical field. Numerous researchers have investigated generalized inverses in fuzzy matrices and intuitionistic matrices. The fuzzy picture is an effective mathematical model for dealing with uncertain realworld issues. The picture fuzzy matrix is a generalization of the classical fuzzy matrix and the intuitionistic fuzzy matrix. In this research, a method for determining the generalized inverse (g-inverse) of a picture fuzzy matrix is implemented. In addition, the concept of a standard basis for picture fuzzy vectors is established. A few results related to the g-inverse of a fuzzy picture matrix are premeditated with relevant examples. An algorithm for evaluating the generalized inverse of a fuzzy picture matrix is provided. This study concludes with an application of the g-inverse of a picture fuzzy matrix.

]]>Khalid M. El-khabeary Afaf El-Dash Nada M. Hafez and Samah M. Abo-El-hadid

A Joint chance-constrained programming (JCCP) technique is regarded as one of the most useful applicable techniques of stochastic programming techniques. It is more suitable for solving uncertain real problems, especially in economics and social problems, where some of model parameters are positive dependent random variables and follow well-known probability distributions. In this paper, we take into account a linear JCCP problem where some right-hand side random parameters are dependent and follow the Dagum distributions. So, firstly we derive a bivariate Dagum distribution with seven parameters with marginals following the Dagum distribution with three parameters. This proposed bivariate Dagum distribution is based on the Farlie-Gumbel-Morgensten copula (as presented in theorem (2.1)). Secondly, the proposed bivariate distribution is used in the context of JCCP technique to transform a linear JCCP model into an exact equivalent deterministic nonlinear programming model through theorem (3.1). Thirdly, through theorem (3.2), we prove that the obtained exact equivalent deterministic nonlinear programming model is a convex model, hence any nonlinear programming method can be used to solve it and find the global optimal solution. Finally, in order to demonstrate how to convert a linear JCCP model into an equivalent deterministic nonlinear programming model and solve it using the cutting plane method, a numerical example is included.

]]>D. Srilatha and V. Kiran

Generalization of metric spaces is always an evergreen topic of interest to many researchers. In order to generalize a metric space, researchers proposed various methods like by weakening any one condition from the definition of a metric or combining the notion of one metric space with the notion of one or more other metric spaces. Recently, S^{JS}–metric space and S_{b}-metric space have been introduced by combining the notion of S - metric space with JS-metric space and b-metric space respectively. Similarly, G_{b}–metric space has been introduced as the generalization of G–metric space using b–metric. This notion motivated the present study. The purpose of this article is to introduce G_{JS}-metric space and to present some fixed point theorems in G_{JS}-metric space. By introducing the idea of G_{JS}-metric space, we combined the notions of two metric spaces namely, G-metric space and JS-metric space. First, we begin with some basic definitions which are useful for the introduction of G_{JS}–metric and then proceed with the necessary standard topological concepts of G_{JS}-metric space. Then, using these topological concepts, we achieve some specific and principal results on G_{JS}-metric space. Further, by providing suitable examples where ever required, we demonstrate the independent nature of G-metric space, G_{JS}-metric space and JS-metric space. Further, we validate the conditions for the presence of tripled fixed point and verify its uniqueness by considering various cases on G_{JS}-metric space.

Siddharth Shah Rudraharsh Tewary Manoj Sahni Ritu Sahni Ernesto Leon Castro and Jose Merigo Lindahl

Recent developments in fuzzy theory have been of great use in providing a framework for the understanding of situations involving decision-making. However, these tools have limitations, such as the fact that multi-attribute decision-making problems cannot be described in a single matrix. Fuzzy and intuitionistic fuzzy matrices are important tools for these types of problems since they can help to solve them. We presented a new super matrix theory in the intuitionistic fuzzy environment in order to overcome these restrictions. This theory is able to readily cope with problems that include numerous attributes while addressing belongingness and non-belonging criteria. Hence, it introduces a fresh perspective into our thinking, which in turn enables us to generalize our findings and arrive at more sound conclusions. For the purpose of theoretical development, we define a variety of different kinds of intuitionistic fuzzy super matrices and present a number of essential algebraic operations in order to make it more applicable to situations that take place in the real world. One multi-criteria decision-making problem based on super matrix theory is discussed here for the sake of validating and illustrating the applicability of the established findings. In addition to this, we suggest a general multi-criteria decision-making algorithm that makes use of intuitionistic fuzzy super matrix theory. This algorithm is more dynamic than both intuitionistic fuzzy matrix and fuzzy super matrix theories, and it can be applied to the resolution of a wide range of issues. The validation of the proposed theory is done by taking a real-world example to show its importance.

]]>Renas T. M.Salim and Nazar H. Shuker

This article presents the concept of a NE-nil clean ring, which is a generalization of the strongly nil clean ring. A ring R is considered NE-nil clean if, for every a in R, there exists a_{1} in R such that aa_{1} = with a − a_{1} = q and a_{1}q = qa_{1}, where q is nilpotent and is idempotent. This article's aim is to introduce a new type of ring, the NE-nil clean ring, and provide the fundamental properties of this ring. We also establish the relationship between NE-nil clean rings and 2-Boolean rings. Additionally, we demonstrate that the Jacobson radical and the right singular ideal over NE-nil clean ring are nil ideals. Among other results, we prove that every strongly nil clean ring and every weak * nil clean ring are NE-nil clean. We establish that a strongly 2-nil clean ring and NE-nil clean ring are equivalent. Furthermore, we introduce and investigate NT-nil clean ring, that is a ring with every a in R, there exists a_{1} in R such that aa_{1} = t with a − a_{1} = q and a_{1}q = qa_{1}, where t is a tripotent and q is nilpotent, by showing that these rings are a generalization of NE-nil clean rings. We provide the basic properties of these rings and explore their relationship with NE-nil clean and Zhou rings.

Ugah Tobias Ejiofor Arum Kingsley Chinedu Charity Uchenna Onwuamaeze Everestus Okafor Ossai Henrrietta Ebele Oranye Nnaemeka Martin Eze Mba Emmanuel Ikechukwu Ifeoma Christy Mba Comfort Njideka Ekene-Okafor Asogwa Oluchukwu Chukwuemeka and Nkechi Grace Okoacha

In this paper, a simple asymptotic test statistic for identifying multiple outliers in linear regression is proposed. Sequential methods of multiple outliers detection test for the presence of a single outlier each time the procedure is applied. That is, the most severe or extreme outlying observation (the observation with the largest absolute internally studentized residual from the original fit of the mode to the entire observations) is tested first. If the test detects this observation as an outlier, then this observation is deleted, and the model is refitted to the remaining (reduced) observations. Then the observation with the next largest absolute internally studentized residual from the reduced sample is tested, and so on. This procedure of deleting observations and recomputing studentized residuals is continued until the null hypothesis of no outliers fails to be rejected. However, in this work our method or procedure entails calculating and uses only one set of internally studentized residuals obtained from fitting the model to the original data throughout the test exercise, and hence the procedure of deleting an observation, refitting the data to the remaining observations (reduced values) and recomputing the absolute internally studentized residuals at each stage of the test is avoided. The test statistic is incorporated into a technique (procedure) that entails a sequential application of a function of the internally studentized residuals. The procedure is a straightforward multistage method and is based on a result giving large sample properties of the internally studentized residuals. Approximate critical values of this test statistic are obtained based on approximations that depend on the application of the Bonferroni inequality since their exact values are not available. The new test statistic is very simple to compute, efficient and effective in large data sets, where more complex methods are difficult to apply because of their enormous computational demands or requirements. The results of the simulation study and numerical examples clearly show that the proposed test statistic is very successful in the identification of outlying observations.

]]>Iakov M. Erusalimskiy and Vladimir A. Skorokhodov

In this paper, graphs called overtrees are introduced and studied. These are connected graphs that contain a single simple cycle. Such graphs are connected graphs following the trees in terms of the number of edges. An overtree can be obtained from a tree by adding an edge to connect two non-adjacent vertices of a tree. The same class of graphs can also be defined as a class of graphs obtained from trees by replacing one vertex of the tree with a simple cycle. The main characteristics of overtrees are , which is the number of vertices, and , which is the number of vertices of a simple cycle (). A formula for the chromatic polynomial of an overtree is obtained, which is determined by the characteristics and only. As a consequence, it is obtained the formula for the chromatic function of a graph which is built from a tree by replacing some of its vertices (possibly all) with simple cycles of arbitrary length. It follows from these formulas that any overtree with an even-length cycle is two-colored, and with an odd-length cycle is three-colored. The same is true for graphs obtained from trees by replacing some vertices with simple cycles.

]]>A. Thirumalai K. Muthunagai and M. Kaliyappan

The skew field of Quaternions is the best known extension of the field of Complex numbers. The beauty of the Quaternions is that they form a field but the handicap is loss of commutativity. Thus the four- dimensional algebra called Bicomplex numbers with the set of all Complex numbers as a subalgebra preserving commutativity came into existence, by considering two imaginary units. The conventional calculus is generalized using Fractional calculus which is useful to extend derivatives of integer order to fractional order. Due to their vast applications to various disciplines of Science and Engineering, Mittag- Leffler functions have become prominent. Our contribution here is a combination of all the three streams mentioned above. In our research findings, bicomplex two-parameter Mittag- Leffler functions have been obtained as the solutions for the set of fractional differential equations that are linear in bicomplex space. A block diagonal of a square matrix is a diagonal matrix whose Principal diagonal elements are square matrices and the diagonal elements of lie along the diagonal of . A Jordan block is a matrix that is upper triangular with in the Principal diagonal, 1s just above the Principal diagonal and all other entries as 0. A Jordan Canonical form is a block diagonal matrix where each block is Jordan. A minimal polynomial of a matrix is a polynomial which is monic in with least degree. By using the methods of the minimal polynomial (eigenpolynomial) and Jordan canonical matrix, we have computed matrix Mittag–Leffler functions. The solutions obtained for the numerical examples have been visualized and interpreted using MATLAB.

]]>S Sinika and G Ramesh

Now in real-life scenarios, indeterminacy arises everywhere in various fields, including physics, mathematics, economics, philosophy, social sciences, etc. It occurs whenever prediction is difficult, when we didn't get a predetermined outcome or obtain fixed or multiple possible outcomes etc. Overcoming indeterminacy is one of the most prominent duties for everyone to lead a confusion-less society. Hence a neutrosophic concept came into force to analyze indeterminacy explicitly. In contrast, a fuzzy set assigns only membership grade, and an intuitionistic set allocates membership and non-membership to elements. Decision-makers can use neutrosophic settings to model uncertainty and ambiguity in complex systems for flexible analysis. The neutrosophic environment with interval numbers makes one handle the situations efficiently. Hence we utilize interval-valued trapezoidal neutrosophic numbers for more flexibility. Trapezoidal number together with interval truth, interval indeterminacy, and interval falsity are the parameters of these neutrosophic numbers. Considering a de-neutrosophication technique in crisp numbers again leads to vagueness in real-life circumstances. Hence our primary goal is to develop a new de-neutrosophication strategy in the form of an interval number instead of the crisp number. This paper provides an overview of the de-neutrosophication and a new ranking technique based on an interval number, and some extended neutrosophic linear programming theorems. Further, an interval version of simplex and Robust Two-Step method (RTSM) are used to answer an interval-valued trapezoidal neutrosophic linear programming problem. Finally, this paper highlights the limitations and advantages of the proposed technique to improve problem-solving in a wide range of fields.

]]>Sabarinsyah Hanni Garminia Pudji Astuti and Zelvin Mutiara Leastari

In this research, it was agreed that a bilinear form is an extension of the inner product since a symmetry bilinear form will be equivalent to the inner product over a field of real numbers. Concepts in bilinear space, such as the concept of orthogonality of two vectors, the concept of orthogonal subspace of a subspace, the concept of adjoint operators of a linear operator and the concept of closed subspace are defined according to those prevailing in the inner product space fact assumed to be extensions of the concepts applicable in the inner product space. In the context of a cap subspace, we can identify the necessary and sufficient conditions for any linear operator in a continuous Hilbert space. These results open up opportunities to introduce the concept of pseudo-continuous linear mapping in bilinear spaces. We have obtained the result that pseudo-continuous linear mapping spaces in bilinear spaces have a relationship with linear mapping spaces that have adjoint mapping. We have also obtained the result that the structure of linear operators limited to Hilbert spaces can be extended to pseudo-continuous operator structures in bilinearal spaces. In this study, we have generalized the properties of self-adjoint operators in product spaces in infinite dimensions to bilinear, including closed properties of addition operations, and scalar multiplication, commutative properties, properties of inverse operators, properties of zero operators, properties of polynomial operators over real fields, and orthogonal properties of eigenspaces of different eigenvalues.

]]>Siti Hajar binti Abu Bakar Muhamad Safiih Bin Lola Anton Abdulbasah Kamil Nurul Hila Zainuddin and Mohd Tajuddin Abdullah

The Spearman rho nonparametric correlation coefficient is widely used to measure the strength and degree of association between two variables. However, outliers in the data can skew the results, leading to inaccurate results as the Spearman correlation coefficient is sensitive toward outliers. Thus, the robust approach is used to construct a robust model which is highly resistant to data contamination. The robustness of an estimator is measured by the breakdown point which is the smallest fraction of outliers in a sample data without affecting the estimator entirely. To overcome this problem, the aim of this study is two-fold. Firstly, researchers have proposed a robust Spearman correlation coefficient model based on the MM-estimator, called the MM-Spearman correlation coefficient. Secondly, to test the performance of the proposed model, it was tested by the Monte Carlo simulation and contaminated air pollution data in Kuala Terengganu, Terengganu, Malaysia. The data have been contaminated from 10% to 50% outliers. The performance of the MM-Spearman correlation coefficient properties was evaluated by statistical measurements such as standard error, mean squared error, root mean squared error and bias. The MM-Spearman correlation coefficient model outperformed the classical model, producing significantly smaller standard error, mean squared error, and root mean squared error values. The robustness of the model was evaluated using the breakdown point, which measures the smallest fraction of outliers that can be present in sample data without entirely affecting the estimator. The hybrid MM-Spearman correlation coefficient model demonstrated high robustness and efficiently handled data contamination up to 50%. However, the study has a limitation in that it can only overcome data contamination up to a maximum of 50%. Despite this limitation, the proposed model provides accurate and efficient results, enabling management authorities to make sound decisions without being affected by contaminated data. The MM-Spearman correlation coefficient model provides a valuable tool for researchers and decision-makers, allowing them to analyze data with a high degree of accuracy and robustness, even in the presence of outliers.

]]>Mashadi Abdul Hadi and Sukono

Recently, there are a lot of arithmetic interval forms. One of them only defines nonnegative interval numbers, whereas another one defines all forms of intervals. However, there are not many differences among the many arithmetic forms that were provided, particularly for addition and subtraction. For multiplication, division or inverse, there are many types of operations offered. But the problem is how to determine inverse of an interval number. There are many alternative offers to determine inverse of an interval number . But only for certain cases and for many cases, we have which is not equal to interval number . Based on these conditions, in this article an analysis of the issues with several existing interval algebras will be given and based on the analysis an alternative will be proposed to determine the form of multiplication and inverse from an interval number, which begins to define the positivity of an interval number with mid-point and then we construct algebra operations especially for multiplication. From the multiplication operation, we can construct the inverse form of an interval number . Furthermore, it is proven that for numbers of interval where , there is an interval number , so that it applies .

]]>Mahfouz Maha I. Rashwan Mahmoud M. and Khadr Zeinab A.

Optimal allocation of stratified sample is obtained either by minimizing the variance of the sample estimate for a fixed total cost of the survey or the total cost of survey for the fixed precision of the estimate. Actually, the survey cost and the variance of the estimate move in opposite directions, that is minimizing any of them results in increasing the other. Moreover, in practice, due to the uncertainty in the population data, the variances as well as the costs should be treated as random variables. In this paper, a multivariate optimal stochastic compromise allocation is proposed using multi-objective mathematical programming model that simultaneously minimizes both of the total cost of the survey as well as the individual variances of the overall stratified mean of each of the characteristics of interest. The proposed Stochastic Programming model is to be solved using the Chance-Constrained Programming technique. The proportional increase in the variance of the estimator under the optimum variance and under the optimum cost is set as a constraint and is upper-bounded by a pre-determined quantity. Simulation-based comparative study is conducted to assess the performance of the proposed allocation versus other optimal allocation techniques. Based on the criteria used for comparison, the findings show that the suggested model produced the highest efficient estimators with the highest precision, and efficient allocation of the sample size to the strata that accounts for the differences in the strata sizes and the variation within strata.

]]>Ridha G. Karem Karwan H. F. Jwamer and Faraidun K. Hamasalh

Although there are theoretical conclusions about the existence, uniqueness, and properties of solutions to ordinary and partial differential equations, only the simplest and most straightforward particular problems can usually be solved explicitly, especially when nonlinear terms are involved, and we typically develop approximation. In order to resolve the form problem of fractional order beginning value (1) by lacunary interpolation with a fractional degree spline function, the main goal of this paper is to investigate and improve some approximate solutions as well as new approximate solution techniques that have been proposed for the first time. From a practical standpoint, the numerical solution of these differential equations is crucial because only a tiny portion of equations can be resolved analytically. For fractional differential equations that are sensitive to the beginning conditions, we provide a fractional spline approach. The polynomial coefficient-based spline interpolation must be constructed using the Caputo fractional integral and derivative. For the given spline function, a stability analysis is completed after investigating error boundaries. The numerical rationale for the suggested technique is thought to use three cases. The outcomes demonstrate how effective the spline fractional technique is in interpolating the coefficient with fractional polynomials. Finally, to demonstrate the effectiveness and correctness of the suggested strategy, general procedure programs are created in MATLAB and used to a number of instructive cases.

]]>B. M. Cerna Maguina and Miguel A. Tarazona Giraldo

In this work, we prove in a very particular way the theorems of Dvoretzky-Roger's, Shur's, Orcliz's and Theorem 14.2 in their versions presented in the text [3]. The demonstrations of these Theorems carried out by us consist in establishing an appropriate link between the object of study and the relation that affirms that, for any real numbers , there exists a unique real number such that . Once the nexus is established, we use the definition of weak or strong convergence together with the Hahn-Banach Theorem to obtain the desired results. The relation is obtained by decomposing the Hilbert space as the direct sum of a closed subspace and its orthogonal complement. Since the dimension of the space is finite, this guarantees that any linear functional defined on the space is continuous, and this guarantees that the kernel of said linear functional is closed in the space . Therefore we have that the space breaks down, as the direct sum of the kernel of the continuous linear functional and its orthogonal complement, that is: , where the dimension of ker and the dimension of .

]]>Akshaya Ramesh and S. Udayabaskaran

A new type of single server queue is considered. In this type, the server asks for an assignment in a multi-level environment and the customer develops impatience during the assignment process. The environment has N levels and the server is assigned to operate in one of these levels with level dependent arrival and service rates. Customers arrive at the system all the time and there is an infinite buffer with the system. The assignment is done by a random switch which can initiate an assignment process only if at least one customer is in the system. The server working in any level of the environment reports to the random switch after serving the last customer in that level. Customers are not flushed out at any time. The random switch initiates an assignment process immediately at the epoch of arrival of a customer to the system. Assignment time is random and during the assignment period, customers are permitted to join the system. Once the assignment process starts, each customer waiting in the buffer clicks on a random impatience timer with him/her and leaves the system in case his/her timer ends before the assignment to the server is made. For this model, steady-state probabilities are found and a performance analysis is also made.

]]>Nor Syahida Mohamad Jumat Sulaiman Azali Saudi and Nur Farah Azira Zainal

In this paper, an efficient and reliable algorithm has been established to solve the second kind of FIE based on the lower-order piecewise polynomial and the lower-order quadrature method, namely Half-sweep Composite Trapezoidal (HSCT), which was used to discretize any integral term. Furthermore, due to the benefit of the complexity reduction technique via the half-sweep iteration concept presented from previous studies based on the cell-centered approach, this paper attempts to derive an HSCT piecewise linear collocation approximation equation generated from the discretization process of the proposed problem by considering the distribution of node points with vertex-centered type. Using half-sweep collocation node points over the linear collocation approximation equation, we could construct a system of HSCT linear collocation approximation equations, whose coefficient matrix is huge-scale and dense. Furthermore, to attain the piecewise linear collocation solution of this linear system, we considered the efficient algorithm of the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method. Therefore, several numerical experiments of the proposed iterative methods have been implemented by solving three tested examples, and the obtained results that were based on three parameters, namely iteration quantity, accomplished time, and maximum absolute error, were recorded and compared against other two iterations, namely Full-Sweep Gauss-Seidel (FSGS) and Half-Sweep Gauss-Seidel (HSGS).

]]>Dlovan Haji Omar Salah Gazi Shareef and Bayda Ghanim Fathi

Optimization refers to the process of finding the best possible solution to a problem within a given set of constraints. It involves maximizing or minimizing a specific objective function while adhering to specific constraints. Optimization is used in various fields, including mathematics, engineering, economics, computer science, and data science, among others. The objective function can be a simple equation, a complex algorithm, or a mathematical model that describes a system or process. There are various optimization techniques available, including linear programming, nonlinear programming, genetic algorithms, simulated annealing, and particle swarm optimization, among others. These techniques use different algorithms to search for the optimal solution to a problem. In this paper, the main goal of unconstrained optimization is to minimize an objective function that uses real variables and has no value restrictions. In this study, based on the modified conjugacy condition, we offer a new conjugate gradient (CG) approach for nonlinear unconstrained problems in optimization. The new method satisfied the descent condition and the sufficient descent condition. We compare the numerical results of the new method with the Hestenes-Stiefel (HS) method. Our novel method is quite effective according to the number of iterations (NOI) and the number of functions (NOF) evaluated, as demonstrated by the numerical results on certain well-known non-linear test functions.

]]>K R Karthikeyan and Senguttuvan Alagiriswamy

The study of Univalent Function Theory is very vast and complicated, so simplifying assumptions were necessary. In view of the Riemann Mapping theorem, the most apt thing would be to replace an analytic function defined on an arbitrary domain with an analytic function defined in the unit disc and having a Taylor's series expansion of the form . The powers of the series are usually integers, so all the prerequisite results also support the study of analytic functions having a series expansion with integers powers. The main deviation presented here is that we have defined a subclass of analytic functions using a Taylor's series whose powers are non-integers. To make this study more comprehensive, Janowski function which maps the unit disc onto a right half plane has been used in conjunction with two primary tools namely Subordination and Hadamard product. Motivated by the well-known class of λ-convex functions, here we have defined a fractional differential operator which is a convex combination of two analytic functions. Using the defined fractional differential operator, we introduce and study a new class of analytic functions involving a conic region impacted by the Janowski function. Necessary and sufficient conditions, coefficient estimates, growth and distortion bounds have been obtained for the defined function class. Since studies of various subclasses of analytic functions with fractional powers are rare, here we have pointed out several closely related studies by various authors. However, the superordinate function is a familiar function which has lots of applications.

]]>Wasan AL Shemmari and Ahmed AL Adilee

Statistical distributions play an essential part in the process of interpreting experimental data; nevertheless, choosing a distribution appropriate for the data that is currently available is not an easy task. Extending a known family of distribution to construct a new one is a long-honored technique. We suggest a new distribution named kumaraswamy generalized half-Logistic distribution (KW-GHLD). This distribution is obtained by adding two parameters to the existing model to raise its ability to fit complex data sets. Many mathematical and statistical properties were investigated, such as the survival function, the hazard function, the moment, the moment generating function, the incomplete moments, the Renyi entropy, the stochastic ordering, the probability-weighted moments, the order statistics, and the quantile function. The maximum likelihood method is utilized to make estimates for the KW-GHL distribution's unknown parameters. We study the efficacy of the desired distribution (KW-GHLD) by applying it to some real data set, which has been discussed within the measures of goodness of fit (AIC, BIC, CAIC, and QHIC) and comparing the outcomes with those obtained by the original distribution (GHLD), which produced best outcomes. This allowed us to determine whether or not the desired distribution is effective. Finally, we present several conclusions related to our findings.

]]>G. Vinitha P. Godhandaraman and V. Poongothai

The two heterogeneous servers of the Markovian retrial queue model with an additional server, impatience behavior and vacation are presented in this research paper. An arriving customer who finds accessible servers gets immediate service. Otherwise, if both servers are engaged, an entering customer will join in the orbit to retry and get their service after some random time. If any customers in the orbit discover that the waiting time is longer than expected, they may leave without receiving the service. We consider two servers with different service rates to provide the service based on "First Come, First Served". When the number of customers in orbit increases occasionally, we will instantly activate an additional server to reduce the queue size. After the orbit becomes null, the server goes for maintenance activity. The practical application is given to justify our model. The proposed model was obtained using the birth-death process and the equations were governed using Chapman-Kolmogorov equations. Finally, we have solved the equations using a recursive approach and the performance indices are derived to improve quality and efficiency.

]]>Zabidin Salleh Muzafar Nurillaev and Che Mohd Imran Che Taib

In topological spaces, although compactness is satisfying the product invariant properties, but for the Lindelöffness, it is not preserved by the product unless one or more factors are assumed to satisfy additional conditions. Similar results yield for the bitopological spaces, that is, the property of pairwise Lindelöf bitopological spaces is not preserved under the product unless one or more factors are assumed to be satisfy additional conditions, for instance, -spaces. The Cartesian product for arbitrarily many bitopological spaces was defined by Datta in 1972. Since then, many researchers have begun their study for the product bitopological spaces for their reason and direction. In this paper, we shall study finite product of pairwise nearly Lindelöf, pairwise almost Lindelöf and pairwise weakly Lindelöf spaces. We show that, all these generalized pairwise Lindelöf spaces are not preserved under a product by some counterexamples provided. Furthermore, we give some necessary conditions for these three bitopological spaces to be preserved under a finite product. Such condition is that one or more of the spaces has to be -space or the product have to be pairwise weak -space. Another interesting result is that the projection of these generalized pairwise Lindelöf spaces product with -space is a closed map.

]]>I.S. Rakhimov

In the paper, we propose three isomorphism criteria for a subclass of finite-dimensional Leibniz algebras. Isomorphism Criterion 1 has been given earlier (see [5]). We introduce notations for new structure constants. Using the new notation, we state the isomorphism criterion 2. To formulate Isomorphism Criterion 3, we introduce "semi-invariant functions" needed. We prove that these three Isomorphism Criteria are equivalent. The isomorphism criterion 3 is convenient to find the invariant functions to represent isomorphism classes. The proof of the isomorphism criteria in the general case is computational and is based on hypothetic convolution identities given in [11]. Therefore, we give details in the ten-dimensional case.

]]>Abdujabar Rasulov

The application of Monte Carlo methods in various fields is constantly growing due to increases in computer capabilities. Increasing speed and memory, and the wide availability of multiprocessor computers, allow us to solve many problems using the "method of statistical sampling", better known as the Monte Carlo method. Monte Carlo methods are known to have particular strengths. These include: Algorithmic simplicity with a strong analogy to the underlying physical processes, solve complex realistic problems that include sophisticated geometry and many physical processes, solve problems with high dimensions, the ability to obtain point solutions or evaluation linear functional of the solution, error estimates can be empirically obtained for all types of problems in parallel way, and ease of efficient parallel implementation. A shortcoming of the method is slow rate of convergence of the error, namely ) where is the number of numerical experiments or realizations of the random variable. In this paper, we will propose Monte Carlo algorithms for the solution of the interior Dirichlet boundary value problem (BVP) for the Helmholtz operator with a polynomial nonlinearity on the right-hand side. The statistical algorithm is justified and complexity of the proposed algorithms is investigated, also the ways of decreasing the computational work are considered.

]]>N. Saranya and K. Suja

This paper aims to explore the fundamental properties of statistical convergence sequences within non-Archimedean fields. In pure mathematics, statistical convergence plays a fundamental role. The idea of statistical convergence is an extension of the concept of convergence. Statistical convergence has been discussed in various fields of mathematics namely ergodic theory , fuzzy set theory, approximation theory, measure theory, probability theory, trigonometric series, number theory, and banach spaces, where problems were resolved using the concept of statistical convergence. Summability theory and functional analysis are two disciplines that heavily rely on the idea of statistical convergence. The study of analysis over non-Archimedean fields is called non-Archimedean analysis. The theory of statistical convergence plays a significant role in the functional analysis and summability theory. The objective of this paper is to expand upon the concepts of statistical convergence and statistically Cauchy sequences in non-Archimedean intuitionistic fuzzy normed spaces, and obtain some relevant results related to them. This article proves that some properties of statistically convergent sequences, which are not true classically, are true in a non-Archimedean field. Furthermore, in these spaces, we defined statistically complete and statistically continuous and established some fundamental facts. Throughout this paper, denotes a complete, non-trivially valued, non-Archimedean field.

]]>Etis Sunandi Khairil Anwar Notodiputro Indahwati and Agus M Soleh

Small Area Estimation (SAE) is a statistical method used to estimate parameters in sub-populations with small samples. This study aims to develop a Beta-Binomial model on SAE with a Hierarchical Likelihood (HL) approach. The model built is called the SAE-BB-HL model. This research begins by deriving a formula for estimating model parameters analytically. A good fit is calculated with the Mean Square Error of Prediction (MSEP) and bias. This study used simulation data and data from the National Socio-Economic Survey (SUSENAS) and Village Potential (PODES) of Bengkulu Province for 2021 collected by Statistics Indonesia (BPS). The simulation study aims to evaluate the SAE-BB-HL model. Simultaneously, the application study aims to predict the illiteracy rate per sub-district in Bengkulu Province. The simulation study results show that the parameter estimates of random area distribution are very close to the actual parameters. It also reveals that the bias and MSEP estimates of the proportion of HL are lower than the direct estimates. In addition, the results of this study show that the SAE-BB-HL model can improve the accuracy and precision of proportion estimation. Applying the SAE-BB_HL model to real data shows that the predictive value of the illiteracy rate tends to be higher when compared to the direct estimator.

]]>Faisal and Andreas Martin

The concept of partition dimension in graph theory was first introduced by Chartrand et al. [1] as a variation of metric dimension. Since then, numerous studies have attempted to determine the partition dimensions of various types of graphs. However, for many types of graphs, their partition dimensions remain unknown as determining a general graph's partition dimension is an NP-complete problem. In this study, we aim to determine the partition dimension of a specific graph, namely the comb product of a wheel and a tree. One approach to finding the partition dimension of a graph is to determine its upper and lower bounds. In this article, we propose an upper bound for the partition dimension of the comb product using number representation for certain bases. We divide the problem into two cases based on the path graph. For the first case, which is the comb product with a path of a single vertex, Tomescu et al. [2] have already provided an upper bound. In the other case, we utilize the bijection property of a number system on the number copy of the tree to find an upper bound. Our results show that the partition dimension of the second case has a smaller upper bound compared to the general upper bound proposed by Chartrand et al. [1].

]]>Muhammad Ashraf Darus Nurul Huda Abdul Aziz Deraman F. Asi Salina M. S. Anuar and Zakaria H. L.

This article presents the numerical approximation of Volterra integro-differential equations (VIDEs) of the second kind using the quadrature rule in the modified block method. The new implementation of new block method which considers the closest point to approximate two solutions of and concurrently was taken into account. This method is said to have an advantage in reducing the number of total steps and function evaluations compared to the classical multistep method. The techniques of quadrature rule which consist of the trapezoidal rule, Simpson's 1/3 rule, Simpson's 3/8 rule and Boole's quadrature rule have been used to approximate the integral parts of Kernel function, for for the case of . The analysis of the order, error constant, consistency and convergence of VIDEs in the proposed method has also been presented. The stability analysis is derived using the specified linear test equation for both approximate solutions until obtained the stability polynomial. To validate the efficiency of the developed method, some of the numerical results are presented and compared with the existing method. It is shown that the modified block method has given better accuracy and efficiency in terms of maximum error and number of steps and function calls.

]]>Sumarni Abu Bakar Noor Syamsiah Mohd Noor Tahir Ahmad and Siti Salwana Mamat

Autocatalytic Set (ACS) is one of the areas of study that can be modelled using graph theory. An Autocatalytic Set (ACS) is defined as a graph, in which there is at least one incoming link for every node in the graph. Past research on ACS tremendously solved many applications including modelling complex systems through integration of ACS with fuzzy theory. Recently, a restricted form of ACS known as Weak Autocatalytic Set (WACS) was established and used to solve multi-criteria decision-making problems (MCDM), in which the related graph is transitive and involves non-cyclic triads. Though, in scenarios that occur in the real world, there exist MCDM problems, in which the related graph is intransitive, involving cyclic triads. Thus, it creates a limitation to used WACS to solve decision-making problems over cyclic triads. This paper introduced another class of ACS known as Bounded Autocatalytic Set (BACS). The concept of BACS provides the ability to represent a relation between one criterion to each other criterion, and the graph involves cyclic triads. Here, the definition of BACS is formed and introduced for the first time, and its basic properties related to edges, paths, and cycles in the form of theorem and propositions are established and presented.

]]>Nazrina Aziz Seu Wen Fei Waqar Hafeez Shazlyn Milleana Shaharudin and Javid Shabbir

The acceptance sampling is a technique for ensuring that both producers and consumers are satisfied with the product's quality. This paper proposes a new group chain sampling plan (NGChSP) using Log-logistic distribution when the life test is truncated at a predetermined time. The minimum number of groups, and the probability of lot acceptance, are determined through satisfying the consumer's risk, under the specified design parameter. This paper shows that the minimum number of groups, decreases when the value of design parameters such as and increases. With the same design parameters, the minimum increases when the shape parameter increases. Moreover, the increases as shape parameter and minimum increases. An illustrative example for NGChSP is provided. The findings suggest that as the test time termination constant decreases, the minimum increases. Furthermore, as the mean ratio, increases, the increases as well. In comparison to GChSP, the NGChSP requires a smaller number of groups, indicating that using the NGChSP for inspection will contribute to lower inspection time and costs. The NGChSP provides a higher probability of lot acceptance than GChSP. This paper concludes that the NGChSP performed better than the GChSP. Therefore, the NGChSP is better equipped for lot inspection in the manufacturing industry.

]]>S. Manicka Vinayagam L. Meenakshi Sundaram and C. Devamanoharan

The purpose of this article is to define and analyse certain new types of a strongly open set namely () in nano ideal topological space and compare it with the other existing sets in nano ideal topology. Here the author uses the lower approximation, upper approximation and boundary region to define nano topology. To emphasize the inclusive relationship of this particular nano ideal set with other existing familiar nano ideal sets like , , and , some counter examples are provided. We have also established the independence of this set with both set and set in nano ideal topological spaces. In addition, , , are introduced, investigated with its basic results and fundamental properties. The Exterior operator plays a vital role in topological spaces. Unless like the interior operator, the exterior operator varies in some cases, for example it reverses inclusions when it comes to the subset property in topological spaces. In the next section, we have defined and analysed some of its basic properties. We have also introduced and discussed its correlation between and . The paper finally concludes with the definition of and describes the relationship of with , and .

]]>Sangeeta A. Parthiban and P. Selvaraju

Researchers have constructed a model to transform "word motion problems into an algorithmic form" in order to be processed by an intelligent tutoring system (ITS). This process has the following steps. Step 1: Categorizing the characteristics of motion problems, step 2: suggesting a model for the categories. "In order to solve all categories of problems, graph theory including backward and forward chaining techniques of artificial intelligence can be utilized". The adoption of graph theory into motion problems has evidence that the model solves almost all of motion problems. Graph labeling is sub field of graph theory which has become the area of interest due to its diversified applications. Formally, if the nodes are labeled under some constraint, the resulting labeling is known as vertex labeling and it will be an edge labeling if the labels are assigned to edges under some conditions. Graph labeling nowadays is one of the rapid growing areas in applied mathematics which has shown its presence in almost every field. The known applications are in Computer Science, Physics, Chemistry, Radar, Coding Theory, Connectomics, Socioloy, x-ray crystallography, Astronomy etc. "For a graph G(V,E) and k > 0, give node labels from {0, 1, . . . , k − 1} such that when the edge labels are induced by the absolute value of the difference of the node labels, the count of nodes labeled with i and the count of nodes labeled with j differ by at most one and the number of lines labeled with i and with j differ by at most 1. So G with such an allocation of labels is k−equitable and becomes 3-equitable labeling, when k = 3". In this paper, the existence and non-existence of 3-equitable labeling of certain graphs are established.

]]>Mimoza Shkembi Stela Ceno and John Shkembi

Integration in Riesz spaces has received significant attention in recent papers. The existing body of literature provides comprehensive analyses of the concepts related to order-type integrals for functions that are defined in ordered vector spaces and Banach lattices, as indicated by the studies covered in [3], [4], [5], [7], [8], [9], and [10]. In our work on strongly order-McShane (Henstock-Kurzweil) equiintegration, we have drawn upon the earlier works of Candeloro and Sambucini [6], as well as Boccuto et al. [1-2], who have conducted investigations in the field of order-type integrals. We have expanded upon their research to develop our own findings. This paper focuses on studying the (o)-McShane integral in ordered spaces, where we emphasize the important fact that investigating the (o)-McShane integral is essential in addition to the (o)-Henstock integral. We highlight that the (o)-McShane integration in Banach lattices has richer properties and is more convenient compared to the (o)-Henstock integral. The properties of (o)-convergence exhibited by ordered McShane integrals are prominently featured in our study. By using (o)-convergence, we have obtained valuable results related to the (o)-McShane integral. We arrive at the same results in Banach lattices as on McShane (Henstock-Kurzweil) norm-integrals, and we demonstrate that the (o)-McShane integral opens up a wide field of study where similar results with Henstock integration can be obtained. The outcomes demonstrate the benefits of utilizing this integration technique in ordered spaces, with potentially significant implications for diverse areas of mathematics and related fields.

]]>Wikanda Phaphan Teerawat Simmachan and Ibrahim Abdullahi

The three-parameter weighted mixture generalized gamma (WMGG) distribution was developed from the four-parameter mixture generalized gamma (MGG) distribution since the parameter estimation of MGG distribution faced with the problem. The estimate of the weighted parameter p was out of the interval [0, 1]. The previous study proposed the maximum likelihood estimators (MLEs) of the WMGG distribution. However, their MLEs were written in nonlinear equations, and certain iterative methods were necessarily needed to solve numerically. The three parameters λ, β, and α were estimated by the quasi-Newton method. Nevertheless, this method performed well only the parameter λ. This motivated the main objective of this work. Consequently, the parameter estimation of the WMGG was further improved. This article developed two maximum likelihood estimation methods: expectation-maximization (EM) algorithm and simulated annealing algorithm of the three parameters of the WMGG distribution. These two methods were compared to the previous study's quasi-Newton method. Monte Carlo simulation technique was employed to assess the algorithm's performance. Sample sizes ranged from small to large as 10, 30, 50, and 100. The simulation was repeated 10,000 rounds in each scenario. Assessment criteria were the mean square error (MSE) and bias. The results revealed that the EM algorithm outperformed the other methods. Furthermore, the quasi-Newton method had the lowest efficiency.

]]>Anvarjon Sharipov and Mukhamedali Keunimjaev

The theory of polyhedra and the geometric methods associated with it are not only interesting in their own right but also have a wide outlet in the general theory of surfaces. Certainly, it is only sometimes possible to obtain the corresponding theorem on surfaces from the theorem on polyhedra by passing to the limit. Still, the theorems on polyhedra give directions for searching for the related theorems on surfaces. In the case of polyhedra, the elementary-geometric basis of more general results is revealed. In the present paper, we study polyhedra of a particular class, i.e., without edges and reference planes perpendicular to a given direction. This work is a logical continuation of the author's work, in which an invariant of convex polyhedra isometric on sections was found. The concept of isometry of surfaces and the concept of isometry on sections of surfaces differ from each other, examples of isometric surfaces that are not isometric on sections and examples of non-isometric surfaces that are isometric on sections. However, they have non-empty intersections, i.e., some surfaces are both isometric and non-isometric on sections. In this paper, we prove the positive definiteness of the found invariant. Further, conditional external curvature is introduced for "basic" sets, open faces, edges, and vertices. It is proved that the conditional curvature of the polyhedral angle considered is monotonicity and positive definiteness. At the end of the article, the problem of the existence and uniqueness of convex polyhedra with given values of conditional curvatures at the vertices is solved.

]]>Muhamad Deni Johansyah Asep Kuswandi Supriatna Endang Rusyaman Salma Az-Zahra Eddy Djauhari and Aceng Sambas

Fractional differential equations (FDEs) are differential equations that involve fractional derivatives. Unlike ordinary derivatives, fractional derivatives are defined by fractional powers of the differentiation operator. FDEs can arise in a variety of contexts, including physics, engineering, biology, and finance. They are typically more complex than ordinary differential equations, and their solutions may exhibit unusual properties such as long-range memory, non-locality, and power-law behavior. The solution of the Riccati Fractional Differential Equation (RFDE) is generally challenging due to its nonlinearity and the presence of the fractional power term. The fractional derivative operators in the RFDE are non-local and involve an integral over a certain range of the independent variable. The non-local nature of the fractional derivatives can make the RFDE harder to handle compared to ordinary differential equations. In this paper, we have examined the Riccati Fractional Differential Equation (RFDE) using the combined theorem of the Adomian Decomposition Method and Laplace Transform (ADM-LT). Furthermore, we have compared with Adomian Decomposition Method and Kashuri-Fundo Transformation (ADM-KFT). It is shown that the ADM-LT is equivalent to the ADM-KFT algorithm for solving the Riccati equation. In addition, we have added new theorem of the relationship between the Kashuri Fundo inverse and the Laplace Transform inverse. The main finding of our study shows that the Adomian Decomposition Method and Laplace Transform (ADM-LT) have a good agreement between numerical simulation and exact solution.

]]>Abdul Hadi Bhatti and Sharmila Binti Karim

Many researchers frequently developed numerical methods to explore the idea of solving ordinary differential equations (ODEs) approximately. Scholars started evolving approximation methods by developing algorithms to improve the accuracy in terms of error for the approximate solution. Polynomials, piece-wise polynomials in the form of Bézier curves, Bernstein polynomials, etc., are frequently used to represent the approximate solution of ODEs. To get the minimum error between the exact and approximate solutions of ODEs, the DP Ball curve (DPBC) using the least squares method (LSM) is proposed to improve the accuracy of the approximate solutions for the initial value problem IVPs. This paper explores the use of control points of the DPBC with error reduction by minimizing the residual function. The residual function is minimized by constructing the objective function by taking the sum of squares of the residue function for the least residual error. Then, by solving the constraint optimization problem, we obtained the best control points of DPBC. Two strategies are employed: investigating DPBC's control points through error reduction with LSM and computing the optimum control points through degree raising of DPBC for the best approximate solution of ODEs. Substituting the values of control points back into the DPBC allows for the best approximate solution to be obtained. Moreover, the convergence of the proposed method to the IVPs is successfully analyzed in this study. The error accuracy of the proposed method is also compared with the existing studies. Numerous numerical examples of first, second, and third orders are presented to illustrate the efficiency of the proposed method in terms of error. The results of the numerical examples are shown in which the error accuracy is considerably improved.

]]>Maizon Mohd Darus Haslinda Ibrahim and Sharmila Karim

A sequence is simply an ordered list of numbers. Sequences exist in mathematics very often. The Fibonacci, Lucas, Perrin, Catalan, and Motzkin sequences are a few that have drawn academics' attention over the years. These sequences have arisen from different perspectives. By investigating the construction of each sequence, these sequences can be classified into three groups, i.e., those that arise from nature, are constructed from other existing sequences, or are generated from geometric representation. This outcome may assist the researchers in adding a new number sequence to the family of sequences. Our observation of the geometric representation of the Motzkin sequence shows that a new sequence can be constructed, namely the Wing sequence. Therefore, we demonstrate the iterations of the Wing sequence for 3≤n≤5. The wings are constructed by classifying them into (n-1) classes and determining the first and second points. It will then provide (n-2) wings in each class. This technique will construct (n-1)(n-2) wings for each n. The iterations may provide a basic technique for researchers to construct a sequence using the technique of geometric representation. The observation of geometric representations can develop people's thinking skills and increase their visual abilities. Hence, the study of geometric representation may lead to new lines of research that go beyond only sequences.

]]>Bernadhita Herindri Samodera Utami Mustofa Usman Warsono and Fitriani

Measure theory is used as the basis for probability theory. One of the most useful measure theories for statistics and probability theory is the concept of distance. The concept of distance introduced in the inner product space is closely related to the order relation in each sequence of elements. In statistics, random variables can be seen as a sequence that can be an object to study, including the partial ordering relation, expectation value, convergence, also infimum and supremum. This study aims to obtain the properties of a partial ordering relation which is useful for forming probability Hilbert spaces, more specifically the σ-algebra. If in ordinary sets, σ-algebra uses the concept of intersection and combination of sets, in probability Hilbert space, σ-algebra uses the concept of partial relation ordering, lattice, and indicator lattice. This research is quantitative research with a method of proof to generalize the concept of the order of elements. The novelty of this research is to find the associative properties of lattice in Hilbert probability space as described in Corollary 1. Furthermore, based on the definition of absolute value in Hilbert probability space, we derive the properties of addition and subtraction of absolute values and find their relationship with the lattice stated in Proposition 1. In the Hilbert probability space, the convergence property of random variables also applies which results in the lattice convergence stated in Proposition 2. Finally, it can be shown that the set of indicators in the Hilbert probability space form the algebra σ which is stated in Proposition 3. This study also gave use of the dataset shares of 42 energy companies in Indonesia in 2022. The results of plotting the data using the probability density function of the Normal distribution, Log-Normal distribution, and Cauchy distribution.

]]>R. Gurusamy A. Meena Kumari and R. Rathajeyalakshmi

The Steiner p-antipodal graph of a connected graph G, has vertex set like G and p number of vertices are adjacent to each other in whenever they are p-antipodal in G. If G has more than one component, then p vertices are adjacent to each other in if at least one vertex from different components. Draw K_{p} related to p-antipodal vertices in . The Steiner antipodal number of a graph G is the smallest natural number p, so that the Steiner p-antipodal graph of G is complete. In this article, Steiner antipodal number has been determined for the generalized corona of graphs and for each natural number p≥2, we can construct many non-isomorphic graphs of order p having Steiner antipodal number p. Also for any pair of natural numbers l,m ≥ 3 with l ≤ m, there is a graph whose Steiner antipodal number is l and Steiner antipodal number of its line graph is m. For every natural number p≥1, there is a graph G whose complement has Steiner antipodal number p.

Shakila Banu P. and Suganthi T.

Rezaei [7], who introduced the first simple graph G, defined it as a quadratic residue graph modulo n if its vertex set is reduced, a residue system modulo n such that two different vertices a and b are nearby, and (mod n). This initiates to study the present article, here we introduce a cartesian product of quadratic residue graphs , where m and n are either prime or composite, and G_{m} and H_{n} are quadratic residue graphs, respectively. The aforementioned work suggests and evaluates the regular graphs that are produced from graph F and its adjacency matrix. In addition, we define and examine their generating matrices with the help of adjacency matrix of F. Also, in this article we define three linear codes that are taken from the graph F and the parameters of codes denotes [N, k, d], where N denotes length, k denotes the dimension which is taken from the number of vertices and d denotes the distance which is taken from the minimum degree. Moreover, we also introduce an encoding and decoding algorithm for the graph using binary bits which is illustrated with a suitable example. Finally, we test the error correction capability of the code by using sphere packing bounds.

Julan HERNADI Ceriawan H. SANTOSO and Iwan T. R. YANTO

Ecological systems can be quite complex, consisting of an interconnected system of plants and animals, predators and prey, flowering plants, seed dispersers, insects, parasites, pollinators, and so on. In the case of the existence of a species affecting the survival of other species and vice versa, it can derive a competitive model in the form of a system of differential equations. A competitive model involves a number of parameters which grow in proportion to the number of interacting species. The resistance of a state variable to tiny disturbances of some parameter is referred to as sensitivity. The competitive model of size N consists of N parameters for intrinsic growth, N parameters for carrying capacity, N^{2} −N parameters for species interaction, and N parameters for initial conditions. As a result, there will be N^{2}(N + 2) distinct values of sensitivity. The purpose of this paper is to derive a general formulation of the sensitivity equations of dynamical system and then apply it to the competitive model. This study also encompasses the formulation of some algorithms and the implementation for solving the sensitivity equation numerically. Finally, the sensitivity functions are employed as qualitative instruments in the optimal design of measurement for parameter estimation through a series of numerical experiments. The results of this study are the ordinary and the generalized sensitivity functions for interacting species. Based on numerical experiments, each group of data provides different information about the existing parameters.

John Abonongo Ivivi J. Mwaniki and Jane A. Aduda

The usefulness of heavy-tailed distributions for modeling insurance loss data is arguably an important subject for actuaries. Appropriate use of trigonometric functions allows a good understanding of the mathematical properties, limits over parameterization, and gives better applicability in modeling different datasets. Thus, the proposed method ensures that no additional parameter(s) is/are introduced in the bit to make a distribution from the F-Loss family of distributions flexible. The purpose of this paper is to improve the flexibility of the F-Loss family of distributions without introducing any additional parameter(s) and to develop heavy-tailed distributions with fewer parameters that give a better parametric fit to a given dataset than other existing distributions. In this paper, a new heavy-tailed distribution known as sine Burr III Loss distribution is proposed using the sine F-Loss generator. This distribution is flexible and able to model varying shapes of the hazard rate compared with the traditional Burr III distribution. The densities exhibit different kinds of decreasing and right-skewed shapes. The hazard rate functions show different kinds of decreasing, increasing constant-decreasing, and upside-down bathtub shapes. The statistical properties and actuarial measures are studied. The skewness is always positive, and the kurtosis is increasing. The numerical values of the actuarial measures show that increasing confidence levels are associated with increasing VaR, TVaR, and TV. The maximum likelihood estimators are studied, and simulations are carried out to ascertain the behavior of the estimators. It is observed that the estimators are consistent. The usefulness of the proposed distribution is demonstrated with two insurance loss datasets and compared with other known classical heavy-tailed distributions. The results show that, the proposed distribution provides the best parametric fit for the two insurance loss datasets. Insurance practitioners can employ the proposed models in modeling insurance loss since they are flexible.

]]>Kalpana R and Shobana L

Molecular descriptors act as an important part in mathematical chemistry, in investigating quantitative structure-property relationship and quantitative structureactivity relationship. A topological descriptors, also called a molecular descriptor, is a mathematical formula applied to any graph which produces new molecular structure. In medicine mathematical model, the chemical compound is represented as an undirected graph, where each vertex represents an atom and each edge indicates a chemical bond between these atoms. The Wiener index is the first topological index to be used in chemistry introduced by Harold Wiener [1947]. It is used to compare the boiling points of some alkane isomers. There are various topological indices which are applied in chemistry. Among them, our interest is on Forgotten index which is degree based topological index introduced by Furtula and Gutman in 2015[2], defined as where d_{u} is the degree of vertex u in G. The mathematicians and chemists have studied several general properties of Forgotten index which may help the chemical and pharmaceutical industry to achieve the significance details by quantitative methods than by experiments.Vaidya et al (2009) proposed the concept of duplication of a vertex by an edge and duplication of an edge by a vertex of graphs. Shobana et al. proposed the double duplication of graphs (2017) [6]. Only connected, simple, undirected and finite graphs are considered throughout this article. Also, some inequalities are obtained by comparing the duplication and double duplication of graphs using Forgotten index which can also be used by chemists to generate new antidrug in future.

Nor Iza Anuar Razak and Zamira Hasanah Zamzuri

The significance of a model is affected by outliers. The outliers can affect the effectiveness of structural equation modeling (SEM). Here we describe and investigate the behavior of the nonparametric single and double residual bootstrap (DRB) methods in the presence of outliers when applied to SEM. Our study also intends to shorten the computational time of the standard double bootstrap by using an alternative double bootstrap approach. We demonstrate our proposed method by conducting a Monte Carlo experiment series for clean normal Gaussian distributions and contaminated data. The simulation studies were manipulated with different sample sizes, effect sizes, and 10% of contamination in the Y direction. The performance of the proposed method is evaluated using standard measurements and the construction of confidence intervals. The reasonably close parameter and bootstrap estimates suggest that the nonparametric single and double residual bootstrap is an excellent method. The DRB method showed a robust declining pattern for standard measurement estimates and shorter confidence intervals compared to the single residual bootstrap method in both normal and contaminated data. Also, the double bootstrap method takes twice as long as the single bootstrap method to compute. The DRB method is straightforward but demands slightly more computational time and better prediction approximation. This study offers additional perspectives to fellow researchers considering using the nonparametric single and alternative DRB methods with contaminated data.

]]>Budi Pratikno Fifthany Marchelina Napitupulu Jajang Agustini Tripena Br. Sb and Mashuri

The research determined the binary response model on logistics regression (LR) and its application. Firstly, we select some eligible factors (predictors,X_{i}, i=1,2,3,4 ) that are involved in the model, namely age ( X_{1} ), sex (X_{2}) , treatment (X_{3}) , and nutrition (X_{4}) , with the response (Y) being the case of tuberculosis (TB). Using the stepwise selection model and odd ratio (OR) interpretation, we have three suspected significant predictors (X_{1} ,X_{3}, and X_{4} ), but we choose two (only) of the significant predictors, which are X_{3} and X_{4}. Therefore, the logistics regression model is written as . To test the goodness of fit of the model, we used deviance test (p-value 0.08). Due to this p-value, we then used the level of significance which is 0.08 (nearly close to 0.05) for obtaining the significant model. For more detailed interpretation, we here noted that the OR of the age (X_{1}) , one of the three suspected significant predictors (X_{1} , X_{3} , and X_{4} ), is close to be one ( 1), so it is an independent predictor (not significant). So, we concluded that the significant predictors are only treatment (X_{3}) and nutrition (X_{4}) . Thus, the linear of the logistics regression model is then given as a So, we noted that TB is only dependent on clinical treatment and providing nutrition.

W. M. Mahmoud M. A. Soliman and Esraa. M. Mohamed

This research aims to study the Sweeping surface which is generated by the motion of the straight line (the profile curve) while this movement of the plane in the space is in the same direction as the normal to a cubic Bezier curve (spine curve). In geometrical modeling, sweeping is an essential and useful tool and has some applications, especially in geometric design. The idea depends on choosing a geometrical object which is the straight line, that is called the generator, and sweeping it along a cubic Bezier curve (spine curve), which is called trajectory, along the Cubic Bezier curve (spine curve) in an isotropic space has produced an Isotropic Bezier Sweeping Surfaces (IBSS). This study discusses Isotropic Bezier Sweeping Surfaces (IBSS) with the Bishop frame. We studied a special case of a surface sweep, which is the cylindrical surface resulting from a path curve that is a straight line. We have calculated the 1st fundamental and 2nd fundamental forms for this surface. The parametric description of the Weingarten Isotropic Bezier Sweeping Surfaces (IBSS) is also calculated in terms of Gaussian and mean curvatures. Mathematica 3D visualizations were used to create these curvatures. Finally, we characterized new associated surfaces according to the Bishop frame on (IBSS), such as studying minimal and developable isotropic Bezier sweeping surfaces (IBSS).

]]>Kamarun Hizam Mansor Oluwaseun Adeyeye and Zurni Omar

The numerical of second order initial value problems (IVPs) has garnered a lot of attention in literature, with recent studies ensuring to develop new methods with better accuracy than previously existing approaches. This led to the introduction of hybrid block methods which is a class of block methods capable of directly solving second order IVPs without reduction to a system of first order IVPs. Its hybrid characteristic features the addition of off-step points in the derivation of this block method, which has shown remarkable improvement in the accuracy of the block method. This article proposes a new three-step hybrid block method with three generalized off-step points to find the direct solution of second order IVPs. To derive the method, a power series is adopted as an approximate solution and is interpolated at the initial point and one off-step point while its second derivative is collocated at all points in the interval to obtain the main continuous scheme. The analysis of the method shows that the developed method is of order 7, zero-stable, consistent, and hence convergent. The numerical results affirm that the new method performs better than the existing methods it is compared with, in terms of error accuracy when solving the same IVPs of second order ordinary differential equations.

]]>D Kavitha K Dhanalakshmi and K Anitha

Normalised Error function has been coined and analyzed in 2018 [13].The concept of normalised error function discussed in [13], motivated us to find the new results of Toeplitz determinant for the subclasses of analytic univalent functions concurrent with error function. By seeing the history of error function in Geometric functions theory, Ramachandran et. al [13] derived the coefficient estimates followed by the Fekete-Szeg¨o problem for the normalised subclasses of starlike and convex functions associated with error function. Finding coefficient estimates is one of the most provoking concepts in geometric function theory. In current scenario scientists are concentrating on special functions which are connected with univalent functions. Based on these concepts, the present paper deals with supremum and infimum of Toeplitz determinant for starlike and convex in terms of error function with convolution product using the concept of subordination. Also, we derive the sharp bounds for probability distribution associated with error starlike and error convex functions.

]]>Banoth Madanlal Naik and V.Naga Raju

Fixed point technique can be considered as one of the most powerful tools to solve problems which occur in several fields like Physics, Chemistry, Computer Science, Economics and other subbranches of Mathematics etc. Banach [3] gave the first result in the field of metric fixed point theory which guarantees the existence and uniqueness of a fixed point in a complete metric space. Thereafter, many Mathematicians replace the notion of metric space and Banach contractive condition with various generalized metric spaces and different contractions to prove fixed point theorems. One such generalized metric space, called G-metric space, was proposed in [6]. Abhijit Pant, R.P.Pant [1] introduced a new type of contraction and obtained some results in metric spaces in the year 2017. The purpose of this paper is to define -complete G-metric space and study three metric fixed point results for such spaces. In the first two fixed point results, we use weaker form of continuity, called m-continuity and new type contractive conditions while in the third result simulation function is used. The results which we obtained will improve, extend and generalize some results in [1] and [2] in the existing literature. In addition to this, we give examples to validate our results.

]]>Rubul Moran Niranjan Bora and Surashmi Bhattacharyya

The windmill graph is the graph formed by joining a common vertex to every vertex of m copies of the complete graph K_{r}. T-coloring of a graph is a map h defined on the set of vertices in such a way that for any edge does not belong to a finite set T of non-negative integers. Strong T-Coloring (ST-coloring) is a particular case of T-coloring and is defined as the map: , for which and for any two distinct edges . Application of T and ST-coloring of graph naturally arises in the modeling of different scientific problems. Frequency assignment problem (FAP) is one of the well known problems in the field of telecommunication, which can be modeled using the concept of T and ST-coloring of graphs. In this paper, we will consider two special types of T-sets. The first one is -initial set, introduced by Cozzens and Roberts, which is of the form where S is any arbitrary set that doesn’t contain any multiple of The second one is λ-multiple of q set, introduced by Raychaudhuri, which is of the form , where S is a subset of the set . We will discuss some parameters related to these two types of colorings viz. T-chromatic number, T-span, T-edge span on the basis of the two T-sets. We will also deduce some generalized results of ST-coloring of any graph based on any T-set, and with the help of these results we will obtain ST-chromatic number and bounds for the ST -span and ST-edge span of windmill graphs.

Sai Lakshmi B. and G. Gajendran

Data classification is a significant task in the field of machine learning. Support vector machine is one of the prominent algorithms in classification. Twin support vector machine is a solitary algorithm evolved from support vector machine which has gained popularity owing to its better generalization ability to a greater extent. Twin support vector machine attains quick training speed by explicitly exploring a pair of non-parallel hyperplanes for imbalanced data. In a Twin support vector machine, choosing numerical values for hyper parameters is challenging. Hyper parameter tuning is a prime factor that enhances the performance of a model. However, randomly preferred hyper parameters in the Twin support vector machine are uncertain. This paper proposes a novel p-dist-based regularized Twin support vector machine for imbalanced binary classification problems. Pairwise distances such as Jaccard and Correlation distances are considered for attuning the hyper parameters. The proposed work has been analyzed on many publicly available real-world benchmark datasets for both linear and non-linear cases. The performance of the p-dist-based regularized Twin support vector machine is computationally tested and compared with existing models. The outcome of the proposed model is validated using quality metrics such as Accuracy, F - mean, G-mean, and Elapsed time. Ultimately, the significant result exhibits better performance with less computational time in comparison to several existing methods.

]]>Akshaya Ramesh and S. Udayabaskaran

In this paper, we consider a single server queueing system operating in a random environment subject to disaster, repair and customer impatience. The random environment resides in any one of N + 1 phases 0, 1, 2, · · · ,N + 1. The queueing system resides in phase k, k = 1, 2, · · · ,N for a random interval of time and the sojourn period ends at the occurrence of a disaster. The sojourn period is exponentially distributed with mean . At the end of the sojourn period, all customers in the system are washed out, the server goes for repair/set up and the system moves to phase 0. During the repair time, customers join the system, become impatient and leave the system. The impatience time is exponentially distributed with mean . Immediately after the repair, the server is ready for offering service in phase i with probability , k = 1, 2, · · · ,N. In the k−level of the environment, customers arrive according to a Poisson process with rate and the service time is exponential with mean . Explicit expressions for time-dependent state probabilities are found and the corresponding steady-state probabilities are deduced. Some new performance measures are also obtained. Choosing arbitrary values of the parameters subject to the stability condition, the behaviour of the system is examined. For the chosen values of the parameters, the performance measures indicated that the system did not exhibit much deviation by the presence of several phases of the environment.

]]>Mahmoud Riad Mahmoud Azza E. Ismail and Moshera A. M. Ahmad

Statistical distributions play a major role in analyzing experimental data, and finding an appropriate one for the data at hand is not an easy task. Extending a known family of distribution to construct a new one is a long honored technique in this regard. The T-X[Y] methodology is utilized to construct a new distribution as described in this study. The T-inverse exponential family of distributions, which was previously introduced by the same authors, is used to examine the exponential-inverse exponential[Weibull] distribution (Exp-IE[Weibull]). Several fundamental properties are explored, including survival function, hazard function, quantile function, median, skewness, kurtosis, moments, Shannon’s entropy, and order statistics. Our distribution exhibits a wide range of shapes with varying skewness and assume most possible forms of hazard rate function. The unknown parameters of the Exp-IE [Weibull] distribution are estimated via the maximum likelihood method for a complete and type II censored samples. We performed two applications on real data. The first one is vinyle chloride data, which is explained by [1] and the second is cancer patients data, which is explained by [2]. The significance of the Exp-IE[Weibull] model in relation to alternative distributions (Fr´echet, Weibull-exponential, logistic-exponential, logistic modified Weibull, Weibull-Lomax [log-logistic] and inverse power logistic exponential) is demonstrated. When using the applied real data, the new distribution (Exp-IE[Weibull]) achieved better results for the AIC and BIC criterion compared to other listed distributions.

]]>Vladimir A. Skorokhodov and Iakov M. Erusalimskiy

The flow control problem in resource networks consists in finding such a set of vertices and capacities of arcs, which go out from these vertices, such that the limit state of the resource network is the closest to the given state . This problem is naturally divided into two subproblems. The first of them is the ”local” subproblem, which consists in determining the capacities of arcs which go out from the vertices of a given subset (hereinafter, the set will be called the set of controlled vertices). The second subproblem is the ”global” subproblem, which consists in finding the optimal set of controlled vertices , consisting of at most s elements. The paper is devoted to the study of the possibility of flows local control in resource networks. Methods for solving a local subproblem for regular resource networks with a low resource allocation are proposed. The conditions for the unreachability of the limit state , which coincides with the given state are obtained. Three cases are considered for the distribution of controlled vertices on a resource network. In each of the considered cases, it is shown that if the condition of unreachability of the limit state is not satisfied, then there is a set of the capacities values of the arcs that go out the controlled vertices, for which the limit state coincides with the state .

]]>Youness Jouilil and Driss Mentagui

Univariate time series forecasting is a crucial machine learning issue across many fields notably sentiment analysis, economy, medicine, agriculture, and finance. In this working paper, we tackled comparing the Support Vector Regression (SVR) to the traditional Autoregressive Integrated Moving Average (ARIMA) algorithms in terms of forecasting through a real case study. In fact, the data set used in this investigation has been extracted from the World Bank. The target time series is the American Foreign direct investment, net outflows (% of GDP) which includes the data for 50 years from 1972 to 2021. For analytical and comparison purposes, all the compilations have been done using the R programming language for Windows 10. The statistical findings revealed that, in short-term prediction, the forecast accuracy of both algorithms reduces in terms of error accuracy, significantly. Comparatively, the analysis conducted in this investigation demonstrates that the machine learning algorithms, especially the SVM one perform better than the ARIMA in short-term forecasting since its accuracy functions are the lowest. Thus, we highly recommend future research to compare the advanced machine learning algorithms especially the recurrent neural network algorithms with the classical algorithms, especially with the ARIMA approach in order to choose the best algorithm in terms of results and predictive performance.

]]>Saritha M.B and R. Varadharajan

Industries are consistently confronted with a myriad of challenges, the most significant of which is the requirement to increase product quality while simultaneously minimising manufacturing costs. Statistical Process Control (SPC) provides quality control charts as one of its primary methods for achieving this goal. When it comes to monitoring the quality features of a process, the control chart is the most popular and widely used kind of statistical analysis tool. It is very necessary to make use of multivariate control charts if the quality of a process is found to be connected with more than one characteristic. The Hotelling- chart is one of the most familiar methods of multivariate control chart. It is used for simultaneously monitoring the process mean and determining whether or not the process mean vector for two or more variables is under control. However, this is applicable only when the data is accurate, determined, and exact. As a result, when the data is vague or ambiguous, the utility of the conventional Hotelling- control chart is limited. Within the scope of this research, we put up a neutrosophic Hotelling- control chart as a potential solution to the issue described above. The performance of the proposed chart is evaluated using simulation at various degrees of shift in process average, with the neutrosophic alarm rate serving as the performance measure. To further investigate the applicability of the suggested chart in the actual world, we made use of a real-world example taken from the chemical sector.

]]>Nur Afza Mat Ali Jumat Sulaiman Azali Saudi and Nor Syahida Mohamad

In this paper, we transformed a two-dimensional unsteady convection-diffusion equation into a two-dimensional steady convection-diffusion equation using the similarity transformation technique. This technique can be easily applied to linear or nonlinear problems and is capable of reducing the size of computational works since the main idea of this technique is to reduce at least one independent variable. The corresponding similarity equation is then solved numerically using an effective numerical technique, namely a new five-point rotated similarity finite difference scheme via half-sweep successive over-relaxation iteration. This work compared the performance of the proposed method with Gauss-Seidel and successive over-relaxation with the full-sweep concept. Numerical tests were carried out to obtain the performance of the proposed method using C simulation. The results revealed that the combination of the five-point rotated similarity finite difference scheme via half-sweep successive over-relaxation iteration is the most superior method in terms of the iteration number and computational time compared to all these methods. Additionally, in terms of accuracy, all three iterative methods are also comparable.

]]>Arwa Salem Maabreh and Mohammad Fraiwan Al-Saleh

It is well known that ranked set sampling (RSS) technique and its variations, when applicable, are more efficient for estimating the population mean than the usual random sampling techniques. Despite the fascinating applications of Cauchy distribution, it has many unusual properties. For example: its moments either don’t exist or exist but are infinite, and its minimal sufficient statistics are just the order statistics. Given that the shape of the Cauchy distribution is similar to the normal one, it would be advantageous to carry out some statistical studies to focus on estimating its parameters; in particular the location parameter which is the median. In this paper, the estimation of the location parameter of the Cauchy distribution using RSS and some of its variations; namely, Double RSS, Median RSS, Multistage RSS, and Steady-State RSS are considered. The estimators are compared with each other and with their counterparts using simple random sampling (SRS). The findings show that RSS or any of its variations, being evaluated in this study, is more efficient in estimating the location parameter compared to SRS. The comparison among the RSS variations reveals that the steady-state RSS is more efficient than other RSS variations. Moreover, to overcome some of the challenges of Cauchy distribution, such as the non-existence of moments, a truncated Cauchy distribution is used. For this distribution, all moments are finite as well as the moments of order statistics. Results show that RSS and Median RSS outperform the SRS in estimating the location parameter, even with the truncated version of Cauchy. Overall, the work of this paper identifies other advantages of RSS techniques.

]]>R Sakthivel and G Vijayalakshmi

In the field of reliability theory, one of the most significant topics to discuss is the process of determining the reliability of a complex system based on the reliabilities of its individual components. The consecutive k-out-of-n:F system is used in telephone networks, photographing in nuclear accelerators, spacecraft relay stations, telecommunication system consisting of relay stations connecting transmitter and receiver, microwave relay stations, the design of integrated circuits, vacuum systems in accelerators, oil pipeline systems and computing networks. The reliability estimation of the consecutive k-out-of-n:F system is studied because it plays an important role in many physical systems. Dynamic Bayesian networks are graphical models for time-varying probabilistic inference and causal analysis under system uncertainty. The dynamic Bayesian network is built for the proposed system since time is continuously measured. The consecutive k-out-of-n:F system depends on the k components, because the system fails when the consecutive k components fail, otherwise the system works. The contributions are the dynamic Bayesian network construction of the proposed system and the reliability analysis of the linear and circular consecutive k-out-of-n:F system. Furthermore, Dynamic Bayesian network- based reliability is shown to be significantly higher than the reliability achieved by Malinowski, Preuss and Gao, Liu, Wang, Peng and Amirian, Khodadadi, Chatrabgoun. The Dynamic Bayesian network- based Reliability of linear and circular consecutive k-out-of-n:F system is also compared.

]]>Ebimene James Mamadu Henrietta Ify Ojarikre and Ignatius Nkonyeasua Njoseh

The role of fractional differential equations in the advancement of science and technology cannot be overemphasized. The time fractional telegraph equation (TFTE) is a hyperbolic partial differential equation (HPDE) with applications in frequency transmission lines such as the telegraph wire, radio frequency, wire radio antenna, telephone lines, and among others. Consequently, numerical procedures (such as finite element method, H^{1} – Galerkin mixed finite element method, finite difference method, and among others) have become essential tools for obtaining approximate solutions for these HPDEs. It is also essential for these numerical techniques to converge to a given analytic solution to certain rate. The Ritz projection is often used in the analysis of stability, error estimation, convergence and superconvergence of many mathematical procedures. Hence, this paper offers a rigorous and comprehensive analysis of convergence of the space discretized time-fractional telegraph equation. To this effect, we define a temporal mesh on [0,T] with a finite element space in Mamadu-Njoseh polynomial space, φ_{m-1}, of degree ≤m-1. An interpolation operator (also of a polynomial space) was introduced along the fractional Ritz projection to prove the convergence theorem. Basically, we have employed both the fractional Ritz projection and interpolation technique as superclose estimate in L_{2} - norm between them to avoid a difficult Ritz operator construction to achieve the convergence of the method.

Embay Rohaeti I Made Sumertajaya Aji Hamim Wigena and Kusman Sadik

Modeling and forecasting multivariate time series (MTS) data with multiple objects may be challenging, especially if the data have volatility and missing data. Several studies on inflation data have been proposed, but these studies either did not use MTS data or did not consider missing data. This study aims to develop an approach that can obtain general models and forecasts for MTS data with volatility and missing data. We proposed Vector Autoregressive Moving Average Imputation Method - Multivariate Time Series Clustering (VAR-IMMA - MTSClust) to group the objects into clusters. The clusters can then be used to obtain general models and forecasts. This study consists of three stages. The first stage is the imputation simulation stage, where 10%, 20%, and 30% of MTS data were randomly removed and imputed using the original VAR-IM and the proposed VAR-IMMA. The second stage is the clustering stage where six clustering methods, i.e., K-means Euclidean, K-means Manhattan, K-means DTW, PAM Euclidean, PAM Manhattan, and PAM DTW, were used on both the completed data and the imputed data from the first stage. The third stage is the modeling and forecasting stage, where clusters from the second stage are used to obtain general models and forecasts for each cluster. The simulations were performed 1000 times and evaluated using RMSE, RMSSTD, R-squared, ARI, and balanced accuracy. The results showed that VAR-IMMA could increase the imputation accuracy by 10% in 50% of cases and even more in another 25% of cases. This increase in imputation accuracy was proven beneficial in the second stage, where clustering on imputed data formed clusters that are still like the completed data clusters despite missing data. K-means Euclidean and PAM Euclidean are two of the best methods. Finally, the use of VAR-IMMA and PAM Euclidean on inflation rate data with missing data was illustrated. The imputed clusters have an ARI score of 0.57 and balanced accuracy of 92%, leading to similar models and forecasts to the ones in the completed data.

]]>Mamane Souleye Ibrahim and Oumarou Abdou Arbi

In this paper, we consider the polytope of all elementary dicycles of a digraph . Dicycles problem, in graph theory and combinatorial optimization, solved by polyhedral approaches has been extensively studied in literature. Therefore cutting plane and branch and cut algorithms are unavoidable to exactly solve such a combinatorial optimization problem. For this purpose, we introduce a new family of valid inequalities called alternating 3-arc path inequalities for the polytope of elementary dicycles . Indeed, these inequalities can be used in cutting plane and branch and cut algorithms to construct strengthened relaxations of a linear formulation of the dicycle problem. To prove the facetness of alternating 3-arc path inequalities, in opposite to what is usually done that consists basically to determine the affine subspace of a linear description of the considered polytope, we resort to constructive algorithms. Given the set of arcs of the digraph , algorithms devised and introduced are based on the fact that from a first elementary dicycle, all other dicycles are iteratively generated by replacing some arcs of previously generated dicycles by others such that the current elementary dicycle contains an arc that does not belong to any other previously generated dicycles. These algorithms generate dicyles with affinely independent incidence vectors that satisfy alternating 3-arc path inequalities with equality. It can easily be verified that all these devised algorithms are polynomial from time complexity point of view.

]]>Rabiatul Adawiah Fadzar and Md Yushalify Misro

The brachistochrone curve is an optimal curve that allows the fastest descent path of an object to slide frictionlessly under the influence of a uniform gravitational field. In this paper, the Brachistochrone curve will be reconstructed using two different basis functions, namely Bézier curve and trigonometric Bézier curve with shape parameters. The Brachistochrone curve between two points will be approximated via a C-shape transition curve. The travel time and curvature will be evaluated and compared for each curve. This research revealed that the trigonometric Bézier curve provides the closest approximation of Brachistochrone curve in terms of travel time estimation, and shape parameters in trigonometric Bézier curve provide better shape adjustability than Bézier curve.

]]>Chatsuda Chanmanee Rukchart Prasertpong Pongpun Julatha U. V. Kalyani T. Eswarlal and Aiyared Iampan

The notion of BP-algebras was introduced by Ahn and Han [2] in 2013, which is related to several classes of algebra. It has been examined by several researchers. In the group, the concept of the direct product (DP) [21] was initially developed and given some features. Then, other algebraic structures are subjected to DP groups. Lingcong and Endam [16] examined the idea of the DP of (0-commutative) B-algebras and B-homomorphisms in 2016 and discovered several related features, one of which is a DP of two Balgebras that is a B-algebra. Later on, the concept of the DP of B-algebra was expanded to include finite family B-algebra, and some of the connected issues were researched. In this work, the external direct product (EDP), a general concept of the DP, is established, and the results of the EDP for certain subsets of BP-algebras are determined. In addition, we define the weak direct product (WDP) of BP-algebras. In light of the EDP BP-algebras, we conclude by presenting numerous essential theorems of (anti-)BP-homomorphisms.

]]>S. Vani Shree and S. Dhanalakshmi

In the midst of the 1960s, a theory by Kotzig-Ringel and a study by Rosa sparked curiosity in graph labeling. Our primary objective is to examine some types of graphs which admit Face Magic Mean Labeling (FMML). A bijection is called a (1,0,0) F-Face magic mean labeling [FMML] of if the induced face labeling A bijection is called a (1,1,0) F-Face magic mean labeling [FMML] of if the induced face labeling In this paper it is being investigated that the (1, 0, 0) – Face Magic Mean Labeling (F-FMML) of Ladder graphs, Tortoise graph and Middle graph of a path graph. Also (1,0,0) and (1,1,0) F-Face Magic Mean Labeling is verified for Ortho Chain Square Cactus graph, Para Chain Square Cactus graph and some snake related graphs like Triangular snake graphs and Quadrilateral snake graphs. For a wide range of applications, including the creation of good kind of codes, synch-set codes, missile guidance codes and convolutional codes with optimal auto correlation characteristics, labeled graphs serve as valuable mathematical models. They aid in the ability to develop the most efficient non-standard integer encodings; labeled graphs have also been used to identify ambiguities in the access protocol of communication networks; data base management to identify the best circuit layouts, etc.

]]>Bayda Ghanim Fathi and Alaa Luqman Ibrahim

The major stationary iterative method used to solve nonlinear optimization problems is the quasi-Newton (QN) method. Symmetric Rank-One (SR1) is a method in the quasi-Newton family. This algorithm converges towards the true Hessian fast and has computational advantages for sparse or partially separable problems [1]. Thus, investigating the efficiency of the SR1 algorithm is significant. It's possible that the matrix generated by SR1 update won't always be positive. The denominator may also vanish or become zero. To overcome the drawbacks of the SR1 method, resulting in better performance than the standard SR1 method, in this work, we derive a new vector depending on the Barzilai-Borwein step size to obtain a new SR1 method. Then using this updating formula with preconditioning conjugate gradient (PCG) method is presented. With the aid of inexact line search procedure by strong Wolfe conditions, the new SR1 method is proposed and its performance is evaluated in comparison to the conventional SR1 method. It is proven that the updated matrix of the new SR1 method, , is symmetric matrix and positive definite matrix, given is initialized to identity matrix. In this study, the proposed method solved 13 problems effectively in terms of the number of iterations (NI) and the number of function evaluations (NF). Regarding NF, the new SR1 method also outperformed the classic SR1 method. The proposed method is shown to be more efficient in solving relatively large-scale problems (5,000 variables) compared to the original method. From the numerical results, the proposed method turned out to be significantly faster, effective and suitable for solving large dimension nonlinear equations.

]]>Noor Julailah Abd Mutalib Norhayati Rosli and Noor Amalina Nisa Ariffin

The stiff stochastic differential equations (SDEs) involve the solution with sharp turning points that permit us to use a very small step size to comprehend its behavior. Since the step size must be set up to be as small as possible, the implementation of the fixed step size method will result in high computational cost. Therefore, the application of variable step size method is needed where in the implementation of variable step size methods, the step size used can be considered more flexible. This paper devotes to the development of an embedded stochastic Runge-Kutta (SRK) pair method for SDEs. The proposed method is an adaptive step size SRK method. The method is constructed by embedding a SRK method of 1.0 order into a SRK method of 1.5 order of convergence. The technique of embedding is applicable for adaptive step size implementation, henceforth an estimate error at each step can be obtained. Numerical experiments are performed to demonstrate the efficiency of the method. The results show that the solution for adaptive step size SRK method of order 1.5(1.0) gives the smallest global error compared to the global error for fix step size SRK4, Euler and Milstein methods. Hence, this method is reliable in approximating the solution of SDEs.

]]>Suzila Mohd Kasim Shaharuddin Cik Soh and Siti Nor Aini Mohd Aslam

Suppose that is a group and is a subset of . Then, the graph of a group , denoted by , is the simple undirected graph in which two distinct vertices are connected to each other by an edge if and only if both vertices satisfy . The main contribution of this paper is to construct the graph using the elements of Mathieu group, . Additionally, the connectivity of has been proven as a connected graph. Finally, an open problem is highlighted in addressing future research.

]]>Jackel Vui Lung Chew Jumat Sulaiman Andang Sunarto and Zurina Patrick

The sole subject of this numerical analysis was the half-sweep modified successive over-relaxation approach (HSMSOR), which takes the form of an iterative formula. This study computed a class of two-dimensional nonlinear parabolic partial differential equations subject to Dirichlet boundary conditions numerically using the implicit-type finite difference scheme. The computational cost optimization was considered by converting the traditional implicit finite difference approximation into the half-sweep finite difference approximation. The implementation required inner-outer iteration cycles, the second-order Newton method, and a linearization technique. The created HSMSOR is utilized to obtain an approximation of the linearized equations system through the inner iteration cycle. In contrast, the problem's numerical solutions are obtained using the outer iteration cycle. The study examined the local truncation error and the stability, convergence, and method analysis. Results from three initial-boundary value issues showed that the proposed method had competitive computational costs compared to the existing method.

]]>Evellin Dewi Lusiana Suci Astutik Nurjannah and Abu Bakar Sambah

The Generalized Dissimilarity Model (GDM) is an extension of Generalized Linear Model (GLM) that is used to describe and estimate biological pairwise dissimilarities following a binomial process in response to environmental gradients. Some improvement has been made to accommodate the uncertainty quantity of GDM by applying resampling scheme such as Bayesian Bootstrap (BBGDM). Because there is an ecological assumption in the GDM, it is reasonable to use a proper Bayesian approach rather than resampling method to obtain better modelling and inference results. Similar to other GLM techniques, the GDM also employs a link function, such as the logit link function that is commonly used for the binomial regression model. By using this link, a Bayesian approach to GDM framework which called Bayesian GDM (BGDM) can be constructed. In this paper, we aim to evaluate the estimators' performance of Bayesian Generalized Dissimilarity Model (BGDM) in relative to BBGDM. Our study revealed that the performance of BGDM estimator outperformed that of BBGDM, especially in term of unbiasedness and efficiency. However, the BGDM estimators failed to meet consistency property. Moreover, the application of the BGDM to a real case study indicates that its inferential abilities are superior to the preceding model.

]]>M. Basim N. Senu A. Ahmadian Z. B. Ibrahim and S. Salahshour

A new differential operators class has been discovered utilising fractional and variable-order fractal Atangana-Baleanu derivatives that have inspired the development of differential equations' new class. Physical phenomena with variable memory and fractal variable dimension can be described using these operators. In addition, the primary goal of this study is to use the operation matrix based on shifted Legendre polynomials to obtain numerical solutions with respect to this new differential equations' class, which will aid us in solving the issue and transforming it into an algebraic equation system. This method is employed in solving two forms of fractal fractional differential equations: non-linear and linear. The suggested strategy is contrasted with the mixture of two-step Lagrange polynomials, the predictor-corrector algorithm, as well as the fractional calculus methods' fundamental theorem, using numerical examples to demonstrate its accuracy and simplicity. The estimation error was proposed to contrast the results of the suggested methods and the exact solution to the problems. The proposed approach could apply to a wider class of biological systems, such as mathematical modelling of infectious disease dynamics and other important areas of study, such as economics, finance, and engineering. We are confident that this paper will open many new avenues of investigation for modelling real-world system problems.

]]>A. K. Awasthi Rachna and Rohit

It cannot be overstated how significant Series Equations are to the fields of pure and applied mathematics respectively. The majority of mathematical topics revolve around the use of series. Virtually, in every subject of mathematics, series play an important role. Series solutions play a major role in the solution of mixed boundary value problems. Dual, triple, and quadruple series equations are useful in finding the solution of four part boundary value problems of electrostatics, elasticity and other fields of Mathematical physics. Cooke devised a method for finding the solution of quadruple series equations involving Fourier-Bessel series and obtained the solution using operator theory. Several workers have devoted considerable attention to the solutions of various equations involving for instance, trigonometric series, The Fourier-Bessel series, The Fourier Legendre series, The Dini series, series of Jacobi and Laguerre polynomials and series equations involving Bateman K-functions. Indeed, many of these problems arise in the investigation of certain classes of mixed boundary value problems in potential theory. There has been less work on quadruple series equations involving various polynomials and functions. In light of the significance of quadruple series solutions, proposed work examines quadruple series equations that include the product of r generalised Bateman K functions. Solution is formal, and there has been no attempt made to rationalise many restricting processes that have been encountered.

]]>Amal HMIMOU M'barek IAOUSSE Soumaia HMIMOU Hanaa HACHIMI and Youssfi EL KETTANI

Missing data is a real problem in all statistical modeling fields, particularly, in structural equation modeling which is a set of statistical techniques used to estimate models with latent concepts. In this research paper, an investigation of the techniques used to handle missing data in structural equation models is elaborated. To clarify this, a presentation of the mechanisms of missing data is made based on the probability distribution. This presentation recognizes three mechanisms: missing completely at random, missing at random, and missing not at random. Ignoring missing data in the statistical analysis may mislead the estimation and generates biased estimates. Many techniques are used to remedy this problem. In the present paper, we have presented three of them, namely, listwise deletion, pairwise deletion, and full information maximum likelihood. To investigate the power of each of these methods while using structural equation models a simulation study is launched. Furthermore, an examination of the correlation between the exogenous latent variables is done to extend the previous studies. We simulated a three latent variable structural model each with three observed variables. Three sample sizes (700, 1000, 1500) are examined accordingly to three missing rates for two specified mechanisms (2%, 10%, 15%). In addition, for each sample hundred other samples were generated and investigated using the same case design. The criteria of examination are a parameter bias calculated for each case design. The results illustrate as theoretically expected the following: (1) the non-convergence of pairwise deletion, (2) a huge loss of information when using listwise deletion, and (3) a relative performance for the full information maximum likelihood compared to listwise deletion when using the parameters bias as a criterion, particularly, for the correlation between the exogenous latent variables. This performance is revealed, chiefly, for larger sample sizes where the multivariate normal distribution occurs.

]]>Muthu Meena Lakshmanan E and Suja K

Non-Archimedean analysis is the study of fields that satisfy the stronger triangular inequality, also known as ultrametric property. The theory of summability has many uses throughout analysis and applied mathematics. The origin of summability methods developed with the study of convergent and divergent series by Euler, Gauss, Cauchy and Abel. There is a good number of special methods of summability such as Abel, Borel, Euler, Taylor, Norlund, Hausdroff in classical Analysis. Norlund, Euler, Taylor and weighted mean methods in Non-Archimedan Analysis have been investigated in detail by Natarajan and Srinivasan. Schoenberg developed some basic properties of statistical convergence and also studied the concept as a summability method. The relationship between the summability theory and statistical convergence has been introduced by Schoenberg. The concept of weighted statistical convergence and its relations of statistical summability were developed by Karakaya and Chishti. Srinivasan introduced some summability methods namely y-method, Norlund method and Weighted mean method in p-adic Fields. The main objective of this work is to explore some important results on statistical convergence and its related concepts in Non-Archimedean fields using summability methods. In this article, Norlund-Euler- statistical convergence, generalized weighted summability using Norlund-Euler- method in an Ultrametric field are defined. The relation between Norlund-Euler- statistical convergence and Statistical Norlund-Euler- summability has been extended to non-Archidemean fields. The notion of Norlund-Euler- statistical convergence and inclusion results of Norlund-Euler statistical convergent sequence has been characterized. Further the relation between Norlund-Euler- statistical convergence of order α & β has been established.

]]>Hussein Ageel Khatab and Salah Gazi Shareef

In application to general function, each of the conjugate gradient and Quasi-Newton methods has particular advantages and disadvantages. Conjugate gradient (CG) techniques are a class of unconstrained optimization algorithms with strong local and global convergence qualities and minimal memory needs. Quasi-Newton methods are reliable and eﬃcient on a wide range of problems and they converge faster than the conjugate gradient method and require fewer function evaluations but they have the disadvantage of requiring substantially more storage and if the problem is ill-conditioned, they may take several iterations. A new class has been developed, termed preconditioned conjugate gradient (PCG) method. It is a method that combines two methods, conjugate gradient and Quasi-Newton. In this work, two new preconditioned conjugate gradient algorithms are proposed namely New PCG1 and New PCG2 to solve nonlinear unconstrained optimization problems. A new PCG1 combines conjugate gradient method Hestenes-Stiefel (HS) with new self-scaling symmetric Rank one (SR1), and a new PCG2 combines conjugate gradient method Hestenes-Stiefel (HS) with new self-scaling Davidon, Flecher and Powell (DFP). The algorithm uses the strong Wolfe line search condition. Numerical comparisons with standard preconditioned conjugate gradient algorithms show that for these new algorithms, computational scheme outperforms the preconditioned conjugate gradient.

]]>Hind K. Al-Jeaid

This paper introduces a new approach to directly solve a system of two coupled partial differential equations (PDEs) subjected to physical conditions describing the diffusion kinetic problem with one delayed neutron precursor concentration in Cartesian geometry. In literature, many difficulties arise when dealing with the current model using various numerical/analytical approaches. Normally, mathematicians search for simple but effective methods to solve their physical models. This work aims to introduce a new approach to directly solve the model under investigation. The present approach suggests to transform the given PDEs to a system of linear ordinary differential equations (ODEs). The solution of this system of ODEs is obtained by a simple analytical procedure. In addition, the solution of the original system of PDEs is determined in explicit form. The main advantage of the current approach is that it avoided the use of any natural transformations such as the Laplace transform in the literature. It also gives the solution in a direct manner; hence, the massive computational work of other numerical/analytical approaches is avoided. Hence, the proposed method is effective and simpler than those previously published in the literature. Moreover, the proposed approach can be further extended and applied to solve other kinds of diffusion kinetic problems.

]]>Asmiati Agus Irawan Aang Nuryaman and Kurnia Muludi

The locating chromatic number introduced by Chartrand et al. in 2002 is the marriage of the partition dimension and graph coloring. The locating chromatic number depends on the minimum number of colors used in the locating coloring and the different color codes in vertices on the graph. There is no algorithm or theorem to determine the locating chromatic number of any graph carried out for each graph class or the resulting graph operation. This research is the development of scientific theory with a focus of the study on developing new ideas to determine the extent to which the locating chromatic number of a graph increases when applied to other operations. The locating chromatic number of the origami graph was obtained. The next exciting thing to know is locating chromatic number for certain operation of origami graphs. This paper discusses locating chromatic number for specific operation of origami graphs. The method used in this study is to determine the upper and lower bound of the locating chromatic number for certain operation of origami graphs. The result obtained is an increase of one color in the locating chromatic number of origami graphs.

]]>Demudu Naganaidu and Zarina Mohd Khalid

Multinomial logistic regression is preferred in the classification of multicategory response data for its ease of interpretation and the ability to identify the associated input variables for each category. However, identifying important input variables in high-dimensional data poses several challenges as the majority of variables are unnecessary in discriminating the categories. Frequently used techniques in identifying important input variables in high-dimensional data include regularisation techniques such as Least Absolute Selection Shrinkage Operator (LASSO) and sure independent screening (SIS) or combinations of both. In this paper, we propose to use ANOVA, to assist the SIS in variable screening for high-dimensional data when the response variable is multicategorical. The new approach is straightforward and computationally effective. Simulated data without and with correlation are generated for numerical studies to illustrate the methodology, and the results of applying the methods on real data are presented. In conclusion, ANOVA performance is comparable with SIS in variable selection for uncorrelated input variables and performs better when used in combination with both ANOVA and SIS for correlated input variables.

]]>Mervat Mahdy Eman Fathy and Dina S. Eltelbany

The objective of this study was to present a novel bivariate distribution, which we denoted as the bivariate odd generalized exponential gompertz(BOGE-G) distribution. Other well-known models included in this one include the gompertz, generalized exponential, odd generalized exponential, and odd generalized exponential gompertz distribution. The model introduced here is of Marshall-Olkin type [16]. The marginals of the new bivariate distribution have odd generalized exponential gompertz distribution which proposed by[7]. Closed forms exist for both the joint probability density function and the joint cumulative distribution function. The bivariate moment generating function, marginal moment generating function, conditional distribution, joint reliability function, marginal hazard rate function, joint mean waiting time, and joint reversed hazard rate function are some of the properties of this distribution that have been discussed. The maximum likelihood approach is used to estimate the model parameters. To demonstrate empirically the significance and adaptability of the new model in fitting and evaluating real lifespan data, two sets of real data are studied using the new bivariate distribution. Using the software Mathcad, a simulation research was conducted to evaluate the bias and mean square error (MSE) characteristics of MLE. We found that the bias and MSE decrease as the sample size increases.

]]>R. Nithya and K. Anitha

The study of set of objects with imprecise knowledge and vague information is known as rough set theory. The diagrammatic representation of this type of information may be handled through graphs for better decision making. Tong He and K. Shi introduced the constructional processes of rough graph in 2006 followed by the notion of edge rough graph. They constructed rough graph through set approximations called upper and lower approximations. He et al developed the concept of weighted rough graph with weighted attributes. Labelling is the process of making the graph into a more sensible way. In this process, integers are assigned for vertices of a graph so that we will be getting distinct weights for edges. Weight of an edge brings the degree of relationship between vertices. In this paper we have considered the rough graph constructed through rough membership values and as well as envisaged a novel type of labeling called Even vertex -graceful labeling as weight value for edges. In case of rough graph, weight of an edge will identify the consistent attribute even though the information system is imprecise. We have investigated this labeling for some special graphs like rough path graph, rough cycle graph, rough comb graph, rough ladder graph and rough star graph etc. This Even vertex -graceful labeling will be useful in feature extraction process and it leads to graph mining.

]]>B. Srirekha Shakeela Sathish and P. Devaki

Rough set theory has a vital role in the mathematical field of knowledge representation problems. Hence, a Rough algebraic structure is defined by Pawlak. Mathematics and Computer Science have many applications in the field of Lattice. The principle of the ordered set has been analyzed in logic programming for crypto-protocols. Iwinski extended an approach towards the lattice set with the rough set theory whereas an algebraic structure based on a rough lattice depends on indiscernibility relation which was established by Chakraborty. Granular means piecewise knowledge, grouping with similar elements. The universe set was partitioned by an indiscernibility relation to form a Granular. This structure was framed to describe the Rough set theory and to study its corresponding Rough approximation space. Analysis of the reduction of granular from the information table is based on object-oriented. An ordered pair of distributive lattices emphasize the congruence class to define its projection. This projection of distributive lattice is analyzed by a lemma defining that the largest and the smallest elements are trivial ordered sets of an index. A Rough approximation space was examined to incorporate with the upper approximation and analysis with various possibilities. The Cartesian product of the distributive lattice was investigated. A Lattice homomorphism was examined with an equivalence relation and its conditions. Hence the approximation space exists in its union and intersection in the upper approximation. The lower approximation in different subsets of the distributive lattice was studied. The generalized lower and upper approximations were established to verify some of the results and their properties.

]]>Jinse Jacob and R. Varadharajan

When adopting the Ordinary Least Squares (OLS) method to compute regression coefficients, the results become unreliable when two or more predictor variables are linearly related to one another. The confidence interval of the estimates becomes longer as a result of the increased variance of the OLS estimator, which also causes test procedures to have the potential to generate deceptive results. Additionally, it is difficult to determine the marginal contribution of the associated predictors since the estimates depend on the other predictor variables that are included in the model. This makes the determination of the marginal contribution difficult. Ridge Regression (RR) is a popular alternative to consider in this scenario; however, doing so impairs the standard approach for statistical testing. The Raise Method (RM) is a technique that was developed to combat multicollinearity while maintaining statistical inference. In this work, we offer a novel approach for determining the raise parameter, because the traditional one is a function of actual coefficients, which limits the use of Raise Method in real-world circumstances. Using simulations, the suggested method was compared to Ordinary Least Squares and Ridge Regression in terms of its capacity to forecast, stability of its coefficients, and probability of obtaining unacceptable coefficients at different levels of sample size, linear dependence, and residual variance. According to the findings, the technique that we designed turns out to be quite effective. Finally, a practical application is discussed.

]]>Wilasinee Peerajit

The cumulative sum (CUSUM) control chart can sensitively detect small-to-moderate shifts in the process mean. The average run length (ARL) is a popular technique used to determine the performance of a control chart. Recently, several researchers investigated the performance of processes on a CUSUM control chart by evaluating the ARL using either Monte Carlo simulation or Markov chain. As these methods only yield approximate results, we developed solutions for the exact ARL by using explicit formulas based on an integral equation (IE) for studying the performance of a CUSUM control chart running a long-memory process with exponential white noise. The long-memory process observations are derived from a seasonal fractionally integrated MAX model while focusing on X. The existence and uniqueness of the solution for calculating the ARL via explicit formulas were proved by using Banach's fixed-point theorem. The accuracy percentage of the explicit formulas against the approximate ARL obtained via the numerical IE method was greater than 99%, which indicates excellent agreement between the two methods. An important conclusion of this study is that the proposed solution for the ARL using explicit formulas could sensitively detect changes in the process mean on a CUSUM control chart in this situation. Finally, an illustrative case study is provided to show the efficacy of the proposed explicit formulas with processes involving real data.

]]>Mashadi Yuliana Safitri and Sukono

Many authors have given the arithmetic form of triangular fuzzy numbers, especially for addition and subtraction; however, there is not much difference. The differences occur for multiplication, division, and inverse operations. Several authors define the inverse form of triangular fuzzy numbers in parametric form. However, it always does not obtain , because we cannot uniquely determine the inverse that obtains the unique identity. We will not be able to directly determine the inverse of any matrix in the form of a triangular fuzzy number. Thus, all problems using the matrix in the form of a triangular fuzzy number cannot be solved directly by determining . In addition, there are various authors who, with various methods, try to determine but still do not produce . Consequently, the solution of a fully fuzzy linear system will produce an incompatible solution, which results in different authors obtaining different solutions for the same fully fuzzy linear system. This paper will promote an alternative method to determine the inverse of a fuzzy triangular number in parametric form. It begins with the construction of a midpoint for any triangular fuzzy number , or in parametric form . Then the multiplication form will be constructed obtaining a unique inverse which produces . The multiplication, division, and inverse forms will be proven to satisfy various algebraic properties. Therefore, if a triangular fuzzy number is used, and also a triangular fuzzy number matrix is used, it can be easily directly applied to produce a unique inverse. At the end of this paper, we will give an example of calculating the inverse of a parametric triangular fuzzy number for various cases. It is expected that the reader can easily develop it in the case of a fuzzy matrix in the form of a triangular fuzzy number.

]]>Jamal Salah Hameed Ur Rehman and Iman Al Buwaiqi

Due to the Mittag-Leffler function's crucial contribution to solving the fractional integral and differential equations, academics have begun to pay more attention to this function. The Mittag-Leffler function naturally appears in the solutions of fractional-order differential and integral equations, particularly in the studies of fractional generalization of kinetic equations, random walks, Levy flights, super-diffusive transport, and complex systems. As an example, it is possible to find certain properties of the Mittag-Leffler functions and generalized Mittag-Leffler functions [4,5]. We consider an additional generalization in this study, , given by Prabhakar [6,7]. We normalize the later to deduce in order to explore the inclusion results in a well-known class of analytic functions, namely and , -uniformly Janowski starlike and k-Janowski convex functions, respectively. Recently, researches on the theory of univalent functions emphasize the crucial role of implementing distributions of random variables such as the negative binomial distribution, the geometric distribution, the hypergeometric distribution, and in this study, the focus is on the Poisson distribution associated with the convolution (Hadamard product) that is applied to define and explore the inclusion results of the followings: and the integral operator . Furthermore, some results of special cases will be also investigated.

]]>Ibrahim S. Hamad

The Integral Theory approach is used to explore the stability and dynamics of a free double-sided symmetric thin liquid film. For a Newtonian liquid with non-variable density and moving viscosity, the flowing in a thinning liquid layer is analyzed in two dimensions. To construct an equation that governs such flow, the Navier and Stokes formulas are utilized with proper boundary conditions of zero shear stress conjointly of normal stress on the bounding free surfaces with dimensionless variables. After that, the equations that are a non-linear evolution structure of layer thickness, local stream rate, and the unknown functions can be solved by using straight stability investigation, and the normal mode strategy can moreover be connected to these conditions to reveal the critical condition. The characteristic equation for the growth rate and wave number can be analyzed by using MATLAM programming to show the region of stable and unstable films. As a result of our research, we are able to demonstrate that the effect of a thin, free, double-sided liquid layer is an unstable component.

]]>Tamara Rezti Syafriana Solimun Ni Wayan Surya Wardhani Atiek Iriany and Adji Achmad Rinaldo Fernandes

Objective: This study aims to determine the development of nonparametric SEM analysis on simulation data using the exponential function. Methodology: This study uses simulation data which is defined as an experimental approach to imitate the behavior of the system using a computer with the appropriate software. This study uses nonparametric structural equation modeling (SEM) analysis. The function used in this study is the exponential function. Results: The results showed that with simulation data all relationships have a significant effect on each other which have formative and reflective indicators. Testing the direct effect of Y2 on Y3 produces a structural coefficient value of 0.255 with a p-value <0.001 which means it is significant. The structural coefficient is positive, indicating that the relationship between the two is positive. This means that the higher Y2, the higher Y3. The results of the measurement model get a coefficient of determination of 0.91. It can be explained that 91% of the diversity of variables Y1, Y2, and Y3 can be explained by the X1 variable while 9% is explained by other variables not used in the model. Novelty: This study uses simulation data that is made very complex to analyze several related system structures at one time and can use a lot of data to get closer to real conditions to obtain comprehensive results, adjust to the criteria to be studied, and meet the following criteria: nonparametric SEM analysis criteria using the exponential function.

]]>R. Aruna Devi and K. Anitha

Rough membership function defines the degree of relationship between conditional and decision attributes of an information system. It is defined by where is the subset of under the relation where is the universe of discourse. It can be expressed in different forms like cardinality form, probabilistic form etc. In cardinality form, it is expressed as where as in probabilistic form it can be denoted as where is the equivalence class of with respect to . This membership function is used to measure the value of uncertainty. In this paper we have introduced the concept of graphical representation of rough sets. Rough graph was introduced by He Tong in 2006. In this paper, we propose a novel method for the construction of rough graph through rough membership function . We propose that there is an edge between vertices if . The rough graph is being constructed for an information system; here objects are considered as vertices. Rough path, rough cycle, rough ladder graph are introduced in this paper. We develop the operations on rough graph and also extend the properties of rough graph.

]]>Rugare Kwashira

The study of Aut(G), the group of automorphisms of G, has been undertaken by various authors. One way to facilitate this study is to investigate the structure of Aut_{c}(G), the subgroup of central automorphisms. For some classes of groups, algebraic properties like solvability, nilpotency, abelian and nilpotency relative to an automorphism can be deduced through the study of the subgroups Aut_{c}(G) and Aut_{c∗} (G) where Aut_{c∗} (G) is the group of central automorphisms that fix Z(G) point-wise. For instance, [6], if Aut_{c}(G) = Aut(G) then G is nilpotent of class 2 and if G is f-nilpotent for Aut_{c∗} (G), then for a group G, the notions of relative nilpotency and nilpotency coincide [8]. The group is abelian if G is identity nilpotent only [8]. For an arbitrary group G, the subgroups Aut_{c}(G) and Aut_{c∗} (G) are trivial, but for the case when G is a p-group, Aut_{c}(G) is non-trivial and the structure of Aut_{c∗} (G) have been described [4]. The study of the influence of types of subgroups on the structure of G is a powerful technique, thus, one can investigate the influence of maximal invariant subgroups of G on the structure of Aut_{c∗} (G). We shall consider a class of finite, non-commutative, n-abelian groups that are not necessarily pgroups. Here, n = 2l + 1 is a positive integer and l is an odd integer. The purpose of this paper is to explicitly describe the central automorphisms of G = G_{l} that fix the center elementwise and consequently the algebraic structure of Aut_{c∗} (G). For this goal, we will study the invariant normal subgroups M of G such that and M is maximal in G. It suffices to study Hom(G/M,Z(G)), the group of homomorphisms from the quotient G/M to the center Z(G). We explore the central automorphism group of pullbacks involving groups of the form G_{l}. We extend our study to central automorphisms in this class of groups G_{l}, in which the mapping is an automorphism. For such groups, Aut_{c∗} (G) can be described through Hom(G/M,Z(G)), where M is normal and a maximal subgroup in G such that the quotient group G/M is abelian. We show that Hom and Aut_{c∗} (G) is isomorphic to the cyclic group of order a prime p. The class of groups studied in our paper falls under a bigger class of groups which have a special characterization that their non normal subgroups are contranormal. The results of this paper can be generalized to this bigger class of groups.

Duduka Venkatesh and V. Naga Raju

Fixed points are also called as invariant points. Invariant point theorems are very essential tools in solving problems arising in different branches of mathematical analysis. In the present paper, we establish three unique common invariant point theorems using two self-mappings, four self-mappings and six self-mappings in the bicomplex valued metric space. In the first theorem, we generate a common invariant point theorem for four self-mappings by using weaker conditions such as weakly compatible, generalized contraction and property. Then, in the second theorem, we generate a common invariant point theorem for six self-mappings by using inclusion relation, generalized contraction, weakly compatible and commuting maps. Further, in the third theorem, we generate a common coupled invariant point for two self mappings using different contractions in the bicomplex valued metric space. The above results are the extention and generalization of the results of [11] in the Bicomplex metric space. Moreover, we provide an example which supports the results.

]]>T. Yogashanthi Shakeela Sathish and K. Ganesan

In this study the intuitionistic fuzzy version of the critical path method has been proposed to solve networking problems with uncertain activity durations. Intuitionistic fuzzy set [1] is an extension of fuzzy set theory [2] unlike fuzzy set, it focuses on degree of belonging, the degree of nonbelonging or non-membership function and the degree of hesitancy which helps the decision maker to adopt the best among the worst cases. Trapezoidal and the triangular intuitionistic fuzzy numbers are utilized to describe the uncertain activity or task durations of the project network. Here trapezoidal and triangular intuitionistic fuzzy numbers are converted into their corresponding parametric form and applying the proposed intuitionistic fuzzy arithmetic operations and a new method of ranking based on the parametric form of intuitionistic fuzzy numbers, the intuitionistic fuzzy critical path with vagueness reduced intuitionistic fuzzy completion duration of the project has been obtained. The authentication of the proposed method can be checked by comparing the obtained results with the results available in pieces of literature.

]]>Mayasar Ahmad Dar Hiral Raja Afshan Butt and Deepmala Sharma

Transparency order is considered to be a cryptographically significant property that characterizes the resistance of S-boxes in opposition to differential power analysis attacks. The S-box having low transparency order is more resistant to these attacks. Until now, little attempts have been noticed to examine theoretically the transparency order and its relationship with other cryptographic properties. All constructions associated with transparency order are relying on search algorithms. In this paper, we discuss the new interpretation of bent functions in terms of their transparency order. Using the concept of vector concatenation and correlation characteristics, we find the transparency order of Boolean functions. The notion of complementary transparency order is given. For a pair of Boolean functions, we interpret complementary transparency order by their Walsh-Hadamard transform. We establish a relationship of transparency order with cross-correlation for a pair of Boolean functions. We find a relationship of transparency order with −variable decomposition bent functions. We generalize the bounds on sum-of-squares of autocorrelation in terms of transparency order of Boolean functions using Walsh-Hadamard spectra. Further the transparency order of a function fulfilling the propagation criterion about a linear subspace is evaluated.

]]>Abdurakhman

The mean-variance portfolio has several weaknesses. It does not accommodate the uncertainty of parameters, tends to be sensitive to the changes of parameter input, and tends to be unreliable on extreme observations. Moreover, it cannot accommodate the changes in investor preferences regarding the evidence of abnormal and asymmetric asset return distribution. To overcome these three weaknesses, we can use the robust mean-variance portfolio that is based on the uncertainty of parameters. However, the robust mean-variance portfolio has not included skewness in its optimization. Therefore, here in this paper, we use the robust mean-variance-skewness portfolio which includes skewness in its optimization. So it can be used for the condition where the data return is skewed asymmetric and contains extreme values. An empirical study of robust mean-variance and robust mean-variance-skewness portfolios has been conducted on four banking stocks in Indonesia, i.e AGRS.JK, BTPN.JK, BBNI.JK, and BBCA.JK. The data used in this study is the daily closing price of the company's stock price for the period January 2, 2020 – January 2, 2022 (489 days) obtained from Yahoo! Finance. From the results of the data analysis, it can be concluded that the variance still plays an important role in determining the weight of the allocation of a portfolio. Meanwhile, the large value of skewness leads to the allocation of the same weight for each stock in a portfolio.

]]>Fatima A. Alshaikh and Ayman Baklizi

We consider maximum likelihood estimation for the parameters and certain functions of the parameters in the Inverse Weibull (IW) distribution based on type II censored data. The functions under consideration are the Mean Residual Life (MRL), which is very important in reliability studies, and Tail Value at Risk (TVaR), which is an important measure of risk in actuarial studies. We investigated the performance of the MLE of the parameters and derived functions under various experimental conditions using simulation techniques. The performance criteria are the bias and the mean squared error of the estimators. Recommendations on the use of the MLE in this model are given. We found that the parameter estimators are almost unbiased, while the MRL and TVaR estimators are asymptotically unbiased. Moreover, the mean squared error of all estimators decreased for larger sample sizes and it increased when the censoring proportion is increased for a fixed sample size. The conclusion is that the maximum likelihood method of estimation works well for the parameters and the derived functions of the parameter like the MRL and TVaR. Two examples on a real data set are presented to illustrate the application of the methods used in this paper. The first one is on survival time of pigs while the other is on fire losses.

]]>Atiek Iriany and Adji Achmad Rinaldo Fernandes

To ascertain whether there is a causal connection between exogenous and endogenous factors, one method is to perform path analysis. The linearity assumption is the one that has the power to alter the model. The model's shape is impacted by the linearity assumption. The path analysis is parametric if the linearity assumption is true, but non-parametric path analysis is used if the non-linear form is unknown and there is no knowledge of the data pattern. If the non-linear form is unknown and there is no knowledge of the data pattern, non-linear path analysis is used. This study's goal was to calculate the nonparametric route function using a combination of truncated spline and Fourier series methods. The findings demonstrated that nonparametric path analysis only in cases where the linearity presumption is violated can one employ the Fourier series and truncated spline. Then, using the Ordinary Least Square (OLS) approach, the estimator of Nonparametric Regression-Based Path Analysis was obtained, delivering an estimation result that is not unique because it makes use of a nonparametric approach. The contribution of this paper can be used as reference material, especially analysis in statistics. With this paper, it is hoped that it can be applied in various fields. Suggestions for further research can develop this research with other models.

]]>Akhmad Fauzy Suparman and Epha Diana Supandi

Piecewise constant (PC) is a stochastic model that can be applied in various fields such as engineering and ecology. The stochastic model contains a noise. The accuracy of the stochastic model in modeling a signal is influenced by the type of noise. This paper aims to propose inverse-gamma noise in the PC model and the procedure for estimating the model parameters. The model parameters are estimated using the Bayes approach. Model parameters have a variable dimension space so that the Bayesian estimator cannot be determined analytically. Therefore, the Bayesian estimator is calculated using the reversible jump Markov Chain Monte Carlo (RJMCMC) algorithm. The performance of the RJMCMC algorithm is validated using data synthesis. The finding is a new PC model in which the noise has an inverse-gamma distribution. In addition, this paper also proposes a parameter estimation procedure for the model based on an RJMCMC. The simulation study shows that the model parameter estimators generated by this algorithm are close to the model parameter values. This paper concludes that inverse gamma noise can be used as an alternative noise in the PC model. The RJMCMC is categorized as a valid algorithm and can estimate the PC model parameters where the noise has an inverse-gamma distribution. The novelty in this paper is the development of a new stochastic model and the procedure for estimating the model parameters. In application, the findings in this paper have the potential to improve the suitability of the stochastic model to the signal.

]]>Khanchit Chuarkham Arthit Intarasit and Pakwan Riyapan

This article presents the probability of ruin for the classical risk process by including the density function of claims which satisfies a mixed linear exponential family. This can be defined as , where , , is a positive integer with with , , , and is the canonical parameter. The main results show that the ordinary differential equation for the probability of ruin in the general case by using chain rule and mathematical induction technique is given in Theorem 2.2, the ordinary differential equation for some mixed linear exponential family when , , , , , is demonstrated in Theorem 2.3, and an explicit solution for the probability of ruin when the mixed linear exponential family satisfies the conditions which are , , with , and is indicated in Theorem 2.4. Finally, we use MATLAB to generate the numerical simulations for the probability of ruin in the risk process that the number of claims is a Poisson process and the density function of claims satisfies a mixed linear exponential family and a gamma distribution under the conditions of Theorem 2.4 with the parameters =1 and =0.2. The numerical results reveal that the relative frequency of the ruin and the ruin probability also satisfy the Lundberg inequality which is the necessary condition for the ruin probability. In addition, the absolute values of its differences are small in order to confirm that the main results are correct.

]]>Hind Y. Saleh Baravan A. Asaad and Ramadhan A. Mohammed

The concept of soft set theory can be used as a mathematical tool for dealing with problems that contain uncertainty. Then, a new mixed mathematical model called the bipolar soft set is created by merging soft sets and bipolarity, which gave the concept of a binary model of grading. Bipolar soft set is characterized by two soft sets, one of which provides positive information and the other negative. Bipolar soft generalized topology is a generalization of bipolar soft topology. The importance of limit points in all branches of mathematics cannot be ignored. It forms one of the most significant and fundamental concepts in topology. On this basis, the derived set concept is required in the establishment and continuation of some properties. Accordingly, the limit point in bipolar soft generalized theory is defined. In this paper, we present the notion of bipolar soft generalized limit points. We explained the relation between the bipolar soft generalized derived and the bipolar soft generalized closure set. Added to that, we discussed some structures of a bipolar soft generalized topological space such as: -interior point, -exterior point, -boundary point, -neighborhood point and basis on . Finally, we give comparisons among these concepts of bipolar soft generalized topological spaces () by using bipolar soft point (). Each concept introduced in this paper is explained with clear examples.

]]>E.-P. Ndong Nguéma and Betrand Fesuh Nono

Restricted Maximum Likelihood (REML) is the most recommended approach for fitting a Linear Mixed Model (LMM) nowadays. Yet, as ML, REML suffers the drawback that it performs such a fitting by assuming normality for both the random effects and the residual errors, a dubious assumption for many real data sets. Now, there have been several attempts at trying to justify the use of the REML likelihood equations outside of the Gaussian world, with varying degrees of success. Recently, a new fitting methodology, code named 3S, was presented for LMMs with only added assumption (to the basic ones) that the residual errors are uncorrelated and homoscedastic. Specifically, the 3S-A1 variant was designed and then shown, for Gaussian LMMs, to differ only slightly from ML estimation. In this article, using the same 3S framework, we develop another iterative nonparametric estimation methodology, code named 3S-A1.RE, for the kind of LMMs just mentioned. However, we show that if the LMM is, indeed, Gaussian with i.i.d. residual errors, then the set of estimating equations defining any 3S-A1.RE iterative procedure is equivalent to the set of REML equations, but while including the nonnegativity constraints on all variance estimates, as well as positive semi-definiteness on all covariance matrices. In numerical tests on some simulated and real world clustered and longitudinal data sets, our new methods proved to be highly competitive when compared to the traditional REML in the R statistical software.

]]>Cylvia Nissa Steffani and Gunardi

Structural Equation Modeling (SEM) is a statistical modeling technique that combines three methods, namely factor analysis, path analysis and regression analysis to test a theoretical model in social science, psychology and management. Covariance-based SEM is a parametric SEM that must meet several parametric assumptions such as, multivariate normally distributed data, large sample sizes and independent observations, so that, variance-based SEM was developed to overcome the problem of covariance SEM, namely the Generalized Structured Component Analysis (GSCA) method. This study aims to implement the GSCA method on factors data that are expected to have an effect on the level of behavioral intention towards online food delivery services and to examine the significance of the mediating variable on the structural relationship. The results of hypothesis testing with a 95% confidence level showed that the quality of convenience motivation, prior online purchase experience, and attitude towards online food delivery services had a significant effect on behavioral intentions towards online food delivery services. The fit value is above 0, 523 which indicates that the model is able to explain around 52,3% of the variation of the data. Furthermore, the hedonic motivation variable has a significant effect on convenience motivation. Post usage usefulness and prior online purchase experience variables significantly affected the attitudes towards online food delivery services. The proposed model using GSCA achieves a much better result (good fit) compared with the previous model using Confirmatory Factor Analysis (CFA) with marginal fit.

]]>Alimzhan A. Ibragimov and Dilafruz N. Khamroeva

In this paper, we consider iterative methods for solving a partial eigenvalue problem for real symmetric interval matrices. Such matrices have applications in modeling many technical problems where a lot of data suffers from limited variation or uncertainty. In modeling most applied problems, when some parameter values fluctuate with a known amplitude, then it can be considered that it is advisable to use interval methods. The algorithms proposed by us are built on the basis of the power method and its modification, the so-called "Method of scalar products" for solving a partial problem of eigenvalues of an interval symmetric matrix. These methods have not yet been studied in detail and are not justified for interval matrices. In the developed algorithms, boundary matrices are first determined by the Deif theorem, and then a partial eigenvalue problem is solved. We also study the problem of convergence of the power method for boundary matrices of a given interval symmetric matrix. The results of the computational experiment show that the interval eigenvalues obtained by the proposed algorithms are in good agreement with the results obtained by other researchers, and in some cases even better. The obtained numerical results are compared by the number of iterations and the width of the interval solution.

]]>Hussein Eledum and Alaa R. El-Alosey

A mixture distribution is a combination of two or more probability distributions; it can be obtained from different distribution families or the same distribution families with different parameters. The underlying distributions may be discrete or continuous, so the resulting mixture probability distribution function should be a mass or density function. In the last few years, there has been great interest in the problem of developing a mixture distribution based on the binomial distribution. This paper uses the probability generating function method to develop a new two-parameter discrete distribution called a binomial-geometric (BG) distribution, a mixture of binomial distribution with the number of trials (parameter ) taken after a geometric distribution. The quantile function, moments, moment generating function, Shannon entropy, order statistics, stress-strength reliability and simulating the random sample are some of the statistical highlights of the BG distribution that are explored. The model's parameters are estimated using the maximum likelihood method. To examine the performance of the accuracy of point estimates for BG distribution parameters, the Monte Carlo simulation is performed with different scenarios. Finally, the BG distribution is fitted to two real lifetime count data sets from the medical field. As a result, the proposed BG distribution is an overdispersed right-skewed and can accommodate a constant hazard rate function. The proposed BG distribution is appropriate for modelling the overdispersed right-skewed real-life count data sets and it can be an alternative to the negative binomial and geometric distributions.

]]>A. Ramachandran and S. Sangeetha

Functional equation plays a very important and interesting role in the area of mathematics, which involves simple algebraic manipulations and through which one can arrive an interesting solution. The theory of functional equations is also used in the development of other areas such as analysis, algebra, Geometry etc., the new methods and techniques are applied in solving problem in Information theory, Finance, Geometry, wireless sensor networks etc., In recent decades, the study of various types of stability of a functional equation such as HUS (Hyers-Ulam stability), HURS (Hyers-Ulam-Rassias stability) and generalized HUS of different types of functional equation and also for mixed type were discussed by many authors in various space. The problem of the stability of different functional equations has been widely studied by many authors, and more interesting results have been proved in the classical case (Archimedean). In recent years, the analogues results of the stability problem of these functional equations were investigated in non-Archimedean space. The aim of this study is to investigate the HUS of a mixed type of general Quadratic-Quartic Cauchy functional equation in non-Archimedean normed space. In this current article, we prove the generalized HUS for the following Quadratic-Quartic Cauchy functional equation over non-Archimedean Normed space.

]]>David S. McLachlan and Godfrey Sauti

This paper presents two new analytical equations, the Two Exponent Phenomenological Percolation Equation (TEPPE) and the Single Exponent Phenomenological Percolation Equation (SEPPE) which, for the proper choice of parameters, approximate the widely used Heaviside Step Function. The plots of the equations presented in the figures in this paper show some, but by no means all, of the step, ramp, delta, and differentiable activation functions that can be obtained using the percolation equations. By adjusting the parameters these equations can give linear, concave, and convex ramp functions, which are basic signals in systems used in engineering and management. The equations are also Analytic Activation Functions, the form or nature of which can be varied by changing the parameters. Differentiating these functions gives delta functions, the height and width of which depend on the parameters used. The TEPPE and SEPPE and their derivatives are presented in terms of the conductivity () owing to their original use in describing the electrical properties of binary composites, but are applicable to other percolative phenomena. The plots in the figures presented are used to show the response (composite conductivity) for the parameters (higher conductivity component of the composite), (lower conductivity component of the composite) and , the volume fraction of the higher conductivity component in the composite. The additional parameters are the critical volume fraction, , which determines the position of the step or delta function on the axis and one or two exponents , and .

]]>Jamal Salah Maryam Al Hashmi Hameed Ur Rehman and Khaled Al Mashrafi

In this article, the researcher considered some well-known mathematical models of ordinary differential equations applied in biology such as the bacterial growth, the natural FC solution models for vegetables, the biological phospholipids pathway, the glucose absorption by the body and the spread of epidemics. The ordinary differential equations for each model are fractionalized by the means of Caputo derivative of a function with respect to certain exponential function. In each model, we embed the concept fractionalization associated with a chosen exponential function in order to modify the given model. Consequently, various propositions are evoked by hypothetically allowing some modifications in several mathematical models of biology. The results are further visualized by providing the graphs of Mittag-Leffler function on various parameters. The graphs' analysis explored the behavior of the solution for every modified model. In this study, the solutions of the modified models are all of the Mittag–Leffler form while all original models are solved by the means of exponential function. Slight changes of the behavior of the solutions are due to the assumptions and the change of parameters.

]]>Abil Mansyur and Elmanani Simamora

In local polynomial regression, prediction confidence interval estimation using standard theory will give coverage probability close to exact coverage probability. However, if the normality assumption is not met, the bootstrap method makes it possible to apply it. The working principle of the bootstrap method uses the resampling method where the sample data becomes a population and there is no need to know the distribution of the sample data is normal or not. Indiscriminate selection of smoothing parameters allows scatterplot results from local polynomial regressions to be rough and can even lead to misleading statistical conclusions. It is necessary to consider the optimal smoothing parameters to get local polynomial regression predictions that are not overfitting or underfitting. We offer two new algorithms based on the nested bootstrap resampling method to determine the bootstrap-t confidence interval in predicting local polynomial regression. Both algorithms consider the search for optimal smoothing parameters. The first algorithm performs paired and residual bootstrap samples, and the second algorithm performs based on residuals with residuals. The first algorithm provides a scatterplot and reasonable coverage probability on relatively large sample data. In contrast, the second algorithm is more powerful for each data size, including for relatively small sample data sizes. The mean of the bootstrap-t confidence interval coverage probability shows that the second algorithm for second-degree local polynomial regression is better than the other three. However, the larger the sample data size gives, the closer the closer the average coverage probability of the two algorithms is to the nominal coverage probability.

]]>Amna R. Ashour Noor A. Ibrahim Mundher A. Khaleel and Pelumi E. Oguntunde

Studies have considered generalizing statistical distributions in the past. These were aimed at making such distributions more flexible and suitable for describing real-world phenomena. In this study, we considered exploring the Weibull Burr Type X distribution, which extends the Burr Type X distribution using the Weibull generator. Particularly, the performance of the maximum likelihood estimators for its parameters encompassing the right censored dataset was explored and compared. On the performance of its estimators with respect to bias and root mean square error, we considered the Monte Carlo simulation study to make a comparison using varying sample sizes and censored percentages. We illustrated the usefulness and potentials of the Weibull Burr Type X distribution using a right censored dataset. We considered comparing the fitness of this model to its sub-models using real world dataset. The result showed that the Weibull Burr Type X distribution provides a better fit than other competing models. This indicates that the distribution is flexible and competitive. The Weibull Burr Type X distribution exhibits unimodal and decreasing shapes. The extra parameter in the distribution varies the model's tail weight and introduces skewness into the model. We introduced this model as an alternative to other existing models for modelling right censored data in various research fields and areas of study.

]]>Chetan Swarup

Many food webs exist in the ecosystem, and their survival is directly dependent on the growth rate of primary prey; it balances the entire ecosystem. The spatiotemporal dynamics of three species' food webs was proposed and analyzed in this paper, where the intermediate predator's predation term follows Holling Type IV and the top predator's predation term follows Holling Type II. To begin, we examine the system's stability using linear stability analysis. We first obtained an equilibrium solution set and then used a Jacobian method to investigate the system's stability at a biologically feasible equilibrium point. We investigate random movement in species in the presence of diffusion, establish conditions for system stability, and derive the Turing instability condition. Following that, the Turing instability condition for a spatial food web system is calculated. Finally, numerical simulations are used to validate the findings. We discovered several intriguing spatial patterns (spots, strip, and mixed patterns) that help us understand the dynamics of the real-world food web. As a result, the Turing instability analysis used in the complex food web system is especially relevant experimentally because the associated consequences can be researched and applied to a wide range of mathematical, ecological, and biological models.

]]>Agung Prabowo and Ngadiman

The 100 and 200- meter running championships for both males and females were first held in 1948 at London Olympic Games 1948. Some time records made by the running champions have been continuously improved. Thus, running championship is not only intended to win the gold medals but also to make new world records. The secondary data were in the form of running championships' time records used to formulate the mathematical models and determine the minimum time limits (fastest). This research used the time-record data of 100 and 200- meter running championships for both males and females winning the gold medals from the Olympic Games held in 1948 to 2020. The mathematical Model for 100 and 200- meter running championships was more appropriately formulated using a logarithmic regression equation. Meanwhile, the time records for running championships of 100 meters for females as well as those of 200 meters for both males and females used a simple linear regression. The world record for running 100 meters for males still belongs to Usain Bolt (9.63 seconds). By using an assumption that the time records are normally distributed, those time records can be broken/improved into 9.53 seconds. Moreover, if the analysis is made using a box-plot diagram, the fastest time can be 9.42 seconds. A similar conclusion was also obtained for the world records of running 100 meters for females and 200 meters for males and females mentioning that the recently achieved time records still can be broken/improved in the future.

]]>Fahimah Fauwziyah Suci Astutik and Henny Pramoedyo

The standard model that is used for count data is Poisson Regression. In fact, most of the count data is overdispersed, which means that the response variable has greater variance than the mean. So the Poisson Regression cannot be used because overdispersion can cause inaccurate parameter estimators. One of the most widely used methods to overcome overdispersion is Negative Binomial Regression. If there are spatial effects such as spatial heterogeneity that are taken into Negative Binomial model, the appropriate method to analyze is Geographically Weighted Negative Binomial Regression (GWNBR). A spatial weighting matrix is required in the GWNBR model. In this study, three weighting functions were used, that is Adaptive Gaussian Kernel, Adaptive Bisquare Kernel, and Adaptive Tricube Kernel. From the three weighting functions, a model will be formed and the best model will be selected based on the smallest AIC. Count data used in this study is maternal deaths during childbirth in West Java Province, which is the highest case in Indonesia. The results of the analysis show that based on the smallest AIC, the best modeling in maternal deaths during childbirth in West Java is the GWNBR model using the Adaptive Gaussian Kernel weight. The results of the best model were obtained from three groups based on the predictor variables that had a significant effect.

]]>Malathy V and Kalyani Desikan

Network equilibrium models are significantly distinct in supply chain networks, traffic networks, and e-waste flow networks. The idea of network equilibrium is strongly perceived while determining the tuner sets of a graph (network). Tuner sets are subsets of vertices of the graph G whose degrees are lower than the average degree of G, d(G) that can compensate or balance the presence of vertices whose degrees are greater than d(G). Generalised core-satellite graph comprises copies of (the satellites) meeting in K_{c} (the core) and it belongs to the family of graphs of diameter two. It has a central core of vertices connected to a few satellites, where all satellite cliques need not be identical and can be of different sizes. Properties like hierarchical structure of large real-world networks, are competently modeled using core-satellite graphs [1, 2, 5]. This family of graphs exhibits the properties similar to scale-free network as they possess anomalous vertex connectivity, where a small fraction of vertices (the core) are densely connected. Since these graphs possess such a structural property, interesting results are obtained for these graphs when tuner sets are determined. In this paper, we have considered the graph , with p > q, a subclass of the generalized core-satellite graph which is a join of η copies of the clique K_{q} and γ copies of the clique K_{p} with the core K_{1}. We have obtained the tuner set for this subclass and established the relation between the Top T(G) and the cardinality of the tuner set through necessary and sufficient conditions. We analyze and characterize these graphs and obtain some interesting results while simultaneously examining the existence of tuner sets.

Aiyared Iampan S. Yamunadevi P. Maragatha Meenakshi and N. Rajesh

The Hilbert algebra, one of several algebraic structures, was first described by Diego in 1966 [7] and has since been extensively studied by other mathematicians. Torra [18] was the first to suggest the idea of hesitant fuzzy sets (HFSs) in 2010, which is a generalization of the fuzzy sets defined by Zadeh [20] in 1965 as a function from a reference set to a power set of the unit interval. The significance of the ideas of hesitant fuzzy subalgebras, ideals, and filters in the study of the different logical algebras aroused our interest in applying these concepts to Hilbert algebras. In this paper, the concepts of HFSs to subalgebras (SAs), ideals (IDs), and deductive systems (DSs) of Hilbert algebras are introduced in terms of anti-types. We call them anti-hesitant fuzzy subalgebras (AHFSAs), anti-hesitant fuzzy ideals (AHFIDs), and anti-hesitant fuzzy deductive systems (AHFDSs). The relationships between AHFSAs, AHFIDs, and AHFDSs and their lower and strong level subsets are provided. As a result of the study, we found their generalization as follows: every AHFID of a Hilbert algebra Ω is an AHFSA and an AHFDS of Ω. We also study and find the conditions for the complement of an HFS to be an AHFSA, an AHFID, and an AHFDS. In addition, the relationships between the complements of AHFSAs, AHFIDs, and AHFDSs and their upper and strong level subsets are also provided.

]]>Iqbal M. Batiha Zainouba Chebana Taki-Eddine Oussaeif Adel Ouannas and Iqbal H. Jebril

Several real-world phenomena emerging in engineering and science fields can be described successfully by developing certain models using fractional-order partial differential equations. The exact, analytical, semi-analytical or even numerical solutions for these models should be examined and investigated by distinguishing between their solvablities and non-solvabilities. In this paper, we aim to establish some sufficient conditions for exploring the existence and uniqueness of solution for a class of initial-boundary value problems with Dirichlet condition. The gained results from this research paper are established for the class of fractional-order partial differential equations by a method based on Lax Milgram theorem, which relies in its construction on properties of the symmetric part of the bilinear form. Lax Milgram theorem is deemed as a mathematical scheme that can be used to examine the existence and uniqueness of weak solutions for fractional-order partial differential equations. These equations are formulated here in view of the Caputo fractional-order derivative operator, which its inverse operator is the Riemann-Louville fractional-order integral one. The results of this paper will be supportive for mathematical analyzers and researchers when a fractional-order partial differential equation is handled in terms of finding its exact, analytical, semi-analytical or numerical solution.

]]>S. Gunavathy R. Alagar Aiyared Iampan and Vediyappan Govindan

This article's goals are to propose a brand-new category of space termed "nano-ideal topological spaces" and to look at how they relate to conventional topological spaces. To determine their relationships in these spaces, we create certain closed sets. These sets' fundamental characteristics and properties are provided. Additionally, we look into two theories of optimal connectivity in nano topological spaces. In particular, we obtain certain features of such spaces and define -connectedness and strongly -connectedness nano-topological spaces in terms of any ideal . This study aims to illustrate a novel kind of nano-topological space called nano--topological space, and we define the relationships between the various classes of open sets. We speak about how we might characterise them. Some of their characterizations are finally supported. The lower and upper approximations are used by the author to define nano topological space. As weak variants of Nano open sets, he also created Nano -open sets, Nano semi-open sets, and Nano pre-open sets. Continuity, the fundamental notion of topology in nano topological space, was also introduced. Also, we introduce the notion of nano -continuity between nano topological spaces and we investigate several properties of this type of near-nano continuity. Finally, we introduce two examples as applications in nano-topological spaces.

]]>Elton Pasku and Anjeza Krakulli

In [5], Squier, Otto and Kobayashi explored a homotopical property for monoids called finite derivation type (FDT) and proved that FDT is a necessary condition that a finitely presented monoid must satisfy if it is to have a finite canonical presentation. In the latter development in [2], Kobayashi proved that the property is equivalent with what is called in [2] finite domination type. It was indicated in the end of [2] that there are monoids which are not even finitely generated, and as a consequence are not of FDT. It was this indication that inspired us to look for the possibility of defining a property of monoids which encapsulates both, FDT and finite domination type. This is realized in the current paper by extending the notion of finite domination from monoids to rewriting systems, and to achieve this, we are based on the approach of Isbell in [1], who defined the notion of the dominion of a subcategory of a category and characterized that dominion in terms of zigzags in over . The reason we followed this approach is that to every rewriting system which gives a monoid , there is always a category associated to it which contains three types of information at the same time: (i) all the possible ways in which the elements of are written in terms of words with letters from , (ii) all the possible ways one can transform a word with letters from into another one representing the same element of by using rewriting rules from . Each of such way gives is in fact a path in the reduction graph of . The last information (iii) encoded in is that contains all the possible ways that two parallel paths of the reduction graph are linked to each other by a series of compositions of whiskerings of other parallel paths. This category turns out to have the advantage that it can "measure" the extent to which a set of parallel paths is sufficient to express any pair of parallel paths by composing whiskers from . The gadget used to measure this, is the Isbell dominion of the whisker category generated by over . We then define the monoid given by to be of finite domination type (FDOT) if both and are finite and there is a finite set of morphisms such that is exactly . The first main result of our paper is that likewise FDT, FDOT is an invariant of the monoid presentation, and the second one is that that FDT implies FDOT, while remains open whether the converse is true or not. The importance of FDOT stands in the fact that not only it generalizes FDT, but the way it is defined has a lot in common with , giving thus hope that FDOT is the right tool to put FDT and into the same framework.

]]>Blerta (Kristo) Nazarko and Ditila Ekmekçiu

In this paper, we will give an approach to the performance evaluation and efficiency value measurements where the gathered real data form a "time series" along a certain period of time (t). Along with the use of the basic models of DEA (Data Envelopment Analysis) method in joint cooperation with Fuzzy DEA models, the impact of the variable factors on DMU's (Decision Making Units) inefficiencies, performance evaluation and ranking is studied by accepting the t period of time as a discreet variable and as a unique "moment". This is determined by the objectives given by the CSAM (Connoisseur-study-analysis model) viewpoint in order to form the most real tableau of the DMUs' performance evaluation by means of an efficiency evaluation chain. For the evaluation of performance as a competitive process, 17 countries from the macroeconomic and financial environment are included (countries from the Western Balkans, the EU, and other countries). The study – a knowledge analysis – is also developed as issues of economic optimization portfolio by going through two steps: First step – (time as a discreet variable) evaluation of DMU's performance by measuring and analysing the efficiency value in the economic environment, with defined goals and criteria, where the concept of differences "deviation" elasticity coefficient (differences between Ef-VRS and Ef-CRS) is also included at the efficiency levels for the inefficient DMUs during the period of time 't'; Second step – (the period of time as a unique "moment") where is operated using the Fuzzy DEA model approach (α-cut) and the coordination of both steps with the DMUs' performance ranking. In addition, the study investigates the impact of the correlative relations between the variables to the DMUs' efficiency values during the evaluation of their performance. The gained results through the methodology followed in this study will give a more real tableau in the study of the performance evaluation and the connoisseur analysis of the DMUs' efficiency value, based on the real data gathered during the period of time (t) ϵ (2015-2019).

]]>Khaldjigitov Abduvali and Djumayozov Umidjon

Usually, the boundary value problems of the theory of elasticity are formulated with respect to displacements, and are reduced to the well-known Lame equations. Strains and stresses can be calculated from displacements as a solution to Lame's equation. Also known are the Beltrami Mitchell equations, which make it possible to formulate the boundary value problem of the theory of elasticity with respect to stresses. Currently, the boundary value problems of the theory of elasticity in stresses are studied in more detail in the two-dimensional case, and usually solved numerically with the introduction of the Airy stress function. But, the direct solution of boundary value problems of elasticity theory with respect to stresses requires further researches. This work, similarly to the boundary value problem in stresses, is devoted to the formulation and numerical solution of boundary value problems of the theory of elasticity with respect to deformations. The proposed boundary value problem consists of six Beltrami-Mitchell-type equations depending on strains and three equations of the equilibrium equation expressed with respect to deformations. As boundary conditions, in addition to the usual conditions for surface forces, three additional conditions are also introduced based on the equilibrium equations. The boundary value problem is considered in detail for a rectangular area. The discrete analogue of the boundary value problem is composed by the finite difference method. The convergence of difference schemes and an iterative method for their solution are studied. Software has been developed in the C++ environment for solving boundary value problems in the theory of elasticity and deformation. A number of boundary value problems on the deformation of a rectangular plate are solved numerically under various boundary conditions. The reliability of the obtained results is substantiated by comparing the numerical results, with the exact solution, as well as with the known solutions of the plate tension problems with parabolic and uniformly distributed edge loads.

]]>Mashadi Abdul Hadi and Sukono

In various articles, fuzzy -normed space concept for is constructed from fuzzy normed space which uses intuitionistic approach or -norm approach concept. However, fuzzy normed space can be approached using fuzzy point too. This paper shows that fuzzy -normed space for can be constructed from fuzzy normed space using fuzzy point approach of fuzzy set. Furthermore, for , it is also discussed how to construct fuzzy ()-normed space from fuzzy -normed space using fuzzy point approach. The method that can be used is as follows. From fuzzy normed space, we construct a norm function that satisfies properties of fuzzy -normed, so that fuzzy -normed space is derived. Conversely, from fuzzy -normed space, we construct a normed function that satisfies properties of fuzzy ()-normed, so that fuzzy ()-normed space is obtained. Finally, we get two new theorems that state that a fuzzy -normed space from any fuzzy normed space and fuzzy ()-normed space for from fuzzy -normed space using fuzzy point of fuzzy set always can be constructed.

]]>Jiabu Ye and Dejian Lai

In clinical trials, practitioners collect baseline covariates for enrolled patients prior to treatment assignment. In recent guidance from Food and Drug Administration and European Medicines Agency, regulators encourage practitioners to utilize baseline information at the analysis stage to improve the efficiency. However, the current guidance focused on linear or non-linear modelling approach. Nonparametric statistical methods were not focus in the guidance. In this article, we conducted simulations of several covariate-adjusted nonparametric statistical tests. Wilcoxon rank sum test is a widely used method for non-normally distributed response variables between two groups but its original form does not take into account the possible effect of covariates. We investigated the empirical power and the type I error of the Wilcoxon type test statistics under various settings of covariate adjustments commonly encountered in clinical trials. In addition to Wilcoxon type test statistics, we also compared simulation results to more advanced nonparametric test statistics such as the aligned rank test and Jaeckel, Hettmansperger-McKean test. The simulation result shows when there is covariate imbalance, applying Wilcoxon rank sum test without adjusting the covariates will become problematic. The survey of the covariate adjustments for varies tests under investigation gives brief guidance to trial practitioners in real practice, particularly whose baseline covariates are not well balanced.

]]>T. G. Thange and S. S. Gangane

In this paper, we have studied the Henstock - Kurzweil integral which is a generalized Riemann integral means. Hen-stock - Kurzweil integral is the natural extension of Riemann integral. We defined Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation which is an extension of real valued Henstock - Kurzweil integral with respect to an increasing function. We investigated elementary properties of the Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation. We proved the convergence theorems and Saks - Henstock lemma of the Henstock - Kurzweil integral of Banach valued functions with respect to a function of bounded vari-ation. Equi-integrability with respect to Banach space valued function is defined and equi-integrable theorem of Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation is proved. Finally Bochner Henstock - Kurzweil integral of Banach valued function with respect to a function of bounded variation is defined and the relation between Bochner Henstock - Kurzweil integral and Henstock - Kurzweil integral is exhibited.

]]>Akbar B. Aliyev and Yeter M. Farhadova

Suspension bridges are a type of construction in which the deck is suspended under a series of suspension cables that are on vertical hangers. The first modern example of this project began to appear in the early 1800s. Modern suspension bridges are lightweight, aesthetically pleasing and can span longer distances than any other bridge form. Many papers have been devoted to the modelling of suspension bridges, for instance, Lazer and McKenna studied the problem of nonlinear oscillation in a suspension bridge. They introduced a (one-dimensional) mathematical model for the bridge that takes into account of the fact that the coupling provided by the stays connecting the main cable to the deck of the road bed is fundamentally nonlinear, that is, they gave rise to the system of semi linear hyperbolic equation, where the first equation describes the vibration of the road bed in the vertical plain and the second equation describes that of the main cable from which the road bed is suspended by the tie cables. Recently, interest in this field has been increasing at a high rate. In this paper, we investigate some mathematical models of suspension bridges with a strong delay in linear aerodynamic resistance force. We establish the exponential decay of the solution for the corresponding homogeneous system and prove the existence of an absorbing set as well as a bounded attractor.

]]>Monica Botros E.A.A.Ziada and I.L. EL-Kalla

In this research, we employ a newly developed strategy based on a modified version of the Adomian decomposition method (ADM) to solve nonlinear fractional differential equations (FDE) with both differential and nondifferential variables. FDE have disturbed the interest of many researchers. This is due to the development of both the theory and applications of fractional calculus. This track from various areas of fractional differential equations can be used to model various fields of science and engineering such as fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many others. Several fractional derivative definitions have been presented, including Riemann–Liouville, Caputo,and Caputo– Fabrizio fractional derivative. We just need to calculate the first Adomain polynomial in this technique avoiding the hurdles in the nondifferentiable nonlinear terms' remaining polynomials. Furthermore, the proposed technique is easy to programme and produces the desired output with minimal work and time on the same processor. When compared to the exact solution, this method has the advantage of reducing calculation steps, while producing accurate results. The supporting evidence proves that modified Adomian decomposition has an advantage over traditional Adomian decomposition method which can be explained very clear with nonlinear fractional differential equations. Our computational examples with difficult issues are used to prove the new algorithm's efficiency. The results show that the modified ADM is powerful, which has a faster convergence solution than the original one. Convergence analysis is discussed, also the uniqueness is explained.

]]>B. M. Cerna Maguiña Dik D. Lujerio Garcia Carlos Reyes Pareja and Torres Dominguez Cinthia

In this article, given a number that ends in one and assuming that there are integer solutions for the equations or or , the straight line was used passing through the center of gravity of the triangle bounded by the vertices . Considering A ≥ 25, we manage to divide the domain of the curve into two disjoint subsets, and using Theorem (2.2) of this article, we find the subset where the integer solution of the equation is found. Similar process is done when , in case P is of the form or . These curves are different and to obtain a process similar to the one carried out previously, we proceeded according to Observation 2.2. Our results allow minimizing the number of operations to perform when our problem requires to be implemented computationally. Furthermore, we obtain some conditions to find the solution of the equations: , where is of class , and is a bounded open domain of with piecewise smooth boundary . All the operations carried out to find the solution have been carried out assuming that these exist, and we have found the conditions that must satisfy for the coefficients . We finish by finding an optimal domain for the real solution of a given polynomial of degree five. This process carried out on said given polynomial can also be carried out to reduce the degree of a given polynomial and thus obtain information about its roots.

]]>Prapart Pue-on

The double integral transform is a robust implementation that is important in handling scientific and engineering problems. Besides its simplicity of use and straightforward application to the issue, the ability to reduce the problems to an algebraic equation that can be easily solved is a substantial advantage of the tool. Among the several integral transforms, the double Sadik transform is acknowledged to be one of the most frequently used in solving differential and integral equations. This work deals with investigating a generalized double integral transform called the double Sadik transform. The proof of the double Sadik transforms for partial fractional derivatives in the Caputo sense is displayed, and the double Sadik transforms method is introduced. The method has been applied to solve the initial boundary value problems for linear space and timefractional telegraph equations. Moreover, the suggested strategy can be used on non-linear problems via an iterative method and a decomposition concept. Some known-solution questions are evaluated with relatively minimal computational cost. The results are represented by utilizing the Mittag-Leffler function and covering the solution of a classical telegraph equation. The obtained exact solutions not only show the accuracy and efficiency of the technique, but also reveal reliability when compared to those obtained using other methods.

]]>R. Sakthipriya and K. Suja

The main aim of this work is to investigate some important properties of statistical convergence sequence in non-Archimedean fields. Statistical convergence has been discussed in various fields of mathematics namely approximation theory, measure theory, probability theory, trigonometric series, number theory, etc. The concept of summability over valued fields is a significant area of mathematics that has many applications in analytic continuation, quantum mechanics, probability theory, Fourier analysis, approximation theory, and fixed point theory. The theory of statistical convergence plays a notable space in the summability theory and functional analysis. The purpose of this work is to provide certain characterizations of ideal statistical convergence of sequence and ideal statistical Cauchy sequence in n-normed spaces and the establishment of relevant results in non-Archimedean fields. The ideal statistical convergence of sequence and ideal statistically Cauchy sequence are defined. A few related theorems are proved in field . The results of this work are extended to establish statistical convergence of double sequences in n-normed space and some new results have been proved. In this work, the main concept is ideal statistical convergence of double sequences in n-normed space over a complete, non-trivially valued, non-Archimedean field. Throughout this article, is a complete, non-trivially valued, non-Archimedean field.

]]>Deepak Gupta Aarti Saini and A.K.Tripathi

One of the most comprehensive theories of stochastic models is queueing theory. Through innovative analytical research with broad applicability, advanced theoretical models are being developed. In the present research, we would like to investigate at a queuing network model with low and high priority users and different server transition probabilities. The two service channels used in this study, and , are connected to the same server, . Customers with low and high priorities are invited by the server . The objective of the research is to design a model that helps in minimizing congestion in different systems. Poisson distribution is used to characterize both the arrival and service patterns. The functioning of this system takes place in a stochastic domain. The differential difference equations have been established, and the consistency of behaviour of the system has been examined. The generating function approach, the law of calculus, and a statistical formula are used to assess the model's performance. Numerical analyses and graphical presentations are used to show the model's outcomes. The results of the model are displayed graphically and through numerical analyses. This model can be used in a number of real situations, including administration, manufacturing, hospitals, banking systems, etc. In such situations, the present study is quite beneficial for understanding the system and redesigning it.

]]>Taoufiq El Harrouti Mourad Azhari Hajar Deqqaq Abdellah Abouabdellah Sanaa El Aidi and Habiba Chaoui

Latent Class Analysis (LCA) and k-Mode Algorithm (K-MA) are two unsupervised machine learning techniques. These methods aim to identify individuals on the basis of their shared traits. They are utilized in the context of categorical data and can be used to detect people's opinions toward green forms of transportation, especially Electric Vehicles (EV) as an alternative to conventional internal combustion engine vehicles. The LCA approach discovers group profiles (clusters) based on observed variables, whereas the K-MA technique is an adaptation of the k-means algorithm for categorical variables. In this study, we apply these two methods to identify Moroccans' preferences for the electrification of their means of transportation. Both algorithms are able to divide the analyzed sample into two groups, with the first group being more interested in EV. The second group consists of individuals who are less concerned about ecologically sustainable transportation. In addition, we conclude that the LCA algorithm performs well and is superior to the K-MA, and that its discrimination power (65% vs 35%) is more than that of the K-MA (52% vs 48%).

]]>Jean-Philippe Tchiekre Christophe Pouet and Armel Fabrice E. Yodé

In the context of non parametric multivariate regression model, we are interested in goodness-of-fit testing for the single-index models. These models are dimension reduction models and are therefore useful in multidimensional nonparametric statistics because of the well-known phenomenon called the curse of dimensionality. Fan and Li [5] have proposed the first consistent test for goodness-of-fit testing of the single-index by using nonparametric kernel estimation method and a central limit theorem for degenerate -statistics of order higher than two. Since then, the minimax properties of this test have not been investigated. Following this work, we use here the asymptotic minimax approach. We are interested in finding the asymptotic minimax rate of testing which gives the minimal distance between the null and alternative hypotheses such that a successful testing is possible. We propose a test procedure of level which can tend to zero when the sample size tends to infinity. We have established the minimax asymptotic properties of our test procedure by showing that it reaches the asymptotic minimax rate for the dimension and there is no test of level reaching this rate for . Because of its minimax asymptotic properties, our test is able to distinguish the null hypothesis of the closest possible alternative. The results obtained were possible thanks to a large deviation result that we established for a degenerate U-statistic of order two appearing in our decision variable.

]]>Nurfaezah Mohd Husin Iskandar Shah Mohd Zawawi Nooraini Zainuddin and Zarina Bibi Ibrahim

In this study, the fully implicit 2-point block backward differentiation formulas (BBDF) method has been successfully utilized for solving stiff ordinary differential equations (ODEs) by taking into account the uses of new starting methods namely, modified Euler's method (MEM), improved modified Euler's method (IMEM), and new Euler's method (NEM). The reason of proposing the BBDF is that the method has been proven useful for stiff ODEs due to its A-stable properties. Furthermore, the method is able to approximate the solutions at two points simultaneously at each step. The proposed method is also implemented through Newton's iteration procedure, which involves the calculation of the Jacobian matrix. Accuracy of the method is evaluated based on its performance in solving linear and non-linear initial value problems (IVPs) of first order stiff ODEs with transient and steady-state solutions. Some comparisons are made with the conventional BBDF approach for indicating the reliability of the proposed method. Numerical results indicate that not only classical Euler's method provides accurate solutions for BBDF, but also the numerous modified versions of Euler's methods improve the accuracy of BBDF, in terms of absolute error at certain step size and stage of iteration.

]]>Moa'ath N. Oqielat

In the current article, a physics-based mathematical model is presented to generate realistic trajectories of water droplets across the Frangipani leaf surface and can be applied on any other kind of leaves, which is the first in the series of two articles that we are going to present the second article later on. In the second article, we will study the collision between the droplet and the liquid streak. The model has many applications in different scientific and engineering fields, such as modelling pesticide movements on leaves surfaces and modeling absorption and nutrition systems. The leaf surface consists of a triangular mesh structure that needs to be constructed using different techniques such as a well-known technique called EasyMesh method. The leaf surface is constructed using surface fitting techniques, such as finite elements methods and Clough-Tocher method, using a set of 3D real-world data points collected by a laser scanner, and the motion of the droplet on each triangle is calculated using a derived equation of motion. The motion of the droplet affected different forces, such as gravity and drag forces. Simulations of the model were verified using Matlab programming, and the results seemed to be real and capture the droplet motion very well.

]]>Isela J. Reyna-Rosas Josué F. Pérez-Sánchez Edgardo Suárez-Domínguez Alejandra Hernández-Alvarado Susana Gonzalez-Santana and F. Izquierdo-Kulich

Electrolytes are of interest because thin plate coatings are normally obtained from aqueous solutions. The properties of the surface are important because various properties such as resistance or durability depend on it. To understand the phenomenological processes, it is better to analyze simpler processes such as sodium chloride. In this paper, a model is proposed to predict the temporal behavior of the fractal dimension of the patterns formed in salts precipitation by solvent evaporation in a scattering surface; for fractal-box counting, ImageJ software was used. The model was obtained by applying stochastic methods and fractal geometry, describing the internal fluctuations caused by precipitation and dissolution on the mesoscopic scale of solid crystalline particles. From adjusting the proposed model to the experimental data, it is possible to estimate the velocity constants related to the microscopic precipitation processes of the particles that form the pattern. The model was validated and used to study the precipitation of carbonate salts and sodium chloride, respectively, obtaining predictions corresponding to the physicochemical properties of these salts. From the adjustment of the proposed models to the observed experimental data, the value of the velocity constants of the precipitation and dissolution processes was also estimated.

]]>Vandana Saini Deepak Gupta and A.K. Tripathi

In this paper, we analyse a feedback queue network in stochastic and in fuzzy environment. We consider a model with three heterogeneous servers which are commonly attached to a server in starting. At the initial stage, all queue performance measures are obtained in steady-state that is in stochastic environment. After that, work is extended to fuzzy environment because practically all characteristics of the system are not exact, they are uncertain in nature. In the present work we use probability generating function technique, triangular fuzzy numbers, classical formulae for the calculation of all queue characteristics and L-R method to calculate queue characteristics in fuzzy environment.

]]>Jatinder Pal Kaur Deepak Gupta Adesh kumar Tripathi and Renuka

Open-shop scheduling problem (OSSP) is a well-known topic with wide industrial applications which belongs to one of the vital issues in the field of engineering. This paper deals with a two-stage open shop scheduling problem in which the processing time of jobs is allied with probabilities. The concept of a string of two job blocks which are disjoint in nature is considered so that the first block covers the jobs with a fixed route and the second block covers the jobs with an arbitrary path. Further, the weights of jobs are also introduced due to their applicability and relative importance in the real world. The objective of this study is to propose a heuristic which on execution, provides an optimal or near-optimal schedule to diminish the makespan. Several numerical illustrations are produced in MATLAB 2018a to demonstrate the effectiveness of the proposed approach, and to confirm the performance, the results are compared with the existing methods developed by Johnson and Palmer.

]]>Kariyam Abdurakhman Subanar and Herni Utami

This research proposed a new algorithm for clustering datasets using the Flexible K-Medoids Partitioning Method. The procedure is divided into two phases, selecting the initial medoids and determining the partitioned dataset. The initial medoids are selected based on the block representation of a combination of the sum and deviation of the variable values. The relative positions of the objects will be separated when the sum of the values of the p variables is different even though these objects have the same variance. The objects are selected flexibly from each block as the initial medoids to construct the initial groups. This process ensures that any identical objects will be in the same group. The candidate of final medoids is determined randomly by selecting objects from each initial group. Then, the final medoids were identified based on the combination of objects that produces the minimum value of the total deviation within the cluster. The proposed method overcomes the empty group that may arise in a simple and fast k-medoids algorithm. In addition, it overcomes identical objects in the different groups that may occur in the initialization of the simple k-medoids algorithm. Furthermore, the artificial data and six real datasets, namely iris, ionosphere, soybean small, primary tumor, heart disease case 1 and zoo were used to evaluate this method, and the results were compared with other algorithms based on the initial and final groups' performance. The experiment results showed that the proposed method ensures that no initial groups are empty. For real datasets, the adjusted Rand index and clustering accuracy of the final groups of the new algorithm outperforms the other methods.

]]>H. S. Abdel-Aziz H. Serry and M. Khalifa Saad

The pseudo spherical images of non-lightlike curves in Minkowski geometry are curves on the unit pseudo sphere, which are intimately related to the curvatures of the original ones. These images are obtained by means of Frenet-Serret frame vector fields associated with the curves. This classical topic is a well-known concept in Lorentzian geometry of curves. In this paper, we introduce the pseudo spherical images for a timelike curve in Minkowski 3-space. Our main purpose of the work is to obtain the time evolution equations of the orthonormal frame and curvatures of these images. The compatibility conditions for the evolutions are used. Finally, the theoretical results obtained through this study are given by some important theorems and explained in two computational examples with the corresponding graphs.

]]>Ajaz Ahmad Pir Tabasum Mushtaq and A. Parthiban

Graph theory plays a significant role in a variety of real-world systems. Graph concepts such as labeling and coloring are used to depict a variety of processes and relationships in material, social, biological, physical, and information systems. Specifically, graph labeling is used in communication network addressing, fault-tolerant system design, automatic channel allocation, etc. 2-odd labeling assigns distinct integers to the nodes of in such a manner, that the positive difference of adjacent nodes is either 2 or an odd integer, , . So, is a 2-odd graph if and only if it permits 2-odd labeling. Studying certain important modifications through various graph operations on a given graph is interesting and challenging. These operations mainly modify the underlying graph's structure, so understanding the complex operations that can be done over a graph or a set of graphs is inevitable. The motivation behind the development of this article is to apply the concept of 2-odd labeling on graphs generated by using various graph operations. Further, certain results on 2-odd labeling are also derived using some well-known number theoretic concepts such as the Twin prime conjecture and Goldbach's conjecture, besides recalling a few interesting applications of graph labeling and graph coloring.

]]>Labiyana Hanif Ali Jumat Sulaiman Azali Saudi and Xu Ming Ming

This paper is concerned with producing an efficient numerical method to solve nonlinear Fredholm integral equations using Half-Sweep Newton-PKSOR (HSNPKSOR) iteration. The computation of numerical methods in solving nonlinear equations usually requires immense amounts of computational complexity. By implementing a Half-Sweep approach, the complexity of the calculation is tried to be reduced to produce a more efficient method. For this purpose, the steps of the solution process are discussed beginning with the derivation of nonlinear Fredholm integral equations using a quadrature scheme to get the half-sweep approximation equation. Then, the generated approximation equation is used to develop a nonlinear system. Following that, the formulation of the HSNPKSOR iterative method is constructed to solve nonlinear Fredholm integral equations. To verify the performance of the proposed method, the experimental results were compared with the Full-Sweep Newton-KSOR (FSNKSOR), Half-Sweep Newton-KSOR (HSNKSOR), and Full-Sweep Newton-PKSOR (FSNPKSOR) using three parameters: number of iteration, iteration time, and maximum absolute error. Several examples are used in this study to illustrate the efficiency of the tested methods. Based on the numerical experiment, the results appear that the HSNPKSOR method is effective in solving nonlinear Fredholm integral equations mainly in terms of iteration time compared to rest tested methods.

]]>Nur Zatul Akmar Hamzah Siti Nurlaili Karim Mathuri Selvarajoo and Noor Azida Sahabudin

Quadratic stochastic operator (QSO) is a branch of nonlinear operator studies initiated by Bernstein in 1924 through his presentation on population genetics. The study of QSO is still ongoing due to the incomplete understanding of the trajectory behavior of such operators given certain conditions and measures. In this paper, we intend to introduce and investigate a class of QSO named Lebesgue QSO which gets its name from the Lebesgue measure as the measure is used to define the probability measure of such QSO. The broad definition of Lebesgue QSO allows the construction of a new measure as its family of probability measure. We construct a class of Lebesgue QSO with exponential measure generated by 3-partition with three different parameters defined on continual state space . Also, we present the dynamics of such QSO by describing the fixed points and periodic points of the system of equations generated by the defined QSO using a functional analysis approach. The investigation is concluded by the regularity of the operator, where such Lebesgue QSO is either regular or nonregular depending on the parameters and defined measurable partitions. The result of this research allows us to define a new family of functions of the probability measure of Lebesgue QSO and compare their dynamics with the existing Lebesgue QSO.

]]>Md. Abdul Mannan Md. Amanat Ullah Uttam Kumar Dey and Mohammad Alauddin

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order of a group, subgroups, along with the associated notions of automorphisms group of the dihedral groups, split extensions of groups and vector spaces arises from the varying properties of real and complex numbers. We must have used the Sylow theorems of this work when it's generalized. Here we discuss possible subgroups of a group in different types of order which will give us a practical knowledge to see the applications of the Sylow theorems. In algebraic structures, we deal with operations of addition and multiplication and in order structures, those of greater than, less than and so on. It is through the study of Sylow theorems that we realize the importance of some definitions as like as the exact sequences and split extensions of groups, Sylow p-subgroup and semi-direct product. Thus it has been found necessary and convenient to study these structures in detail. In situations, where it was found that a given situation satisfies the basic axioms of structure and having already known the properties of that structure. Finally, we find out possible subgroups of a group in different types of order for abelian and non-abelian cases.

]]>Budi Pratikno Nailatul Azizah and Avita Nur Azizah

We determined the power and its graph simulations on the discrete Poisson and Chi-square distributions. There are four important steps of the research methodology summarized as follow: (1) determine the sufficient statistics (if possible), (2) create the rejection area (UMPT test is sometime used), (3) derive the formula of the power, and (4) determine the graphs using the data (in simulation). The formula of the power and their curves are then created using code. The result showed that the power of the illustration of the discrete (Binomial distribution) depended on the number of trials and bound of the rejection area. The curve of the power is sigmoid (-curve) and tends to be zero when parameter shape () is greater than 0.4. It decreases (started from = 0.2) as the parameter theta increases. In the Poisson context, the curve of the power of the Poisson distribution is not -curve, and it only depends on the parameter shape . We note that the curve of the power of the Poisson is quickly to be one for greater than 2 and less than 10. In this case, the size of the Poisson distribution is greater than 0.05, so it is not a reasonable thing even the power is close to be one. In this context, we have to choose the maximum power and minimum size. In the context of Chi-square distribution, the graph of the power and size functions depend on rejection region boundary (). Here, we note that skewness of the -curve is positive as the increases. Similarly, the size also depends on the (and constant), and it decrease as the increases. We here also noted that the power is quickly to be one for large degree of freedom ().

]]>Rasha A. Farghali and Samah M. Abo-El-Hadid

Beta regression model is used for modeling proportions measured on a continuous scale; its parameters are estimated with the maximum likelihood method. Classical regression models, such as linear regression model and nonlinear regression models like logistic regression are not suitable for such situations. As in linear regression model, the independent variables are assumed to be uncorrelated if this assumption is not met, then the multicollinearity appears. Multicollinearity problem means that there is a near dependency between the independent variables. Biased estimators are commonly used for correcting the multicollinearity problem. In this study, we propose a generalized biased estimator for correcting multicollinearity in beta regression that is generalize beta ridge regression estimator (GBRRE). The performance of the proposed generalized biased estimator is evaluated theoretically via the matrix mean squared errors and the scalar mean squared errors; and practically using a Monte Carlo simulation study. The simulation results show that the optimal shrinkage estimator is K1 and the worst one is K2. Also, the proposed generalized estimator is applied to a real data set of pre-university education students in Egypt during the academic year (2018/2019) and we found the application results agree with the simulation results. Finally based on the results of the simulation study and the application the performance of the suggested generalized biased estimator is better than maximum likelihood estimators.

]]>A. Dinesh Kumar and R. Sivaraman

In this paper, we have determined the limit of ratio of (n+1)th term to the nth term of famous sequences in mathematics like Fibonacci Sequence, Fibonacci – Like Sequence, Pell's Sequence, Generalized Fibonacci Sequence, Padovan Sequence, Generalized Padovan Sequence, Narayana Sequence, Generalized Narayana Sequence, Generalized Recurrence Relations of Fibonacci – Type sequence, Polygonal Numbers, Catalan Sequence, Cayley numbers, Harmonic Numbers and Partition Numbers. We define this ratio as limiting ratio of the corresponding sequence. Sixteen different classes of special sequences are considered in this paper and we have determined the limiting ratios for each one of them. In particular, we have shown that the limiting ratios of Fibonacci sequence and Fibonacci – Like sequence is the fascinating real number called Golden Ratio which is 1.618 approximately. We have shown that the limiting ratio of Pell's sequence is a real number called Silver Ratio and the limiting ratios for generalized Fibonacci sequence are metallic ratios. We have also obtained the limiting ratios of Padovan and generalized Padovan sequence. The limiting ratio of Narayana sequence happens to be a number called super Golden Ratio which is 1.4655 approximately. We have shown that the limiting ratios of Generalized Narayana sequence are the numbers known as super Metallic Ratios. We have also shown that the limiting ratio of generalized recurrence relation of Fibonacci type is 2 and that of Polygonal numbers and Harmonic numbers are 1. We have proved that the limiting ratio of the famous Catalan sequence and Cayley numbers are 4. Finally, assuming Rademacher's Formula, we have shown that the limiting ratio of Partition numbers is the natural logarithmic base e. We have proved fourteen theorems to derive limiting ratios of various well known sequences in this paper. From these limiting ratio values, we can understand the asymptotic behavior of the terms of all these amusing sequences of numbers in mathematics. The limiting ratio values also provide an opportunity to apply in lots of counting and practical problems.

]]>V. Vidhya and K. Ganesan

In any decision-making process, imprecision is a significant issue. To deal with the ambiguous environment of collective decision-making, various tools and approaches have been created. Fuzzy set theory is one of the most recent approaches for coping with imprecision. The Fuzzy Transportation Problem (FTP) is a well-known network planned linear programming problem which exists in a variety of situations and has received a lot of attention recently. Many authors defined and solved the fuzzy transportation problem with frequently utilized fuzzy numbers such as triangular fuzzy numbers or trapezoidal fuzzy numbers. On the other hand, real-world problems usually involve more than four variables. To tackle these concerns, the pentagonal fuzzy number is applied to the problems. This article proposes an approach to solving transportation problems whose parameters are pentagonal fuzzy numbers without requiring an initial feasible solution. An algorithm based on the core and spread method and an extended MODI method is developed to determine the optimal solution to the problem. The proposed process is based on the approximation method and gives a more efficient result. An illustrated example is used to validate the model. As a result, the proposed methodology is both simpler and more computationally efficient than the existing approaches.

]]>Maheswaran Srinivasan

Many a time, items can be classified as defective or non-defective and the objective is to identify all the defective items, if any, in the population. The concept of group testing deals with identifying all such defective items using a minimum number of tests. This paper proposes probabilistic group testing through a subset intersection group testing strategy. The proposed algorithm 'Subset Intersection Group Testing Strategy' deals with dividing the whole population, if it is positive, into different rows and columns and individually testing all the defective rows and columns. Through this proposed strategy, the number of group tests is either always one when no defective is found or 1+r+c, where r and c denote the number of rows and columns, when at least one defective is found. The proposed algorithms are validated using simulation for different combinations of group size and the incidence probability of an item being defective (p) and implications are drawn. The results indicate that the average number of total tests required is smaller when p is small and considerably increases as p increases. Therefore, for the smaller values of p, this proposed strategy is more effective. Also, an attempt is made to estimate an upper bound for the number of tests through this strategy in various scenarios.

]]>D. Senthilkumar and P. Sabarish

There are more sampling concepts active in production Industries, for inspecting the samples and analysing performance of the population. Also the sampling plans reduce errors in the production and produce the error free products. In this study, construction and selection of Double Inspection with reference to Single Sampling Plan i.e., DISSP, by attribute are investigated by using the Bivariate Poisson distribution. The Methodology, DISSP, was proposed based on two quality characteristics of the same sample size, and the planning parameters (n, C_{1}, C_{2}) are based on the operating characteristics, the conventional two-point condition by the planning table parameters (AQL and LQL). It is based on selected quality requirements and risks designed to allow manufacturers to easily determine the required sample size and corresponding acceptance criteria. A Comparison was done based on the efficiency of the plan with an existing single sampling plan and gave a numerical example to expose the operating tables. Also, the study shows the advantages of the proposed plan, and performance of the curves like, Operating characteristics, Average Outgoing Quality, and Average Total Inspection to expose the proposed double inspection sampling plan.

Nahashon Mwirigi Prof. Richard Simwa Dr. Mary Wainaina and Dr. Stanley Sewe

In modeling HIV/AIDS progression, we carried out a comprehensive investigation into the risk factors for state-specific-failure rates to identify the influential co-variates using Bayesian Model averaging method (BMA). BMA provides a posterior probability via Markov Chain Monte Carlo (MCMC) for each variable that belongs to the model. It accounts for model uncertainty by averaging all plausible models using their posterior probabilities as the weights for model-averaged predictions and estimates of the required parameters. Patients' age, and gender, among other co-variates, have been found to influence the state-specific-failure rates highly. However, the impact of each of the factors on the state specific-failure was not quantified. This paper seeks to evaluate and quantify the contribution of the patient's age and gender, CD4 cell count during any two consecutive visits, and state movement on the state-specific-failure rates for patients transiting either to the same, better or worse state. We used R Studio statistical Programming software to implement the method by applying BMS and BMA packages. State movement had a comparatively large coefficient with a posterior inclusion probability (PIP) of 0.8788 (87.88%). Hence, the most critical variable followed by observation-two-CD4-cell-count with a PIP of 0.1416 (14.16%), age and gender were the last with a PIP of 0.0556 (5.56%) and 0.0510 (5.10%) respectively for patients transiting to the same state. For patients transiting to a better state, the patients' age group dominated with a PIP of 0.9969 (99.69%), followed by patients' gender with a PIP of 0.0608 (6.08%). Patients' CD4 cell count during the second observation had the least PIP of 0.0399 (3.99%). For patients transiting to a worse disease state, patients CD4 cell count during the second observation proved to be the most important, with a PIP of 0.6179(61.79%) followed by state movement with a PIP of 0.2599 (25.99%), patients gender tailed with a PIP of 0.0467 (4.67%).

]]>R.Kuppan and L.Shobana

Let be a simple, finite, connected, plane graph with the vertex set , the edge set and the face set ). Martin Baca [1] defined, a connected plane graph with vertex set , edge set and face set to be face antimagic if there exists positive integers and and a bijection : such that the induced mapping : , where for a face , is the sum of all for all edges surrounding is also a bijection. This paper proves the existence of face antimagic labeling for the double duplication of all vertices by edges of gear graph for , grid graph for , where even, prism graph for and the double duplication of all vertices by edges of strong face of triangular snake graph for . The face antimagic labeling for double duplication of special graphs can be used to encrypt and decrypt the messages, which is used as a real time application. In [3], we used face antimagic labeling of strong face of duplication of all vertices by edges of a tree for to encrypt and decrypt thirteen secret numbers which can be extended to double duplication of graphs to encode and decode the numbers, which in turn can be used in military base, ATM and so on.

]]>Arvind Kumar Sinha and Pradeep Shende

Often the information in the surrounding world is incomplete, and such incomplete information gives rise to uncertainties. Pawlak's rough set model is an approach to approximation under uncertainty. It uses a tolerance relation to obtain single granulation of the incomplete information system for approximation. In this work, we extend the single granulation rough set for the incomplete information system to an uncertainty optimization-based rough set (UOBRS). The proposed approach is used to minimize the uncertainty using multiple tolerance relations. We list properties of the UOBRS for incomplete information systems. We compare UOBRS with the classical single granulation rough set (SGRS) and multi-granular rough set (MGRS). We list the basic properties of the UOBMGRS. We introduce the application of the UOBRS for attribute subset selection in case of incomplete information system. We use the measure of approximation quality to assess the uncertainties of the attributes. We compare the approximation quality of the attributes using UOBRS with the approximation quality using SGRS and MGRS. We get higher approximation quality with the less number of attributes using UOBRS as compared to SGRS and MGRS. The proposed method is a novel approach to dealing with incomplete information systems for more effective dataset analysis.

]]>Mohammad Hassan Mudaber Nor Haniza Sarmin and Ibrahim Gambo

The unity product graph of a ring is a graph which is obtained by setting the set of unit elements of as the vertex set. The two distinct vertices and are joined by an edge if and only if . The subgraphs of a unity product graph which are obtained by the vertex and edge deletions are said to be its induced and spanning subgraphs, respectively. A subset of the vertex set of induced (spanning) subgraph of a unity product graph is called perfect code if the closed neighbourhood of , forms a partition of the vertex set as runs through . In this paper, we determine the perfect codes in the induced and spanning subgraphs of the unity product graphs associated with some commutative rings with identity. As a result, we characterize the rings in such a way that the spanning subgraphs admit a perfect code of order cardinality of the vertex set. In addition, we establish some sharp lower and upper bounds for the order of to be a perfect code admitted by the induced and spanning subgraphs of the unity product graphs.

]]>Solimun Solimun and Adji Achmad Rinaldo Fernades

This study aims to examine the differences in various cluster validity indexes in the grouping of credit customers at Bank X Malang City, Indonesia using the average linkage and Euclidean distance methods. This study uses primary data with the variables used are service quality, environment, mode, willingness to pay, and obedient paying behavior obtained through a questionnaire with a Likert scale through purposive sampling distributed to 100 respondents. The data are then analyzed by clusters using the ward linkage and Euclidean distance methods on various validity cluster indexes, including the Silhouette Index, Krzanowski-Lai, Dunn, Gap, Davies-Bouldin, Index C, Global Sillhouette, Goodman-Kruskal in this study used as a tool analysis. This study uses R software. The results show that the Krzanowski-Lai, Dunn, Gap, Global Sillhouette, and Goodman-Kruskal indices have the same cluster members, as well as the Silhouette and Davies-Bouldin indices. The best cluster indexes are the Silhouette and Davies-Bouldin indexes. All validity indices produce variance between and within the same cluster. The novelty of this study is to compare 8 validity indices, namely Sillhouette Index, Krzanowski-Lai, Dunn, Gap, Davies-Bouldin, Index C, Global Sillhouette, and Goodman-Kruskal simultaneously.

]]>S. K. Sharma and Keshav Goel

The supply, demand and transportation cost in transportation problem cannot be obtained by all existing methods directly. In the existing literature, various methods have been proposed for calculating transportation cost. In this paper, we are comparing various methods for measuring the optimal cost. The objective of this paper is obtaining IBFS of real-life problems by various methods. In this paper, we include various methods such as AMM (Arithmetic Mean Method), ASM (Assigning Shortest Minimax Method) etc. The Initial Basic Feasible solution is one of the most important parts for analyzing the optimal cost of transportation Problem. For many applications of transportation problem such as image registration and wrapping, reflector design seismic tomography and reflection seismology etc, we analyze the transportation cost. TP is used to find the best solution in such a way in which product produced at several sources (origins) are supply to the various destinations. To fulfil all requirement of destination at lowest cost possible is the main objective of a transportation problem. All transport companies are looking forward to adopting a new approach for minimizing the cost. Along these lines, it is essential just as an adequate condition for the transportation problem to have an attainable arrangement. A numerical example is solved by different approaches for obtaining IBFS.

]]>Dwi Endah Kusrini Setiawan Heri Kuswanto and Budi Nurani Ruchjana

This research paper aims to form and estimate the spatial dynamic panel simultaneous equations models (SDPS) with fixed time effect that potentially have heteroscedasticity cases. The model formed with the individual effect is not eliminated but placed in the error model to accommodate cases of heteroscedasticity in the model. GMM with two stages least square (2SLS) method for the single equation is deliberately chosen as the estimation method for the SDPS model because it can eliminate heterogeneity cases in the model. The effectiveness of the estimate is seen based on the value of RMSE (Root Mean Square Error), mean and standard deviation (SD) of bias estimate by simulating Monte Carlo 100 times with different parameter pairs and different pairs N and T can also be concluded that parameter scenario changes do not give much effect on the mean bias value and SD bias. The SDPS model shows that the consistency of the estimated parameter values can be achieved easily if the number of T is added. Changes in the number of N and T indicate that the greater the N and T, the smaller RMSE value tends to be.

]]>N. Adan N. Razali N. A. Zainuri N. A. Ismail A. Gorgey and N. I. Hamdan

Runge-Kutta is a widely used numerical method for solving the non-linear Lorenz system. This study focuses on solving the Lorenz equations with the classical parameter values by using the lower order symmetrized Runge-Kutta methods, Implicit Midpoint Rule (IMR), and Implicit Trapezoidal Rule (ITR). We show the construction of the symmetrical method and present the numerical experiments based on the two methods without symmetrization, with one- and two-step active symmetrization in a constant step size setting. For our numerical experiments, we use MATLAB software to solve and plot the graphical solutions of the Lorenz system. We compare the oscillatory behaviour of the solutions and it appears that IMR and two-step active IMR turn out to be chaotic while the rest turn out to be non-chaotic. We also compare the accuracy and efficiency of the methods and the result shows that IMR performs better than the symmetrizers, while two-step active ITR performs better than ITR and one-step active ITR. Based on the results, we conclude that different implicit numerical methods with different steps of active symmetrization can significantly impact the solutions of the non-linear Lorenz system. Since most study on solving the Lorenz system is based on explicit time schemes, we hope this study can motivate other researchers to analyze the Lorenz equations further by using Runge-Kutta methods based on implicit time schemes.

]]>Ahssaine Bourakadi Naima Soukher Baraka Achraf Chakir and Driss Mentagui

In this paper, we will combine random set theory and portfolio theory, through the estimation of the lower bound of the Markowitz random set based on the Mean-Variance Analysis of Asset Portfolios Approach, which represents the efficient frontier of a portfolio. There are several Markowitz optimization approaches, of which we denote the most known and used in the modern theory of portfolio, namely, the Markowitz's approach, the Markowitz Sharpe's approach and the Markowitz and Perold's approach, generally these methods are based on the minimization of the variance of the return of a portfolio. On the other hand, the method used in this paper is completely different from those denoted above, because it is based on the theory of random sets, which allowed us to have the mathematical structure and the graphic of the Markowitz set. The graphical representation of the Markowitz set gives us an idea of the investment region. This region, called the investment zone, contains the stocks in which the rational investor can choose to invest. Mathematical and statistical estimation techniques are used in this paper to find the explicit form of the Markowitz random set, and to study its elements in function of the signs of the estimated parameters. Finally, we will apply the results found to the case of the returns of a portfolio composed of 200 assets from the Paris Stock Market Prices. The results obtained by this simulation allow us to have an idea on the stocks to recommend to the investors. In order to optimize their choices, these stocks are those which will be located above the curve of the hyperbola which represents the Markowitz set.

]]>Henrik Bernshausen Christoph Fuhrmann Hanns-Ludwig Harney Klaus Harney and Andreas Muller

The measurement problem of item response theory is the question of how to assign ability parameters to persons and difficulty parameters to items such that the comparison of abilities is independent of the specific set of difficulties . Correspondingly, the comparison of difficulties should be independent of the specific set of abilities . These requirements are called specific objectivity. They are the basis of the Rasch model. It measures and on one and the same scale. The present paper asks the different question of how to assign ability parameters to persons in a way that the comparison of abilities is independent of the position on the scale where the measurement takes place. Correspondingly, the comparison of difficulties should also be independent of the position on the scale where the calibration of difficulties takes place. Again, and measured on one and the same scale. These requirements are called form invariance. They lead to an item response function (IRF) different from that of the Rasch model. It integrates information from and beyond the mere score dependence and also shows specific objectivity (in a generalized mathematical form). The properties of the form invariant item response function are compared to that of the Rasch model, and related to previous work by Warm, Jaynes and Samejima. Moreover, several numerical examples for the use of it are provided.

]]>J.Catherine Grace John and B.Elavarasan

A relation is a mathematical tool for describing set relationships. Relationships are common in databases and scheduling applications. Science and engineering are designed to help humans make better decisions. To make these choices, we must first understand human expectations, the outcomes of various options, and the degree of confidence. With all of these data, partial orders will be generated. In several fields of engineering and computer science, partial order and lattice theory are now widely used. To mention a few, they are used in cloud computing (vector clocks, global predicate detection), concurrency theory (pomsets, occurrence nets), programming language semantics (fixed-point semantics), and data mining (concept analysis). Other theoretical disciplines benefit from them as well, such as combinatorics, number theory, and group theory. Partially ordered sets emerge naturally when dealing with multidimensional systems of qualitative ordinal variables in social science, especially to solve ranking, prioritising, and assessment concerns. As an alternative to standard techniques, partial order theory and partially ordered sets can be used to generate composite indicators for evaluating well-being, quality of life, and multidimensional poverty. They can be applied in multi-criteria analysis or for decision-making purposes in the study of individual and social desires, including in social choice theory. They're also valuable in social network analysis, where they may be utilized to apply mathematics to explore network topologies and dynamics. The Hasse diagram method, for example, produces a partial order with multiple incomparabilities (lack of order) between pairs of items. This is a common problem in ranking studies, and it can often be avoided by combining object attributes that lead to a complete order. However, such a mix introduces subjectivity and prejudice into the rating process. This work discusses the notion of a -prime radical of a partially ordered set with respect to ideal. In posets, we investigated the concept of -primary ideals. It is investigated how to characterise -primary ideals in relation to -prime radicals. In addition, an ideal's -primary decomposition is constructed.

]]>Vasile-Alexandru Suchar and Luis Gustavo Nardin

Non-stationarity potentially comes from many sources and they impact the analysis of a wide range of systems in various fields. There is a large set of statistical tests for checking specific departures from stationarity. This study uses Monte Carlo simulations over artificially generated time series data to assess the effectiveness of 16 statistical tests to detect the real state of a wide variety of time series (i.e., stationary or non-stationary) and to identify their source of non-stationarity, if applicable. Our results show that these tests have a low statistical power outside their scope of operation. Our results also corroborate with previous studies showing that there are effective individual statistical tests to detect stationary time series, but no effective individual tests for detecting non-stationary time series. For example, Dickey-Fuller (DF) family tests are effective in detecting stationary time series or non-stationarity time series with positive unit root, but fail to detect negative unit root as well as trend and break in the mean, variance, and autocorrelation. Stationarity and change point detection tests usually misclassify stationary time series as non-stationary. The Breusch-Pagan BG serial correlation test, the ARCH homoscedasticity test, and the structural change SC tests can help to identify the source of non-stationarity to some extent. This outcome reinforces the current practice of running several tests to determine the real state of a time series, thus highlighting the importance of the selection of complementary statistical tests to correctly identifying the source of non-stationarity.

]]>Fathy H. Riad

This paper deals with obtaining the interval and point estimation to Modified Kies exponential distribution in case of progressive first failure (PFF) censored data. It uses two approaches, classical and non-classical methods of estimation, including the highest posterior density (HPD). We obtained the Maximum Likelihood Estimation of the parameters and the logarithm likelihood function, and we used the maximum likelihood estimation of the parameters as a classical approach. We calculated the confidence intervals for the parameters and the Bootstrap confidence Intervals. We employed the posterior distribution and the Bayesian estimation (BE) under different loss functions (Symmetric loss function, The MCMC usage, and The M-H algorithm). Some results depending on simulation data are adopted to explain estimation methods. We used various censoring schemes and various sample sizes to determine whether the sample size affects the estimation measures. We used different confidence intervals to determine the best and shortest intervals. Also, the major findings in the paper are remarked on in the conclusion section.

]]>Raghad S.Shamsah

With a novel generation operator known as Spherical Scaling Wavelet Projection Operator, this study proposes new strategies for achieving the scaling wavelet expansion's convergence of functions with respect to almost everywhere under generic hypotheses. Hypotheses of results are based on three types of conditions: Space's function f, Kind of Wavelet functions (spherical) and Wavelet Conditions. The results showed that in the case of and under the assumption that scaling wavelet function of a given multiresolution analysis is spherical wavelet with 0-regularity, the convergence of ) expansions almost everywhere will be achieved under a new kind of partial sums operator. We can examine some properties of spherical scaling wavelet functions like rapidity of decreasing and boundedness. After estimating the bounds of spherical scaling wavelet expansions, we examined the limited (bounds) of this operator. The results are established on the almost everywhere wavelet expansions convergence of space functions. Several techniques were followed to achieve this convergence, such as the bounded condition of the Spherical Hardy-Littlewood maximal operator is achieved using the maximal inequality and Riesz basis functions conditions. The general wavelet expansions' convergence was demonstrated using the spherical scaling wavelet function and several of its fundamental features. In fact, the partial sums in these expansions are dominated in their magnitude by the maximal function operator, which may be applied to establish convergence. The convergence here may be obtained by assuming minimal regularity for a spherical scaling wavelet function . The focus of this research is on recent advances in convergence theory issues with respect to spherical wavelet expansions' partial sums operators. The employment of scaling wavelet basis functions defined on is regarded to be a key in solving convergence problems that occur inside spaces dimension .

]]>K.Marimuthu and J.Uma

Geometric Function Theory is one of the major area of mathematics which suggests the significance of geometric ideas and problems in complex analysis. Recently, the univalent functions are given particular attention and they are used to construct linear operators that preserve the class of univalent functions and some of its subclasses. Also, similar attention has been given to distribution series. Many authors have studied about certain subclasses of univalent and bi-univalent functions connected with distribution series like Pascal distribution, Binomial distribution, Poisson distribution, Mittag-Leffler-type Poisson distribution, Geometric distribution, Exponential distribution, Borel distribution, Generalized distribution and Generalized discrete probability distribution to name few. Some of the important results on Uniformly convex spirallike functions (UCF) and Uniformly spirallike functions (USF) related with such a distribution series are also of interest. The main aim of the present investigation is to obtain the necessary and sufficient conditions for Poisson distribution series to belong to the classes and . The inclusion properties associated with Poisson distribution series are taken up for study in this article. Proof of some inequalities on integral function connected to Poisson distribution series has also been discussed. Further, some corollaries and results that follow consequently from the theorems are also analysed.

]]>S. Aarthi and M. Shanmugasundari

This study provides non-preemptive priority fuzzy and intuitionistic fuzzy queuing models with unequal service rates. For performance evaluations of the industrial, supply chain, stock management, workstations, data exchange, and telecommunications equipment, non-preemptive priority queues are appropriate. The parameters of non-preemptive priority queues may be fuzzy due to unpredictable causes. The primary goal of this research is to compare the performance of a non-preemptive queuing model applying fuzzy queuing theory and intuitionistic fuzzy queuing theory. The performance metrics in the fuzzy queuing theory model are given as a range of values, however, the intuitionistic fuzzy queuing theory model offers a multitude of values. Both the arrival rate and the service rate are triangular and intuitionistic triangular fuzzy numbers in this case. An analysis is provided to identify the quality metrics by using a developed methodology through which without converting into crisp values we are taking the fuzzy values as it is and to demonstrate the viability of the suggested method, two numerical problems are solved.

]]>Wannaphon Suriyakat and Kanita Petcharat

The exponentially weighted moving average (EWMA) control chart is a popular tool used to monitor and identify slight unnatural variations in the manufacturing, industrial, and service processes. In general, control charts operate under the assumption of normality observation of the attention quality feature, but it is not easy to maintain this assumption in practice. In such situations, the data of random processes are correlated data, such as stock price in the economic field or air pollution data in the environment field. The characteristics and performance of the control chart are measured by the average run length (ARL). In this article, we present the new explicit formula of ARL for EWMA control chart based on MAX(q,r) process. The proposed explicit formula of ARL for the MAX(q,r) process is proved using the Fredholm integral equation technique. Moreover, ARL values are also assessed using the numerical integral equations method based on Gaussian, midpoint, and trapezoidal rules. Banach's fixed point theorem guarantees the existence and uniqueness of the solution. Furthermore, the accuracy of the proposed explicit formula is assessed in absolute percentage relative error compared with the numerical integral equations method. The results found that the explicit formula's ARL values are similar to those obtained using the numerical integral equation method; the absolute percentage relative errors are less than 0.0001 percent. As a result, the essential conclusion is that the explicit formula outperforms the numerical method in computational time. Consequently, the proposed explicit formula and the numerical integral equation have been the alternative approaches for computing ARL values of the EWMA control chart. They would be applied in various fields, including economics, environment, biology, engineering, and others.

]]>A. Baklizi A. Saadati Nik and A. Asgharzadeh

The Lomax distribution has been used as a statistical model in several fields, especially for business failure data and reliability engineering. Accurate parameter estimation is very important because it is the base for most inferences from this model. In this paper, we shall study this problem in detail. We developed several points and interval estimators for the parameters of this model assuming the data are type II progressively censored. Specifically, we derive the maximum likelihood estimator and the associated Wald interval. Bayesian point and interval estimators were considered. Since they can't be obtained in a closed form, we used a Markov chains Monte Carlo technique, the so called the Metropolis – Hastings algorithm to obtain approximate Bayes estimators and credible intervals. The asymptotic approximation of Lindley to the Bayes estimator is obtained for the present problem. Moreover, we obtained the least squares and the weighted least squares estimators for the parameters of the Lomax model. Simulation techniques were used to investigate and compare the performance of the various estimators and intervals developed in this paper. We found that the Lindley's approximation to the Bayes estimator has the least mean squared error among all estimators and that the Bayes interval obtained using the Metropolis – Hastings to have better overall performance than the Wald intervals in terms of coverage probabilities and expected interval lengths. Therefore, Bayesian techniques are recommended for inference in this model. An example of real data on total rain volume is given to illustrate the application of the methods developed in this paper.

]]>Salah Gazi Shareef

Iterative methods such as the conjugate gradient method are well known methods for solving non-linear unconstrained minimization problems partially because of their capacity to handle large-scale unconstrained optimization problems rapidly, and partly due to their algebraic representation and implementation in computer programs. The conjugate gradient method has wide applications in a lot of fields such as machine learning, neural networks and many other fields. Fletcher and Reeves [1] expanded the approach to nonlinear problems in 1964. It is considered to be the first nonlinear conjugate gradient technique. Since then, lots of new other conjugate gradient methods have been proposed. In this work, we will propose a new coefficient conjugate gradient method to find the minimum of the non-linear unconstrained optimization problems based on parameter of Hestenes Stiefel. Section one in this work contains the derivative of new method. In section two, we will satisfy the descent and sufficient descent conditions. In section three, we will study the property of the global convergence of the new proposed. In the fourth section, we will give some numerical results by using some known test functions and compare the new method with Hestenes S. to demonstrate the effectiveness of the suggestion method. Finally, we will give conclusions.

]]>Nur Kamilah Sa'diyah Ani Budi Astuti and Maria Bernadetha T. Mitakda

One regression model to explain the relationship between predictor and response variable in the form of count is Poisson regression. In the case of certain Poisson with the presence of many zero values, causing overdispersion can be overcome with the Poisson Hurdle model. There is a good method for estimating the parameters on small sample sizes for all distributions, namely the Bayesian method. The response variable of the original data does not follow Poisson distribution, so parameter will be estimated by Bayesian method. The performance of the Bayesian Hurdle Poisson regression can be seen from simulation data on various sample sizes and overdispersion levels generated based on the parameters of original data showing that the Bayesian Hurdle Poisson regression model proposed in this study is suitable for large sample sizes or with varying levels of overdispersion due or because normal distribution is used as prior. Even though the response variable of the simulation data is generated with a Poisson distribution, it still does not follow a Poisson distribution because it's in accordance with the original data. The parameter estimated based on the simulation data is similar to the parameter estimated on the original data (both the estimator of the MLE Hurdle Poisson regression parameter and the parameter estimator of the Bayesian Hurdle Poisson regression). This indicates that the simulation scenario is appropriate.

]]>Ugwu Samson O. Nduka Uchenna C. Eze Nnaemeka M. Odoh Paschal N. and Ugwu Gibson C.

Using spread-charts to monitor process variation and thereafter using the -chart to monitor the process mean after is a common practice. To apply these charts independently using estimated 3-sigma limits is common. Recently, some authors considered the application of and R-charts together as a charting scheme, -chart when the standards are known, Case KK, only the mean standard is known, Case KU and both standards unknown, Case UU. The average run length (ARL) performance criterion was used. However, because of the skewed nature of the run length (RL) distribution, many authors have frowned at the use of ARL as a sole performance measure and encouraged the percentiles of the RL distribution instead. Therefore, the cdfs of the RLs of the chart under the cases mentioned will be derived in this work, and the percentiles are used to look at the chart for Case KU and the yet to be considered case of the chart, Case UK where only the process variance is known is included for comparison. These are the contribution to the existing literature. -chart performed better in Case KU than in Case UK and the unconditional in-control median run length described the behavior of the chart better than the in-control ARL.

]]>B. M. Cerna Maguiña Dik D. Lujerio Garcia Héctor F. Maguiña and Miguel A. Tarazona Giraldo

In this work, we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the form where is a given natural number that ends in one. This allows us to decide the primality of a natural number that ends in one. Also we get some results on twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension , in which the relation is obtained: , where is a real number for . When or , we manage to address Fermat's Last Theorem and the equation , proving that both equations do not have positive integer solutions. For , the Cauchy-Schwartz Theorem and Young's inequality were proved in an original way.

]]>Athirah Zulkarnain Hazzirah Izzati Mat Hassim Nor Haniza Sarmin and Ahmad Erfanian

A set of vertices and edges forms a graph. A graph can be associated with groups using the groups' properties for its vertices and edges. The set of vertices of the graph comprises the elements of the group, while the set of edges of the graph is the properties and requirements for the graph. A non-abelian tensor square graph of a group is defined when its vertex set represents the non-tensor centre elements' set of G. Then, two distinguished vertices are connected by an edge if and only if the non-abelian tensor square of these two elements is not equal to the identity of the non-abelian tensor square. This study investigates the non-abelian tensor square graph for the symmetric group of order six. In addition, some properties of this group's non-abelian tensor square graph are computed, including the diameter, the dominating number, and the chromatic number. The perfect code for the non-abelian tensor square graph for a symmetric group of order six is also found in this paper.

]]>Nahashon Mwirigi Stanley Sewe Mary Wainaina and Richard Simwa

This study considered the problem of selecting the best single model for modeling state-specific failure rates in HIV/AIDS progression for patients on antiretroviral therapy with age and gender as risk factors using exponential, twoparameter, and three-parameter Weibull distributions. CD4 count changes in any two consecutive visits, the mean waiting time (μ), and transitional rates (λ) for remaining in the same state or transiting to a better or a worse state were analyzed. Various model selection criteria, namely, Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), and Log-Likelihood (LL), were used in each specific disease state. The Maximum Likelihood Estimation (MLE) method was applied to obtain the parameters of the distributions used. Plots of State-specific transition rates (λ) depicted constant, increasing, decreasing, and unimodal trends. Three-parameter Weibull distribution was the best for male patients and patients aged (40-69) years transiting in the states 1-2, 3-4, and 4-5, and 1-2, 3-4, and 5-6, respectively, and for male, female patients, and patients aged (40-69), remaining in the same state. Two-parameter Weibull distribution was the best for female patients and patients aged (20-39) years transiting in the states 1-2, 2-3, 4-5, and 1-2, 2-3, 3-4, respectively. Exponential distribution proved inferior to the other two distributions used.

]]>Munirah Rossdy Rashidah Omar and Shaharuddin Cik Soh

Let denote the functions' class that is normalized, analytic, as well as univalent in the unit disc given by . Convex, starlike, as well as close-to-convex functions resemble the main subclasses of , expressed by , as well as , accordingly. Many mathematicians have recently studied radius problems for various classes of functions contained in . The determination of the univalence radius, starlikeness, and convexity for specific special functions in is a relatively new topic in geometric function theory. The problem of determining the radius has been initiated since the 1920s. Mathematicians are still very interested in this, particularly when it comes to certain special functions in . Indeed, many papers investigate the radius of starlikeness for numerous functions. With respect to the open unit disc and class , the class of concave functions , known as , is defined. It is identified as a normalised analytic function , which meets the requirement of having the opening angle of at . A univalent function is known as concave provided that is concave. In other words, we have that is also convex. There is no literature to date on determining the radius of starlikeness for concave univalent functions related to certain rational functions, lune, cardioid, and the exponential equation. Hence, by employing the subordination method, we present new findings on determining several radii of starlikeness for different subclasses of starlike functions for the class of concave univalent functions .

]]>Said Taoufiki and Jamal El Achky

Each of us has had the experience of being overtaken by another less demanding customer in a queue. And each of us got behind a demanding customer and had to wait a long time. The frequencies of discrimination that appear here are overruns and heavy work. These are two phenomena that accompany queues, and have a great impact on customer satisfaction. Recently, authors have turned to measure queuing fairness based on the idea that a customer may feel anger towards the queuing system, even if he does not stay long on hold if he had one of the two experiences. We have found that this type of approach is more in line with studies provided by sociologists and psychologists. The frequencies of discrimination in a queue are studied for certain models of a single server. But for the case of multi-servers, there is only one study of a two-server Markovian queue. In this article, we wish to generalize this last study and we demonstrate that the result found in the case of two servers remains valid after comparing the discrimination frequencies of two Markov queueing systems to several servers.

]]>David Kwamena Mensah Michael Arthur Ofori and Nathaniel Howard

Physiological vital signs acquired during traumatic events are informative on the dynamics of the trauma and their relationship with other features such as sample-specific covariates. Non-time dependent covariates may introduce extra challenges in the Gaussian Process () regression, as their main predictors are functions of time. In this regard, the paper introduces the use of Orthogonalized Gnanadesikan-Kettering covariates for handling such predictors within the Gaussian process regression framework. Spectral Bayesian regression is usually based on symmetric spectral frequencies and this may be too restrictive in some applications, especially physiological vital signs modeling. This paper builds on a fast non-standard variational Bayes method using a modified Van der Waerden sparse spectral approximation that allows uncertainty in covariance function hyperparameters to be handled in a standard way. This allows easy extension of Bayesian methods to complex models where non-time dependent predictors are available and the relationship between the smoothness of trend and covariates is of interest. The utility of the methods is illustrated using both simulations and real traumatic systolic blood pressure time series data.

]]>Triastuti Wuryandari Gunardi and Danardono

The Cox regression model is widely used for survival data analysis. The Cox model requires a proportional hazard. If the proportional hazard assumption is doubtful, then the additive hazard model can be used, where the covariates act in an additively to the baseline hazard function. If the observed survival time is more than once for one individual during the observation, it is called a recurrent event. The additive hazard model measures risk difference to the effect of a covariate in absolutely, while the proportional hazards model measure hazard ratio in relatively. The risk coefficients estimation in the additive hazard model mimics the multiplicative hazard model, using partial likelihood methods. The derivation of these estimators, outlined in the technical notes, is based on the counting process approach. The counting process approach was first developed by Aalen on 1975 which combines elements of stochastic integration, martingale theory and counting process theory. The method is applied to study about the effect of supplementation on infant growth and development. Based on the processing results, the factors that affect the growth and development of the infant are gender, treatment and mother's education.

]]>Said Taoufiki and Driss Gretete

Volatility occupies a strategic place in the financial markets. In this context of crisis, and with the great movements of the markets, traders have been forced to turn to volatility trading for the potential gain it provides. The Black-Scholes formula for the value of a European option to purchase the underlying depends on a few parameters which are more or less easy to calculate, except for the realized volatility at maturity which makes a problem, because there is no single value, nor an established way to calculate it. In this article, we exploit the Martingale pricing method to find the expected present value of a given asset relative to a riskneutral probability measure. We consider a bond-stock market that evolves according to the dynamics of the Black-Scholes model, with a risk-free interest rate varying with time. Our methodology has effectively directed us towards interesting formulas that we have derived from the exact calculation, giving the present value of the volatility realized over a period of maturity for a European option in a stochastic volatility model.

]]>Deepmala Sharma and Sampada Tiwari

From the last few years, generalized bent functions gain a lot of attention in research as they have many applications in various fields such as combinatorial design, sequence design theory, cryptography, CDMA communication, etc. A deep and broad study of generalized bent functions with their properties is done in literature. Kumar et al.[11] first gave the concept of generalized bent function. Many researchers studied the properties and characterizations of generalized bent functions. In [2] authors introduced the concept of generalized (-ary) negabent functions and studied some properties of generalized (-ary) negabent functions. In this paper, we study the generalized (-ary) bent functions , where is the ring of integers with mod , is the vector space of dimension over and ≥2 is any positive integer. We discuss several properties of generalized (-ary) bent functions with respect to their nega-Hadamard transform. We also study the relation between generalized nega-Hadamard transforms and generalized nega-autocorrelations. Furthermore, we prove the necessary and sufficient conditions for the bentness and negabentness of generalized (-ary) bent function generated by the secondary construction for , where .

]]>Ahmad Fadly Nurullah Rasedee Mohammad Hasan Abdul Sathar Najwa Najib Nurhidaya Mohamad Jan Siti Munirah Mohd and Siti Nor Aini Mohd Aslam

The current study was conducted to establish a new numerical method for solving Duffing type differential equations. Duffing type differential equations are often linked to damping issues in physical systems, which can be found in control process problems. The proposed method is developed using a three-point block method in backward difference form, which offers an accurate approximation of Duffing type differential equations with less computational cost. Applying an Adam's like predictor-corrector formulation, the three point block method is programmed with a recursive relationship between explicit and implicit coefficients to reduce computational cost. By establishing this recursive relationship, we established a corrector algorithm in terms of the predictor. This eliminates any undesired redundancy in the calculation when obtaining the corrector. The proposed method allows a more efficient solution without any significant loss of accuracy. Four types of Duffing differential equations are selected to test the viability of the method. Numerical results will show efficiency of the three-point block method compared against conventional and more established methods. The outcome of this research is a new method for successfully solving Duffing type differential equation and other ordinary differential equations that are found in the field of science and engineering. An added advantage of the three-point block method is its adaptability to parallel programming.

]]>Sharipov Anvarjon Soliyevich and Topvoldiyev Fayzulla Foziljonovich

In one of the directions of classical differential geometry, the properties of geometric objects are studied in their entire range, which is called geometry "in large". Many problems of geometry "in large" are connected with the existence and uniqueness of surfaces with given characteristics. Geometric features can be intrinsic curvature, extrinsic or Gaussian curvature, and other features associated with the surface. The existence of a polyhedron with given curvatures of vertices or with a given development is also a problem of geometry "in large". Therefore, the problem of finding invariants of polyhedra of a certain class and the solution of the problem of the existence and uniqueness of polyhedra with given values of the invariant are relevant. This work is devoted to finding invariants, surfaces isometric on sections. In particular, we study the expansion properties of convex polyhedra that preserve isometry on sections. For such polyhedra, an invariant associated with the vertex of a convex polyhedral angle is found. Using this invariant, we can consider the question of restoring a convex polyhedron with given values of conditional curvature at the vertices. The isometry on section differs from the isometry of surfaces. The isometry of surfaces does not imply the isometry in sections, and vice versa. One of the invariants of surfaces isometric in cross sections is the area of the cylindrical image. This paper presents the properties of the area of a cylindrical image.

]]>Aiyared Iampan M. Balamurugan and V. Govindan

Among many algebraic structures, algebras of logic form an essential class of algebras. BCK and BCI-algebras are two classes of logical algebras. They were introduced by Imai and Iséki [6, 7] in 1966 and have been extensively investigated by many researchers. The concept of fuzzy soft sets is introduced in [17] to generalize standard soft sets [21]. The concept of intuitionistic fuzzy soft sets is introduced by Maji et al. [18], which is based on a combination of the intuitionistic fuzzy set [2] and soft set models. The first section will discuss the origins and importance of studies in this article. Section 2 will review the definitions of a BCK/BCI-algebra, a soft set, a fuzzy soft set, and an intuitionistic fuzzy soft set and show the essential properties of BCK/BCI-algebras to be applied in the next section. In Section 3, the concept of an anti-intuitionistic fuzzy soft b-ideal (AIFSBI) is discussed in BCK/BCI-algebras, and essential properties are provided. A set of conditions is provided for an AIFSBI to be an anti-intuitionistic fuzzy soft ideal (AIFSI). The definition of quasi-coincidence of an intuitionistic fuzzy soft point with an intuitionistic fuzzy soft set (IFSS) is considered in a more general form. In Section 4, the concepts of an ()-AFSBI and an ()-AIFSBI of are introduced, and some characterizations of ()-AIFSBI are discussed using the concept of an AIFSBI with thresholds. Finally, conditions are given for a ()-AIFSBI to be a (∈,∈)-AIFSBI.

]]>Sri Maryani Bambang H Guswanto and Hendra Gunawan

Recently, we have seen the phenomena in use of partial differential equations (PDEs) especially in fluid dynamic area. The classical approach of the analysis of PDEs were dominated in early nineteenth century. As we know that for PDEs the fundamental theoretical question is whether the model problem consists of equation and its associated side condition is well-posed. There are many ways to investigate that the model problems are well-posed. Because of that reason, in this paper we consider the -boundedness of the solution operator families for Navier-Lamé equation by taking into account the surface tension in a bounded domain of -dimensional Euclidean space (≥ 2) as one way to study the well-posedess. We investigate the -boundedness in half-space domain case. The -boundedness implies not only the generation of analytic semigroup but also the maximal regularity for the initial boundary value problem by using Weis's operator valued Fourier multiplier theorem for time dependent problem. It was known that the maximal regularity class is the powerful tool to prove the well-posesness of the model problem. This result can be used for further research for example to analyze the boundedness of the solution operators of the model problem in bent-half space or general domain case.

]]>Ming-Ming Xu Jumat Sulaiman and Nur Afza Mat Ali

The numerical solutions of the second-order linear Fredholm integro-differential equations have been considered and discussed based on several discretization schemes. In this paper, the new schemes are developed derived on the hybrid of the three-point half-sweep linear rational finite difference (3HSLRFD) approaches with the half-sweep composite trapezoidal (HSCT) approach. The main advantage of the established schemes is that they discretize the differential terms and integral term of second-order linear Fredholm integro-differential equations into the algebraic equations and generate the corresponding linear system. Furthermore, the half-sweep (HS) concept is combined with the refinement of the successive over-relaxation (RSOR) iterative method to create the new half-sweep successive over-relaxation (HSRSOR) iterative method, which is implemented to get the numerical solution of a system of linear algebraic equations. Apart from that, the classical or full-sweep Gauss-Seidel (FSGS) and full-sweep successive over-relaxation iterative (FSSOR) methods are presented, which serve as the control method in this paper. In the end, we employed FSGS, FSRSOR and HSRSOR methods to obtain numerical solutions of three examples and make a detailed comparison from three aspects of the number of iterations, elapsed time and maximum absolute error. Numerical results demonstrate that FSRSOR and HSRSOR methods have lesser iterations, faster elapsed time, and are more accurate than FSGS. In addition, HSRSOR is the most effective of the three methods. To sum up, this paper has successfully proposed the applicability and superiority of the new HSRSOR method based on 3HSLRFD-HSCT schemes.

]]>A. Dinesh Kumar and R. Sivaraman

Among several properties that real numbers possess, this paper deals with the exciting formation of positive rational numbers constructed in the form of a Tree, in which every number has two branches to the left and right from the root number. This tree possesses all positive rational numbers. Hence it consists of infinite numbers. We call this tree "Fraction Tree". We will formally introduce the Fraction Tree and discuss several fascinating properties including proving the one-one correspondence between natural numbers and the entries of the Fraction Tree. In this paper, we shall provide the connection between the entries of the fraction tree and Fibonacci numbers through some specified paths. We have also provided ideas relating the terms of the Fraction Tree with that of continued fractions. Five interesting theorems related to the entries of the Fraction Tree are proved in this paper. The simple rule that is used to construct the Fraction Tree enables us to prove many mathematical properties in this paper. In this sense, one can witness the simplicity and beauty of making deep mathematics through simple and elegant formulations. The Fraction Tree discussed in this paper which is technically called Stern-Brocot Tree has profound applications in Science as diverse as in clock manufacturing in the early days. In particular, Brocot used the entries of the Fraction Tree to decide the gear ratios of mechanical clocks used several decades ago. A simple construction rule provides us with a mathematical structure that is worthy of so many properties and applications. This is the real beauty and charm of mathematics.

]]>M. Kozae Samar A. Abo Quota and Alaa H. N.

This paper introduces a new idea in the unital involutive Banach algebras and its closed subset. This paper aims to study the cohomology theory of operator algebra. We will study the Banach algebra as an applied example of operator algebra, and the Banach algebra will be denoted by . The definitions of cyclic, simplicial, and dihedral cohomology group of will be introduced. We presented the definition of -relative dihedral cohomology group that is given by: , and we will show that the relation between dihedral and -relative dihedral cohomology group can be obtained from the sequence . Among the principal results that we will explain is the study of some theorems in the relative dihedral cohomology of Banach algebra as a Connes-Tsygan exact sequence, since the relation between the relative Banach dihedral and cyclic cohomology group ( and ) of will be proved as the sequence . Also, we studied and proved some basic notations in the relative cohomology of Banach algebra with unity and defined its properties. So, we showed the Morita invariance theorem in a relative case with maps and , and proved the Connes-Tsygan exact sequence that relates the relative cyclic and dihedral (co)homology of . We proved the Mayer-Vietoris sequence of in a new form in the Banach B-relative dihedral cohomology: . It should be borne in mind that the study of the cohomology theory of operator algebra is concerned with studying the spread of Covid 19.

]]>Kamola Saxibovna Ablazova

In the statistical management of processes in the initial phase, the stability of the technological process is determined based on the available samples. If the process is not stable, eliminating possible causes is brought into a statistically controlled position. At the same time, simple Shewhart control charts are used. In practice, some methods bring the process to a stable state (ISO standards, standards of various states). After the process has become stable, the boundaries of control charts are found for further management. Then, with the help of new samples, the process is managed. The article considers a process modeled by a two-dimensional normal distribution. New control charts have been found to check the normality and correlation of two-dimensional random variable components. The process is regulated using these charts, preserving the shape of the density of the individual components of the normal vector and linearity of these components. When constructing control charts, the Kolmogorov-Smirnov type agreement criterion and the Fisher criterion on the strength of the linear coupling of components were used. A concrete example shows the course of the introduction of these charts in production. The work results can be used in the initial phase of regulation and during the control check of the process under study. We used these control charts to assess product quality and quality control coming from the machine that produces the sleeves. It presents statistical methods for analyzing problems in factory practice and solutions for their elimination.

]]>B. Vasuki L. Shobana and B. Roopa

An approach to encrypt and decrypt messages is obtained by relating the concepts of graph labeling and cryptography. Among the various types of labelings given in [3], our interest is on face antimagic labeling introduced by Mirka Miller in 2003 [1]. Baca [2] defines a connected plane graph with edge set and face set as face antimagic if there exist positive integers and and a bijection such that the induced mapping , where for a face , is the sum of all for all edges surrounding is also a bijection. In cryptography there are many cryptosystems such as affine cipher, Hill cipher, RSA, knapsack and so on. Amongst these, Hill cipher is chosen for our encryption and decryption. In Hill cipher [8], plaintext letters are grouped into two-letter blocks, with a dummy letter X inserted at the end if needed to make all blocks of the same length, and then replace each letter with its respective ordinal number. Each plaintext block is then replaced by a numeric ciphertext block , where and are different linear combinations of and modulo 26: (mod 26) and (mod 26) with condition as is one. Each number is translated into a cipher text letter which results in cipher text. In this paper, face antimagic labeling on double duplication of graphs along with Hill cipher is used to encrypt and decrypt the message.

]]>Toru Ogura and Shin-ichi Tsukada

Missing data occur in various fields, such as clinical trials and social science. Canonical correlation analysis often used to analyze the correlation between two random vectors, cannot be performed on a dataset with missing data. Canonical correlation coefficients (CCCs) can also be calculated from a covariance matrix. When the covariance matrix can be estimated by excluding (complete-case and available-case analyses) or imputing (multivariate imputation by chained equations, k-nearest neighbor (kNN), and iterative robust model-based imputation) missing data, CCCs are estimated from this covariance matrix. CCCs have bias even with all observation data. Usually, estimated CCCs are even larger than population CCCs when a covariance matrix estimated from a dataset with missing data is used. The purpose is to bring the CCCs estimated from the dataset with missing data close to the population CCCs. The procedure involves three steps. First, principal component analysis is performed on the covariance matrix from the dataset with missing data to obtain the eigenvectors. Second, the covariance matrix is transformed using first to fourth eigenvectors. Finally, the CCCs are calculated from the transformed covariance matrix. CCCs derived using with this procedure are called the principal CCCs (PCCCs), and simulation studies and numerical examples confirmed the effectiveness of the PCCCs estimated from the dataset with missing data. There were many cases in the simulation results where the bias and root-mean-squared error of the PCCC estimated from the missing data based on kNN were the smallest. In the numerical example results, the first PCCC estimated from the missing data based on kNN is close to the first CCC estimated from the dataset comprising all observation data when the correlation between two vectors is low. Therefore, PCCCs based on kNN were recommended.

]]>Jamal Salah Hameed Ur Rehman and Iman Al- Buwaiqi

The Riemann zeta function is valid for all complex number , for the line = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, ... (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines = 0 and = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line . As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10^{13}, non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of , and omitting the impact of an indeterminate form , that appears in Riemann's approach. In this study, we also consider the simultaneous occurrence but we adopt an element-wise approach of the Taylor series by expanding for all = 1, 2, 3, ... at the real parts of the non-trivial zeta zeros lying in the critical strip for is a non-trivial zero of , we first expand each term at then at . Then in this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

Ugwu Samson. O Uchenna Nduka .C Ezra Precious .N Ugwu Gibson .C Odoh Paschal .N and Nwafor Cynthia. N

It is well known that the median is a better measure of location in skewed distributions. Run-length (RL) distribution is a skewed distribution, hence, median run-length measures chart performance better than the average run length. Some authors have advocated examination of the entire percentiles of the RL distribution in assessing chart performance. Such works already exist for Shewhart −chart, CUSUM chart, CUSUM and EWMA charts, Hotelling's chi-square, and the two simple Shewhart multivariate non-parametric charts. Similar work on -chart for one- and two-sided lacks in the literature. This work stands in the gap. Therefore, a detailed and comparative study of the one-sided upper and the two-sided -control charts for some m reference samples at fixed sample size and false alarm rate will be considered here using the information from the unconditional RL cdf curve and its percentiles (mainly median). The order of the RL cdf curves of the one-sided upper -chart is independent of the state of the process unlike in the two-sided one. The one-sided upper chart outperformed the two-sided one both in the in-control and in detecting positive shifts. The two-sided -chart is more sensitive in detecting incremental shifts than to decremental shifts.

]]>Duygu Dönmez Demir and Gülsüm Şanal

The theory of inequality is in a process of continuous development and has become a quite effective and powerful tool in various branches of mathematics to solve many problems. Convex functions are closely related to the theory of inequality, and many important inequalities are the results of the applications of convex functions. Recently, the results obtained for convex functions have been tried to be extended for strongly convex functions. In our previous studies, the perturbed trapezoid inequality obtained for convex functions has been extended to the functions that can be differentiated -times. This study deals with some general identities introduced for -times differentiable strongly convex functions. Besides, new inequalities related to general perturbed trapezoid inequality are constructed. These inequalities are obtained for the classes of functions which ^{ th} derivatives of absolute values of the mentioned functions are strongly convex. It is seen that new classes of strongly convex functions turn into those obtained for convex functions under certain conditions. Considering the upper bounds obtained for strongly convex functions, it is concluded that it is better than those obtained for convex functions.

Maisoun Sewailem and Ayman Baklizi

In this paper, we consider improving maximum likelihood inference for the scale parameter of the Lomax distribution. The improvement is based on using modifications to the maximum likelihood estimator based on the Barndorff-Nielsen modification of the profile likelihood function. We apply these modifications to obtain improved estimators for the scale parameter of the Lomax distribution in the presence of a nuisance shape parameter. Due to the complicated expression for the Barndorff-Nielsen's modification, several approximations to this modification are considered in this paper, including the modification based on the empirical covariances and the approximation based on using suitably derived approximate ancillary statistics. We obtained the approximations for the Lomax profile likelihood function and the corresponding modified maximum likelihood estimators. They are not available in simple closed forms and can be obtained numerically as roots of some complicated likelihood equations. Comparisons between maximum profile likelihood estimator and modified profile likelihood estimators in terms of their biases and mean squared errors were carried out using simulation techniques. We found that the approximation based on the empirical covariances to have the best performance according to the criteria used. Therefore we recommend to use this modified version of the maximum likelihood estimator for the Lomax scale parameter, especially for small sample sizes with heavy censoring, which is quite common in industrial life testing experiments and reliability studies. An example based on real data is given to illustrate the methods considered in this paper.

]]>Jonathan Tsetimi and Ebimene James Mamadu

The use of orthogonal polynomials as basis functions via a suitable approximation scheme for the solution of many problems in science and technology has been on the increase and quite fascinating. In many numerical schemes, the convergence depends solely on the nature of the basis function adopted. The Mamadu-Njoseh polynomials are orthogonal polynomials developed in 2016 with reference to the weight function, which bears the same convergence rate as that of Chebyshev polynomials. Thus, in this paper, the fractional variational orthogonal collocation method (FVOCM) is proposed for the solution of fractional Fredholm integro-differential equation using Mamadu-Njoseh polynomials (MNP) as basis functions. Here, the proposed method is an elegant mixture of the variational iteration method (VIM) and the orthogonal collocation method (OCM). The VIM is one of the popular methods available to researchers in seeking the solution to both linear and nonlinear differential problems requiring neither linearization nor perturbation to arrive at the required solution. Collocating at the roots of orthogonal polynomials gives birth to the OCM. For the proposed method, the VIM is initiated to generate the required approximations whereby producing the series which is collocated orthogonally to derive the unknown parameters. The numerical results show that the method derives a high accurate and reliable approximation with a high convergence rate. We have also presented the existence and uniqueness of solution of the method. All computational frameworks in this research are performed via MAPLE 18 software.

]]>Rootvesh Mehta Sandeep Malhotra Dhiren Pandit and Manoj Sahni

A stiff equation is a differential equation for which certain numerical methods are not stable, unless the step length is taken to be extraordinarily small. The stiff differential equation includes few terms that could result in speedy variation in the solution. When integrating a differential equation numerically, the requisite step length should be incredibly small. In the solution curve, much variation can be observed where the solution curve straightens out to approach a line with slope almost zero. The phenomenon of stiffness is observed when the step-size is unacceptably small in a region where the solution curve is very smooth. A lot of work on solving the stiff ordinary differential equations (ODEs) have been done by researchers with numbers of numerical methods that currently exist. Extensive research has been done to unveil the comparison between their rate of convergence, number of computations, accuracy, and capability to solve certain type of test problems. In the present work, an advanced Feed Forward Neural Network (FFNN) and Bayesian regularization algorithm-based method is implemented to solve first order stiff ordinary differential equations and system of ordinary differential equations. Using proposed method, the problems are solved for various time steps and comparisons are made with available analytical solutions and other existing methods. A problem is simulated using proposed FFNN model and accuracy has been acquired with less calculation efforts and time. The outcome of the work is showing good result to use artificial neural network methods to solve various types of stiff differential equations in near future.

]]>S. Dhanasekar Saroj Kumar Dash and Neena Uthaman

The core of the theoretical Computing science and mathematics is computational complexity theory. It is usually concerned with the classification of computational problems in to P and NP problems by using their inherent challenges. There is no efficient algorithms for these problems. Travelling Salesman Problem is one of the most discussed problems in Combinatorial Mathematics. To deduct a Hamiltonian cycle in which the cost or time is minimum is the main objective of the TSP. There exist many algorithms to solve it. Since all the existing algorithms are not efficient to solve it, still many researchers are working to produce efficient algorithms. If the description of the parameters is vague, then fuzzy notions which include membership value are applied to model the parameters. Still the modeling does not give the exact representation of the vagueness. The Intuitionistic fuzzy set which includes non-membership value along with membership values in its domain is applied to model the parameters. The decision variables in the TSP, the cost, time or distance are modeled as intuitionistic fuzzy numbers, then the TSP is named as Intuitionistic fuzzy TSP (InFTSP). We develop the intuitionistic fuzzified version of littlewood's formula or branch and bound method to solve the Intuitionistic fuzzy TSP. This method is effective because it involves the simple arithmetic operation of Intuitionistic fuzzy numbers and ranking of intuitionistic fuzzy numbers. Ordering of intuitionistic fuzzy numbers is vital in optimization problems since it is equivalent to the ordering of alternatives. In this article, we used weighted arithmetic mean method to order the fuzzy numbers. Weighted arithmetic mean method satisfies linear property which is a very important characteristic of ranking function. Numerical examples are solved to validate the given algorithm and the results are discussed.

]]>Mohamed Khalifa Ahmed Issa

Time series forecasting is the main objective in many life applications such as weather prediction, natural phenomena analysis, financial or economic analysis, etc. In real-life data analysis, missing data can be considered as a feature that the researcher faces because of human error, technical damage, or catastrophic natural phenomena, etc. When one or more observations are missing, it might be urgent to estimate the model as well as to estimate the missing values which lead to a better understanding of the data, and more accurate prediction. Different time series require different effective techniques to have better estimates for those missing values. Traditionally, the missing values are simply replaced by mean and mode imputation, deleted or handled using other methods, which are not convenient enough to address missing values, as those methods can cause bias. One of the most popular models used in estimating time-series data is autoregressive models. Autoregressive models forecast the future values in terms of the previous ones. The first-order autoregressive AR (1) model is the one which the current value is based on the immediately preceding value, then estimating parameters of AR (1) with missing observations is an urgent topic in time series analysis. Many approaches have been developed to address the estimation problems in time series such as ordinary least square (OLS), Yule Walker (YW). Therefore, a suggested method will be introduced to estimate the parameter of the model by using weighted least squares. The properties of the (WLS) estimator are investigated. Moreover, a comparison between those methods using AR (1) model with missing observations is conducted through a Monte Carlo simulation at various sample sizes and different proportions of missing observations, this comparison is conducted in terms of mean square error (MSE) and mean absolute error (MAE). The results of the simulation study state that (WLS) estimator can be considered as the preferable method of estimation. Also, time series real data with missing observations were estimated.

]]>Elvis Adam Alhassan Kaiyu Tian and Adjabui Michael

This chapter review presents two ideas and techniques in solving Systems of Linear Equations in the most simple minded straightforward manner to enable the student as well as the instructor to follow it independently with very little guidance. The focus is on using simpler and easier approaches such as Determinants; and Elementary Row Operations to solve Systems of Linear Equations. We found the solution set of a few systems of linear equations by a successive ratio of the determinant of all the matrices formed from replacing each column of the coefficient matrix by the right hand side vector and the determinant of the coefficient matrix repeatedly giving the values of the variables in the system in the order in which they appeared. Similarly, we also used the three types of elementary row operations namely; Row Swap; Scalar Multiplication; and Row Sum to find the solution set of systems of linear equations through row echelon form to reduced row echelon form to find the solution set of some systems of linear equations. Technical forms of systems of linear equations were used to illustrate the two approaches in finding their solution sets. In each approach we started by finding the coefficient matrices from the systems of linear equations.

]]>Veninstine Vivik J Sheeba Merlin G P. Xavier and Nila Prem JL

The idea of graph coloring problem (GCP) plays a vital role in allotment of resources resulting in its proper utilization in saving labor, space, time and cost effective, etc. The concept of GCP for graph is assigning minimum number of colors to its nodes such that adjacent nodes are allotted a different color, the smallest of which is known as its chromatic number . This work considers the approach of taking the tensor product between two graphs which emerges as a complex graph and it drives the idea of dealing with complexity. The load balancing on such complex networks is a hefty task. Amidst the various methods in graph theory the coloring is a quite simpler tool to unveil the intricate challenging networks. Further the node coloring helps in classifying the nodes with least number of classes in any network. So coloring is applied to balance the allocations in such complex network. We construct the tensor product between two graphs like path with wheel and helm, cycle with sunlet and closed helm graphs then structured their nature. The coloring is then applied for the nodes of the extended new graph to determine their optimal bounds. Hence we obtain the chromatic number for the tensor product of , , and .

]]>Mikhail Ivanovich Popov Aleksey Vasilyevich Skrypnikov Vyacheslav Gennadievich Kozlov Alexey Viktorovich Chernyshov Alexander Danilovich Chernyshov Sergey Yurievich Sablin Vladimir Valentinovich Nikitin and Roman Alexandrovich Druzhinin

In the paper, numerical and approximate analytical solutions for the problem of the motion of a spacecraft from a starting point to a final point during a certain time are obtained. The unpowered and powered portions of the flight are considered. For a numerical solution, a finite-difference scheme of the second order of accuracy is constructed. The space-related problem considered in the study is essentially nonlinear, which necessitates the use of trigonometric interpolation methods to replace the task of calculating the Fourier coefficients with the integral formulas by solving the interpolation system. One of the simplest options for trigonometric sine interpolation on a semi-closed segment [–a, a), where the right end is not included in the general system of interpolation points, is considered. In order to maintain the conditions of orthogonality of sines, an even number of 2M calculation points is uniformly applied to the segment. The sine interpolation theorem is proved and a compact formula is given for calculating the interpolation coefficients. A general theory of fast sine expansion is given. It is shown that in this case, the Fourier coefficients decrease much faster with the increase in serial number compared to the Fourier coefficients in the classical case. This property allows reducing the number of terms taken into account in the Fourier series, as well as the amount of computer calculations, and increasing the accuracy of calculations. The analysis of the obtained solutions is carried out, and their comparison with the exact solution of the test problem is proposed. With the same calculation error, the time spent on a computer using the fast expansion method is hundreds of times less than the time spent on classical finite-difference method.

]]>Mohd Azri Pawan Teh Nazrina Aziz and Zakiyah Zain

This research develops a generalized family of group chain sampling plans using the minimum angle method (MAM). The MAM is a method whereby both the producer's and consumer's risks are considered when designing the sampling plans. There are three sampling plans nested under the family of group chain acceptance sampling which are group chain sampling plans (GChSP-1), new two-sided group chain sampling plans (NTSGChSP-1), and two-sided group chain sampling plans (TSGChSP-1). The methodology applied is random values of the fraction defectives for both producer and consumer, and the optimal number of groups, is obtained using the Scilab software. The findings reveal that some of the design parameters manage to obtain the corresponding to the smallest angle, and some of the values fail to get the . The obtained in this research guarantees that the producer and the consumer are protected at most 10% from having defective items.

]]>Nazrina Aziz Tan Jia Xin Zakiyah Zain and Mohd Azri Pawan Teh

Acceptance criteria are the conditions imposed on any sampling plan to determine whether the lot is accepted or rejected. Group chain sampling plan (GChSP-1) was constructed according to the 5 acceptance criteria; modified group chain sampling plan (MGChSP-1) was derived with 3 acceptance criteria; later new group chain sampling plan (NGChSP-1) was introduced with 4 acceptance criteria where the NGChSP-1 balances the acceptance criteria between the GChSP-1 and MGChSP-1. Producers favor a sampling plan with more acceptance criteria because it reduces the probability of rejecting a good lot (producer risk), whereas consumers may prefer a sampling plan with fewer acceptance criteria as it reduces the probability of accepting a bad lot (consumer risk). The disparity in acceptance criteria creates a conflict between the two main stakeholders in acceptance sampling. In the literature, there are numerous methods available for developing sampling plans. To date, NGChSP-1 was developed using the minimum angle method. In this paper, NGChSP-1 was constructed with the minimizing consumer's risk method for generalized exponential distribution where mean product lifetime is used as quality parameter. There are six phases involved to develop the NGChSP-1 for different design parameters. Result shows the minimum number of groups decrease when the value of design parameters increases. The results of the performance comparison show that the NGChSP-1 is a better sampling plan than the GChSP-1 because it has a smaller number of groups and lower probability of lot acceptance than the GChSP-1. NGChSP-1 should offer better alternatives to industrial practitioners in sectors involving product life test.

]]>Suparman Hery Suharna Mahyudin Ritonga Fitriana Ibrahim Tedy Machmud Mohd Saifullah Rusiman Yahya Hairun and Idrus Alhaddad

Autoregressive (AR) model is applied to model various types of data. For confidential data, data confusion is very important to protect the data from being known by other unauthorized parties. This paper aims to find data modeling with transformations in the AR model. In this AR model, the noise has a Laplace distribution. AR model parameters include order, coefficients, and variance of the noise. The estimation of the AR model parameter is proposed in a Bayesian method by using the reversible jump Markov Chain Monte Carlo (MCMC) algorithm. This paper shows that the posterior distribution of AR model parameters has a complicated equation, so the Bayes estimator cannot be determined analytically. Bayes estimators for AR model parameters are calculated using the reversible jump MCMC algorithm. This algorithm was validated through a simulation study. This algorithm can accurately estimate the parameters of the transformed AR model with Laplacian noise. This algorithm produces an AR model that satisfies the stationary conditions. The novelty in this paper is the use of transformations in the Laplacian AR model to secure research data when the research results are published in a scientific journal. As an example application, the Laplacian AR model was used to model CO_{2} emission data. The results of this paper can be applied to modeling and forecasting confidential data in various sectors.

Ahmed Anwer Mustafa

CJG is a nonlinear conjugation gradient. Algorithms have been used to solve large-scale unconstrained enhancement problems. Because of their minimal memory needs and global convergence qualities, they are widely used in a variety of fields. This approach has lately undergone many investigations and modifications to enhance it. In our daily lives, the conjugate gradient is incredibly significant. For example, whatever we do, we strive for the best outcomes, such as the highest profit, the lowest loss, the shortest road, or the shortest time, which are referred to as the minimum and maximum in mathematics, and one of these ways is the process of spectral gradient descent. For multidimensional unbounded objective function, the spectrum conjugated gradient (SCJG) approach is a strong tool. In this study, we describe a revolutionary SCG technique in which performance is quantified. Based on assumptions, we constructed the descent condition, sufficient descent theorem, conjugacy condition, and global convergence criteria using a robust Wolfe and Powell line search. Numerical data and graphs were constructed utilizing benchmark functions, which are often used in many classical functions, to demonstrate the efficacy of the recommended approach. According to numerical statistics, the suggested strategy is more efficient than some current techniques. In addition, we show how the unique method may be utilized to improve solutions and outcomes.

]]>Nawal Adlina Mohd Ikbal Syafrina Abdul Halim and Norhaslinda Ali

Inefficient estimation of distribution parameters for current climate will lead to misleading results in future climate. Maximum likelihood estimation (MLE) is widely used to estimate the parameters. However, MLE is not well performed for the small size. Hence, the objective of this study is to compare the efficiency of MLE with ordinary least squares (OLS) through the simulation study and real data application on wind speed data based on model selection criteria, Akaike information criterion (AIC) and Bayesian information criterion (BIC) values. The Anderson-Darling (AD) test is also performed to validate the proposed distribution. In summary, OLS is better than MLE when dealing with small sample sizes of data and estimating the shape parameter, while MLE is capable of estimating the value of scale parameter. However, both methods are well performed at a large sample size.

]]>Azam A. Imomov and Zuhriddin A. Nazarov

In this paper, we consider some functionals of the sums of independent identically distributed random variables. The functionals of the sums are important in probabilistic models and stochastic branching systems. In connection with the application in various probabilistic models and stochastic branching systems, we are interested in the fulfillment of the law of large numbers and the Central limit theorem for these sums. The main hypotheses of the paper are the presence of second order moments of the variables and the fulfillment of the Lindeberg condition is considered. The research object and subject of this paper consists of specially generated random variables using the sums of non-bound random variables. In total, 6 different sums in a special form were studied in the paper and this sum was not previously studied by other scientists. The purpose of the paper is to examine whether these sums in a special form satisfy the terms of the law of large numbers and the Central limit theorem. The main result of the paper is to show that the law of large numbers and the terms of the classical limit theorem are fulfilled in some cases. The results obtained in the paper are of theoretical importance, The Central limit theorem analogues proved here are applications of Lindeberg theorem. The results can be applied to the determination of the fluctuation of immigration branching systems as well as the asymptotic state of autoregression processes. At the same time, from the main results obtained in the paper it can be used in practical lessons conducted on the theory of probability. The results of the paper will be an important guide for young researchers. Important theorems proved in the paper can be used in probability theory, stochastic branching systems and other practical problems.

]]>K.Geetha and S.P.Reshma

In this study we have discussed a fuzzy eoq model for deteriorating products with time varying deterioration under inflation and exponential time dependent demand rate. Shortages are not allowed in this fuzzy eoq model and the impact of inflation is investigated. An inventory model is used to determine whether the order quantity is more than or equal to a predetermined quantity for declining items.The optimal solution for the existing model is derived by taking truncated taylor’s series approximation for finding closed form optimal solution. The cost of deterioration, cost of ordering, cost of holding and the time taken to settle the delay in account are considered using triangular fuzzy numbers. In this study the fuzzy triangular numbers are used to estimate the optimal order quantity and cycle duration. Furthermore we have used graded mean integration method and signed distance approach to defuzzify these values. To validate our model numerical examples are discussed for all cases with the help of sensitivity analysis for different parameters. Finally, a higher decay rate results in a shorter ideal cycle time as well as higher overall relevant cost is established. The presented model can be used to predict demand as a quadratic function of time,stock level time dependent demand, selling price, and other variables.

]]>Labiyana Hanif Ali Jumat Sulaiman and Azali Saudi

In this study, we applied Newton method with a new version of KSOR, called PKSOR to form NPKSOR in solving nonlinear second kind Fredholm integral equations. A new version of KSOR is an update to the KSOR method with two relaxation parameters. The properties of KSOR helps in enlargement of the solution domain so the relaxation parameter can take the value . With PKSOR, the relaxation parameter in KSOR is treated into two different relaxation paramaters as and which resulting lower number of iteration compared to the KSOR method. By combining the Newton method with PKSOR, we intend to from more efficient method to solve the nonlinear Fredholm integral equations. The discretization part of this study is done using first-order quadrature scheme to develop a nonlinear system. We formulate the solution of the nonlinear system using the given approach by reducing it to a linear system and then solving it using iterative methods to obtain an approximate solution. Furthermore, we compare the results of the proposed methods with NKSOR and NGS methods on three examples. Based on our findings, the NPKSOR method is more efficient than NKSOR and NGS methods. By implementing the NPKSOR method, we can boost the convergence rate of the iteration by considering two relaxation parameters, resulting in a lower number of iteration and computational time.

]]>Nimitha John and Balakrishna Narayana

Two or more non-stationary time series are said to be co-integrated if a certain linear combination of them becomes stationary. Identification of co-integrating relationships among the relevant time series helps the researchers to develop efficient forecasting methods. The classical approach of analyzing such series is to express the co-integrating time series in the form of error correction models with Gaussian errors. However, the modeling and analysis of cointegration in the presence of non-normal errors needs to be developed as most of the real time series in the field of finance and economics deviates from the assumption of normality. This paper focuses on modeling of a bivariate cointegration with a student's-t distributed error. The co-integrating vector obtained from the error correction equation is estimated using the method of maximum likelihood. A unit root test of first order non stationary process with student's t-errors is also defined. The resulting estimators are used to construct test procedures for testing the unit root and cointegration associated with two time series. The likelihood equations are all solved using numerical approaches because the estimating equations do not have an explicit solution. A simulation study is carried out to illustrate the finite sample properties of the model. The simulation experiments show that the estimates perform reasonably well. The applicability of the model is illustrated by analyzing the data on time series of Bombay stock exchange indices and crude oil prices and found that the proposed model is a good fit for the data sets.

]]>Nur Azulia Kamarudin Suzilah Ismail and Norhayati Yusof

Model selection is the process of choosing a model from a set of possible models. The model's ability to generalise means it can fit both current and future data. Despite numerous emergences of procedures in selecting models automatically, there has been a lack of studies on procedures in selecting multiple equations models, particularly seemingly unrelated regression equations (SURE) models. Hence, this study concentrates on an automated model selection procedure for the SURE model by integrating the expectation-maximization (EM) algorithm estimation method, named SURE(EM)-Autometrics. This extension procedure was originally initiated from Autometrics, which is only applicable for a single equation. To assess the performance of SURE(EM)-Autometrics, simulation analysis was conducted under two strengths of correlation among equations and two levels of significance for a two-equation model with up to 18 variables in the initial general unrestricted model (GUM). Three econometric models have been utilised as a testbed for true specification search. The results were divided into four categories where a tight significance level of 1% had contributed a high percentage of all equations in the model containing variables precisely comparable to the true specifications. Then, an empirical comparison of four model selection techniques was conducted using water quality index (WQI) data. System selection to select all equations in the model simultaneously proved to be more efficient than single equation selection. SURE(EM)-Autometrics dominated the comparison by being at the top of the rankings for most of the error measures. Hence, the integration of EM algorithm estimation is appropriate in improving the performance of automated model selection procedures for multiple equations models.

]]>Bahtiar Jamili Zaini and Shamshuritawati Md Sharif

Jennrich statistic is a method that can be used to test the equality of 2 or more independent correlation matrices. However, Jennrich statistic begins to be problematic when there is presence of outliers that could lead to invalid results. When exiting outliers in data, Jennrich statistic implications will affect Type I errors and will reduce the power of test. To overcome the presence of outliers, this study suggests the use of robust methods as an alternative method and therefore, it will integrate the estimator into Jennrich statistic. Thus, it can improve the testing performance of correlation matrix hypotheses in relation to outlier problems. Therefore, this study proposes 3 statistical tests, namely Js-statistic, Jm-statistic, and Jmad-statistic that can be used to test the equation of 2 or more correlation matrices. The performance of the proposed method is assessed using the power of test. The results show that Jm-statistic and Jmad-statistic can overcome outlier problems into Jennrich statistic in testing the correlation matrix hypothesis. Jmad-statistic is also superior in testing the correlation matrix hypothesis for different sample sizes, especially those involving 10% outliers.

]]>Sara Abdel Baset Ramadan Hamed Maha El-Ashram and Zakaria Abdel Samea

Response surface methodology (RSM) is a group of mathematical and statistical techniques helpful for improving, developing and optimizing processes. It also has important uses in the design, development and formulation of new products. Moreover, it has a great help in the enhancement of existing products. (RSM) is a method used to discover response functions, which meet and fulfill all quality diagnostics simultaneously. Most applications have more than one response; the main problem is multi-response optimization (MRO). The classical methods used to solve the Multi-Response Optimization problem do not guarantee optimal designs and solutions. Besides, they take a long time and depend on the researcher's judgment. Therefore, some researchers used a Goal Programming-based method; however, they still do not guarantee an optimal solution. This study aims to form a goal programming model derived from a chance constrained approach using quantile regression to deal with outliers not normal and errors. It describes the relationship between responses and control variables at distinctive points in the response conditional distribution; it also considers the uncertainty problem and presents an illustrative example and simulation study for the suggested model.

]]>Ahmed Talip Hussein and Emad Allawi Shallal

Y. Imai, K. Iseki [4], and K. Iseki [5] presented types from summary algebras which are called BCK-algebras and BCI-algebras. It is known that the brand of BCK algebras is a suitable subtype from the type from BCI-algebras. The researchers Q. P. Hu [2] & X. Li [3] presented a width type from essence algebras: BCH- algebras. They have exhibited that the type of BCI-algebras is a suitable subtype of the type of BCH-algebras. Moreover, J. Neggers and H. S. K [9] presented the connotation from d - algebras that are else popularization from BCK-algebras, inspected kinsmen amidst d-algebras & BCK-algebras. They calculated diversified topologies to research from lattices but they did not discuss the experience of making the binary operation of d- algebra continuous. Topological set notions are famous and yet accurate by numerous mathematicians. Even global topographical algebraic structure is sought by several writers. We realize a Tb-algebra, get it several ownerships of such build, the generality significant flavors and arrive to realize a new gender of spaces designated BP- space, where we arrived the results. Let be B-space and is periodic proportional. Then is a compact set in and = , . Also If is an invariant under , then , and are invariant under for every Q in if is also. If the function is closed (one to one) then , () is invariant under and the set of interior points of is invariant under , if the function is open and .

]]>Muhamad Deni Johansyah Asep K Supriatna Endang Rusyaman and Jumadil Saputra

The differential equation is an equation that involves the derivative (derivatives) of the dependent variable with respect to the independent variable (variables). The derivative represents nothing but a rate of change, and the differential equation helps us present a relationship between the changing quantity with respect to the change in another quantity. The Adomian decomposition method is one of the iterative methods that can be used to solve differential equations, both integer and fractional order, linear or nonlinear, ordinary or partial. This method can be combined with integral transformations, such as Laplace, Sumudu, Natural, Elzaki, Mohand, Kashuri-Fundo, and Kamal. The main objective of this research is to solve differential equations of fractional order using a combination of the Adomian decomposition method with the Kamal integral transformation. Furthermore, the solution of the fractional differential equation using the combined method of the Adomian decomposition method and the Kamal integral transformation was investigated. The main finding of our study shows that the combined method of the Adomian decomposition method and the Kamal integral transformation is very accurate in solving differential equations of fractional order. The present results are original and new for solving differential equations of fractional order. The results attained in this paper confirm the illustrative example has been solved to show the efficiency of the proposed method.

]]>Vijay C. Makwana Vijay. P. Soni Nayan I. Patel and Manoj Sahni

In this paper, a new hypothesis of fuzzy number has been proposed which is more precise and direct. This new proposed approach is considered as an equivalence class on set of real numbers R with its algebraic structure and its properties along with theoretical study and computational results. Newly defined hypothesis provides a well-structured summary that offers both a deeper knowledge about the theory of fuzzy numbers and an extensive view on its algebra. We defined field of newly defined fuzzy numbers which opens new era in future for fuzzy mathematics. It is shown that, by using newly defined fuzzy number and its membership function, we are able to solve fuzzy equations in an uncertain environment. We have illustrated solution of fuzzy linear and quadratic equations using the defined new fuzzy number. This can be extended to higher order polynomial equations in future. The linear fuzzy equations have numerous applications in science and engineering. We may develop some iterative methods for system of fuzzy linear equations in a very simple and ordinary way by using this new methodology. This is an innovative and purposefulness study of fuzzy numbers along with replacement of this newly defined fuzzy number with ordinary fuzzy number.

]]>A. K. Awasthi Rachna and Harpreet Kaur

In the past 53 years, many efforts have been contributed to develop and demonstrate the properties of reinforced composite materials. The ever-increasing use of composite materials through engineering structures needs the proper analysis of the mechanical response of these structures. In the proposed work, we have an exact form of Stress components and Displacement components to a Griffith crack at the interface of an Isotropic and Orthotropic half-space bounded together. The expression was evaluated in the vicinity of crack tips by using Fourier transform method but here these components have been evaluated with the help of Fredholm integral equations and then reduce to the coupled Fredholm integral equations. In this paper, we use the problem of Lowengrub and Sneddon and reduce it to dual integral equations. Solution of these equations through the use of the method of Srivastava and Lowengrub is reduced to coupled Fredholm integral equation. Further reduces the problem to decoupled Fredholm integral equation of 2nd kind. We get the solution of dual integral equations and the problem is reduced to coupled Fredholm integral equation. We find the solution of the Fredholm integral equation and reduce it to decoupled Fredholm integral equation of 2nd kind. The Physical interest in fracture design criterion is due to Stress and crack opening Displacement components. In the end, we can easily calculate the Stress components and Displacement components in the exact form.

]]>Katta.Mallaiah and Veladi Srinivas

In the present paper, we establish the existence of two unique common fixed point theorems with a new contractive condition for four self-mappings in the S-metric space. First, we establish a common fixed-point theorem by using weaker conditions such as compatible mappings of type-(E) and subsequentially continuous mappings. Further, in the next theorem, we use another set of weaker conditions like sub-compatible and sub-sequentially continuous mappings, which are weaker than occasionally weak compatible mappings. Moreover, it is observed that the mappings in these two theorems are sub-sequentially continuous, but these mappings are neither continuous nor reciprocally continuous mappings. These two results will extend and generalize the existing results of [7] and [9] in the S-metric space. Furthermore, we also provide some suitable examples to justify our outcomes.

]]>K. Deva and S. Mohanaselvi

A picture fuzzy set is a more powerful tool to deal with uncertainties in the given information as compared to fuzzy set and intuitionistic fuzzy set and has energetic applications in decision-making. The aim of this study is to develop a new possibility measure for ranking picture fuzzy numbers and then some of its basic properties are proved. The proposed method provides the same ranking order as the score function in the literature. Moreover, the new possibility measure can provide additional information for the relative comparison of the picture fuzzy numbers. A picture fuzzy multi attribute decision-making problem is solved based on the possibility matrix generated by the proposed method after being aggregated using picture fuzzy Einstein weighted averaging aggregation operator. To verify the importance of the proposed method, an picture fuzzy multi attribute decision-making strategy is presented along with an application for selecting suitable alternative. The superiority of the proposed method and limitations of the existing methods are discussed with the help of a comparative study. Finally, a numerical example and comparative analysis are provided to illustrate the practicality and feasibility of the proposed method.

]]>Arash Pourkia

Braid groups and their representations are at the center of study, not only in low-dimensional topology, but also in many other branches of mathematics and theoretical physics. Burau representation of the Artin braid group which has two versions, reduced and unreduced, has been the focus of extensive study and research since its discovery in 1930's. It remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for knots and links. In the present work, we show that interesting representations of braid group could be achieved using a simple and intuitive approach, where we simply analyse the path of strands in a braid and encode the over-crossings, under-crossings or no-crossings into some parameters. More precisely, at each crossing, where, for example, the strand crosses over the strand we assign t to the top strand and b to the bottom strand. We consider the parameter t as a relative weight given to strand relative to , hence the position for t in the matrix representation. Similarly, the parameter b is a relative weight given to strand relative to , hence the position for b in the matrix representation. We show this simple path analyzing approach that leads us to an interesting simple representation. Next, we show that following the same intuitive approach, only by introducing an additional parameter, we can greatly improve the representation into the one with much smaller kernel. This more general representation includes the unreduced Burau representation, as a special case. Our new path analyzing approach has the advantage that it applies a very simple and intuitive method capturing the fundamental interactions of the strands in a braid. In this approach we intuitively follow each strand in a braid and create a history for the strand as it interacts with other strands via over-crossings, under-crossings or no-crossings. This, directly, leads us to the desired representations.

]]>Vishally Sharma and A. Parthiban

An assignment of intergers to the vertices of a graph subject to certain constraints is called a vertex labeling of . Different types of graph labeling techniques are used in the field of coding theory, cryptography, radar, missile guidance, -ray crystallography etc. A DCL of is a bijective function from node set of to such that for each edge , we allot 1 if divides or divides & 0 otherwise, then the absolute difference between the number of edges having 1 & the number of edges having 0 do not exceed 1, i.e., . If permits a DCL, then it is called a DCG. A complete graph , is a graph on nodes in which any 2 nodes are adjacent and lilly graph is formed by joining , sharing a common node. i.e., , where is a complete bipartite graph & is a path on nodes. In this paper, we propose an interesting conjecture concerning DCL for a given , besides, discussing certain general results concerning DCL of complete graph -related graphs. We also prove that admits a DCL for all . Further, we establish the DCL of some -related graphs in the context of some graph operations such as duplication of a node by an edge, node by a node, extension of a node by a node, switching of a node, degree splitting graph, & barycentric subdivision of the given .

]]>Endang Rusyaman Kankan Parmikanti Diah Chaerani and Khoirunnisa Rohadatul Aisy Muslihin

Lubricating oil is still a primary need for people dealing with machines. The important thing of lubricating oil is viscosity which is closely related to surface tension. Fluid viscosity states the measure of friction in the fluid, while surface tension is the tendency of the fluid to stretch due to attractive forces between the molecules (cohesion). We want to know how and to what extent the relationship between viscosity and surface tension of the lubricating oil is. This paper will discuss the analysis of a model in the form of an exponential fractional differential equation that states the relationship between surface tension and viscosity of lubricating oil. The Modified Homotopy Perturbation Method (MHPM) will be used to determine the solution of the fractional differential equation. This study indicates a relationship between viscosity and surface tension in the form of fractional differential equation in which the existence and uniqueness of the solution are guaranteed. From the analysis of the solution function both analytically and geometrically supported by empirical data, it can be concluded that there is a strong exponential relationship between viscosity and surface tension in lubricating oil.

]]>Thakhani Ravele Caston Sigauke and Lordwell Jhamba

Solar power poses challenges to the management of grid energy due to its intermittency. To have an optimal integration of solar power on the electricity grid it is important to have accurate forecasts. This study discusses the comparative analysis of semi-parametric extremal mixture (SPEM), generalised additive extreme value (GAEV) or quantile regression via asymmetric Laplace distribution (QR-ALD), additive quantile regression (AQR-1), additive quantile regression with temperature variable (AQR-2) and penalised cubic regression smoothing spline (benchmark) models for probabilistic forecasting of hourly global horizontal irradiance (GHI) at extremely high quantiles ( = 0.95, 0.97, 0.99, 0.999 and 0.9999). The data used are from the University of Venda radiometric in South Africa and are from the period 1 January 2020 to 31 December 2020. Empirical results from the study showed that the AQR-2 is the best fitting model and gives the most accurate prediction of quantiles at = 0.95, 0.97, 0.99 and 0.999, while at 0.9999-quantile the GAEV model has the most accurate predictions. Based on these results it is recommended that the AQR-2 and GAEV models be used for predicting extremely high quantiles of hourly GHI in South Africa. The predictions from this study are valuable to power utility decision-makers and system operators when making high-risk decisions and regulatory frameworks that require high-security levels. This is the first application to conduct a comparative analysis of the proposed models using South African solar irradiance data, to the best of our knowledge.

]]>Siham Rabee Ramadan Hamed Ragaa Kassem and Mahmoud Rashwaan

Calibration estimation approach is a widely used method for increasing the precision of the estimates of population parameters. It works by modifying the design weights as little as possible by minimizing a given distance function to the calibrated weights respecting a set of constraints related to specified auxiliary variables. This paper proposes a goal programming approach for generalized calibration estimation. In the generalized calibration estimation, multi study variables will be considered by incorporating multi auxiliary variables. Almost all calibration estimation's literature proposed calibrated estimators for the population mean of only one study variable. And nevertheless, up to researcher's knowledge, there is no study that considers calibration estimation approach for multi study variables. According to the correlation structure between the study variables, estimating the calibrated weights will be formulated in two different models. The theory of the proposed approach is presented and the calibrated weights are estimated. A simulation study is conducted in order to evaluate the performance of the proposed approach in the different scenarios compared by some existing calibration estimators. The Simulation results of the four generated populations show that the proposed approach is more flexible and efficient compared to classical methods.

]]>Suvimol Phanyaem

The Cumulative Sum (CUSUM) chart is widely used and has many applications in different fields such as finance, medical, engineering, and other fields. In real applications, there are many situations in which the observations of random processes are serially correlated, such as a hospital admission in the medical field, a share price in the economic field, or a daily rainfall in the environmental field. The common characteristic of control charts that has been used to evaluate the performance of control charts is the Average Run Length (ARL). The primary goals of this paper are to derive the explicit formula and develop the numerical integral equation of the ARL for the CUSUM chart when observations are seasonal autoregressive models with exogenous variable, SARX(P,r)_{L} with exponential white noise. The Fredholm Integral Equation has been used for solving the explicit formula of ARL, and we used numerical methods including the Midpoint rule, the Trapezoidal rule, the Simpson's rule, and the Gaussian rule to approximate the numerical integral equation of ARL. The uniqueness of solutions is guaranteed by using Banach's Fixed Point Theorem. In addition, the proposed explicit formula was compared with their numerical methods in terms of the absolute percentage difference to verify the accuracy of the ARL results and the computational time (CPU). The results obtained indicate that the ARL from the explicit formula is close to the numerical integral equation with an absolute percentage difference of less than 1%. We found an excellent agreement between the explicit formulas and the numerical integral equation solutions. An important conclusion of this study was that the explicit formulas outperformed the numerical integral equation methods in terms of CPU time. Consequently, the proposed explicit formulas and the numerical integral equation have been the alternative methods for finding the ARL of the CUSUM control chart and would be of use in fields like biology, engineering, physics, medical, and social sciences, among others.

Abdul Hadi Bhatti and Sharmila Binti Karim

Numerical methods are regularly established for the better approximate solutions of the ordinary differential equations (ODEs). The best approximate solution of ODEs can be obtained by error reduction between the approximate solution and exact solution. To improve the error accuracy, the representations of Wang Ball curves are proposed through the investigation of their control points by using the Least Square Method (LSM). The control points of Wang Ball curves are calculated by minimizing the residual function using LSM. The residual function is minimized by reducing the residual error where it is measured by the sum of the square of the residual function of the Wang Ball curve's control points. The approximate solution of ODEs is obtained by exploring and determining the control points of Wang Ball curves. Two numerical examples of initial value problem (IVP) and boundary value problem (BVP) are illustrated to demonstrate the proposed method in terms of error. The results of the numerical examples by using the proposed method show that the error accuracy is improved compared to the existing study of Bézier curves. Successfully, the convergence analysis is conducted with a two-point boundary value problem for the proposed method.

]]>Abimibola Victoria Oladugba and Brenda Mbouamba Yankam

The variance dispersion graphs (VDGs) and the fraction of design space (FDS) graphs are two graphical methods that effectively describe and evaluate the points of best and worst prediction capability of a design using the scaled prediction variance properties. These graphs are often utilized as an alternative to the single-value criteria such as D- and E- when they fail to describe the true nature of designs. In this paper, the VDGs and FDS graphs of third-order orthogonal uniform composite designs (OUCD_{4}) and orthogonal array composite designs (OACD_{4}) using the scaled-prediction variance properties in the spherical region for 2 to 7 factors are studied throughout the design region and over a fraction of space. Single-valued criteria such as D-, A- and G-optimality are also studied. The results obtained show that the OUCD_{4} is more optimal than the OACD_{4} in terms of D-, A- and G-optimality. The OUCD_{4} was shown to possess a more stable and uniform scaled-prediction variance throughout the design region and over a fraction of design space than the OACD_{4} although the stability of both designs slightly deteriorated towards the extremes.

Noura S. Mohamed Moshira A. Ismail and Sanaa A. Ismail

Finite mixture models have been used in many fields of statistical analysis such as pattern recognition, clustering and survival analysis, and have been extensively applied in different scientific areas such as marketing, economics, medicine, genetics and social sciences. Introducing mixtures of new generalized lifetime distributions that exhibit important hazard shapes is a major field of research aiming at fitting and analyzing a wider variety of data sets. The main objective of this article is to present a full mathematical study of the properties of the new finite mixture of the three-parameter Weibull extension model, considered as a generalization of the standard Weibull distribution. The new proposed mixture model exhibits a bathtub-shaped hazard rate among other important shapes in reliability applications. We analytically prove the identifiability of the new mixture and investigate its mathematical properties and hazard rate function. Maximum likelihood estimation of the model parameters is considered. The Kolmogrov-Smirnov test statistic is used to fit two famous data sets from mechanical engineering to the proposed model, the Aarset data and the Meeker and Escobar datasets. Results show that the two-component version of the proposed mixture is a superior fit compared to various lifetime distributions, either one-component or two-component lifetime distributions. The new proposed mixture is a significant statistical tool to study lifetime data sets in numerous fields of study.

]]>Saimir Tola Alfred Daci and Gentian Zavalani

This paper presents numerical simulations and comparisons between different approaches concerning elastic thin rods. Elastic rods are ideal for modeling the stretching, bending, and twisting deformations of such long and thin elastic materials. The static solution of Kirchhoff's equations [2] is produced using ODE45 solver where Kirchhoff and reference system equations are combined instantaneously. Solutions using formulations are based on Euler's elastica theory [1] which determines the deformed centerline of the rod by solving a boundary-value problem, on the Discreet Elastic Rod method using Bishop frame (DER) [5,6] which is based on discrete differential geometry, it starts with a discrete energy formulation and obtains the forces and equations of motion by taking the derivative of energies. Instead of discretizing smooth equations, DER solves discrete equations and obeys geometrical exactness. Using DER we measure torsion as the difference of angles between the material and the Bishop frame of the rod so that no additional degree of freedom is needed to represent the torsional behavior. We found excellent agreement between our Kirchhoff-based solution and numerical results obtained by the other methods. In our numerical results, we include simulation of the rod under the action of the terminal moment and illustrations of the gravity effects.

]]>Bhuwaneshwar Kumar Gupt Mankupar Swer Md. Irphan Ahamed B. K. Singh and Kh. Herachandra Singh

In this paper, the problem of optimum stratification of heteroscedastic populations in stratified sampling is considered for a known allocation under Simple Random Sampling With and Without Replacement (SRSWR & SRSWOR) design. The known allocation used in the problem is one of the model-based allocations proposed by Gupt [1,2] under a superpopulation model considered by Hanurav [3], Rao [4], and Gupt and Rao [5] which was modified by the author (Gupt [1,2]) to a more general form. The problem of finding optimum boundary points of stratification (OBPS) in stratifying populations considered here is based on an auxiliary variable which is highly correlated with the study variable. Equations giving the OBPS have been derived by minimizing the variance of estimator of the population mean. Since the equations giving OBPS are implicit and difficult for solving, some methods of finding approximately optimum boundary points of stratification (AOBPS) have also been obtained as the solutions of the equations giving OBPS. While deriving equations giving OBPS and methods of finding AOBPS, basic statistical definitions, tools of calculus, analytic functions and tools of algebra are used. While examining the efficiencies of the proposed methods of stratification, they are tested in a few generated populations and a live population. All the proposed methods of stratification are found to be efficient and suitable for practical applications. In this study, although the proposed methods are obtained under a heteroscedastic superpopulation model for level of heteroscedasticity one, the methods have shown robustness in empirical investigation in varied levels of heteroscedastic populations. The stratification methods proposed here are new as they are derived for an allocation, under the superpopulation model, which has never been used earlier by any researcher in the field of construction of strata in stratified sampling. The proposed methods may be a fascinating piece of work for researchers amidst the vigorously progressing theoretical research in the area of stratified sampling. Besides, by virtue of exhibiting high efficiencies in the performance of the methods, the work may provide a practically feasible solution in the planning of socio-economic survey.

]]>M. F. Zairul Fuaad N. Razali H. Hishamuddin and A. Jedi

The accuracy and efficiency of water tank system problems can be determined by comparing the Symmetrized Implicit Midpoint Rule (IMR) with the IMR. Static and dynamic analyses are part of a mathematical model that uses energy conservation to generate a nonlinear Ordinary Differential Equation. Static analysis provides optimal working points, while dynamic analysis outputs an overview of the system behaviour. The procedure mentioned is tested on two water tank designs, namely, cylindrical and rectangular tanks with two different parameters. The Symmetrized IMR is used in this study. Results show that the two-step active Symmetrized IMR applied on the proposed mathematical model is precise and efficient and can be used for the design of appropriate controls. The cylindrical water tank model takes the fastest time in emptying the water tank. The approach of the various water tank models shows an increase in accuracy and efficiency in the range of parameters used for practical model applications. The results of the numerical method show that the two-step Symmetrized IMR provides more precise stability, accuracy and efficiency for the fixed step size measurements compared with other numerical methods.

]]>Habti Abeida

Absolutely Continuous non-singular complex elliptically symmetric distributions (referred to as the nonsingular CES distributions) have been extensively studied in various applications under the assumption of nonsingularity of the scatter matrix for which the probability density functions (p.d.f's) exist. These p.d.f's, however, can not be used to characterize the CES distributions with a singular scatter matrix (referred to as the singular CES distributions). This paper presents a generalization of the singular real elliptically symmetric (RES) distributions studied by Díaz-García et al. to singular CES distributions. An explicit expression of the p.d.f of a multivariate non-circular complex random vector with singular CES distribution is derived. The stochastic representation of the singular non-circular CES (NC-CES) distributions and the quadratic forms in NC-CES random vector are proved. As special cases, explicit expressions for the p.d.f's of multivariate complex random vectors with singular non-circular complex normal (NC-CN) and singular non-circular complex Compound-Gaussian (NC-CCG) distributions are also derived. Some useful properties of singular NC-CES distributions and their conditional distributions are derived. Based on these results, the p.d.f's of non-circular complex t-distribution, K-distribution, and generalized Gaussian distribution under singularity are presented. These general results degenerate to those of singular circular CES (C-CES) distributions when the pseudo-scatter matrix is equal to the zero matrix. Finally, these results are applied to the problem of estimating the parameters of a complex-valued non-circular multivariate linear model in the presence either of singular NC-CES or C-CES distributed noise terms by proposing widely linear estimators

]]>H.Priya and B. Srutha Keerthi

The aim of the paper is to obtain the First Hankel Determinant and the Second Hankel determinant. We shall make use of few lemmas which are based on Caratheodory's class of analytic functions. We establish a new Sakaguchi class of univalent function, further we estimate the sharp bound for initial coefficients and using the Bessel function expansion. We have discussed about the coefficient as well for the Second Hankel Determinant. The results are obtained for Sakaguchi kind. Our results travel along exploring the stages of Hankel Determinants. Various types of technologies like wire, optical or other electromagnetic systems are used for the transmission of data in one device to another. Filters play an important role in the process that can remove disorted signals. By using different parameter values for the function belongs to Sakaguchi kind of functions the Low pass filter and High pass filter can be designed and that can be done by the coefficient estimates.

]]>Ugah Tobias Ejiofor Mba Emmanuel Ikechukwu Eze Micheal Chinonso Arum Kingsley Chinedu Mba Ifeoma Christy Urama Chinasa and Comfort Njideka Ekene-Okafor

It is not uncommon to find an outlier in the response variable in linear regression. Such a deviant value needs to be detected and scrutinized to find out why it is not in agreement with its fitted value. Srikantan [1] has developed a test statistic for detecting the presence of an outlier in the response variable in a multiple linear regression model. Approximate critical values of this test statistic are available and are obtained based on the first-order Bonferroni upper bound. The exact critical values are not available and a result of that, tests carried out on the basis of this approximate critical values may not be very accurate. In this paper, we obtained more accurate and precise critical values of this test statistic for large sample sizes (herein called asymptotic critical values) to improve on the tests that use these critical values. The procedure involved using the exact probability density function of this test statistic to obtain its asymptotic critical values. We then compared these asymptotic critical values with the approximate critical values obtained. An application to simulation results for linear regression models was used to examine the power of this test statistic. The asymptotic critical values obtained were found to be more accurate and precise. Also, the test performed better under these asymptotic values (the power performance of this test statistic was found to better when the asymptotic critical values were used).

]]>Ivana Mala Vaclav Sladek and Diana Bilkova

Normality tests are used in the statistical analysis to determine whether a normal distribution is acceptable as a model for the data analysed. A wide range of available tests employs different properties of normal distribution to compare empirical and theoretical distributions. In the present paper, we perform the Monte Carlo simulation to analyse test power. We compare commonly known and applied tests (standard and robust versions of the Jarque-Bera test, Lilliefors test, chi-square goodness-of-fit test, Shapiro-Francia test, Cramer-von Mises goodness-of-fit test, Shapiro-Wilk test, D'Agostino test, and Anderson-Darling test) to the test based on robust L-moments. In the text, in Jarque-Bera type test the moment characteristics of skewness and kurtosis are replaced with their robust versions - L-skewness and L-kurtosis. The distributions with heavy tails (lognormal, Weibull, loglogistic and Student) are used to draw random samples to show the performance of tests when applied on data with outliers. Small sample properties (from 10 observations) are analysed up to large samples of 200 observations. Our results concerning the properties of the classical tests are in line with the conclusion of other recent articles. We concentrate on properties of the test based on L-moments. This normality test is comparable to well-performing and reliable tests; however, it is outperformed by the most powerful Shapiro-Wilks and Shapiro-Francia tests. It works well for Student (symmetric) distribution, comparably with the most frequently used Jarque-Berra tests. As expected, the test is robust to the presence of outliers in comparison with sensitive tests based on product moments or correlations. The test turns out to be very universally reliable.

]]>B. M. Cerna Maguiña Dik D. Lujerio Garcia and Héctor F. Maguiña

In this work, using the basic tools of functional analysis, we obtain a technique that allows us to obtain important results, related to quadratic equations in two variables that represent a natural number and differential equations. We show the possible ways to write an even number that ends in six, as the sum of two odd numbers and we establish conditions for said odd numbers to be prime, also making use of a suitable linear functional we obtain representations of natural numbers of the form in order to obtain positive integer solutions of the equation quadratic where is a natural number given that it ends with one. And finally, we show with three examples the use of the proposed technique to solve some ordinary and partial linear differential equations. We believe that the third corollary of our first result of this investigation can help to demonstrate the strong Goldbach conjecture.

]]>Betty Subartini Ira Sumiati Sukono Riaman and Ibrahim Mohammed Sulaiman

At present, three numerical solution methods have mainly been used to solve fractional-order chaotic systems in the literature: frequency domain approximation, predictor–corrector approach and Adomian decomposition method (ADM). Based on the literature, ADM is capable of dealing with linear and nonlinear problems in a time domain. Also, the Adomian decomposition method (ADM) is among the efficient approaches for solving linear and non-linear equations. Numerical solution method is one of the critical problems in theoretical research and in the applications of fractional-order systems. The solution is decomposed into an infinite series and the integral transformation to a differential equation is implemented in this work. Furthermore, the solution can be thought of as an infinite series that converges to an exact solution. The aim of this study is to combine the Adomian decomposition approach with a different integral transformation, including Laplace, Sumudu, Natural, Elzaki, Mohand, and Kashuri-Fundo. The study's key finding is that employing the combined method to solve fractional ordinary differential equations yields good results. The main contribution of our study shows that the combined numerical methods considered produce excellent numerical performance for solving fractional ordinary differential equations. Therefore, the proposed combined method has practical implications in solving fractional order differential equations in science and social sciences, such as finding analytical and numerical solutions for secure communication system, biological system, financial risk models, physics phenomenon, neuron models and engineering application.

]]>Ni Made Ayu Astari Badung Adji Achmad Rinaldo Fernandes and Waego Hadi Nugroho

This study aims to compare the size of distance (Euclidean distance, Manhattan distance, and Mahalanobis distance) and linkage (average linkage, single linkage, and complete linkage) in integrated cluster analysis with Multiple Discriminant Analysis on Home Ownership Credit Bank consumers in Indonesia. The data used are secondary data from the 5C assessment on Bank consumers in Indonesia. The data contain notes on the 5 C assessment as well as 3 credit collectability (current, special mention, and substandard) from Home Ownership Credit customers. The population in this study were all Home Ownership Credit customers in all banks in Indonesia. The sampling technique used was purposive random sampling. The sample size is 300 customers from customer data at three branches of Bank in Indonesia. This research is a quantitative study using cluster analysis integrated with multiple discriminant analysis. The best method for classifying Home Ownership Credit Bank customers based on the 5C variable assessment is an integrated cluster analysis with Multiple Discriminant Analysis based on the Mahalanobis distance with 2 clusters, namely the high cluster and the low cluster. Use of an integrated cluster with Multiple Discriminant Analysis to compare distance and linkage measures. In addition, the objects used are Home Ownership Credit Bank customers in Indonesia.

]]>Erlinda Citra Lucki Efendi Adji Achmad Rinaldo Fernandes and Maria Bernadetha Theresia Mitakda

This study aims to estimate the nonparametric truncated spline path functions of linear, quadratic, and cubic orders at one and two knot points and determine the best model on the variables that affect the timely payment of House Ownership Credit (HOC). In addition, this study aims to test the hypothesis to determine the variables that have a significant effect on punctuality in paying House Ownership Credit (HOC). The data used in this study are primary data. The variables used are service quality and lifestyle as exogenous variables, willingness to pay as mediating variables and on time to pay as endogenous variables. Analysis of the data used in this study is a nonparametric path using R software. The results showed that the best model was obtained on a nonparametric truncated spline linear path model with 2 knot points. The model has the smallest GCV value of 25.9059 and R^{2} value of 96.96%. In addition, the results of hypothesis testing on function estimation have a significant effect on the relationship between service quality and willingness to pay, the relationship between service quality and on time to pay, the relationship between lifestyle and willingness to pay, and the relationship between lifestyle and on time pay. The novelty of this research is to model and test the hypothesis of nonparametric regression development, namely nonparametric truncated spline paths of linear, quadratic and cubic orders.

Jetsada Singthongchai Noppakun Thongmual and Nirun Nitisuk

This research is about estimating parameters in simple linear regression model. Regression model is applied for predictive in many filed. Ordinary lest square (OLS) approach and Maximum likelihood (ML) approach are employed for estimating parameter in simple linear regression model when the assumption is not violated. This research interested in simple linear regression model when the assumption is violated. Simple Averaging (SA) approach is an alternative for estimating parameters in simple linear regression model where assumptions are not successfully used. We improved SA approach based on the median which is called the improved Simple Averaging (ISA) approach. For comparing the two approaches for estimating parameter in simple linear regression model, ISA approach is compared with SA approach under Root Mean Square Error (RMSE) which reflected accuracy of prediction in simple linear regression. By using the sample, the results showed that ISA approach is better than SA approach where the value of RMSE of ISA approach is less than the value of RMSE of SA approach. Therefore, ISA approach is better than SA approach. Our study suggests ISA approach to estimating parameter on simple linear regression because ISA approach accuracy than SA approach and ISA approach simplify the estimation of parameters in the simple linear regression model. Hence, ISA approach an alternative for estimating parameters in simple linear regression model when the assumptions are not successfully used.

]]>B. M. Cerna Maguiña and Janet Mamani Ramos

Although it is true that there are several articles that study quadratic equations in two variables, they do so in a general way. We focus on the study of natural numbers ending in one, because the other cases can be studied in a similar way. We have given the subject a different approach, that is why our bibliographic citations are few. In this work, using basic tools of functional analysis, we achieve some results in the study of integer solutions of quadratic polynomials in two variables that represent a given natural number. To determine if a natural number ending in one is prime, we must solve equations (i) , (ii) , (iii) . If these equations do not have an integer solution, then the number P is prime. The advantage of this technique is that, to determine if a natural number p is prime, it is not necessary to know the prime numbers less than or equal to the square root of p. The objective of this work was to reduce the number of possibilities assumed by the integer variables in the equation (i), (ii), (iii) respectively. Although it is true that this objective was achieved, we believe that the lower limits for the sums of the solutions of equations (i), (ii), (iii), were not optimal, since in our recent research we have managed to obtain limits lower, which reduce the domain of the integer variables solve equations (i), (ii), (iii), respectively. In a future article we will show the results obtained. The methodology used was deductive and inductive. We would have liked to have a supercomputer, to build or determine prime numbers of many millions of digits, but this is not possible, since we do not have the support of our respective authorities. We believe that the contribution of this work to number theory is the creation of linear functionals for the study of integer solutions of quadratic polynomials in two variables, which represent a given natural number. The utility of large prime numbers can be used to encode any type of information safely, and the scheme shown in this article could be useful for this process.

]]>Malik Saad Al-Muhja Habibulla Akhadkulov and Nazihah Ahmad

Approximation Theory is a branch of analysis and applied mathematics requiring that the approximation process preserves certain -shaped properties defined at a finite interval , such as convexity in all or parts of the interval. The (Co)convex and Unconstrained Polynomial (COCUNP) approximation is one of the key estimations of the approximation theory that Kopotun has recently raised for ten years. Numerous studies have been conducted on modern methods of weighted approximation to construct the best degree of approximation. In developing COCUNP a novel technique, the Lebesgue Stieltjes integral-i technique is used to resolve certain disadvantages, such as Riemann's integrable functions, which do not have the degree of the best approximation in norm space. In order to achieve the main goal, Derivation of New Degree (DOND) of the best COCUNP approximation was constructions. The theoretical results revealed that, in general, the new degrees of best approximation were able to smaller errors compared to the existing literature in the same estimating. In conclusion, this study has successfully developed DOND for the best (Co)convex Polynomial (COCP) weighted approximation.

]]>Magi P M Sr.Magie Jose and Anjaly Kishore

Let be a simple graph of order and let be the Seidel matrix of , defined as where if the vertices and are adjacent and if the vertices and are not adjacent and if . Let be the diagonal matrix where denotes the degree of the vertex of . The Seidel Laplacian matrix of a graph is defined as and the Seidel signless Laplacian matrix of a graph is defined as . The zero-divisor graph of a commutative ring , denoted by , is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices are adjacent if and only if . In this paper, we find the Seidel polynomial and Seidel Laplacian polynomial of the join of two regular graphs using the concept of schur complement and coronal of a square matrix. Also we describe the computation of the Seidel Laplacian and Seidel signless Laplacian eigenvalues of the join of more than two regular graphs, using the well known Fiedler's lemma and apply these results to describe these eigenvalues for the zero-divisor graph on . Further we find the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of for some values of , say , where are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of , for any .

]]>Sidite Duraj Eriola Sila and Elida Hoxha

The study of fixed points in the metric spaces plays a crucial role in the development of Functional Analysis. It is evolved by generalizing the metric space or improving the contractive conditions. Recently, the partial rectangular metric space and its topology have been the center of study for many researchers. They have defined open and closed balls the equivalent Cauchy sequences and Cauchy sequences, convergent sequences which are used as tools in many achieved results. In this paper, two facts for equivalent Cauchy sequences in a partial rectangular metric space are provided by using an ultra - altering distance function. Furthermore, some results of Cauchy sequences in a partial rectangular metric space are highlighted. There is proved that under some conditions the equivalent Cauchy sequences are Cauchy sequences in a partial rectangular metric space. Some fixed point results have been taken as applications of our new conditions of Cauchy sequences and equivalent Cauchy sequences in a partial rectangular metric space for orbitally continuous functions . To illustrate the obtained results some examples are given.

]]>Sameen Ahmed Khan

The main aim of this article is to start with an expository introduction to the trigonometric ratios and then proceed to the latest results in the field. Historically, the exact ratios were obtained using geometric constructions. The geometric methods have their own limitations arising from certain theorems. In view of the certain limitations of the geometric methods, we shall focus on the powerful techniques of equations in deriving the exact trigonometric ratios using surds. The cubic and higher-order equations naturally arise while deriving the exact trigonometric ratios. These equations are best expressed using the expansions of the cosines and sine of multiple angles using the Chebyshev polynomials of the first and second kind respectively. So, we briefly present the essential properties of the Chebyshev polynomials. The equations lead to the question of reduced polynomials. This question of the reduced polynomials is addressed using the Euler's totient function. So, we describe the techniques from theory of equations and reduced polynomials. The trigonometric ratios of certain rational angles (when measured in degrees) give rise to rational trigonometric ratios. We shall discuss these along with the related theorems. This is a frontline area of research connecting trigonometry and number theory. Results from number theory and theory of equations are presented wherever required.

]]>Vladislav V. Lyubimov

The aim of this paper is to obtain three types of expressions for calculating the probability of implementing palindromic digit combinations on a finite equally possible combination of zeros and ones. When calculating the probability of implementation of palindromic digit combinations, the classical definition of probability is applied. The main results of the paper are formulated in the form of three theorems. Moreover, the consequences of these theorems and typical examples of calculating the probability of implementing palindromic digit combinations in a data string of binary code are considered. All formulated theorems and their consequences are accompanied by proofs. The obtained numerical results of the paper can be used in the analysis of numerical computer data written as a binary code string in BIN format files. It should also be noted that the combinatorial expressions described in the article for calculating the number of palindromic combinations of digits in the binary number system can be used in number theory and in various branches of computer science. The development of these results from the point of view of obtaining an expression for calculating the number of palindromic combinations of digits in the binary number system contained in two-dimensional data arrays is also of immediate theoretical and practical interest. However, these results are not presented in this work, but they can be considered in subsequent publications.

]]>Mardeen Sh. Taher and Salah G. Shareef

It is known that the conjugate gradient method is still a popular method for many researchers who are focused in solving the large-scale unconstrained optimization problems and nonlinear equations because the method avoids the computation and storage of some matrices so the memory's requirements of the method are very small. In this work, a modified Perry conjugate gradient method which fulfills a global convergence with standard assumptions is shown and analyzed. The idea of new method is based on Perry method by using the equation which is founded via Powell in 1978. The weak Wolfe–Powell search conditions are used to choose the optimal line search, under the line search and suitable conditions, we prove both descent and sufficient descent conditions. In particular, numerical results show that the new conjugate gradient method is more effective and competitive when compared to other of standard conjugate gradient methods including: - CG- Hestenes and Stiefel (H/S) method, CG-Perry method, CG- Dai and Yuan (D/Y) method. The comparison is completed under a group of standard test problems with various dimensions from the CUTEst test library and the comparative performances of the methods are evaluated by total the number of iterations and the total number of function evaluations.

]]>Ridhwan Reyaz Ahmad Qushairi Mohamad Yeaou Jiann Lim Muhammad Saqib Zaiton Mat Isa and Sharidan Shafie

Studies on Casson fluid are essential in the development of the manufacturing and engineering fields since it is widely used there. Meanwhile, fractional derivative has been known to be a constructive paradox that can be beneficial in the future. In this study, the development fractional derivative on Casson fluid flow is investigated. A fractional Casson fluid model with effect of thermal radiation is derived together with momentum and energy equations. The Caputo definition of fractional derivative is used in the mathematical formulation. Casson fluid with constant wall temperature over an oscillating plate in the presence of thermal radiation is considered. Solutions were obtained by using Laplace transform and are presented in the form of Wright function. Graphical analysis on velocity and temperature profiles was conducted with variations in parametric values such as fractional parameter, Grashof number, Prandtl number and radiation parameter. Numerical computations were carried out to investigate behaviours of skin friction and Nusselt number. It is found that when the fractional parameter is increased, the velocity and temperature profiles will also increase. Existence of fractional parameter in both velocity and temperature profiles shows the transitional phenomenon of both profiles from an unsteady state to steady state, providing a new perspective on Casson fluid flow. An increment in both profiles is also observed when the thermal radiation parameter is increased. The present results are also validated with published results, and it is found that they are in agreement with each other.

]]>Nurfarah Zulkifli and Nor Muhainiah Mohd Ali

Let be a finite group. The probability that two selected elements from and from are chosen at random in a way that the greatest common divisor also known as gcd, of the order of and , which is equal to one, is called as the relative coprime probability. Meanwhile, another definition states that the vertices or nodes are the elements of a group and two distinct vertices or nodes are adjacent if and only if their orders are coprime and any of them is in the subgroup of the group and this is called as the relative coprime graph. This research focuses on determining the relative coprime probability and graph for cyclic subgroups of some nonabelian groups of small order and their associated graph properties by referring to the definitions and theorems given by previous researchers. Besides, various results of the relative coprime probability for nonabelian groups of small order are obtained. As for the relative coprime graph, the result shows that the domination number for each group is one whereas the number of edges and the independence number for each group vary. Types of graphs that can be formed are either star graph, planar graph or complete subgraph depending on the order of the subgroup of a group.

]]>Domenico P.L. Castrigiano

Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. — The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it furnishes some new results recalling the close relationship to Wiener-Hopf operators and, in case of semiboundedness, to singular operators of Hilbert transformation type. Specific symbols considered in the literature admit further analysis. Some conclusions are drawn for semibounded integrable and real square-integrable symbols. There is an approach to semibounded TO, which starts from closable semibounded forms related to a Toeplitz matrix. The Friedrichs extension of the TO associated with such a form is studied. Finally, analytic TO and Toeplitz-like operators are briefly examined, which in general differ from the TO treated here.

]]>K. Kumara Swamy Swatmaram Bipan Hazarika and P. Sumati Kumari

It has been a century since the Banach fixed point theorem was established, and because of this, the result is the progenitor in some ways. This seems essential to revisit fixed point theorems in specific and in light of most of those. Those are numerous and prevalent in mathematics, as we will demonstrate. Fixed point theorems can be noticed in advanced mathematics, economics, micro-structures, geometry, dynamics, computational mathematics, and differential equations. space is to broaden and extrapolate the paradigm of the concept of metric space. The characteristic of a space, in essence, is to comprehend the topological features of three points rather than two points via the perimeter of a triangle, where the metric indicates the distance between two points. The domain of space is significantly larger than that of the class of space. Hence we utilised this generalized space in order to obtain common tripled fixed point for three mappings using rational type contractions in the setting of spaces. Recently, Khomadram et al have developed coupled fixed point theorems in spaces via rational type contractions. The main aim of our paper is to broaden and extrapolate the paradigm of Khomadram's results into tripled fixed point theorems. Therefore, examples are offered to support our findings.

]]>Nik Nur Amiza Nik Ismail Azwani Alias and Fatimah Noor Harun

Internal solitary waves have been documented in several parts of the world. This paper intends to look at the effects of the variable topography and rotation on the evolution of the internal waves of depression. Here, the wave is considered to be propagating in a two-layer fluid system with the background topography is assumed to be rapidly and slowly varying. Therefore, the appropriate mathematical model to describe this situation is the variable-coefficient Ostrovsky equation. In particular, the study is interested in the transition of the internal solitary wave of depression when there is a polarity change under the influence of background rotation. The numerical results using the Pseudospectral method show that, over time, the internal solitary wave of elevation transforms into the internal solitary wave of depression as it propagates down a decreasing slope and changes its polarity. However, if the background rotation is considered, the internal solitary waves decompose and form a wave packet and its envelope amplitude decreases slowly due to the decreasing bottom surface. The numerical solutions show that the combination effect of variable topography and rotation when passing through the critical point affected the features and speed of the travelling solitary waves.

]]>Robert Reynolds and Allan Stauffer

Carl Johan Malmsten (1846) and David Beirens de Haan (1847) published work containing some interesting integrals. While no formal derivations of the integrals in his book Nouvelles Tables d'Intègrales Dèfines are available in current literature deriving and evaluating such formulae are useful in all aspects of science and engineering whenever such formulae are used. Formulae in the book of Bierens de Haan are used in connection with certain potential problems where there is the need to determine the vector potential of two parallel, infinitely long, tubular rectangular conductors carrying cur-rents in opposite directions. In this current work we supply formal derivations for some of these integrals along with deriving some special cases as new integrals in order to expand upon the book of Bierens de haan to aid in potential research where these formulae are applicable. Updating book of integrals is always a useful exercise as it keeps the volume accurate and more useful for potential readers and researchers. Formal derivations are also useful as they help in verifying the correctness of integrals in such volumes. The definite integral we derived in this work is given by (1) in terms of the Lerch function, where the parameters a; k; m; and p are general complex numbers subject to their restrictions. This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.

]]>P Jamsheena and A V Chithra

Let be a commutative ring with unity. The essential ideal graph of , denoted by , is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices and are adjacent whenever is an essential ideal. An essential ideal of a ring is an ideal of (), having nonzero intersection with every other ideal of . The set contains all the maximal ideals of . The Jacobson radical of , , is the set of intersection of all maximal ideals of . The comaximal ideal graph of , denoted by , is a simple graph with vertices as proper ideals of A not contained in and the vertices and are associated with an edge whenever . In this paper, we study the structural properties of the graph by using the ring theoretic concepts. We obtain a characterization for to be isomorphic to the comaximal ideal graph . Moreover, we derive the structure theorem of and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of and determine the values of for which is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to for some .

]]>Adeniji A A Noufe H. A Mkolesia A C and Shatalov M Y

Predator-prey models are the building blocks of the ecosystems as biomasses are grown out of their resource masses. Different relationships exist between these models as different interacting species compete, metamorphosis occurs and migrate strategically aiming for resources to sustain their struggle to exist. To numerically investigate these assumptions, ordinary differential equations are formulated, and a variety of methods are used to obtain and compare approximate solutions against exact solutions, although most numerical methods often require heavy computations that are time-consuming. In this paper, the traditional differential transform (DTM) method is implemented to obtain a numerical approximate solution to prey-predator models. The solution obtained with DTM is convergent locally within a small domain. The multi-step differential transform method (MSDTM) is a technique that improves DTM in the sense that it increases its interval of convergence of the series expansion. One predator-one prey and two-predator-one prey models are considered with a quadratic term which signifies other food sources for its feeding. The result obtained numerically and graphically shows point DTM diverges. The advantage of the new algorithm is that the obtained series solution converges for wide time regions and the solutions obtained from DTM and MSDTM are compared with solutions obtained using the classical Runge-Kutta method of order four. The results demonstrated is that MSDTM computes fast, is reliable and gives good results compared to the solutions obtained using the classical Runge-Kutta method.

]]>Prapart Pue-on

In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations. The significant role of this system in various fields has attracted the attention of researchers for a decade. The definition of fractional derivative here is described in the Caputo sense. The proposed method relies on the generalized Taylor series expansion as well as the fact that the fractional derivative of stationary is zero. The process starts by constructing a residual function by supposing the finite order of an approximate power series solution that prescribes the initial conditions. Then, utilizing some conditions, the residual functions are converted to a linear system for the power series coefficients. Solving the linear system reveals the coefficients of the fractional power series solution. Finally, by substituting these coefficients into the supposed form of a solution, the approximate fractional power series solutions are derived. This technique has the advantage of being able to be applied directly to the problem and spending less time on computation. It is not only an easy method for implementation of the problem, but also provides productive results after a few iterations. Some problems with known solutions emphasize the procedure's simplicity and reliability. Moreover, the obtained exact solution demonstrated the efficiency and accuracy of the presented method.

]]>Mahmoud Riad Mahmoud Moshera. A. M. Ahmad and Badiaa. S. Kh. Mohamed

The Lomax distribution (or Pareto II) was first introduced by K. S. Lomax in 1954. It can be readily applied to a wide range of situations including applications in the analysis of the business failure life time data, economics, and actuarial science, income and wealth inequality, size of cities, engineering, lifetime, and reliability modeling. In his pioneering paper, Shannon 1948 defined the notion of entropy as a mathematical measure of information, which is sometimes called Shannon entropy in his honor. He laid the groundwork for a new branch of mathematics in which the notion of entropy plays a fundamental role over different areas of applications such as statistics, information theory, financial analysis, and data compression. [Ebrahimi and Pellerey 14] introduced the residual entropy function because the entropy shouldn't be applied to a system that has survived for some units of time, and therefore, the residual entropy is used to measure the ageing and characterize, classify and order lifetime distributions. In this paper, the estimation of the entropy and residual entropy of a two parameter Lomax distribution under a generalized Type-II hybrid censoring scheme are introduced. The maximum likelihood estimation for the entropy is provided and the Bayes estimation for the residual entropy is obtained. Simulation studies to assess the performance of the estimates with different sample sizes are described, finally conclusions are discussed.

]]>Dilcu Barnes and Saeed Maghsoodloo

This paper focuses on the renewal function which is simply the mathematical expectation of number of renewals in a stochastic process. Renewal functions are important, and they have various applications in many fields. However, obtaining an analytical expression for the renewal function may be very complicated and even impossible. Therefore, researchers focused on developing approximation methods for them. The purpose of this paper is to explore the renewal functions for non-negligible repair for the most common reliability underlying distributions using the first four raw moments of the failure and repair distributions. This article gives the approximate number of cycles, number of failures and the resulting availability for particular distributions assuming Mean Time to Repair is not negligible and that Time to Restore, or repair has a probability density function denoted as r(t). When Mean Time to Repair is not negligible and Time to Restore has a probability density function denoted as r(t), the expected number of failures, cycles and the resulting availability were obtained by taking the Laplace transforms of corresponding renewal functions. An approximation method for obtaining the expected number of cycles, number of failures and availability using raw moments of failure and repair distributions are provided. Results show that the method produces very accurate results for especially large values of time t.

]]>Brenda Mbouamba Yankam and Abimibola Victoria Oladugba

Experimenters often evaluate the steadiness and consistency of designs over the region of interest by means of the prediction variance capabilities using the variance dispersion graph and the fraction of design space graph. The variance dispersion graph and the fraction of design space graph effectively describe the prediction variance capabilities of a design in the region of interest. However, the prediction variance capabilities of third-order response surface designs have not been studied in the literature. In this paper, the prediction variance capabilities of two third-order response surface designs term augmented orthogonal uniform composite designs and orthogonal array composite designs in the cuboidal region for 3≤k≤7 with center points are examined. The prediction variance capabilities are evaluated using the variance dispersion graph and the fraction of design space graph. Also, D-, E-, G-and T-optimality criteria are used in evaluating these designs in terms of single-value criterion. The results obtained show that the augmented orthogonal uniform composite designs have better prediction variance capabilities in the cuboidal region in the terms of the variance dispersion graphs for factors 3 and 4. The augmented orthogonal uniform composite designs also have better prediction variance capabilities for 3≤k≤7 compare to the orthogonal array composite designs in terms of the fraction of design space graph. The augmented orthogonal uniform composite designs are shown to be superior over the orthogonal array composite designs in terms of D-, E-, G-and T-optimality criteria for single-value criterion. This shows that the performances of the prediction variance capabilities of third-order response surface designs can be clearly visualized by means of the variance dispersion graph and fraction of design space and should be consider over the single-value criteria even though the single value-criteria show some degree of design performance. The augmented orthogonal uniform composite design is should often be considered in experimentation over the orthogonal array composite design since the augmented orthogonal uniform composite design performance better.

]]>Veronika Starodub Ruslan V. Skuratovskii and Sergii S. Podpriatov

We research triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article is development of a method for creating curves that pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This is the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new fascinating geometrical object. During the research number of new triangle conics and cubics are derived, their properties in Euclidian space are considered. In addition, it is discussed corollaries of the obtained theorems in projective geometry, which proves that all of the discovered results could be transferred to the projective plane. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. We investigate the class of curves applicable in cryptography.

]]>Fitriani Indah Emilia Wijayanti Budi Surodjo Sri Wahyuni and Ahmad Faisol

Let R be a ring, K,M be R-modules, L a uniserial R-module, and X a submodule of L. The triple (K,L,M) is said to be X-sub-exact at L if the sequence K→X→M is exact. Let σ(K,L,M) is a set of all submodules Y of L such that (K,L,M) is Y -sub-exact. The sub-exact sequence is a generalization of an exact sequence. We collect all triple (K,L,M) such that (K,L,M) is an X-sub exact sequence, where X is a maximal element of σ(K,L,M). In a uniserial module, all submodules can be compared under inclusion. So, we can find the maximal element of σ(K,L,M). In this paper, we prove that the set σ(K,L,M) form a category, and we denoted it by C_{L}. Furthermore, we prove that C_{Y} is a full subcategory of C_{L}, for every submodule Y of L. Next, we show that if L is a uniserial module, then C_{L} is a pre-additive category. Every morphism in C_{L} has kernel under some conditions. Since a module factor of L is not a submodule of L, every morphism in a category C_{L} does not have a cokernel. So, C_{L} is not an abelian category. Moreover, we investigate a monic X-sub-exact and an epic X-sub-exact sequence. We prove that the triple (K,L,M) is a monic X-sub-exact if and only if the triple Z-modules (, , ) is a monic -sub-exact sequence, for all R-modules N. Furthermore, the triple (K,L,M) is an epic X-sub-exact if and only if the triple Z-modules (, , ) is a monic -sub-exact, for all R-module N.

Tserenbat Oirov Gereltuya Terbish and Nyamsuren Dorj

The paper focuses on the estimation of the force of mortality of living time distribution. We use a third-order B-spline function to construct the logarithm for force of mortality of living time. The number of the knots, their locations and B-spline coefficients based on a sample of observations are estimated by the maximum likelihood estimation method. Evaluation of B-spline parameters estimated by maximum likelihood estimation tested with criteria of the modified chi-squared goodness of the fit statistic. An algorithm developed to calculate Sequential Procedure for the modified chi-squared goodness of the fit testing. The Matlab code was written using the algorithm. Within this evaluation, the number of knots in the model has significantly reduced. The developed method was used to explain the mortality rate of women aged 0 to 69 among the Mongolian population in 2019 and estimate the life expectancy of Mongolians. The results of this experiment provided an excellent estimation of the force of mortality. Construction of a mortality rate estimation gives possibilities to determine mortality trends and force of mortality. Here, force of mortality is further used to construct a survival function, a lifetime distribution function, and a lifetime distribution probability density function. The method can also be used in financial market models and in models that estimate the useful life of equipment.

]]>Robert Reynolds and Allan Stauffer

The aim of this paper is to provide a table of definite integrals which includes both known and new integrals. This work is important because we provide a formal derivation for integrals in [7] not currently present in literature along with new integrals. By deriving new integrals we hope to expand the current list of integral formulae which could assist in research where applicable. The authors apply their contour integral method [9] to an integral in [8] to achieve this new integral formula in terms of the Lerch function. In this present work, the authors provide a formal derivation for an interesting Exponential Fourier transform and express it in terms of the Lerch function. The Exponential Fourier transform has many real world applications namely, in the field of Electrical engineering, in the work of electrical transients by [10] and in the field of Civil engineering, in the work of stress analysis of boundary load on soil by [11]. The definite integral we derived in this work is given by (1) where the variables . This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.

]]>Kittisak Tinpun

Let S be a semigroup and let G be a subset of S. A set G is a generating set G of S which is denoted by . The rank of S is the minimal size or the minimal cardinality of a generating set of S, i.e. rank . In last twenty years, the rank of semigroups is worldwide studied by many researchers. Then it lead to a new definition of rank that is called the relative rank of S modulo U is the minimal size of a subset such that generates S, i.e. rank . A set with is called generating set of S modulo U. The idea of the relative rank was generalized from the concept of the rank of a semigroup and it was firstly introduced by Howie, Ruskuc and Higgins in 1998. Let X be a finite chain and let Y be a subchain of X. We denote the semigroup of full transformations on X under the composition of functions. Let be the set of all transformations from X to Y which is so-called the transformation semigroup with restricted range Y. It was firstly introduced and studied by Symons in 1975. Many results in were extended to results in . In this paper, we focus on the relative rank of semigroup and the semigroup of all orientation-preserving transformations in . In Section 2.1, we determine the relative rank of modulo the semigroup of all order-preserving or order-reversing transformations. In Section 2.2, we describe the results of the relative rank of modulo the semigroup . In Section 2.3, we determine the relative rank of modulo the semigroup of all orientation-preserving or orientation-reversing transformations. Moreover, we obtain that the relative rank modulo and modulo are equal.

]]>Bhagwan Dass Vijay Prakash Tomar Krishan Kumar and Vikas Ranga

The concept of fuzzy sets presented by Zadeh has conquered an enormous achievement in numerous fields. Uncertainty in real world is ubiquitous. Entropy is an important tool with uncertainty and fuzziness. In this article, we propose new measure of directed divergence on fuzzy set. The extension of the fuzzy sets and one that integrated with other theories have been applied by some researchers. To prove the validity of measure, some axioms are proved. Using the proposed measure, we generate a method about decision making criteria and give a suitable method. In this article, we describe directed divergence measure for fuzzy set. Properties of proposed measure are discussed. In the real world, the multicriteria decision making is a very practical method and has a wide range of uses. By using multicriteria decision making, we can find best choice among the given criteria. In recent years, many researchers extensively apply fuzzy directed divergence for multicriteria decision making. Also some researchers defined the application of parameterized Hesitant Fuzzy Soft Set theory in decision making. In this article, we shall investigate the multiple criteria decision making problem under fuzzy environment. Application of introduced measure is given for decision making problem. A numerical example is given for decision making problem. In a fuzzy multicriteria problem, the analysis is given by an illustration example of the new define approach regarding admission preference of a student for post graduate course of science stream.

]]>Bhuwaneshwar Kumar Gupt F. Lalthlamuanpuii and Md. Irphan Ahamed

In survey planning, sometimes, there arises situation to use cluster sampling because of nature of spatial relationship between elements of population or physical feature of land over which elements are dispersed or unavailability of reliable list of elements. At the same time, there requires technique and strategy for ensuring precision of the sample in representing the parent population. Although several theoretical cum practical works have been done in cluster sampling, stratified sampling and stratified cluster sampling, so far, the problem of stratified cluster sampling for a study variable based on an auxiliary variable, which is required in practice, has never been approached. For the first time, this paper deals with the problem of optimum stratification of population of clusters in cluster sampling with clusters of equal size of a characteristic y under study based on highly correlated concomitant variable x for allocation proportional to stratum cluster totals under a super population model. Equations giving optimum strata boundaries (OSB) for dividing population, in which sampling unit of the population is a cluster, are obtained by minimising sampling variance of the estimator of population mean. As the equations are implicit in nature, a few methods of finding approximately optimum strata boundaries (AOSB) are deduced from the equations giving OSB. In deriving the equations, mathematical tools of calculus and algebra are used in addition to statistical methods of finding conditional expectation of variance. All the proposed methods of stratification are empirically examined by illustrating in live data, population of villages in Lunglei and Serchhip districts of Mizoram State, India, and found to perform efficiently in stratifying the population. The proposed methods may provide practically feasible solution in planning socio-economic survey.

]]>Piyatida Phanthuna and Yupaporn Areepong

A modified exponentially weighted moving average (EWMA) scheme expanded from an EWMA chart is an instrument for immediate detection on a small shifted size. The objective of this research is to suggest the average run length (ARL) with the explicit formula on a modified EWMA control chart for observations of a seasonal autoregressive model of order p^{th} (SAR(p)_{L}) with exponential residual. A numerical integral equation method is brought to approximate ARL for checking an accuracy of explicit formulas. The results of two methods show that their ARL solutions are close and the percentage of the absolute relative change (ARC) is obtained to less than 0.002. Furthermore, the modified EWMA chart with the SAR(p)_{L} model is tested to shift detection when the parameters c and are changed. The ARL and the relative mean index (RMI) results are found to be better when c and are increased. In addition, the modified EWMA control chart is compared to performance with the EWMA scheme and such that their results encourage the modified EWMA chart for a small shift. Finally, this explicit formula can be applied to various real-world data. For example, two data about information and communication technology are used for the validation and the capability of our techniques.

Shahira Shafie and Abdul Malek Yaakob

Networked rule bases in fuzzy system, acknowledged as fuzzy network, carries multiple stages of development in decision making processes that involves the uncertainty in the data used as medium in various field. Fuzzy network promotes transparency in multicriteria decision making (MCDM) whereby the criteria are divided into subsystems of cost and benefit to ensure good assessment performance. By considering Hesitant fuzzy sets (HFS), which gives the permission of a set of possible values to present the membership degree of an element, we develop a novel approach that applies fuzzy network and the maximizing deviation method in solving MCDM problem. Fuzzy network addresses transparency in the formulation and maximizing deviation method can restore weight information in MCDM problems whether partially known or fully unknown. The proposed method is applied in case study of stock evaluation that carries opinion evaluated by several decision makers and compared in terms of performance using Spearman rho correlation.

]]>Sharmila Karim and Haslinda Ibrahim

Permutation is an interesting subject to explore until today where it is widely applied in many areas. This paper presents the use of factorial numbers for generating starter sets where starter sets are used for listing permutation. Previously starter sets are generated by using their permutation under exchange-based and cycling based. However, in the new algorithm, this process is replaced by factorial numbers. The base theory is there are number of distinct starter sets. Every permutation has its decimal number from zero until for Lexicographic order permutation only. From a decimal number, it will be converted to a factorial number. Then the factorial number will be mapped to its corresponding starter sets. After that, the Half Wing of Butterfly will be presented. The advantage of the use of factorial numbers is the avoidance of the recursive call function for starter set generation. In other words, any starter set can be generated by calling any decimal number. This new algorithm is still in the early stage and under development for the generation of the half wing of butterfly representation. Case n=5 is demonstrated for a new algorithm for lexicographic order permutation. In conclusion, this new development is only applicable for generating starter sets as a lexicographic order permutation due to factorial numbers is applicable for lexicographic order permutation.

]]>Yik-Siong Pang Nor Aishah Ahad and Sharipah Soaad Syed Yahaya

Multivariate outliers can exist in two forms, casewise and cellwise. Data collection typically contains unknown proportion and types of outliers which can jeopardize the location estimation and affect research findings. In cases where the two coexist in the same data set, traditional distance-based trimmed mean and coordinate-wise trimmed mean are unable to perform well in estimating location measurement. Distance-based trimmed mean suffers from leftover cellwise outliers after the trimming whereas coordinate-wise trimmed mean is affected by extra casewise outliers. Thus, this paper proposes new robust multivariate location estimation known as α-distance-based trimmed median () to deal with both types of outliers simultaneously in a data set. Simulated data were used to illustrate the feasibility of the new procedure by comparing with the classical mean, classical median and α-distance-based trimmed mean. Undeniably, the classical mean performed the best when dealing with clean data, but contrarily on contaminated data. Meanwhile, classical median outperformed distance-based trimmed mean when dealing with both casewise and cellwise outliers, but still affected by the combined outliers' effect. Based on the simulation results, the proposed yields better location estimation on contaminated data compared to the other three estimators considered in this paper. Thus, the proposed can mitigate the issues of outliers and provide a better location estimation.

]]>Mans L Mananohas Charles E Mongi Dolfie Pandara Chriestie E J C Montolalu and Muhammad P M Mo'o

The weight enumerator of a code is a homogeneous polynomial that provides a lot of information about the code. In this case, for the development of a code, research on the weight enumerator is very important. In this study, we focus on the code . Let be the weight enumerator of the code . Fujii and Oura showed that is generated by and . Indeed, we show that is an element of the polynomial ring . We know that the weight enumerator of all self-dual double-even (Type II) code is generated by and . Recall is a type II code. Thus, is an element of the polynomial ring and . One of the motivations of this research is to investigate the connection between these two polynomial rings in representing . Let and be the coefficients of polynomial that represent as an element of and , respectively. We find that is an element of the polynomial . In addition, we also show that there are no weight enumerators of Type II code generated by and that can be written uniquely as isobaric polynomials in five homogeneous polynomial elements of degrees 8, 24, 24, 24, 24.

]]>Alaa Hassan Noreldeen Wageeda M. M. and O. H. Fathy

Polynomial: algebra is essential in commutative algebra since it can serve as a fundamental model for differentiation. For module differentials and Loday's differential commutative graded algebra, simplified homology for polynomial algebra was defined. In this article, the definitions of the simplicial, the cyclic, and the dihedral homology of pure algebra are presented. The definition of the simplicial and the cyclic homology is presented in the Algebra of Polynomials and Laurent's Polynomials. The long exact sequence of both cyclic homology and simplicial homology is presented. The Morita invariance property of cyclic homology was submitted. The relationship was introduced, representing the relationship between dihedral and cyclic (co)homology in polynomial algebra. Besides, a relationship , was examined, defining the relationship between dihedral and cyclic (co)homology of Laurent polynomials algebra. Furthermore, the Morita invariance property of dihedral homology in polynomial algebra was investigated. Also, the Morita property of dihedral homology in Laurent polynomials was studied. For the dihedral homology, the long exact sequence was obtained of the short sequence . The long exact sequence of the short sequence was obtained from the reflexive (co)homology of polynomial algebra. Studying polynomial algebra helps calculate COVID-19 vaccines.

]]>Hussein Eledum and Hytham Hussein Awadallah

In the multiple linear regression model, the problem of multicollinearity may come together with autocorrelation; therefore, several methods of estimation are developed to deal with this case; Two-Stage Ridge Regression (TR) is one of them. This article's main objective is to run a Monte Carlo simulation to investigate the impact of both problems, Multicollinearity and Autocorrelation, in multiple linear regression model on the performance of the TR. The simulation is carried out under different levels of multicollinearity, and sets of autocorrelation coefficient, taking into account different sample sizes. Some new properties for the TR method, including expectation, variance and mean square error, are droved. In contrast, the study also has developed some techniques to estimate the biasing parameter for the TR by modifying some popular techniques used in ridge regression (RR). Moreover, Mean Square Error is used as a base for evaluation and comparison. The empirical findings from the simulations revealed that the TR estimator performs better than the RR, and the values of the biasing parameter under the TR are always less than that under the RR. This paper contributes to the existing literature on developing new estimation methods used to overcome the presence of mixed problems in a linear regression model and studying their properties.

]]>Md. Irphan Ahamed Bhuwaneshwar Kumar Gupt and Manoshi Phukon

In stratified sampling, ever since Dalenius [1] undertook the problem of optimum stratification, the research in the area has been progressing in various perspectives and dimensions till date. Amidst the multifaceted developments in the trend of the research, consideration of the topic by taking into account various aspects such as different sample selection methods and allocations, study variable based stratification, auxiliary variable based stratification, superpopulation models, extension to two study variables for a single auxiliary variable, extension to two stratification variables for a single study variable etc., are a few noteworthy ones. However, with regard to considering optimum stratification of heteroscedastic populations, as live populations are generally heteroscedastic, it was Gupt and Ahamed [2,3] who considered the problem for a few allocations under a heteroscedastic regression superpopulation (HRS) model. As a sequel to the work of the authors, in this paper, the problem of optimum stratification for an objective variable y based on a concomitant variable x under the HRS model is considered for an allocation proposed by Gupt [4,5] and termed as Generalised Auxiliary Variable Optimum Allocation (GAVOA). Methods of stratification in the form of equations and approximate solutions to the equations which stratify populations at optimum strata boundaries (OSB) and approximately optimum strata boundaries (AOSB) respectively are obtained. Mathematical analysis is used in minimizing sampling variance of the estimator of population mean and deriving all the proposed methods of stratification. The proposed equations divide heteroscedastic populations, symmetrical or moderately skewed or highly skewed, at OSB, but, the equations are implicit in nature and not easy in solving. Therefore, a few methods of finding AOSB are deduced from the equations through analytically justified steps of approximation. The methods may provide practically feasible solutions in survey planning in stratifying heteroscedastic population of any level of heteroscedasticity and the work may contribute, to some extent, theoretically in the research area. The methods are empirically examined in a few generated heteroscedastic data of varied shapes with some assumed levels of heteroscedasticity and found to perform with high efficiency. The proposed methods of stratification are restricted to the particular allocation used.

]]>Pradeep Shende and Arvind Kumar Sinha

Data is generating at an exponential pace with the advancement in information technology. Such data highly contain uncertain and vague information. The rough set approximation is a way to find information in the data-set under uncertainty and to classify objects of the dataset. This work presents a mathematical approach to evaluate the data-sets uncertainties and their application to data reduction. In this work, we have extended the multi-granulation variable precision rough set in the context of uncertainty optimization. We develop an uncertainty optimization-based multi-granular rough set (UOMGRS) to minimize the uncertainties in the data set more effectively. Using UOMGRS, we find the most informative attribute in the feature space. It is desirable to minimize the rough set boundary region using the attribute having the highest approximation quality. Thus we group the attributes whose relative quality of approximation is the maximum to maximize the positive region and to minimize the uncertain region. We compare the UOMGRS with the single granulation rough set (SGRS) and the multi-granular rough set (MGRS). By our proposed method, we require only an average of 62% attributes for approximation whereas, SGRS and MGRS need an average of at least 72% of attributes in the data set for approximation of the concepts in the data-set. Our proposed method requires less amount of data for the classification of objects in the dataset. The method helps minimize the uncertainties in the dataset in a more efficient way.

]]>Samsul Arifin Hanni Garminia and Pudji Astuti

Not a long time ago, Ghorbani and Nazemian [2015] introduced the concept of dimension of valuation which measures how much does the ring differ from the valuation. They've shown that every Artinian ring has a finite valuation dimensions. Further, any comutative ring with a finite valuation dimension is semiperfect. However, there is a semiperfect ring which has an infinite valuation dimension. With those facts, it is of interest to further investigate property of rings that has a finite dimension of valuation. In this article we define conditions that a Noetherian ring requires and suffices to have a finite valuation dimension. In particular we prove that, if and only if it is Artinian or valuation, a Noetherian ring has its finite valuation dimension. In view of the fact that a ring needs a semi perfect dimension in terms of valuation, our investigation is confined on semiperfect Noetherian rings. Furthermore, as a finite product of local rings is a semi perfect ring, the inquiry into our outcome is divided into two cases, the case of the examined ring being local and the case where the investigated ring is a product of at least two local rings. This is, first of all, that every local Noetherian ring possesses a finite valuation dimension, if and only if it is Artinian or valuation. Secondly, any Notherian Ring generated by two or more local rings is shown to have a finite valuation dimension, if and only if it is an Artinian.

]]>Robert Reynolds and Allan Stauffer

It is always useful to improve the catalogue of definite integrals available in tables. In this paper we use our previous work on Lobachevsky integrals to derive entries in the tables by Bierens De Haan and Anatolli Prudnikov featuring errata results and new integral formula for interested readers. In this work we derive a definite integral given by (1) in terms of the Lerch function. The importance of this work lies in the derivation of known and new results not presently found in current literature. We used our contour integral method and applied it to an integral in Prudnikov and used it to derive a closed form solution in terms of a special function. The advantage of using a special function is the added benefit of analytic continuation which widens the range of computation of the parameters. Special functions have significance in mathematical analysis, functional analysis, geometry, physics, and other applications. Special functions are used in the solutions of differential equations or integrals of elementary functions. Special functions are linked to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.

]]>Ali Naji Shaker

A partial differential equation has been using the various boundary elements techniques for getting the solution to eigenvalue problem. A number of mathematical concepts were enlightened in this paper in relation with eigenvalue problem. Initially, we studied the basic approaches such as Dirichlet distribution, Dirichlet process and the Model of mixed Dirichlet. Four different eigenvalue problems were summarized, viz. Dirichlet eigenvalue problems, Neumann eigenvalue problems, Mixed Dirichlet-Neumann eigenvalue problem and periodic eigenvalue problem. Dirichlet eigenvalue problem was analyzed briefly for three different cases of value of λ. We put the result for multinomial as its prior is Dirichlet distribution. The result of eigenvalues for the ordinary differential equation was extrapolated. The Basic mathematics was also performed for λ calculations which follow iterative method.

]]>Jamila Jawdat and Ayat Kamal

This paper deals with Quasi-Chebyshevity in the Bochner function spaces , where X is a Banach space. For W a nonempty closed subset of X and x ∊ X, an element w0 in W is called "best approximation" to x from W, if , for all w in W. All best approximation points of x from W form a set usually denoted by P_{W} (x). The set W is called "proximinal" in X if P_{W} (x) is non empty, for each x in X. Now, W is said to be "Quasi-Chebyshev" in X whenever, for each x in X, the set P_{W} (x) is nonempty and compact in X. This subject was studied in general Banach spaces by several authors and some results had been obtained. In this work, we study Quasi-Chebyshevity in the Bochner L^{p}- spaces. The main result in this paper is that: given W a Quasi-Chebyshev subspace in X then L^{p}(μ, W) is Quasi-Chebyshev in , if and only if L^{1} (μ, W) is Quasi-Chebyshev in L^{1}(μ, X). As a consequence, one gets that if W is reflexive in X such that X satisfies the sequential KK-property then L^{p}(μ, W) is Quasi-Chebyshev in .

Faiz Zulkifli Zulkifley Mohamed Nor Afzalina Azmee and Rozaimah Zainal Abidin

Ordinal regression is used to model the ordinal response variable as functions of several explanatory variables. The most commonly used model for ordinal regression is the proportional odds model (POM). The classical technique for estimating the unknown parameters of this model is the maximum likelihood (ML) estimator. However, this method is not suitable for solving problems with extreme observations. A robust regression method is needed to handle the problem of extreme points in the data. This study proposes Huber M-estimator as a robust method to estimate the parameters of the POM with a logistic link function and polytomous explanatory variables. This study assesses ML estimator performance and the robust method proposed through an extensive Monte Carlo simulation study conducted using statistical software, R. Measurement for comparisons are bias, RMSE, and Lipsitzs' goodness of fit test. Various sample sizes, percentages of contamination, and residual standard deviations are considered in the simulation study. Preliminary results show that Huber estimates provide the best results for parameter estimation and overall model fitting. Huber's estimator has reached a 50% breakdown point for data containing extreme points that are quite far from most points. In addition, the presence of extreme points that have only a distance of two times far from most points has no major impact on ML estimates. This means that the estimates for ML and Huber may yield the same results if the model's residual values are between -2 and 2. This situation may also occur for data with a percentage of contamination below 5%.

]]>S. Nasrin R. N. Mondal and M. M. Alam

Riga plate is the span wise array of electrodes and permanent magnets that creates a plane surface and produced the electromagnetic hydrodynamic fluid behavior and mostly used in industrial processes with fluid flow affairs. In cases where an external application of a magnetic or electric field is required, better flow is obtained by the involvement of the Riga plate. Riga plate acts as an agent to reduce the skin friction and enhance the heat transfer phenomena. It also diminishes the turbulent effects, so that it is possible to get an efficient flow control and it increases the performance of the machine. So the numerical investigation of the unsteady Couette flow with Hall and ion-slip current effects past between two Riga plates has been studied and the numerical solutions are acquired by using explicit finite difference method and estimated results have been gained for several values of the dimensionless parameter such as pressure gradient parameter, Hall and Ion-slip parameters, modified Hartmann number, Prandtl number, and Eckert number. In this article, the importance of the modified Hartmann number on the flow profiles is immense owing to the Riga plate. The expression of skin friction and Nusselt number has been computed and the outcomes of the relevant parameters on various distributions have been sketched and presented as well as graphically.

]]>Christoph Fuhrmann Hanns-Ludwig Harney Klaus Harney and Andreas M¨uller

The present article derives the minimal number N of observations needed to approximate a Bayesian posterior distribution by a Gaussian. The derivation is based on an invariance requirement for the likelihood . This requirement is defined by a Lie group that leaves the unchanged, when applied both to the observation(s) and to the parameter to be estimated. It leads, in turn, to a class of specific priors. In general, the criterion for the Gaussian approximation is found to depend on (i) the Fisher information related to the likelihood , and (ii) on the lowest non-vanishing order in the Taylor expansion of the Kullback-Leibler distance between and , where is the maximum-likelihood estimator of , given by the observations . Two examples are presented, widespread in various statistical analyses. In the first one, a chi-squared distribution, both the observations and the parameter are defined all over the real axis. In the other one, the binomial distribution, the observation is a binary number, while the parameter is defined on a finite interval of the real axis. Analytic expressions for the required minimal N are given in both cases. The necessary N is an order of magnitude larger for the chi-squared model (continuous ) than for the binomial model (binary ). The difference is traced back to symmetry properties of the likelihood function . We see considerable practical interest in our results since the normal distribution is the basis of parametric methods of applied statistics widely used in diverse areas of research (education, medicine, physics, astronomy etc.). To have an analytical criterion whether the normal distribution is applicable or not, appears relevant for practitioners in these fields.

]]>Reza Pakyari

Geometric Extreme Exponential Distribution (GEE) is one of the statistical models that can be useful in fitting and describing lifetime data. In this paper, the problem of estimation of the reliability R = P(Y < X) when X and Y are independent GEE random variables with common scale parameter but different shape parameters has been considered. The probability R = P(Y < X) is also known as stress-strength reliability parameter and demonstrates the case where a component has stress X and is subjected to strength Y. The reliability R = P(Y < X) has applications in engineering, finance and biomedical sciences. We present the maximum likelihood estimator of R and study its asymptotic behavior. We first study the asymptotic distribution of the maximum likelihood estimators of the GEE parameters. We prove that the maximum likelihood estimators and so the reliability R have asymptotic normal distribution. A bootstrap confidence interval for R is also presented. Monte Carlo simulations are performed to assess he performance of the proposed estimation method and validity of the confidence interval. We found that the performance of the maximum likelihood estimator and also the bootstrap confidence interval is satisfactory even for small sample sizes. Analysis of a dataset has been given for illustrative purposes.

]]>Samsul Arifin Hanni Garminia and Pudji Astuti

We present some methods for calculating the module's uniserial dimension that finitely generated over a DVD in this article. The idea of a module's uniserial dimension over a commutative ring, which defines how far the module deviates from being uniserial, was recently proposed by Nazemian etc. They show that if R is Noetherian commutative ring, which implies that every finitely generated module over R has uniserial dimension. Ghorbani and Nazemians have shown that R is Noetherian (resp. Artinian) ring if only the ring R X R has (resp. finite) valuation dimension. The finitely generated modules over valuation domain are further examined from here. However, since the region remains too broad, further research into the module's uniserial dimensions that finitely generated over a DVD is needed. In the case of a DVD R, a finitely generated module over R can, as is well-known, be divided into a direct sum of torsion and a free module. Therefore, first, we present methods for determining the primary module's uniserial dimension, and then followed by methods for the general finitely generated module. As can be observed, the module's uniserial dimension is a function of the elementary divisors and the rank of the non torsion module item, which is the major finding of this work.

]]>Hermansah Dedi Rosadi Abdurakhman and Herni Utami

In this research, we propose a Nonlinear Auto-Regressive network with exogenous inputs (NARX) model with a different approach, namely the determination of the main input variables using a stepwise regression and exogenous input using a deterministic seasonal dummy. There are two approaches in making a deterministic seasonal dummy, namely the binary and the sine-cosine dummy variables. Approximately half the number of input variables plus one is contained in the neurons of the hidden layer. Furthermore, the resilient backpropagation learning algorithm and the tangent hyperbolic activation function were used to train each network. Three ensemble operators are used, namely mean, median, and mode, to solve the overfitting problem and the single NARX model's weakness. Furthermore, we provide an empirical study using actual data, where forecasting accuracy is determined by Mean Absolute Percent Error (MAPE). The empirical study results show that the NARX model with binary dummy exogenous is the most accurate for trend and seasonal with multiplicative properties data patterns. For trend and seasonal with additive properties data patterns, the NARX model with sine-cosine dummy exogenous is more accurate, except the fact that the NARX model uses the mean ensemble operator. Besides, for trend and non-seasonal data patterns, the most accurate NARX model is obtained using the mean ensemble operator. This research also shows that the median and mode ensemble operators, which are rarely used, are more accurate than the mean ensemble operator for data that have trend and seasonal patterns. The median ensemble operator requires the least average computation time, followed by the mode ensemble operator. On the other hand, all of our proposed NARX models' accuracy consistently outperforms the exponential smoothing method and the ARIMA method.

]]>M. Fariz Fadillah Mardianto Gunardi and Herni Utami

Fourier series is a function that is often used Mathematically and Statistically especially for modeling. Here, Fourier series can be constructed as an estimator in nonparametric regression. Nonparametric regression is not only using cross section data, but also longitudinal data. Some of nonparametric regression estimators have been developed for longitudinal data case, such as kernel, and spline. In this study, we concentrate to develop an inference analysis that related to Fourier series estimator in nonparametric regression for longitudinal data. Nonparametric regression based on Fourier series is capable to model data relationship with fluctuation or oscillation pattern that represents with sine and cosine functions. For point estimation analysis, Penalized Weighted Least Square (PWLS) is used to determine an estimator for parameter vector in nonparametric regression. Different with previous studies, PWLS is used to get smooth estimator. The result is an estimator for nonparametric regression curve for longitudinal data based on Fourier series approach. In addition, this study also investigated the asymptotic properties of the nonparametric regression curve estimators using the Fourier series approach for longitudinal data, especially linearity and consistency. Some study cases based on previous research and a new study case is given to make sure that Fourier series estimator in nonparametric regression has good performance in longitudinal data modeling. This study is important in order to develop further inferences Statistics, such as interval estimation and test hypothesis that related nonparametric regression with Fourier series estimator for longitudinal data.

]]>Jewgeni H. Dshalalow Kizza Nandyose and Ryan T. White

This paper deals with a class of antagonistic stochastic games of three players A, B, and C, of whom the first two are active players and the third is a passive player. The active players exchange hostile attacks at random times of random magnitudes with each other and also with player C. Player C does not respond to any attacks (that are regarded as a collateral damage). There are two sustainability thresholds M and T are set so that when the total damages to players A and B cross M and T, respectively, the underlying player is ruined. At some point (ruin time), one of the two active players will be ruined. Player C's damages are sustainable and some rebuilt. Of interest are the ruin time and the status of all three players upon as well as at any time t prior to . We obtain an analytic formula for the joint distribution of the named processes and demonstrate its closed form in various analytic and computational examples. In some situations pertaining to stock option trading, stock prices (player C) can fluctuate. So in this case, it is of interest to predict the first time when an underlying stock price drops or significantly drops so that the trader can exercise the call option prior to the drop and before maturity T. Player A monitors the prices upon times assigning 0 damage to itself if the stock price appreciates or does not change and assumes a positive integer if the price drops. The times are themselves damages to player B with threshold T. The "ruin" time is when threshold M is crossed (i.e., there is a big price drop or a series of drops) or when the maturity T expires whichever comes first. Thus a prior action is needed and its time is predicted. We illustrate the applicability of the game on a number of other practical models, including queueing systems with vacations and (N,T)-policy.

]]>Arvind Kumar Sinha and Srikumar Panda

The main objective of the paper is to study the three-dimensional fractional Fourier Mellin transforms (3DFRFMT), their basic properties and applicability due to mainly use in the radar system, reconstruction of grayscale images, in the detection of the human face, etc. Only the fractional Fourier transform is based on time-frequency distribution, whereas only the fractional Mellin transform is on scale covariant transformation. Both transforms can discover action in the definite assortment. The fractional Fourier transform is applicable for controlling the range of shift, whereas the fractional Mellin transform is accustomed to managing the range of rotation and scaling of the function. So, combining both transformations, we get an elegant expression for 3DFRFMT, which can be used in several fields. The paper introduces the concept of three-dimensional fractional Fourier Mellin transforms and their applications. Modulation property is the most useful concept in the signal system, radar technology, pattern reorganization, and many more in the integral transform. Parseval's identity applies to the conservation of energy in the universe. Thus we establish the modulation theorem, Parseval's theorem, scaling theorem, analytic theorem for three-dimensional fractional Fourier Mellin transform. We also give some examples of three-dimensional fractional Fourier-Mellin transform on some functions. Finally, we provide three-dimensional fractional Fourier-Mellin transform applications for solving homogeneous and non-homogeneous Mboctara partial differential equations that we can apply with advantages to solve the different types of problems in signal processing systems. The transform is beneficial in a maritime strategy as a co-realtor to control moments in any specific three-dimensional space. The concept is the most powerful tool to deal with any information system problems. After obtaining the generalization, we can explore many more ideas in applying three-dimensional fractional Fourier-Mellin transformations in many real word problems.

]]>S. A. Ojobor and A. Obihia

The aim of this paper is to solve numerically the Cauchy problems of nonlinear partial differential equation (PDE) in a modified variational iteration approach. The standard variational iteration method (VIM) is first studied before modifying it using the standard Adomian polynomials in decomposing the nonlinear terms of the PDE to attain the new iterative scheme modified variational iteration method (MVIM). The VIM was used to iteratively determine the nonlinear parabolic partial differential equation to obtain some results. Also, the modified VIM was used to solve the nonlinear PDEs with the aid of Maple 18 software. The results show that the new scheme MVIM encourages rapid convergence for the problem under consideration. From the results, it is observed that for the values the MVIM converges faster to exact result than the VIM though both of them attained a maximum error of order 10^{-9}. The resulting numerical evidences were competing with the standard VIM as to the convergence, accuracy and effectiveness. The results obtained show that the modified VIM is a better approximant of the above nonlinear equation than the traditional VIM. On the basis of the analysis and computation we strongly advocate that the modified with finite Adomian polynomials as decomposer of nonlinear terms in partial differential equations and any other mathematical equation be encouraged as a numerical method.

Zahari Md Rodzi Abd Ghafur Ahmad Norul Fadhilah Ismail and Nur Lina Abdullah

The hesitant fuzzy set (HFS) concept as an extension of fuzzy set (FS) in which the membership degree of a given element, called the hesitant fuzzy element (HFE), is defined as a set of possible values. A large number of studies are concentrating on HFE and HFS measurements. It is not just because of their crucial importance in theoretical studies, but also because they are required for almost any application field. The score function of HFE is a useful method for converting data into a single value. Moreover, the scoring function provides a much easier way to determine each alternative's ranking order for multi-criteria decision-making (MCDM). This study introduces a new hesitant degree of HFE and the z-score function of HFE, which consists of z-arithmetic mean, z-geometric mean, and z-harmonic mean. The z-score function is developed with four main bases: a hesitant degree of HFE, deviation value of HFE, the importance of the hesitant degree of HFE, α, and importance of the deviation value of HFE, β. These three proposed scores are compared with the existing scores functions to identify the proposed z-score function's flexibility. An algorithm based on the z-score function was developed to create an algorithm solution to MCDM. Example of secondary data on supplier selection for automated companies is used to prove the algorithms' capability in ranking order for MCDM.

]]>Nazrina Aziz Zahirah Hasim and Zakiyah Zain

Acceptance sampling is an important technique in quality assurance; its main goal is to achieve the most accurate decision in accepting lot using minimum resources. In practice, this often translates into minimizing the required sample sizes for the inspection, while satisfying the maximum allowable risks by consumer and producer. Numerous sampling plans have been developed over the past decades, the most recent being the incorporation of grouping to enable simultaneous inspection in the two-sided chain sampling which considers information from preceding and succeeding samples. This combination offers improved decision accuracy with reduced inspection resources. To-date, two-sided group chain sampling plan (TSGCh) for characteristic based on truncated lifetime has only been explored for Pareto distribution of the 2^{nd} kind. This article introduces TSGCh sampling plan for products with lifetime that follows generalized exponential distribution. It focuses on minimizing consumer's risk and operates with three acceptance criteria. The equations that derived from the set conditions involving generalized exponential and binomial distributions are mathematically solved to develop this sampling plan. Its performance is measured on the probability of lot acceptance and number of minimum groups. A comparison with the established new two-sided group chain (NTSGCh) indicates that the proposed TSGCh sampling plan performs better in terms of sample size requirement and consumers' protection. Thus, this new acceptance sampling plan can reduce the inspection time, resources, and costs via smaller sample size (number of groups), while providing the desired consumers' protection.

Sardar G Amen Ali F Jameel and Abdul Malek Yaakob

The Bezier curve is a parametric curve used in the graphics of a computer and related areas. This curve, connected to the polynomials of Bernstein, is named after the design curves of Renault's cars by Pierre Bézier in the 1960s. There has recently been considerable focus on finding reliable and more effective approximate methods for solving different mathematical problems with differential equations. Fuzzy differential equations (known as FDEs) make extensive use of various scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under uncertainty. This article discusses the use of Bezier curves for solving elevated order fuzzy initial value problems (FIVPs) in the form of ordinary differential equation. A Bezier curve approach is analyzed and updated with concepts and properties of the fuzzy set theory for solving fuzzy linear problems. The control points on Bezier curve are obtained by minimizing the residual function based on the least square method. Numerical examples involving the second and third order linear FIVPs are presented and compared with the exact solution to show the capability of the method in the form of tables and two dimensional shapes. Such findings show that the proposed method is exceptionally viable and is straightforward to apply.

]]>Iskandar Shah Mohd Zawawi Zarina Bibi Ibrahim and Khairil Iskandar Othman

Block methods that approximate the solution at several points in block form are commonly used to solve higher order differential equations. Inspired by the literature and ongoing research in this field, this paper intends to explore a new derivation of block backward differentiation formula that employs independent parameter to provide sufficient accuracy when solving second order ordinary differential equations directly. The use of three backward steps and five independent parameters are considered adequately in generating the variable coefficients of the formulas. To ascertain only one parameter exists in the derived formula, the order of the method is determined. Such independent parameter retains the favorable convergence properties although the values of parameter will affect the zero stability and truncation error. An ability of the method to compute the approximated solutions at two points concurrently is undeniable. Another advantage of the method is being able to solve the second order problems directly without recourse to the technique of reducing it to a system of first order equations. The essential of the error analysis is to observe the effect of independent parameter on the accuracy, in the sense that with certain appropriate values of parameter, the accuracy is improved. The performance of the method is tested with some initial value problems and the numerical results confirm that the maximum error and average error obtained by the proposed method are smaller at certain step size compared to the other conventional direct methods.

]]>V. I. Struchenkov and D. A. Karpov

Being a continuation of the paper published in Mathematics and Statistics, vol. 7, No. 5, 2019, this article describes the algorithm for the first stage of spline- approximation with an unknown number of elements of the spline and constraints on its parameters. Such problems arise in the computer-aided design of road routes and other linear structures. In this article we consider the problem of a discrete sequence approximation of points on a plane by a spline consisting of line segments conjugated by circular arcs. This problem occurs when designing the longitudinal profile of new and reconstructed railways and highways. At the first stage, using a special dynamic programming algorithm, the number of elements of the spline and the approximate values of its parameters that satisfy all the constraints are determined. At the second stage, this result is used as an initial approximation for optimizing the spline parameters using a special nonlinear programming algorithm. The dynamic programming algorithm is practically the same as in the mentioned article published earlier, with significant simplifications due to the absence of clothoids when connecting straight lines and curves. The need for the second stage is due to the fact that when designing new roads, it is impossible to implement dynamic programming due to the need to take into account the relationship of spline elements in fills and in cuts, if fills will be constructed from soils of cuts. The nonlinear programming algorithm is based on constructing a basis in zero spaces of matrices of active constraints and adjusting this basis when changing the set of active constraints in an iterative process. This allows finding the direction of descent and solving the problem of excluding constraints from the active set without solving systems of linear equations in general or by solving linear systems of low dimension. As an objective function, instead of the traditionally used sum of squares of the deviations of the approximated points from the spline, the article proposes other functions, taking into account the specifics of a specific project task.

]]>Shih Yu Chang and Hsiao-Chun Wu

In linear algebra, the trace of a square matrix is defined as the sum of elements on the main diagonal. The trace of a matrix is the sum of its eigenvalues (counted with multiplicities), and it is invariant under the change of basis. This characterization can be used to define the trace of a tensor in general. Trace inequalities are mathematical relations between different multivariate trace functionals involving linear operators. These relations are straightforward equalities if the involved linear operators commute, however, they can be difficult to prove when the non-commuting linear operators are involved. Given two Hermitian tensors H_{1} and H_{2} that do not commute. Does there exist a method to transform one of the two tensors such that they commute without completely destroying the structure of the original tensor? The spectral pinching method is a tool to resolve this problem. In this work, we will apply such spectral pinching method to prove several trace inequalities that extend the Araki–Lieb–Thirring (ALT) inequality, Golden–Thompson(GT) inequality and logarithmic trace inequality to arbitrary many tensors. Our approaches rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent mechanism to treat generic tensor multivariate trace inequalities. As an example application of our tensor extension of the Golden–Thompson inequality, we give the tail bound for the independent sum of tensors. Such bound will play a fundamental role in high-dimensional probability and statistical data analysis.

Terry E. Moschandreou

The problem to The Clay Math Institute "Navier-Stokes, breakdown of smooth solutions here on an arbitrary cube subset of three dimensional space with periodic boundary conditions is examined. The incompressible Navier-Stokes Equations are presented in a new and conventionally different way here, by naturally reducing them to an operator form which is then further analyzed. It is shown that a reduction to a general 2D N-S system decoupled from a 1D non-linear partial differential equation is possible to obtain. This is executed using integration over n-dimensional compact intervals which allows decoupling. The operator form is considered in a physical geometric vorticity case, and a more general case. In the general case, the solution is revealed to have smooth solutions which exhibit finite-time blowup on a fine measure zero set and using the Prékopa-Leindler and Gagliardo-Nirenberg inequalities it is shown that for any non zero measure set in the form of cube subset of 3D there is no finite time blowup for the starred velocity for large dimension of cube and small d. In particular vortices are shown to exist and it is shown that zero is in the attractor of the 3D Navier-Stokes equations.

]]>Sharmeen Binti Syazwan Lai Nur Huda Nabihan Binti Md Shahri Mazni Binti Mohamad Hezlin Aryani Binti Abdul Rahman and Adzhar Bin Rambli

An imbalanced data problem occurs in the absence of a good class distribution between classes. Imbalanced data will cause the classifier to be biased to the majority class as the standard classification algorithms are based on the belief that the training set is balanced. Therefore, it is crucial to find a classifier that can deal with imbalanced data for any given classification task. The aim of this research is to find the best method among AdaBoost, XGBoost, and Logistic Regression to deal with imbalanced simulated datasets and real datasets. The performances of these three methods in both simulated and real imbalanced datasets are compared using five performance measures, namely sensitivity, specificity, precision, F1-score, and g-mean. The results of the simulated datasets show that logistic regression performs better than AdaBoost and XGBoost in highly imbalanced datasets, whereas in the real imbalanced datasets, AdaBoost and logistic regression demonstrated similarly good performance. All methods seem to perform well in datasets that are not severely imbalanced. Compared to AdaBoost and XGBoost, logistic regression is found to predict better for datasets with severe imbalanced ratios. However, all three methods perform poorly for data with a 5% minority, with a sample size of n = 100. In this study, it is found that different methods perform the best for data with different minority percentages.

]]>Adeyeye Oluwaseun and Omar Zurni

Some of the issues relating to the human immunodeficiency virus (HIV) epidemic can be expressed as a system of nonlinear first order ordinary differential equations. This includes modelling the spread of the HIV virus in infecting CD4+T cells that help the human immune system to fight diseases. However, real life differential equation models usually fail to have an exact solution, which is also the case with the nonlinear model considered in this article. Thus, an approximate method, known as the block method, is developed to solve the system of first order nonlinear differential equation. To develop the block method, a linear block approach was adopted, and the basic properties required to classify the method as convergent were investigated. The block method was found to be convergent, which ascertained its usability for the solution of the model. The solution obtained from the newly developed method in this article was compared to previous methods that have been adopted to solve same model. In order to have a justifiable basis of comparison, two-step length values were substituted to obtain a one-step and two-step block method. The results show the newly developed block method obtaining accurate results in comparison to previous studies. Hence, this article has introduced a new method suitable for the direct solution of first order differential equation models without the need to simplify to a system of linear algebraic equations. Likewise, its convergent properties and accuracy also give the block method an edge over existing methods.

]]>Siti Aisyah Zakaria Nor Azrita Mohd Amin Noor Fadhilah Ahmad Radi and Nasrul Hamidin

High ground-level ozone (GLO) concentrations will adversely affect human health, vegetations as well as the ecosystem. Therefore, continuous monitoring for GLO trends is a good practice to address issues related to air quality based on high concentrations of GLO. The purpose of this study is to introduce stationary and non-stationary model of extreme GLO. The method is applied to 25 selected stations in Peninsular Malaysia. The maximum daily GLO concentration data over 8 hours from year 2000 to 2016 are used. The factors of this distribution are anticipated using maximum likelihood estimation. A comparison between stationary (constant model) and non-stationary (linear and cyclic model) is performed using the likelihood ratio test (LRT). The LRT is based on the larger value of deviance statistics compared to a chi-square distribution providing the significance evidence to non-stationary model either there is linear trend or cyclic trend. The best fit model between selected models is tested by Akaike's Information Criterion. The results show that 25 stations conform to the non-stationary model either linear or cyclic model, with 14 stations showing significant improvement over the linear model in location parameter while 11 stations follow the cyclic model. This study is important to identify the trends of ozone phenomenon for better quality risk management.

]]>Rawaa Ibrahim Esa Rasha H Ibraheem and Al.i F Jameel

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

]]>Hafed H Saleh Azmi A. and Ali. F. Jameel

There has recently been considerable focus on finding reliable and more effective approximate methods for solving biological mathematical models in the form of differential equations. One of the well-known approximate or semi-analytical methods for solving linear, nonlinear differential well as partial differential equations within various fields of mathematics is the Variational Iteration Method (VIM). This paper looks at the use of fuzzy differential equations in human immunodeficiency virus (HIV) infection modeling. The main advantage of the method lies in its flexibility and ability to solve nonlinear equations easily. VIM is introduced to provide approximate solutions for linear ordinary differential equation system including the fuzzy HIV infection model. The model explains the amount of undefined immune cells, and the immune system viral load intensity intrinsic that will trigger fuzziness in patients infected by HIV. CD4+T-cells and cytototoxic T-lymphocytes (CTLs) are known for the immune cells concerned. The dynamics of the immune cell level and viral burden are analyzed and compared across three classes of patients with low, moderate and high immune systems. A modification and formulation of the VIM in the fuzzy domain based on the use of the properties of fuzzy set theory are presented. A model was established in this regard, accompanied by plots that demonstrate the reliability and simplicity of the methods. The numerical results of the model indicate that this approach is effective and easily used in fuzzy domain.

]]>E. N. Sinyukova S. V. Drahanyuk and O. O. Chepok

All-round development of the everyday logic of students should be considered as one of the most important tasks of general secondary education on the whole and general secondary mathematics education in particular. We discuss the problem of organization in teachers' training institutions of higher education and the expedient training of the future math teachers at institutions of general secondary education. The main goal is to ensure their ability to realize all their future professional activities and the necessary participation in forming the everyday logic of their pupils. The authors think that vocational educational program of training is that the future secondary school math teachers must contain a separate course of mathematical logic including at least 90 training hours (3 credits ECTS). Although the content filling of the course cannot be irrespective of the general level of arrangement of mathematics education in the corresponding country, it ought to be a subject of discussion of the international mathematics community and managers in the sphere of higher mathematics education. Simultaneously, the role, the place, and the expedient structure of such a course in the corresponding training programs should be under discussion. The article represents the authors' point of view on the problems indicated above. The research has a qualitative characteristic as a whole. Only some of its conclusions have statistical corroboration.

]]>Siham Rabee Ramadan Hamed Ragaa Kassem and Mahmoud Rashwaan

Calibration estimation is one of the most important ways to improve the precision of the survey estimates. It is a method in which the designs weights are modified as little as possible by minimizing a given distance measure to the calibrated weights respecting a set of constraints related to suitable auxiliary information. This paper proposes a new approach for Multivariate Calibration Estimation (MCE) of the population mean of a study variable under stratified random sampling scheme using two auxiliary variables. Almost all literature on calibration estimation used Lagrange multiplier technique in order to estimate the calibrated weights. While Lagrange multiplier technique requires all equations included in the model to be differentiable functions, some un- differentiable functions may be faced in some cases. Hence, it is essential to look for using another technique that can provide more flexibility in dealing with the problem. Accordingly, in this paper, using goal programming approach is newly suggested as a different approach for MCE. The theory of the proposed calibration estimation is presented and the calibrated weights are estimated. A comparison study is conducted using actual and generated data to evaluate the performance of the proposed approach for multivariate calibration estimator with other existing calibration estimators. The results of this study prove that using the proposed GP approach for MCE is more flexible and efficient compared to other calibration estimation methods of the population mean.

]]>Luthfatul Amaliana Ani Budi Astuti and Nur Silviyah Rahmi

Per capita expenditure of an area is a welfare indicator of the community. It is also a reflection of the economic capacity in meeting basic needs. Bali is the second richest province in Indonesia. This study aims to model the per capita expenditure of Bali at the sub-district level using Spatial-EBLUP (SEBLUP) approach in SAE. Small area estimation (SAE) modeling is an indirect estimation approach capable of increasing the effectiveness of sample sizes and minimizing variance. The heterogeneity of an area is influenced by other areas around. Everything is related to one another, but something closer will be more influential than something far away. Therefore, the spatial effect can be included in the random effect of a model small area, which is called as SEBLUP model. The selection of a spatial weights matrix is very important in spatial data modeling. It represents the neighborhood relationship of each spatial observation unit. A SEBLUP model needs a spatial weights matrix, which can be based on distance (radial distance and power distance), contiguity (queen), and a combination of distance and contiguity (radial distance and queen contiguity). The result of the implementation of the SEBLUP approach in per capita expenditure of Bali shows that the SEBLUP model with radial distance spatial weights matrix is the best model with the smallest ARMSE. South Denpasar Sub-district is the most prosperous sub-district with the highest per capita expenditure in Bali. Meanwhile, Abang Sub-district is the smallest per capita expenditure.

]]>A. Torres-Hernandez and F. Brambila-Paz

In this paper an approximation to the zeros of the Riemann zeta function has been obtained for the first time using a fractional iterative method which originates from a unique feature of the fractional calculus. This iterative method, valid for one and several variables, uses the property that the fractional derivative of constants are not always zero. This allows us to construct a fractional iterative method to find the zeros of functions in which it is possible to avoid expressions that involve hypergeometric functions, Mittag-Leffler functions or infinite series. Furthermore, we can find multiple zeros of a function using a singe initial condition. This partially solves the intrinsic problem of iterative methods, which in general is necessary to provide N initial conditions to find N solutions. Consequently the method is suitable for approximating nontrivial zeros of the Riemann zeta function when the absolute value of its imaginary part tends to infinity. Some examples of its implementation are presented, and finally 53 different values near to the zeros of the Riemann zeta function are shown.

]]>Wichayaporn Jantanan Anusorn Simuen Winita Yonthanthum and Ronnason Chinram

Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Г-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Г-semigroups. Many results in ideal theory of Г-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Г-semigroups. In Section 2, we define almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups by using the concept ideas of interior Г-ideals and almost Г-ideals of Г-semigroups. Every almost interior Г-ideal of a Г-semigroup S is clearly a weakly almost interior Г-ideal of S but the converse is not true in general. The notions of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups are generalizations of the notion of interior Г-ideal of a Г-semigroup S. We investigate basic properties of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Г-semigroups. In Section 3, we introduce fuzzy almost interior Г-ideals and fuzzy weakly almost interior Г-ideals of Г-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Г-ideals [weakly almost interior Г-ideals] and fuzzy almost interior Г-ideals [fuzzy weakly almost interior Г-ideals] of Г-semigroups.

]]>Nurfa Risha and Muhammad Farchani Rosyid

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e., on the standard sphere and Gromoll-Meyer exotic sphere . In this case, and are homeomorphic but not diffeomorphic. The standard sphere can be constructed as the quotient manifold with the so-called -action of S^{3}, whereas the Gromoll-Meyer exotic sphere as the quotient manifold with respect to the so-called -action of S^{3}. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism . The stochastic flow can be regarded as the same stochastic flow on S^{7}, but viewed in Gromoll-Meyer differential structure. The flow on and the corresponding flow on constructed in this paper have the same regularities. There is no difference between the stochastic flow's appearance on S^{7} viewed in standard differential structure and the appearance of the same stochastic flow viewed in the Gromoll-Meyer differential structure. Furthermore, since the inverse mapping h^{-1} is differentiable on , the Riemannian metric tensor on , i.e., the pull-back of the Riemannian metric tensor G on the standard sphere , is also differentiable. This fact implies, for instance, the fact that the Fokker-Planck equation associated with the stochastic flow and the Fokker-Planck equation associated with the stochastic differential equation have the same regularities provided that the function β is C^{1}-differentiable. Therefore both differential structures on S^{7} give the same description of the dynamics of the distribution function of the stochastic process understudy on seven spheres.

Jayanta Biswas Pritam Kayal and Debabrata Samanta

Non-Negative Matrix Factorization (NMF) is utilized in many important applications. This paper presents development of an efficient low rank approximate NMF algorithm for feature extraction related to text mining and spectral data analysis. NMF can be used for clustering. NMF factorizes a positive matrix A to two positive matrices W and H matrices where A=WH. The proposal uses k-means clustering algorithm to determine the centroid of each cluster and assigns the centroid coordinates of each cluster as one column for W matrix. The initial choice of W matrix is positive. The H matrix is determined with gradient descent algorithm based on thin QR optimization. The performance comparison of the proposed NMF algorithm is illustrated with results. The accurate choice of initial positive W matrix reduces approximation error and the use of thin QR algorithm in combination with gradient descent approach provides rapid convergence rate for NMF. The proposed algorithm is implemented with the randomly generated matrix in MATLAB environment. The number of significant singular values of the generated matrix is selected as the number of clusters. The error and convergence rate comparison of the proposed algorithm with the current algorithms are demonstrated in this research. The accurate measurement of execution time for individual program is not possible in MATLAB. The average time execution over 200 iterations is therefore calculated with an increasing iteration count of the proposed algorithm and the comparative results are presented.

]]>Alanazi Talal Abdulrahman Randa Alharbi Osama Alamri Dalia Alnagar and Bader Alruwaili

A supersaturated design is an important method that relies on factorial designs whose number of factors is greater than experiments' number. The analysis of supersaturated designs is challenging due to the complexity of the design matrix. This problem is challenging due to the fact that the design matrix has a complicated structure. Identification of the variable including the active factor plays an essential role when supersaturated design is used to analyse the data. A variable selection technique to screen active effects in the SSDs and regression analysis are applied to our case study. This study set out to examine the actual reasons for the spread of electronic games statistically such as Saudi society. An online survey provided quantitative data from 200 participants. Respondents were randomly divided into two conditions (Yes+, No-) and asked to respond to one of two sets of the causes of electronic games. The responses was analysed using contrast method with supersaturated designs and regression methods using the SPSS computer software to determine the actual causes that led to the spread of electronic games. The findings indicated that because of their constant preoccupation, some parents resort to such games in order to get rid of the child's inconvenience and insufficient awareness among parents of the dangers of these games, and excessive pampering is the factor that led to the spread of electronic games in Saudi society statistically. On this basis, it is recommended that Saudi government professionals develop an operational plan to study these causes to take actions. In future investigations, no recent studies address the external environmental aspects that could influence gaming among individuals, and hence further research is required in this field.

]]>Rejula Mercy. J and S. Elizabeth Amudhini Stephen

Springs are important members often used in machines to exert force, absorb energy and provide flexibility. In mechanical systems, wherever flexibility or relatively a large load under the given circumstances is required, some form of spring is used. In this paper, non-traditional optimization algorithms, namely, Ant Lion Optimizer, Grey Wolf Optimizer, Dragonfly optimization algorithm, Firefly algorithm, Flower Pollination Algorithm, Whale Optimization Algorithm, Cat Swarm Optimization, Bat Algorithm, Particle Swarm Optimization, Gravitational Search Algorithm are proposed to get the global optimal solution for the closed coil helical spring design problem. The problem has three design variables and eight inequality constraints and three bounds. The mathematical formulation of the objective function U is to minimize the volume of closed coil helical spring subject to constraints. The design variables considered are Wire diameter d, Mean coil diameter D, Number of active coils N of the spring. The proposed methods are tested and the performance is evaluated. Ten non-traditional optimization methods are used to find the minimum volume. The problem is computed in the MATLAB environment. The experimental results show that Particle Swarm Optimization outperforms other methods. The results show that PSO gives better results in terms of consistency and minimum value in terms of time and volume of a closed coil helical spring compared to other methods. When compared to other Optimization methods, PSO has few advantages like simplicity and efficiency. In the future, PSO could be extended to solve other mechanical element problems.

]]>Adeyinka Solomon Ogunsanya Waheed Babatunde Yahya Taiwo Mobolaji Adegoke Christiana Iluno Oluwaseun R. Aderele and Matthew Iwada Ekum

In this work, a three-parameter Weibull Inverse Rayleigh (WIR) distribution is proposed. The new WIR distribution is an extension of a one-parameter Inverse Rayleigh distribution that incorporated a transformation of the Weibull distribution and Log-logistic as quantile function. The statistical properties such as quantile function, order statistic, monotone likelihood ratio property, hazard, reverse hazard functions, moments, skewness, kurtosis, and linear representation of the new proposed distribution were studied theoretically. The maximum likelihood estimators cannot be derived in an explicit form. So we employed the iterative procedure called Newton Raphson method to obtain the maximum likelihood estimators. The Bayes estimators for the scale and shape parameters for the WIR distribution under squared error, Linex, and Entropy loss functions are provided. The Bayes estimators cannot be obtained explicitly. Hence we adopted a numerical approximation method known as Lindley's approximation in other to obtain the Bayes estimators. Simulation procedures were adopted to see the effectiveness of different estimators. The applications of the new WIR distribution were demonstrated on three real-life data sets. Further results showed that the new WIR distribution performed credibly well when compared with five of the related existing skewed distributions. It was observed that the Bayesian estimates derived performs better than the classical method.

]]>Nurul Sima Mohamad Shariff and Waznatul Widad Mohamad Ishak

Retirement savings decision is related to the individual judgment on savings planning, and preparation for the retirement. Several factors may affect this decision towards retirement savings. Some of them are demographic factors and other determinants, such as financial knowledge and management, future expectation, social influences and risk tolerance. Due to this interest, this study aims to impact of such factors on retirement savings decision. Furthermore, this study will also discuss the retirement savings decision among Malaysians at different age groups. The data were collected through a survey strategy by using a set of questionnaires. The questions were divided into several sections on the demographic profile, Likert-scale questions on the factors, and the retirement savings decisions. The technique sampling used in this study is a random sampling with 385 respondents. As such, several statistical procedures will be utilized such as the reliability test, Kruskal-Wallis H test, and the ordered probit model. The results of this study found that age, financial knowledge and management, future expectation, and social influences were the significant determinants towards retirement savings decision in Malaysia.

]]>Harliza Mohd Hanif Daud Mohamad and Rosma Mohd Dom

The complexity of a method has been discussed in the decision-making area since complexity may impose some disadvantages such as loss of information and a high degree of uncertainty. However, there is no empirical justification to determine the complexity level of a method. This paper focuses on introducing a method of measuring the complexity of the decision-making method. In the computational area, there is an established method of measuring complexity named Big-O Notation. This paper adopts the method for determining the complexity level of the decision-making method. However, there is a lack of applying Big-O in the decision-making method. Applying Big-O in decision-making may not be able to differentiate the complexity level of two different decision-making methods. Hence, this paper introduces a Relative Complexity Index (RCI) to cater to this problem. The basic properties of the Relative Complexity Index are also discussed. After the introduction of the Relative Complexity Index, the method is implemented in Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method.

]]>Zahari Md Rodzi Abd Ghafur Ahmad Nur Sa’aidah Ismail Wan Normila Mohamad and Sarahiza Mohmad

Dual hesitant fuzzy set (DHFS) consists of two parts: membership hesitant function and non-membership hesitant function. This set supports more exemplary and flexible access to set degrees for each element in the domain and can address two types of hesitant in this situation. It can be considered a powerful tool for expressing uncertain information in the decision-making process. The function of z-score, namely z-arithmetic mean, z-geometric mean, and z-harmonic mean, has been proposed with five important bases, these bases are hesitant degree for dual hesitant fuzzy element (DHFE), DHFE deviation degree, parameter α (the importance of the hesitant degree), parameter β (the importance of the deviation degree) and parameter ϑ (the importance of membership (positive view) or non-membership (negative view). A comparison of the z-score with the existing score function was made to show some of their drawbacks. Next, the z-score function is then applied to solve multi-criteria decision making (MCDM) problems. To illustrate the proposed method's effectiveness, an example of MCDM specifically in pattern recognition has been shown.

]]>Oluremi Davies Ogun

The contents of this paper apply to researches in the fields of economics, statistics – physical or life sciences, other social sciences, accounting and finance, business management and mathematics – core and applied. First, I discussed the misconception and the implications thereof, inherent in the conventional practice of entering interest rates as natural or untransformed series in data analysis most especially, regression models. The trends and variabilities of both transformed and untransformed interest rate series were shown to be similar thereby enhancing the likelihood of similar performances in regressions. By extension therefore, the indicated conventional practice unnecessarily and unjustifiably precluded elasticity inference on the coefficients of interest rates and summing up to procedural inefficiency as an independent computation of elasticity became the only available option. Percentages were not the equivalence of percentage changes and thus only series in growth terms hence, percentage changes should be spared log transformation. Secondly, the paper stressed the imperative to avoid unwieldy and theory incongruent expressions in post preliminary data analysis, by flagging the idea that regression models, in particular, of the growth varieties, should as much as practicable, sync with the dictates of modern time series econometrics in the specification of final equations.

]]>R. Sivaraman

One of the Greatest mathematicians of all time, Gotfried Leibniz, introduced amusing triangular array of numbers called Leibniz's Harmonic triangle similar to that of Pascal's triangle but with different properties. I had introduced entries of Leibniz's triangle through Beta Integrals. In this paper, I have proved that the Beta Integral assumption is exactly same as that of entries obtained through Pascal's triangle. The Beta Integral formulation leads us to establish several significant properties related to Leibniz's triangle in quite elegant way. I have shown that the sum of alternating terms in any row of Leibniz's triangle is either zero or a Harmonic number. A separate section is devoted in this paper to prove interesting results regarding centralized Leibniz's triangle numbers including obtaining a closed expression, the asymptotic behavior of successive centralized Leibniz's triangle numbers, connection between centralized Leibniz's triangle numbers and Catalan numbers as well as centralized binomial coefficients, convergence of series whose terms are centralized Leibniz's triangle numbers. All the results discussed in this section are new and proved for the first time. Finally, I have proved two exceedingly important theorems namely Infinite Hockey Stick theorem and Infinite Triangle Sum theorem. Though these two theorems were known in literature, the way of proving them using Beta Integral formulation is quite new and makes the proof short and elegant. Thus, by simple re-formulation of entries of Leibniz's triangle through Beta Integrals, I have proved existing as well as new theorems in much compact way. These ideas will throw a new light upon understanding the fabulous Leibniz's number triangle.

]]>Anne M. Fernando Ana Vivas Barber and Sunmi Lee

Understanding the dynamics of Malaria can help in reducing the impact of the disease. Previous research proved that including animals in the human transmission model, or 'zooprophylaxis', is effective in reducing transmission of malaria in the human population. This model studies plasmodium vivax malaria and has variables for animal population and mosquito attraction to animals. The existing time-independent Malaria population ODE model is extended to time-dependent model with the differences explored. We introduce the seasonal mosquito population, a Gaussian profile based on data, as a variant for the previous models. The seasonal reproduction number is found using the next generation matrix, endemic and stability analysis is carried out using dynamical systems theory. The model includes short and long term human incubation periods and sensitivity analysis on parameters and all simulations are over three year period. Simulations show for each year larger peaks in the infected populations and seasonal reproduction number during the summer months and we analyze which parameters have more sensitivity in the model and in the seasonal reproduction number. Analysis provides conditions for disease free equilibrium (DFE) and the system is found to be locally asymptotically stable around the DFE when R_{0}<1, furthermore we find the uniqueness of the endemic equilibrium point. The sensitivity analysis for the parameters shows that the model was not sensitive to the exact values of the long or short term periods as it was to the average number of contacts between host and mosquito or rate of disease progression for mosquitoes. This model shows that inclusion of variable mosquito population informs how domestic animals in the human population can be more effectively used as a method of reducing the transmission of malaria. The most relevant contribution of this work is including the time evolution of mosquito population and simulations show how this feature affects human infection dynamics. An analytical expression for the endemic equilibrium point is provided for future work to establish conditions over which an epidemic may be prevented.

Kuntida Kawinwit Akapak Charoenloedmongkhon and Sanoe Koonprasert

Integral equations are essential tools in various areas of applied mathematics. A computational approach to solving an integral equation is important in scientific research. The Haar wavelet collocation method (HWCM) with operational matrices of integration is one famous method which has been applied to solve systems of linear integral equations. In this paper, an approximated analytical method based on the Haar wavelet collocation method is applied to the system of diffusion convection partial differential equations with initial and boundary conditions. This system determines the enzymatic glucose fuel cell with the chemical reaction rate of the Morrison equation. The enzymatic glucose fuel cell model describes the concentration of glucose and hydrogen ion that can be converted into energy. During the process, the model reduces to the linear integral equation system including computational Haar matrices. The computational Haar matrices can be computed by HWCM coding in the Maple program. Illustrated examples are provided to demonstrate the preciseness and effectiveness of the proposed method. The results are shown as numerical solutions of glucose and hydrogen ion.

]]>Juan Carlos Ferrando

If T is a (densely defined) self-adjoint operator acting on a complex Hilbert space H and I stands for the identity operator, we introduce the delta function operator at T. When T is a bounded operator, then is an operator-valued distribution. If T is unbounded, is a more general object that still retains some properties of distributions. We provide an explicit representation of in some particular cases, derive various operative formulas involving and give several applications of its usage in Spectral Theory as well as in Quantum Mechanics.

]]>Ida Kurnia Waliyanti Indah Emilia Wijayanti and M. Farchani Rosyid

Jordan ring is one example of the non-associative rings. We can construct a Jordan ring from an associative ring by defining the Jordan product. In this paper, we discuss the properties of non-associative rings by studying the properties of the Jordan rings. All of the ideals of a non-associative ring R are non-associative, except the ideal generated by the associator in R. Hence, a quotient ring can be constructed, where is the ideal generated by associators in R. The fundamental theorem of the homomorphism ring can be applied to the non-associative rings. By a little modification, we can find that is isomorphic to . Furthermore, we define a module over a non-associative ring and investigate its properties. We also give some examples of such modules. We show if M is a module over a non-associative ring R, then M is also a module over if is contained in the annihilator of R. Moreover, we define the tensor product of modules over a non-associative ring. The tensor product of the modules over a non-associative ring is commutative and associative up to isomorphism but not element by element.

]]>Jackel Vui Lung Chew Jumat Sulaiman and Andang Sunarto

A porous medium equation is a nonlinear parabolic partial differential equation that presents many physical occurrences. The solutions of the porous medium equation are important to facilitate the investigation on nonlinear processes involving fluid flow, heat transfer, diffusion of gas-particles or population dynamics. As part of the development of a family of efficient iterative methods to solve the porous medium equation, the Half-Sweep technique has been adopted. Prior works in the existing literature on the application of Half-Sweep to successfully approximate the solutions of several types of mathematical problems are the underlying motivation of this research. This work aims to solve the one-dimensional porous medium equation efficiently by incorporating the Half-Sweep technique in the formulation of an unconditionally-stable implicit finite difference scheme. The noticeable unique property of Half-Sweep is its ability to secure a low computational complexity in computing numerical solutions. This work involves the application of the Half-Sweep finite difference scheme on the general porous medium equation, until the formulation of a nonlinear approximation function. The Newton method is used to linearize the formulated Half-Sweep finite difference approximation, so that the linear system in the form of a matrix can be constructed. Next, the Successive Over Relaxation method with a single parameter was applied to efficiently solve the generated linear system per time step. Next, to evaluate the efficiency of the developed method, deemed as the Half-Sweep Newton Successive Over Relaxation (HSNSOR) method, the criteria such as the number of iterations, the program execution time and the magnitude of absolute errors were investigated. According to the numerical results, the numerical solutions obtained by the HSNSOR are as accurate as those of the Half-Sweep Newton Gauss-Seidel (HSNGS), which is under the same family of Half-Sweep iterations, and the benchmark, Newton-Gauss-Seidel (NGS) method. The improvement in the numerical results produced by the HSNSOR is significant, and requires a lesser number of iterations and a shorter program execution time, as compared to the HSNGS and NGS methods.

]]>Jamal Salah

In 1859, Bernhard Riemann, a German mathematician, published a paper to the Berlin Academy that would change mathematics forever. The mystery of prime numbers was the focus. At the core of the presentation was indeed a concept that had not yet been proven by Riemann, one that to this day baffles mathematicians. The way we do business could have been changed if the Riemann hypothesis holds true, which is because prime numbers are the key element for banking and e-commerce security. It will also have a significant influence, impacting quantum mechanics, chaos theory, and the future of computation, on the cutting edge of science. In this article, we look at some well-known results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. Initially, we observe omitting a logical undefined term in the integral representation of Zeta function by the means of Gamma function. For that we propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: s= 1/2 by assuming that ζ(s) and ζ(1-s) simultaneously equal zero. Consequently, we conditionally prove Riemann Hypothesis.

]]>Ftameh Khaled and Pah Chin Hee

It is widely recognized that the theory of quadratic stochastic operator frequently arises due to its enormous contribution as a source of analysis for the investigation of dynamical properties and modeling in diverse domains. In this paper, we are motivated to construct a class of quadratic stochastic operators called mixing quadratic stochastic operators generated by geometric distribution on infinite state space . We also study regularity of such operators by investigating of the limit behavior for each case of the parameter. Some of non-regular cases proved for a new definition of mixing operators by using the shifting definition, where the new parameters satisfy the shifted conditions. A mixing quadratic stochastic operator was established on 3-partitions of the state space and considered for a special case of the parameter Ɛ. We found that the mixing quadratic stochastic operator is a regular transformation for and is a non-regular for . Also, the trajectories converge to one of the fixed points. Stability and instability of the fixed points were investigated by finding of the eigenvalues of Jacobian matrix at these fixed points. We approximate the parameter Ɛ by the parameter , where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy . We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular.

]]>Norshakila Abd Rasid Zarina Bibi Ibrahim Zanariah Abdul Majid and Fudziah Ismail

This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one.

]]>Samingun Handoyo Ying-Ping Chen Gugus Irianto and Agus Widodo

The aim of the research is to find the best performance both of logistic regression and linear discriminant which their threshold uses some various values. The performance tools used for evaluating classifier model are confusion matrix, precision-recall, F1 score and receiver operation characteristic (ROC) curve. The Audit-risk data set are used for the implementation of the proposed method. The screening data and dimension reduction by using principal component analysis (PCA) are the first step that must be conducted before the data are divided into the training and testing set. After the training process for obtaining the classifier model parameters has been completed, the calculation of performance measures is done only on the testing set where the various constants are added to the threshold value of both classifier models. The logistic regression classifier has the best performance of 94% on the precision-recall, 91.7% on the F1-score, and 0.906 on the area under curve (AUC) where the threshold values are on the interval between 0.002 and 0.018. On the other hand, the linear discriminant classifier has the best performance when the threshold value is 0.035 and its performance value is respectively the precision-recall of 94%, the F1-score of 91.7%, and the AUC of 0.846.

]]>Dwi Sulistyaningsih Eko Andy Purnomo and Purnomo

This study is focused on investigating errors made by students and the various causal factors in working on trigonometry problems by applying sine and cosine rules. Samples were taken randomly from high school students. Data were collected in two ways, namely a written test that was referred to Polya's strategy and interviews with students who made mistakes. Students' errors were analyzed with the Newman concept. The results show that all types of errors occurred with a distribution of 3.83, 19.15, 24.74, 24.89 and 27.39% for reading errors (RE), comprehension error (CE), transformation errors (TE), process skill errors (PSE), and encoding errors (EE), respectively. The RE, CE, TE, PSE, and EE are marked by errors in reading symbols or important information, misunderstanding information and not understanding what is known and questioned, cannot change problems into mathematical models and also incorrectly use signs in arithmetic operations, student inaccuracies in the process of answering and also their lack of understanding in fraction operations, and the inability to deduce answers, respectively. An anomaly occurs because it turns out students who have medium trigonometry achievements make more mistakes than students who have low achievement.

]]>Evgjeni Xhafaj Daniela Halidini Qendraj Alban Xhafaj and Etleva Halidini

The study explores the number of factors that affect the use of Google Classroom in Albanian universities, by using the methodological developments of partial least squares structural equation modelling technique (PLS– SEM). This technique has been used because it allows flexibility in modelling the relationship between constructs (or factors) and in exploring theoretical concepts. An alternative model is introduced by extending the Unified Theory of Acceptance and Use of Technology (UTAUT2) and by integrating new relation between constructs. Our data are from a study of 528 students from 4 Albanian universities during the year 2020. Using Importance-Performance Matrix Analysis (IPMA) our analysis suggest that Habit is the construct that has the greatest importance in determining the Behavioral Intention towards Google Classroom, whereas Behavioral Intention has the greatest importance related to Use Behavior of Google Classroom. The results of the study show that Habit, and Hedonic Motivation have a greater impact on the Behavioral Intention to use Google Classroom. Additionally, we find that all constructs of the alternative model have an important influence to Behavioral Intention to Google Classroom and explain 65.3 per cent of its variance. This study will help the Higher Educations Institutions in assessing the factors that influence the use of Google Classroom, in such a way that this platform should be used as a support tool in the future.

]]>Viktor Pandra Badrun Kartowagiran and Sugiman

The aims of this research are: 1) producing the test instrument of mathematics skill on elementary school which is valid and reliable, 2) finding out the characteristics of the test instrument of mathematics skill on elementary school. The instrument test development in this research uses the development model of Wilson, Oriondo and Antonio which is modified. The number of testing sample in this research is 160 students in each class. This research results: 1) the validity index of aiken v is 0.979 in grade IV and 0.988 in grade V. The coefficient of instrument skill in class IV and V are 0.883 and 0.954. 2) the compatibility model in this research is it is suitable for 1PL model or parameter b (difficulty level). The result of parameter analysis of test item in class IV and V, shows that the overall item is in good category which is between -2 to 2. The case indicates that the overall item is accepted and reliable to be used for measuring the development of mathematics skill of elementary school students.

]]>A. A. Dahalan and J. Sulaiman

The computational technique has become a significant area of study in physics and engineering. The first method to evaluate the problems numerically was a finite difference. In 2002, a computational approach, an explicit finite difference technique, was used to overcome the fuzzy partial differential equation (FPDE) based on the Seikkala derivative. The application of the iterative technique, in particular the Two Parameter Alternating Group Explicit (TAGE) method, is employed to resolve the finite difference approximation resulting after the fuzzy heat equation is investigated in this article. This article broadens the use of the TAGE iterative technique to solve fuzzy problems due to the reliability of the approaches. The development and execution of the TAGE technique towards the full-sweep (FS) and half-sweep (HS) techniques are also presented. The idea of using the HS scheme is to reduce the computational complexity of the iterative methods by nearly/more than half. Additionally, numerical outcomes from the solution of two experimental problems are included and compared with the Alternating Group Explicit (AGE) approaches to clarify their feasibility. In conclusion, the families of the TAGE technique have been used to overcome the linear system structure through a one-dimensional fuzzy diffusion (1D-FD) discretization using a finite difference scheme. The findings suggest that the HSTAGE approach is surpassing in terms of iteration counts, time taken, and Hausdorff distance relative to the FSTAGE and AGE approaches. It demonstrates that the number of iterations for HSTAGE approach has decreased by approximately 71.60-72.95%, whereas for the execution time, the implementation of HSTAGE method is between 74.05-86.42% better. Since TAGE is ideal for concurrent processing, this method has been seen as the key benefit as it consumes sets of independent tasks that can be performed at the same time. The ability of the suggested technique is projected to be useful for the advanced exploration in solving any multi-dimensional FPDEs.

]]>Restituto M. Llagas Jr.

Studying mathematics comprises acquiring a positive disposition toward mathematics and seeing mathematics as an effective way of looking at real-life situations. This study aimed to correlate the disposition to Mathematics of prospective Filipino teachers to some teacher-related variables. The participants were the prospective Filipino teachers at the University of Northern Philippines (UNP) and at the Divine Word College of Vigan (DWCV). Two sets of instruments were utilized in the study – the self-report questionnaire and the Mathematics Dispositional Functioning Inventory developed by Beyers [1]. Frequency and percentage, weighted mean, and chi-square were utilized for data analysis. Results show that the overall disposition to mathematics of the participants is "Positive". The cognitive, affective, and conative aspects received a positive disposition. However, some items show an uncertain disposition to mathematics. The participants' profile variables have no significant relationship with their cognitive and conative disposition to mathematics. A training plan was conceptualized to provide information on the results of the study, to enhance the awareness and understanding of dispositions, to equip appropriate methods in solving mathematical problems, and to provide enrichment activities that will foster a positive disposition to mathematics and consequently will improve prospective teachers' and students' performance. Teachers are influential to the development of the students of effective ways of learning, doing, and thinking about mathematics. Understanding how attitudes are learned to establish an association between the teacher's disposition and students' attitude and performance. Thus, fostering dispositions to mathematics through training improves prospective Filipino teachers' and students' performance.

]]>A. A. Aminu S. E. Olowo I. M. Sulaiman N. Abu Bakar and M. Mamat

Max-plus algebra is a discrete algebraic system developed on the operations max () and plus (), where the max and plus operations are defined as addition and multiplication in conventional algebra. This algebraic structure is a semi-ring with its elements being real numbers along with ε=-∞ and e=0. On the other hand, the synchronized discrete event problem is a problem in which an event is scheduled to meet a deadline. There are two aspects of this problem. They include the events running simultaneously and the completion of the lengthiest event at the deadline. A recent survey on max-plus linear algebra shows that the operations max () and plus () play a significant role in modeling of human activities. However, numerous studies have shown that there are very limited literatures on the application of the max-plus algebra to real-life problems. This idea motivates the basic algebraic results and techniques of this research. This paper proposed the discrepancy method of max-plus for solving n×n system of linear equations with n≤n, and further show that an nxn linear system of equations will have either a unique solution, an infinitely many solutions or no solution whiles nxn linear system of equations has either an infinitely many solutions or no solution in (). Also, the proposed concept was extended to the job-shop problem in a synchronized event. The results obtained have shown that the method is very efficient for solving n×n system of linear equations and is also applicable to job-shop problems.

]]>Pakwan Riyapan Sherif Eneye Shuaib Arthit Intarasit and Khanchit Chuarkham

Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S; V; I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods

]]>B. K. Buzdov

When cooling living biological tissue (active, non-inert medium), cryomedicine uses cryo-instruments with various forms of cooling surface. Cryoinstruments are located on the surface of biological tissue or completely penetrate into it. With a decrease in the temperature of the cooling surface, an unsteady temperature field appears in the tissue, which in the general case depends on three spatial coordinates and time. To date, there are a large number of scientific publications that consider mathematical models of cryodestruction of biological tissue. However, in the overwhelming majority of them, the Pennes equation (or some of its modifications) is taken as the basis of the mathematical model, from which the linear nature of the dependence of heat sources of biological tissue on the desired temperature field is visible. This character of the dependence does not allow one to describe the actually observed spatial localization of heat. In addition, Pennes' model does not take into account the fact that the freezing of the intercellular fluid occurs much earlier than the freezing of the intracellular fluid and the heat corresponding to these two processes is released at different times. In the proposed work, a new mathematical model of cooling and freezing of living biological tissue are built with a flat rectangular applicator located on its surface. The model takes into account the above features and is a three-dimensional boundary-value problem of the Stefan type with nonlinear heat sources of a special type and has applications in cryosurgery. A method is proposed for the numerical study of the problem posed, based on the use of locally one-dimensional difference schemes without explicitly separating the boundary of the influence of cold and the boundaries of the phase transition. The method was previously successfully tested by the author in solving other two-dimensional problems arising in cryomedicine.

]]>J. Uma Maheswari A. Anbarasan and M. Ravichandran

The notion of complex valued metric spaces proved the common fixed point theorem that satisfies rational mapping of contraction. In the contraction mapping theory, several researchers demonstrated many fixed-point theorems, common fixed-point theorems and coupled fixed-point theorems by using complex valued metric spaces. The idea of b-metric spaces proved the fixed point theorem by the principle of contraction mapping. The notion of complex valued b-metric spaces, and this metric space was the generalization of complex valued metric spaces. They explained the fixed point theorem by using the rational contraction. In the metric spaces, we refer to this metric space as a quasi-metric space, the symmetric condition d(x, y) = d(y, x) is ignored. Metric space is a special kind of space that is quasi-metric. The Quasi metric spaces were discussed by many researchers. Banach introduced the theory of contraction mapping and proved the theorem of fixed points in metric spaces. We are now introducing the new notion of complex quasi b-metric spaces involving rational type contraction which proved the unique fixed point theorems with continuous as well as non-continuous functions. Illustrate this with example.

]]>Vipin Verma and Mannu Arya

Many researchers have been working on recurrence relation which is an important topic not only in mathematics but also in physics, economics and various applications in computer science. There are many useful results on recurrence relation sequence but there main problem to find any term of recurrence relation sequence we need to find all previous terms of recurrence relation sequence. There were many important theorems obtained on recurrence relations. In this paper we have given special identity for generalized kth order recurrence relation. These identities are very useful for finding any term of any order of recurrence relation sequence. Authors define a special formula in this paper by this we can find direct any term of a recurrence relation sequence. In this recurrence relation sequence to find any terms we need to find all previous terms so this result is very important. There is important property of a relation between coefficients of recurrence relation terms and roots of a polynomial for second order relation but in this paper, we gave this same property of recurrence relation of all higher order recurrence relation. So finally, we can say that this theorem is valid all order of recurrence relation only condition that roots are distinct. So, we can say that this paper is generalization of property of a relation between coefficients of recurrence relation terms and roots of a polynomial. Theorem: - Let C_{1} and C_{2} are arbitrary real numbers and suppose the equation (1) Has X_{1} and X_{2} are distinct roots. Then the sequence is a solution of the recurrence relation (2) . For n= 0, 1, 2 …where β_{1} and β_{2} are arbitrary constants. Proof: - First suppose that of type we shall prove is a solution of recurrence relation (2). Since X_{1}, X_{2} and X_{3} are roots of equation (1) so all are satisfied equation (1) so we have, . Consider . This implies . So the sequence is a solution of the recurrence relation. Now we will prove the second part of theorem. Let is a sequence with three . Let . So (3). (4). Multiply by X_{1} to (3) and subtracts from (4). We have similarly we can find . So we can say that values of β_{1} and β_{2} are defined as roots are distinct. So non- trivial values ofβ_{1} and β_{2} can find and we can say that result is valid. Example: Let be any sequence such that n≥3 and a_{0}=0, a_{1}=1, a_{2}=2. Then find a_{10} for above sequence. Solution: The polynomial of above sequence is . Solving this equation we have roots are 1, 2, and 3 using above theorem we have (7). Using a_{0}=0, a_{1}=1, a_{2}=2 in (7) we have β_{1}+β_{2}+β_{3}=0 (8). β_{1}+2β_{2}+3β_{2}=1 (9).β_{1}+4β_{2}+9β_{3}=2 (10) Solving (8), (9) and (10) we have , , . This implies . Now put n=10 we have a_{10}=-27478. Recurrence relation is a very useful topic of mathematics, many problems of real life may be solved by recurrence relations, but in recurrence relation there is a major difficulty in the recurrence relation. If we want to find 100th term of sequence, then we need to find all previous 99 terms of given sequence, then we can get 100th term of sequence but above theorem is very useful if coefficients of recurrence relation of given sequence satisfies the condition of the above theorem, then we can apply above theorem and we can find direct any term of sequence without finding all previous terms.

Nik Muhammad Farhan Hakim Nik Badrul Alam Nazirah Ramli and Norhuda Mohammed

Fuzzy time series is a powerful tool to forecast the time series data under uncertainty. Fuzzy time series was first initiated with fuzzy sets and then generalized by intuitionistic fuzzy sets. The intuitionistic fuzzy sets consider the degree of hesitation in which the degree of non-membership is incorporated. In this paper, a fuzzy set time series forecasting model based on intuitionistic fuzzy sets via delegation of hesitancy degree to the major grade de-i-fuzzification approach was developed. The proposed model was implemented on the data of student enrollments at the University of Alabama. The forecasted output was obtained using the fuzzy logical relationships of the output, and the performance of the forecasted output was compared with the fuzzy time series forecasting model based on fuzzy sets using the mean square error, root mean square error, mean absolute error, and mean absolute percentage error. The results showed that the forecasting model based on induced fuzzy sets from intuitionistic fuzzy sets performs better compared to the fuzzy time series forecasting model based on fuzzy sets.

]]>Auni Aslah Mat Daud

An important part of the study of epidemic models is the local stability analysis of the equilibrium points. The linear algebra method which is commonly employed is the well-known Routh-Hurwitz criteria. The criteria give necessary and sufficient conditions for all of the roots of the characteristic polynomial to be negative or have negative real parts. To date, there are no epidemic models in the literature which employ Lienard-Chipart criteria. This note recommends an alternative linear algebra method namely Lienard-Chipart criteria, to significantly simplify the local stability analysis of epidemic models. Although Routh-Hurwitz criteria are a correct method for local stability analysis, Lienard-Chipart criteria have advantages over Routh-Hurwitz criteria. Using Lienard-Chipart criteria, only about half of the Hurwitz determinants inequalities are required, with the remaining conditions of each set concern with only the sign of the alternate coefficients of the characteristic polynomial. The Lienard-Chipart criteria are especially useful for polynomials with symbolic coefficients, as the determinants are usually significantly more complicated than original coefficients as degree of the polynomial increases. Lienard-Chipart criteria and Routh-Hurwitz criteria have similar performance for systems of dimension five or less. Theoretically, for systems of dimension higher than five, verifying Lienard-Chipart criteria should be much easier than verifying Routh-Hurwitz criteria and the advantage of Lienard-Chipart criteria may become clear. Examples of local stability analysis using Lienard-Chipart criteria for two recently proposed models are demonstrated to show the advantages of simplified Lienard-Chipart criteria over Routh-Hurwitz criteria.

]]>Muhammad Ammar Shafi Mohd Saifullah Rusiman and Siti Nabilah Syuhada Abdullah

The colon and rectum is the final portion of the digestive tube in the human body. Colorectal cancer (CRC) occurs due to bacteria produced from undigested food in the body. However, factors and symptoms needed to predict tumor size of colorectal cancer are still ambiguous. The problem of using linear regression arises with the use of uncertain and imprecise data. Since the fuzzy set theory's concept can deal with data not to a precise point value (uncertainty data), this study applied the latest fuzzy linear regression to predict tumor size of CRC. Other than that, the parameter, error and explanation for the both models were included. Furthermore, secondary data of 180 colorectal cancer patients who received treatment in general hospital with twenty five independent variables with different combination of variable types were considered to find the best models to predict the tumor size of CRC. Two models; fuzzy linear regression (FLR) and fuzzy linear regression with symmetric parameter (FLRWSP) were compared to get the best model in predicting tumor size of colorectal cancer using two measurement statistical errors. FLRWSP was found to be the best model with least value of mean square error (MSE) and root mean square error (RMSE) followed by the methodology stated.

]]>Navya Pratyusha M Rajyalakshmi K Apparao B V and Charankumar G

Pittsburgh Sleep Quality Index (PSQI) Scoring (Buysse et al. 1989) is a powerful method to measure the sleep quality index based on the scores of various factors namely duration of sleep, sleep disturbance, sleep latency, day dysfunction due to sleepiness, sleep efficiency, need meds to sleep and overall sleep quality. Mainly we focused on the smart phones' usage and its impact on the quality of sleep at the bed time. Many studies have proved that the usage of smart phones at bed time affects the sleep quality, health and productivity. In the present study, we have collected data randomly from the middle-aged adults and observed the relation between gender and the quality of sleep using phi coefficient. It is clearly observed that as we move from males to females, we move negatively from good sleep quality to poor sleep quality. It indicates that males have poor sleep quality than females. We also performed an analysis of variance to test the hypothesis that there is any association between the smart phones' usage and its impact on quality of sleep at bed time.

]]>Leontiev V. L.

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

]]>I M Sulaiman M Mamat M Y Waziri U A Yakubu and M Malik

Conjugate Gradient (CG) method is the most prominent iterative mathematical technique that can be useful for the optimization of both linear and non-linear systems due to its simplicity, low memory requirement, computational cost, and global convergence properties. However, some of the classical CG methods have some drawbacks which include weak global convergence, poor numerical performance both in terms of number of iterations and the CPU time. To overcome these drawbacks, researchers proposed new variants of the CG parameters with efficient numerical results and nice convergence properties. Some of the variants of the CG method include the scale CG method, hybrid CG method, spectral CG method, three-term CG method, and many more. The hybrid conjugate gradient (CG) algorithm is among the efficient variant in the class of the conjugate gradient methods mentioned above. Some interesting features of the hybrid modifications include inherenting the nice convergence properties and efficient numerical performance of the existing CG methods. In this paper, we proposed a new hybrid CG algorithm that inherits the features of the Rivaie et al. (RMIL*) and Dai (RMIL+) conjugate gradient methods. The proposed algorithm generates a descent direction under the strong Wolfe line search conditions. Preliminary results on some benchmark problems show that the proposed method efficient and promising.

]]>Deepshikha Deka Bhanita Das Bhupen K Baruah and Bhupen Baruah

Research, development and extensive use of generalized form of distributions in order to analyze and modeling of applied sciences research data has been growing tremendously. Weibull and Fréchet distribution are widely discussed for reliability and survival analysis using experimental data from physical, chemical, environmental and engineering sciences. Both the distributions are applicable to extreme value theory as well as small and large data sets. Recently researchers develop several probability distributions to model experimental data as these parent models are not adequate to fit in some experiments. Modified forms of the Weibull distribution and Fréchet distribution are more flexible distributions for modeling experimental data. This article aims to introduce a generalize form of Weibull distribution known as Fréchet-Weibull Distribution (FWD) by using the T-X family which extends a more flexible distribution for modeling experimental data. Here the pdf and cdf with survival function [S(t)], hazard rate function [h(t)] and asymptotic behaviour of pdf and survival function and the possible shapes of pdf, cdf, S(t) and h(t) of FWD have been studied and the parameters are estimated using maximum livelihood method (MLM). Some statistical properties of FWD such as mode, moments, skewness, kurtosis, variation, quantile function, moment generating function, characteristic function and entropies are investigated. Finally the FWD has been applied to two sets of observations from mechanical engineering and shows the superiority of FWD over other related distributions. This study will provide a useful tool to analyze and modeling of datasets in Mechanical Engineering sciences and other related field.

]]>S. Padmashini and S. Pethanachi Selvam

Domination in graphs is to dominate the graph G by a set of vertices , vertex set of G) when each vertex in G is either in D or adjoining to a vertex in D. D is called a perfect dominating set if for each vertex v is not in D, which is adjacent to exactly one vertex of D. We consider the subset C which consists of both vertices and edges. Let denote the set of all vertices V and the edges E of the graph G. Then is said to be a corporate dominating set if every vertex v not in is adjacent to exactly one vertex of , where the set P consists of all vertices in the vertex set of an edge induced sub graph , (E_{1} a subset of E) such that there should be maximum one vertex common to any two open neighborhood of different vertices in V(G[E_{1}]) and Q, the set consists of all vertices in the vertex set V_{1}, a subset of V such that there exists no vertex common to any two open neighborhood of different vertices in V_{1}. The corporate domination number of G, denoted by , is the minimum cardinality of elements in C. In this paper, we intend to determine the exact value of corporate domination number for the Cartesian product of the Cycle and Path .

Oluwaseun Adeyeye and Zurni Omar

Various algorithms have been proposed for developing block methods where the most adopted approach is the numerical integration and collocation approaches. However, there is another conventional approach known as the Taylor series approach, although it was utilised at inception for the development of linear multistep methods for first order differential equations. Thus, this article explores the adoption of this approach through the modification of the aforementioned conventional Taylor series approach. A new methodology is then presented for developing block methods, which is a more accurate method for solving second order ordinary differential equations, coined as the Modified Taylor Series (MTS) Approach. A further step is taken by presenting a generalised form of the MTS Approach that produces any k-step block method for solving second order ordinary differential equations. The computational complexity of this approach after being generalised to develop k-step block method for second order ordinary differential equations is calculated and the result shows that the generalised algorithm involves less computational burden, and hence is suitable for adoption when developing block methods for solving second order ordinary differential equations. Specifically, an alternate and easy-to-adopt approach to developing k-step block methods for solving second order ODEs with fewer computations has been introduced in this article with the developed block methods being suitable for solving second order differential equations directly.

]]>Jasmine Lee Jia Min and Syafrina Abdul Halim

Increased flood risk is recognized as one of the most significant threats in most parts of the world, resulting in severe flooding events which have caused significant property and human life losses. As there is an increase in the number of extreme flash flood events observed in Klang Valley, Malaysia recently, this paper focuses on modelling extreme daily rainfall within 30 years from year 1975 toyear 2005 in Klang Valley using generalized extreme value (GEV) distribution. Cyclic covariate is introduced in the distribution because of the seasonal rainfall variation in the series. One stationary (GEV) and three nonstationary models (NSGEV1, NSGEV2, and NSGEV3) are constructed to assess the impact of cyclic covariates on the extreme daily rainfall events. The better GEV model is selected using Akaike's information criterion (AIC), bayesian information criterion (BIC) and likelihood ratio test (LRT). The return level is then computed using the selected fitted GEV model. Results indicate that the NSGEV3 model with cyclic covariate trend presented in location and scale parameters provides better fits the extreme rainfall data. The results showed the capability of the nonstationary GEV with cyclic covariates in capturing the extreme rainfall events. The findings would be useful for engineering design and flood risk management purposes.

]]>Nurhaida Subanar Abdurakhman and Agus Maman Abadi

This article deals with problems of detecting abrupt changes in time series ba Change Point Model (CPM) framework. We propose a fuzzification in a Fuzzy Time Series (FTS) model to eliminate a trend in a contaminated dependent series. The independent residuals are then inputed on the CPM method. In simulating an abrupt change, an ARIMA(1,1,1) and variance of the model are considered. The abrupt change is modelled as an AO (Additive Outlier) type of outliers. The minimum weight or breaksize of the abrupt change is defined based on the ARIMA variance formulated in this article. The percentage of uncorrelated residuals obtained by the FTS model and the percentage of correct detection of the proposed procedure are shown by simulation. The proposed detecting algorithm is implemented to detect abrupt changes in monthly tourism series in literature, i.e., in Taiwan and in Bali. The first series shows a slowly increasing trend with one abrupt change while the second series exhibits not only a slowly increasing trend but also a strong seasonal pattern with two abrupt changes. For comparison, we detect the changes in the empirical examples on an existing automatic detection procedure using tso package in R. For the first example, the results show that both detecting procedures give exactly a similar location of one change point where the package recognises it as an AO type of outliers. The abrupt change is related to the period of SARS outbreak in Taiwan. On the second example, the proposed procedure locates 4 change points which form two locations of changes, i.e., the first two change points are within 2 time points so do the last two change points. The locations are closed to times of Bali Bombing events. Meanwhile, the automatic procedure recognizes only one AO outlier on the series.

]]>Kusno

Formulation of developable patches is beneficial for modeling of the plate-metal sheet in the based-metal-industries objects. Meanwhile, installing the developable patches on a frame of the items and making a hole on these objects surface still need some practical techniques for developing. For these reasons, this research aims to introduce some methods for fitting a curve segment, cutting the developable patches, and adjusting their formulas. Using these methods can design various profile shapes of rubber filer installed on a frame of the objects and create a fissure or hole on the patches' surface. The steps are as follows. First, we define the planes containing the patches' generatrixes and orthogonal to the boundary curves. Then, it fits the Hermite and Bézier curve, via arranging some control points data on these planes, to model the rubber filler shapes. Second, we numerically evaluate a method for cutting the patches with a plane and adjusting the patches' form by modifying their formula from a linear interpolation form into a combination of curve and vectors forms. As a result, it can present some equations and procedures for plotting required curves, cutting surfaces, and modifying the extensible or narrowable shape of Hermite patches. These methods offer some advantages and contribute to designing the based-metal-sheets' object surfaces, especially modeling various forms of rubber filer profiles installed on a frame of the objects and making hole shapes on the plate-metal sheets.

]]>Viliam Ďuriš

Various problems in the real world can be viewed as the Constraint Satisfaction Problem (CSP) based on several mathematical principles. This paper is a guideline for complete automation of the Timetable Problem (TTP) formulated as CSP, which we are able to solve algorithmically, and so the advantage is the possibility to solve the problem on a computer. The theory presents fundamental concepts and characteristics of CSP along with an overview of basic algorithms used in terms of its solution and forms the TTP as CSP and delineates the basic properties and requirements to be met in the timetable. The theory in our paper is mostly based on the Jeavons, Cohen, Gyssens, Cooper, and Koubarakis work, on the basis of which we've constructed a computer programme, which verifies the validity and functionality of the Constraint satisfaction method for solving the Timetable Problem. The solution of the TTP, which is characterized by its basic characteristics and requirements, was implemented by a tree-based search algorithm to a program and our main contribution is an algorithmic verification of constraints abilities and reliability when solving a TTP by means of constraints. The created program was also used to verify the time complexity of the algorithmic solution.

]]>Suparman Abdellah Salhi and Mohd Saifullah Rusiman

Moving average (MA) is a time series model often used for pattern forecasting and recognition. It contains a noise that is often assumed to have a Gaussian distribution. However, in various applications, noise often does not have this distribution. This paper suggests using Laplacian noise in the MA model, instead. The comparison of Gaussian and Laplacian noises was also investigated to ascertain the right noise for the model. Moreover, the Bayesian method was used to estimate the parameters, such as the order and coefficient of the model, as well as noise variance. The posterior distribution has a complex form because the parameters are concerened with the combination of spaces of different dimensions. Therefore, to overcome this problem, the Markov Chain Monte Carlo (MCMC) reversible jump algorithm is adopted. A simulation study was conducted to evaluate its performance. After it has worked properly, it was applied to model human heart rate data. The results showed that the MCMC algorithm can estimate the parameters of the MA model. This was developed using Laplace distributed noise. Moreover, when compared with the Gaussian, the Laplacian noise resulted in a higher order model and produced a smaller variance.

]]>Wilhemina Adoma Pels Atinuke Olusola Adebanji and Sampson Twumasi-Ankrah

The study focused on the Generalized Pareto Distribution (GPD) under the Peak Over Threshold approach (POT). Twenty-one estimation methods were considered for extreme value modeling and their performances were compared. Our goal is to identify the best method in various conditions by the use of a systematic simulation study. Some other estimators which were initially not created under the POT framework (NON-POT) were also compared concurrently with the ones under the POT framework. The simulation results under varying shape parameters showed the Zhang Estimator as "best" in performance for NON-POT in estimating both the shape and scale parameter for heavy-tailed cases. In the POT framework, the Zhang Estimator again performed "best" in estimating very heavy tails for the shape and very short tails for the scale regardless of the value of the scale parameter. When varying sample size, under the NON-POT framework, the Zhang estimator performed as "best" heavy-tailed whiles for the POT framework, the Pickands Estimator was "best" in performance at estimating the shape parameter for large sample sizes and the Zhang, small sample sizes.

]]>Yakhshiboev M. U.

The case of one-dimensional and multidimensional non-convolutional integral operators in Lebesgue spaces is considered in this paper. The convergence in the norm and almost everywhere of non-convolution integral operators in Lebesgue spaces was insufficiently studied. The kernels of non-convolutional integral operators do not need to have a monotone majorant, so the well-known results on the convergence almost everywhere of convolutional averages are not applicable here. The kernels of nonconvolutional integral operators take into account different behaviors at and depending on (which is important in applications) and cover the situation in the particular case of convolutional and non-convolutional integral operators. We are interested in the behavior of function as . Theorems on convergence almost everywhere in the case of one-dimensional and multidimensional nonconvolution integral operators in Lebesgue spaces are proved. The theorems proved are more general ones (including for convolutional integral operators) and cover a wide class of kernels.

]]>Vladimir A. Skorokhodov

The problem of reachability on graphs with restriction is studied. Such restrictions mean that only those paths that satisfy certain conditions are valid paths on the graph. Because of this, for classical optimization problems one has to consider only a subset of feasible paths on the graph, which significantly complicates their solution. Reachability constraints arise naturally in various applied problems, for example, in the problem of navigation in telecommunication networks with areas of strong signal attenuation or when modeling technological processes in which there is a condition for the order of actions or the compatibility of operations. General concepts of a graph with non-standard reachability and a valid path on it are introduced. It is shown that the classical graphs, as well as graphs with restrictions on passing through the selected arcs subsets are special cases of graphs with non-standard reachability. General approach to solving the shortest path problem on a graph with non-standard achievability is developed. This approach consists in constructing an auxiliary graph and reducing the shortest path problem on a graph with non-standard reachability to a similar problem on an auxiliary graph. The theorem on the correspondence of the paths of the original and auxiliary graphs is proved.

]]>E. N. Sinyukova and O. L. Chepok

It is well known that concepts of a geodesic line and a geodesic mapping are among the most fundamental concepts of classical theory of Riemannian spaces. In geometry, concept of Riemannian space has been formed as a generalization of the concept of a smooth surface in a three-dimensional Euclidean space. It has turned out to be possible to extend to Riemannian space the concept of a geodesic point of a curve and to represent a geodesic line of Riemannian space as a curve that consists exclusively of geodesic points. The fact has allowed understanding not only the local but also the global character of basic equations of geodesic mappings' theory of Riemannian spaces that have been originally received as a result of local investigations. An example of the global solution of the so-called new form of basic equations in the theory of geodesic mappings of Riemannian spaces is built in the article. Sphere that is considered as a subset of Euclidean space , forms its topological background. Investigations are based on the concept of equidistant Riemannian space. They are carried out according to the atlas that consists of two charts, obtained with the help of a stereographic projection.

]]>Adejumo T. Joel. Omonijo D. Ojo Owolabi A. Timothy Okegbade A. Ibukun Odukoya A. Jonathan and Ayedun C. Ayedun

Over the years, non-parametric test statistics have been the only solution to solve data that do not follow a normal distribution. However, giving statistical interpretation used to be a great challenge to some researchers. Hence, to overcome these hurdles, another test statistics was proposed called Rank transformation test statistics so as to close the gap between parametric and non-parametric test statistics. The purpose of this study is to compare the conclusion statement of Rank transformation test statistics with its equivalent non parametric test statistics in both one and two samples problems using real-life data. In this study, (2018/2019) Post Unified Tertiary Matriculation Examinations (UTME) results of prospective students of Ladoke Akintola University of Technology (LAUTECH) Ogbomoso across all faculties of the institution were used for the analysis. The data were subjected to nonparametric test statistics which include; Asymptotic Wilcoxon sign test and Wilcoxon sum Rank (both Asymptotic and Distribution) using Statistical Packages for Social Sciences (SPSS). In the same vein, R-statistical programming codes were written for Rank Transformation test statistics. Their P-values were extracted and compared with each other with respect to the pre-selected alpha level (α) = 0.05. Results in both cases revealed that there is a significant difference in the median of the scores across all faculties since their type I error rate are less than the preselected alpha level 0.05. Therefore, Rank transformation test statistics is recommended as alternative test statistics to non-parametric test in both one sample and two-sample problems.

]]>Retno Ayu Cahyoningtyas Solimun and Adji Achmad Rinaldo Fernandes

The purpose of this research is to develop structural modeling with metric and nonmetric measurement scales. Also, this study compares the level of efficiency between the first order and second-order models. The application of structural modeling in agriculture is the satisfaction of farmers in East Java. The data used in this study are about perceptions by distributing questionnaires to farmers in East Java Province in 2020. The respondents in this study were 155 districts in East Java Province. Therefore, the sampling technique chosen is probability sampling, which is a proportional area random sampling. The results are obtained that the first-order model is better than the second-order model because it has the lowest MSE value and the highest R2. The results of the path analysis for the first order and second-order models produce the same results that there is a significant positive effect between the gratitude variables on the farmer satisfaction variable. That is, the more gratitude felt by farmers, the satisfaction will be increased by East Java Farmers. On the other hand, the test results showed that demographic variables did not significantly influence gratitude variables.

]]>Priya Arora and V. P Tomar

Background: Measuring the information and removal of uncertainty are the essential nature of human thinking and many world objectives. Information is well used and beneficial if it is free from uncertainty and fuzziness. Shannon was the primitive who coined the term entropy for measure of uncertainty. He also gave an expression of entropy based on probability distribution. Zadeh used the idea of Shannon to develop the concept of fuzzy sets. Later on, Atanassov generalized the concept of fuzzy set and developed intuitionistic fuzzy sets. Purpose: Sometimes we do not have complete information about fuzzy set or intuitionistic fuzzy sets. Some partial information is known about them i.e either only few values of membership function or non membership function are known or a relationship between them is known or some inequalities governing these parameters are known. Kapur has measured the partial information given by a fuzzy set. In this paper, we have attempted to quantify partial information given by intuitionistic fuzzy sets by considering all the cases. Methodologies: We analyze some well-known definitions and axioms used in the field of fuzzy theory. Principal Results: We have devised methods to measure the incomplete information given about intuitionistic fuzzy sets. Major Conclusions: By devising the methods of measuring partial information about IFS, we can use this information to get an idea about the given set and use this information wisely to make a good decision.

]]>Jonathan Kwaku Afriyie Sampson Twumasi-Ankrah Kwasi Baah Gyamfi Doris Arthur and Wilhemina Adoma Pels

Unit root tests for stationarity have relevancy in almost every practical time series analysis. Deciding on which unit root test to use is a topic of active interest. In this study, we compare the performance of the three commonly used unit root tests (i.e., Augmented Dickey-Fuller (ADF), Phillips-Perron (PP), and Kwiatkowski Phillips Schmidt and Shin (KPSS)) in time series. Based on literature, these unit root tests sometimes disagree in selecting the appropriate order of integration for a given series. Therefore, the decision to use a unit root test relies essentially on the judgment of the researcher. Suppose we wish to annul the subjective decision. In that case, we have to locate an objective basis that unmistakably characterizes which test is the most appropriate for a particular time series type. Thus, this study seeks to unravel this problem by providing a guide on which unit root tests to utilize when there is a disagreement between them. A simulation study of eight (8) univariate time series models with eight (8) different sample sizes, three (3) differencing orders, and nine different parameter values were performed. It was observed from the results that the performance of the three tests improved as the sample size increased. Based on comparing the overall performance, the KPSS was the "best" unit root test to use when there is disagreement.

]]>Jirapud Limthanakul and Nopparat Pochai

Chloride is a well-known chemical compound that is very useful in industry and agricultural, chloride can be transformed to hypochlorite, chlorite, chlorate and perchlorate, chloride and their substances are not dangerous if we used in the optimal level. Groundwater that contaminated chloride and their substances impacts human health, for an example, if we drink water that contaminated chloride exceed 250 mg/L it can cause heart problems and contribute to high blood pressure. to avoid this problem, we used mathematical models to explain groundwater contamination with chloride and their substances. Transient groundwater flow model provides the hydraulic head of groundwater, in this model we will get the level of groundwater, next, we need to find its velocity and direction by using the result in first model put into second model. Groundwater velocity model provides x- and z-direction vector in groundwater, after computation we will plugin the result into the last model to approximated the chloride concentration in groundwater. Groundwater contamination dispersion model provides chloride, hypochlorite, chlorite, chlorate and perchlorate concentration. The proposed explicit finite difference techniques are used to approximate the model solution. Explicit method was used to solved hydraulic head model. Forward space described groundwater velocity model. Forward time and central space used to predict transient groundwater contaminated models. The simulations can be used to indicate when each simulated zone becomes a hazardous zone or a protection zone.

]]>Aleksandr Bochkov Dmitrii Pervukhin Aleksandr Grafov and Veronika Nikitina

The quality of construction of Lorenz curves depends on the features of the information used. As a rule, information is represented by a sample of values of the studied indicator, which is checked for unevenness. Economic indicators of income and cost, and features of their samples are considered. The feature of the cost economic indicator associated with the presence in the sample of its values of the clot is highlighted (the concentration of values on a small segment of the entire range of sample). It is shown that the established order of constructing empirical laws based on such samples does not give the desired effect when constructing Lorenz curves due to the loss of information content of the sample in the places of the clot. The purpose of this article is to improve the quality of the Lorenz curve by increasing the information content of the sample with a clot by applying the clustering procedure when constructing an empirical law. A step-by-step clustering procedure is proposed for dividing the entire range of sample into intervals to construct an empirical distribution law, which is an element of the novelty of this study. A specific example shows how to improve the quality of building a Lorenz curve using this procedure. In addition, it is shown that Lorenz curves for economic indicators can be constructed directly on the basis of the empirical distribution law and at the same time take into account its features.

]]>Shams A. Ahmed and Mohamed Elbadri

Newell Whitehead Segal (NWS) equation has been used in describing many natural phenomena arising in fluid mechanics and hence acquired more attention. Studies in the past gave importance to obtaining numerical or analytical solutions of this kind of equations by employing methods like Modified Homotopy Analysis Transform method (MHATM), Adomian Decomposition method (ADM), Homotopy Analysis Sumudu Transform method (HASTM), Fractional Complex Transform (FCT) coupled with He's polynomials method (FCT-HPM) and Fractional Residual Power Series method (FRPSM). This research aims to demonstrate an efficient analytical method called the Sumudu Decomposition Method (SDM) for the study of analytical and numerical solutions of the NWS of fractional order. The coupling of Adomian Decomposition method with Sumudu transform method simplifies the calculation. From the numerical results obtained, it is evident that SDM is easy to execute and offers accurate results for the NWS equation than with other methods such as FCT-HPM and FRPSM. Therefore, it is easy to apply the coupling of Adomian Decomposition technique with Sumudu transform method, and when applied to nonlinear differential equations of fractional order, it yields accurate results.

]]>Temitope Olu Ogunlade Oluwatayo Michael Ogunmiloro Segun Nathaniel Ogunyebi Grace Ebunoluwa Fatoyinbo Joshua Otonritse Okoro Opeyemi Roselyn Akindutire Omobolaji Yusuf Halid and Adenike Oluwafunmilola Olubiyi

This work concerns a deterministic and stochastic model describing the transmission of typhoid fever infection in human host community, where the vaccination of susceptible births and immigrants as well as screening and treatment of carriers and infected individuals are considered in the model build - up. The well-posedness and computation of the basic reproduction number R_{typ} of the deterministic model are obtained and analysed. The deterministic model is further transformed into a stochastic model, where the drift and diffusion parts of the model are obtained, and the existence and uniqueness of the stochastic model are discussed. Numerical simulations involving the model parameters of R_{typ} showed that vaccination of susceptible births and influx of immigrants as well as screening and treatment of carriers and infected humans are effective in bringing the threshold R_{typ}(R_{typ})≈0.7944) below 1, and the results of other simulations suggest more health policies are to be implemented, as low R_{typ} may not be guaranteed because vaccination wanes over time. In addition, the numerical simulations of the stochastic model equations describing the sub - population of human individuals in the total human host community are carried out using the computational software MATLAB.

Chatarina Enny Murwaningtyas Sri Haryatmi Kartiko Gunardi and Herry Pribawanto Suryawan

This paper deals with an Indonesian option pricing using mixed fractional Brownian motion to model the underlying stock price. There have been researched on the Indonesian option pricing by using Brownian motion. Another research states that logarithmic returns of the Jakarta composite index have long-range dependence. Motivated by the fact that there is long-range dependence on logarithmic returns of Indonesian stock prices, we use mixed fractional Brownian motion to model on logarithmic returns of stock prices. The Indonesian option is different from other options in terms of its exercise time. The option can be exercised at maturity or at any time before maturity with profit less than ten percent of the strike price. Also, the option will be exercised automatically if the stock price hits a barrier price. Therefore, the mathematical model is unique, and we apply the method of the partial differential equation to study it. An implicit finite difference scheme has been developed to solve the partial differential equation that is used to obtain Indonesian option prices. We study the stability and convergence of the implicit finite difference scheme. We also present several examples of numerical solutions. Based on theoretical analysis and the numerical solutions, the scheme proposed in this paper is efficient and reliable.

]]>Piyali Mallick and Lakshmi Narayan De

In this work, we propose a stochastic inventory model under the situations that delay in imbursement is acceptable. Most of the inventory model on this topic supposed that the supplier would offer the retailer a fixed delay period and the retailer could sell the goods and accumulate revenue and earn interest with in the credit period. They also assumed that the trade credit period is independent of the order quantity. Limited investigators developed EOQ model under permissible delay in payments, where trade credit is connected with the order quantity. When the order quantity is a lesser amount of the quantity at which the delay in payment is not permitted, the payments for the items must be made immediately. Otherwise, the fixed credit period is permitted. However, all these models were completely deterministic in nature. In reality, this trade credit period cannot be fixed. If it is fixed, then retailer will not be interested to buy higher quantity than the fixed quantity at which delay in payment is permitted. To reflect this situation, we assumed that trade credit period is not static but fluctuates with the ordering quantity. The demand throughout any arrangement period follows a probability distribution. We have calculated the total variable cost for every unit of time. The optimum ordering policy of the scheme can be found with the aid of three theorems (proofs are provided). An algorithm to determine the best ordering rule with the assistance of the propositions is established and numerical instances are provided for clarification. Sensitivity investigation of all the parameters of the model is presented and deliberated. Some previously published results are special cases of the consequences gotten in this paper.

]]>R. Sivaraman

Computation of day of a week from given date belonging to any century has been a great quest among astronomers and mathematicians for long time. In recent centuries, thanks to efforts of some great mathematicians we now know methods of accomplishing this task. In doing so, people have developed various methods, some of which are very concise and compact but not much accessible explanation is provided. The chief purpose of this paper is to address this issue. Also, almost all known calculations involve either usage of tables or some pre-determined codes usually assigned for months, years or centuries. In this paper, I had established the mathematical proof of determining the day of any given date which is applicable for any number of years even to the time of BCE. I had provided the detailed mathematical derivation of month codes which were key factors in determining the day of any given date. Though the procedures for determining the day of given date are quite well known, the way in which they arrived is not so well known. This paper will throw great detail in that aspect. To be precise, I had explained the formula obtained by German Mathematician Zeller in detail and tried to simplify it further which will reduce its complexity and at the same time, would be as effective as the original formula. The explanations for Leap Years and other astronomical facts were clearly presented in this paper to aid the derivation of the compact form of Zeller's Formula. Some special cases and illustrations are provided wherever necessary to clarify the computations for better understanding of the concepts.

]]>Hani Syahida Zulkafli George Streftaris and Gavin J. Gibson

Hypoglycaemia is a condition when blood sugar levels in body are too low. This condition is usually a side effect of insulin treatment in diabetic patients. Symptoms of hypoglycaemia vary not only between individuals but also within individuals making it difficult for the patients to recognize their hypoglycaemia episodes. Given this condition, and because the symptoms are not exclusive to only hypoglycaemia, it is very important for patients to be able to identify that they are having a hypoglycaemia episode. Consistency models are statistical models that quantify the consistency of individual symptoms reported during hypoglycaemia. Because there are variations of consistency model, it is important to identify which model best fits the data. The aim of this paper is to asses and verify the models. We developed an assessment method based on stochastic latent residuals and performed posterior predictive checking as the model verification. It was found that a grouped symptom consistency model with multiplicative form of symptom propensity and episode intensity threshold ﬁts the data better and has more reliable predictive ability as compared to other models. This model can be used in assisting patients and medical practitioners to quantify patients' reporting symptoms capability, hence promote awareness of their hypoglycaemia episodes so that corrective actions can be quickly taken.

]]>Edy Nurfalah Irvana Arofah Ika Yuniwati Andi Haslinah and Dwi Retno Lestari

This work is a research development of two-tier multiples choice diagnostic test instruments on calculus material. The purpose of this study is; 1) Obtaining the construction of a two-tier multiples choice diagnostic test based on the validity of the contents and Constable, 2) obtaining the quality of two-tier multiples choice diagnostic tests based on the reliability value. The method used is focused on the construction of diagnostic tests. The development research was adapted from the Retnawati development model. The research generated: 1) Construction of a two-tier multiples choice diagnostic test based on the validity of the contents and the construction obtained that the two-tier multiples choice diagnostic test is proven valid. 2) The quality of two-tier multiples choice diagnostic tests based on the reliability value gained that the compiled two-tier diagnostic test instruments. The validity of the content is evidenced by the average validity index (V), for the two-tier multiples choice diagnostic test instrument obtained an average validity index (V) of 0.9333 and for an interview guideline instrument acquired the validity index (V) 0.7556 in which both the validity index (V) approaches the value 1. Whereas for the validity of the construction acquired three dominant factors based on the scree-plot and corresponds to many factors on the calculus material examined in this study. The quality of two-tier multiples choice diagnostic tests is compiled of two-tier diagnostic test instruments based on the reliability value gained.

]]>N. A. Abdul Rahman

Fuzzy delay differential equation has always been a tremendous way to model real-life problems. It has been developed throughout the last decade. Many types of fuzzy derivatives have been considered, including the recently introduced concept of strongly generalized differentiability. However, considering this interpretation, very few methods have been introduced, obstructing the potential of fuzzy delay differential equations to be developed further. This paper aims to provide solution for fuzzy nonlinear delay differential equations and the derivatives considered in this paper is interpreted using the concept of strongly generalized differentiability. Under this method, the calculations will lead to two cases i.e. two solutions, and one of the solutions is decreasing in the diameter. To fulfil this, a method resulting from the elegant combination of fuzzy Sumudu transform and Adomian decomposition method is used, it is termed as fuzzy Sumudu decomposition method. A detailed procedure for solving fuzzy nonlinear delay differential equations with the mentioned type of derivatives is constructed in detail. A numerical example is provided afterwards to demonstrate the applicability of the method. It is shown that the solution is not unique, and this is in accord with the concept of strongly generalized differentiability. The two solutions can later be chosen by researcher with regards to the characteristic of the problems. Finally, conclusion is drawn.

]]>Andy Liew Pik Hern Aini Janteng and Rashidah Omar

Let S to be the class of functions which are analytic, normalized and univalent in the unit disk . The main subclasses of S are starlike functions, convex functions, close-to-convex functions, quasiconvex functions, starlike functions with respect to (w.r.t.) symmetric points and convex functions w.r.t. symmetric points which are denoted by , and K_{S} respectively. In recent past, a lot of mathematicians studied about Hankel determinant for numerous classes of functions contained in S. The qth Hankel determinant for and is defined by . is greatly familiar so called Fekete-Szeg¨o functional. It has been discussed since 1930's. Mathematicians still have lots of interest to this, especially in an altered version of . Indeed, there are many papers explore the determinants H_{2}(2) and H_{3}(1). From the explicit form of the functional H_{3}(1), it holds H_{2}(k) provided k from 1-3. Exceptionally, one of the determinant that is has not been discussed in many times yet. In this article, we deal with this Hankel determinant . From this determinant, it consists of coefficients of function f which belongs to the classes and K_{S} so we may find the bounds of for these classes. Likewise, we got the sharp results for and K_{s} for which a_{2} = 0 are obtained.

Siti Hajar Khairuddin Mohd Hilmi Hasan and Manzoor Ahmed Hashmani

Fuzzy C-Means (FCM) is one of the mostly used techniques for fuzzy clustering and proven to be robust and more efficient based on various applications. Image segmentation, stock market and web analytics are examples of popular applications which use FCM. One limitation of FCM is that it only produces Gaussian membership function (MF). The literature shows that different types of membership functions may perform better than other types based on the data used. This means that, by only having Gaussian membership function as an option, it limits the capability of fuzzy systems to produce accurate outcomes. Hence, this paper presents a method to generate another popular shape of MF, the trapezoidal shape (trapMF) from FCM to allow more flexibility to FCM in producing outputs. The construction of trapMF is using mathematical theory of Gaussian distributions, confidence interval and inflection points. The cluster centers or mean (μ) and standard deviation (σ) from the Gaussian output are fully used to determine four trapezoidal parameters; lower limit a, upper limit d, lower support limit b, and upper support limit c with the assistance of function trapmf() in Matlab fuzzy toolbox. The result shows that the mathematical theory of Gaussian distributions can be applied to generate trapMF from FCM.

]]>Ali F Jameel Sardar G Amen Azizan Saaban Noraziah H Man and Fathilah M Alipiah

Delay differential equations (known as DDEs) are a broad use of many scientific researches and engineering applications. They come because the pace of the shift in their mathematical models relies all the basis not just on their present condition, but also on a certain past cases. In this work, we propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using the Homotopy Perturbation Method with double parametric form fuzzy numbers. The detailed algorithm of the approach to fuzzification and defuzzificationis analysis is provided. In the initial conditions of the proposed problem there are uncertainties with regard to the triangular fuzzy number. A double parametric form of fuzzy numbers is defined and applied for the first time in this topic for the present analysis. This method's simplicity and ability to overcome delay differential equations without complicating Adomian polynomials or incorrect nonlinear assumptions. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show the features of this proposed method, a numerical example is illustrated, involving first order fuzzy delay differential equation. These findings indicate that the suggested approach is very successful and simple to implement.

]]>O. S. Deepa

The reliability of the product has developed a dynamic issue in a worldwide business market. Generally acceptance sampling guarantees the superiority of the product. In acceptance sampling plan, increasing the sample size may lead to minimization of customers' risk of accepting bad lots and producers' risk of rejecting good lots to a certain level but will increase the cost of inspection. Hence truncation of life test time may be introduced to reduce the cost of inspection. Modified Average Sample Number (MASN) for Improved Double Sampling Plan (IDSP) based on truncated life test for popular exponentiated family such as exponentiated gamma, exponentiated lomax and exonentiated Weibull distribution are considered. The modified ASN creates a band width for average sample number which is much useful for the consumer and producer. The interval for average sample number makes the choice of consumer with a maximum and minimum sample size which is of much benefit without any loss for the producer. The probability of acceptance and average sample number based on modified double sampling plan for lower and upper limit is computed for the exponentiated family. Optimal parameters of IDSP under various exponentiated families with different shape parameters were computed. The proposed plan is compared over traditional double sampling and modified double sampling using Gamma distribution, Weibull distribution and Birnbaum-Saunders distribution and shows that the proposed plan with respect to exponentiated family performs better than all other plans. The tables were provided for all distributions. Comparative study of tables based on proposed exponentiated family and earlier existing plan are also done.

]]>Nor Syahmina Kamarudin and Syahida Che Dzul-Kifli

The dynamics of a multidimensional dynamical system may sometimes be inherited from the dynamics of its classical dynamical system. In a multidimensional case, we introduce a new map called a -action on space X induced by a continuous map as such that, where, and is a map of the form . We then look at how topological transitivity of f effects the behaviour of k-type transitivity of the -action, . To verify this, we look specifically at spaces called 1-step shifts of finite type over two symbols which are equipped with a map called the shift map, . We apply some topological theories to prove the -action on 1-step shifts of finite type over two symbols induced by the shift map, is k-type transitive for all whenever is topologically transitive. We found a counterexample which shows that not all maps are k-type transitive for all . However, we have also found some sufficient conditions for k-type transitivity for all. In conclusions, the map on 1-step shifts of finite type over two symbols induced by the shift map is k-type transitive for all whenever either the shift map is topologically transitive or satisfies the sufficient conditions. This study helps to develop the study of k-chaotic behaviours of -action on the multidimensional dynamical system, contributions, and its application towards symbolic dynamics.

]]>Ali F Jameel Akram H. Shather N.R. Anakira A. K. Alomari and Azizan Saaban

This research focuses on the approximate solutions of second-order fuzzy differential equations with fuzzy initial condition with two different methods depending on the properties of the fuzzy set theory. The methods in this research based on the Optimum homotopy asymptotic method (OHAM) and homotopy analysis method (HAM) are used implemented and analyzed to obtain the approximate solution of second-order nonlinear fuzzy differential equation. The concept of topology homotopy is used in both methods to produce a convergent series solution for the propped problem. Nevertheless, in contrast to other destructive approaches, these methods do not rely upon tiny or large parameters. This way we can easily monitor the convergence of approximation series. Furthermore, these techniques do not require any discretization and linearization relative with numerical methods and thus decrease calculations more that can solve high order problems without reducing it into a first-order system of equations. The obtained results of the proposed problem are presented, followed by a comparative study of the two implemented methods. The use of the methods investigated and the validity and applicability of the methods in the fuzzy domain are illustrated by a numerical example. Finally, the convergence and accuracy of the proposed methods of the provided example are presented through the error estimates between the exact solutions displayed in the form of tables and figures.

]]>Sirasak Sasiwannapong Saowanit Sukparungsee Piyapatr Busababodhin and Yupaporn Areepong

The control chart is an important tool in multivariate statistical process control (MSPC), which for monitoring, control, and improvement of the process control. In this paper, we propose six types of copula combinations for use on a Multivariate Exponentially Weighted Moving Average (MEWMA) control chart. Observations from an exponential distribution with dependence measured with Kendall's tau for moderate and strong positive and negative dependence (where ) among the observations were generated by using Monte Carlo simulations to measure the Average Run Length (ARL) as the performance metric and should be sufficiently large when the process is in-control on a MEWMA control chart. In this study, we develop an approach performance on the MEWMA control chart based on copula combinations by using the Monte Carlo simulations.The results show that the out-of-control (ARL_{1}) values for were less than for in almost all cases. The performances of the Farlie-Gumbel-Morgenstern×Ali-Mikhail-Haq copula combination was superior to the others for all shifts with strong positive dependence among the observations and . Moreover, when the magnitudes of the shift were very large, the performance metric values for observations with moderate and strong positive and negative dependence followed the same pattern.

Diah Ayu Widyastuti Adji Achmad Rinaldo Fernandes Henny Pramoedyo Nurjannah and Solimun

Regression analysis has three approaches in estimating the regression curve, namely: parametric, nonparametric, and semiparametric approaches. Several studies have discussed modeling with the three approaches in cross-section data, where observations are assumed to be independent of each other. In this study, we propose a new method for estimating parametric, nonparametric, and semiparametric regression curves in spatial data. Spatial data states that at each point of observation has coordinates that indicate the position of the observation, so between observations are assumed to have different variations. The model developed in this research is to accommodate the influence of predictor variables on the response variable globally for all observations, as well as adding coordinates at each observation point locally. Based on the value of Mean Square Error (MSE) as the best model selection criteria, the results are obtained that modeling with a nonparametric approach produces the smallest MSE value. So this application data is more precise if it is modeled by the nonparametric truncated spline approach. There are eight possible models formed in this research, and the nonparametric model is better than the parametric model, because the MSE value in the nonparametric model is smaller. As for the semiparametric regression model that is formed, it is obtained that the variable X_{2} is a parametric component while X_{1} and X_{3} are the nonparametric components (Model 2). The regression curve estimation model with a nonparametric approach tends to be more efficient than Model 2 because the linearity assumption test results show that the relationship of all the predictor variables to the response variable shows a non-linear relationship. So in this study, spatial data that has a non-linear relationship between predictor variables and responses tends to be better modeled with a nonparametric approach.

Habshah Midi and Jama Mohamed

The support vector regression (SVR) model is currently a very popular non-parametric method used for estimating linear and non-linear relationships between response and predictor variables. However, there is a possibility of selecting vertical outliers as support vectors that can unduly affect the estimates of regression. Outliers from abnormal data points may result in bad predictions. In addition, when both vertical outliers and high leverage points are present in the data, the problem is further complicated. In this paper, we introduced a modified robust SVR technique in the simultaneous presence of these two problems. Three types of SVR models, i.e. eps-regression (ε-SVR), nu-regression (v-SVR) and bound constraint eps-regression (ε-BSVR), with eight different kernel functions are integrated into the new proposed algorithm. Based on 10-fold cross-validation and some model performance measures, the best model with a suitable kernel function is selected. To make the selected model robust, we developed a new double SVR (DSVR) technique based on fixed parameters. This can be used to detect and reduce the weight of influential observations or anomalous points in the data set. The effectiveness of the proposed technique is verified by using a simulation study and some well-known contaminated data sets.

]]>Luthfatul Amaliana Solimun Adji Achmad Rinaldo Fernandes and Nurjannah

WarpPLS analysis has three algorithms, namely the outer model parameter estimation algorithm, the inner model, and the hypothesis testing algorithm which consists of several choices of resampling methods namely Stable1, Stable2, Stable3, Bootstrap, Jackknife, and Blindfolding. The purpose of this study is to apply the WarpPLS analysis by comparing the six resampling methods based on the relative efficiency of the parameter estimates in the six methods. This study uses secondary data from the questionnaire with 1 variable being formative and 2 variables being reflective. Secondary data for the Infrastructure Service Satisfaction Index (IKLI) were obtained from the Study Report on the Regional Development Planning for Economic Growth and the Malang City Gini Index in 2018, while secondary data for the Social Capital Index (IMS) and Community Development Index (IPMas) were obtained from the Research Report on Performance Indicators Regional Human Development Index and Poverty Rate of Malang City in 2018. The results of this study indicate that based on two criteria used, namely the calculation of relative efficiency and measure of fit as a model good, it can be concluded that the Jackknife resampling method is the most efficient, followed with the Stable1, Bootstrap, Stable3, Stable2, and Blindfolding methods.

]]>Azumah Karim Ananda Omutokoh Kube and Bashiru Imoro Ibn Saeed

Global temperature change is an important indicator of climate change. Climate time series data are characterized by trend, seasonal/cyclical as well as irregular components. Adequately modeling these components cannot be overemphasized. In this paper, we have proposed an approach of modeling temperature data using semiparametric additive generalized linear model. We have derived a penalized maximum likelihood estimation of the additive component of the semiparametric generalized linear models, that is, of regression coefficients and smooth functions. A statistical modeling with real time series data set was conducted on temperature data. The study has provided indications on the gain of using semiparametric modeling in situations where a signal component can be additively decomposed in to trend, cyclical and irregular components. Thus, we recommend semiparametric additive penalized models as an option to fit time series data sets in modelling the different component with different functions to adequately explain the relation inherent in data.

]]>S. Al-Ahmad I. M. Sulaiman M. Mamat and L. G. Puspa

The method of differential transform (DTM) is among the famous mathematical approaches for obtaining the differential equations solutions. This is due to its simplicity and efficient numerical performance. However, the major drawback of the DTM is obtaining a truncated series solution which is often a good approximation to the true solution of the equation in a specified region. In this study, a modification of DMT scheme known as MDTM is proposed for obtaining an accurate approximation of ordinary differential equations of second order. The scheme whose procedure is designed via DTM, the Laplace transforms and finally Padé approximation, gives a good approximate for the true solution of the equations in a large region. The proposed approach would be able to overcome the difficulty encountered using the classical DTM, and thus, can serve as an alternative approach for obtaining the solutions of these problems. Preliminary results are presented based on some examples which illustrate the strength and application of the defined scheme. Also, all the obtained results corresponded to exact solutions.

]]>Noraishikin Zulkarnain Noorhelyna Razali Nuryazmin Ahmat Zainuri Haliza Othman and Alias Jedi

Mathematics is one of the major subjects that every engineering student needs to learn. However, every student may have different views and interests on Mathematics subjects because of their different levels of thinking. To foster the appreciation of engineering students on the applications of Mathematics in engineering courses and help them apply and enhance their mathematical knowledge, the Fundamental Engineering Unit at the Faculty of Engineering and Built Environments, Universiti Kebangsaan Malaysia (UKM), organised the first ‘Mathematics Day' on Thursday, May 4, 2017. For their final year project, 12 students participated in a competition where they used mathematical or statistical applications to create a poster. The competition was judged by the academic assessors, industry and UKM alumni. This study examines the mathematical elements and applications in students' posters. The relevance of the elements and topics in the Engineering Mathematics course in the posters is reviewed. Reports from students who were present during the competition are also analysed to determine the effectiveness of the activity. The expected outcome of the student reports is interpreted using a statistical descriptive method, and results indicate that the students had a positive reaction to the activity.

]]>Amit Kumar Rana

Fuzzy sets theory is a very useful technique to increase effectiveness and efficiency of forecasting. The conventional time series is not applicable when the variable of time series are words variables i.e. variables with linguistic terms. As India and most of the Asian countries are of agriculture-based economy with very smaller farmer land holding area in comparison to America, Australia and Europe counterparts, it becomes more important for these countries to have an approximate idea regarding future crop production. It not only will help in planning policies for future but also will be a great help for farmers and agro based companies for their future managements. For small area production, soft computing technique is an important and effective tool for predicting production, as agriculture production involve a high degree of uncertainties in many parameters. In the present study, 21 years agricultural crop yield data is used and a comparative analysis of forecast is done with three fuzzy models. The robustness of the model is tested on real time agricultural farm production data of wheat crop of G.B. Pant University of Agriculture and Technology Pantnagar, India. As soft computing techniques involve uncertainty of the system under study, it becomes more and more important for forecasting models to be accurate with the prediction. The efficiency of the three models is examined on the basis of statistical errors. The models under study are judged on the basis of Mean Square Error and average percentage error. The results of the study are in case of small area production prediction and will encourage for predicting large scale production.

]]>Mahesh Puri Goswami and Naveen Jha

In this article, we investigate bicomplex triple Laplace transform in the framework of bicomplexified frequency domain with Region of Convergence (ROC), which is generalization of complex triple Laplace transform. Bicomplex numbers are pairs of complex numbers with commutative ring with unity and zero-divisors, which describe physical interpretation in four dimensional spaces and provide large class of frequency domain. Also, we derive some basic properties and inversion theorem of triple Laplace transform in bicomplex space. In this technique, we use idempotent representation methodology of bicomplex numbers, which play vital role in proving our results. Consequently, the obtained results can be highly applicable in the fields of Quantum Mechanics, Signal Processing, Electric Circuit Theory, Control Engineering, and solving differential equations. Application of bicomplex triple Laplace transform has been discussed in finding the solution of third-order partial differential equation of bicomplex-valued function.

]]>Solimun Adji Achmad Rinaldo Fernandes and Retno Ayu Cahyoningtyas

Nonlinear principal component analysis is used for data that has a mixed scale. This study uses a formative measurement model by combining metric and nonmetric data scales. The variable used in this study is the demographic variable. This study aims to obtain the principal component of the latent demographic variable and to identify the strongest indicators of demographic formers with mixed scales using samples of students of Brawijaya University based on predetermined indicators. The data used in this study are primary data with research instruments in the form of questionnaires distributed to research respondents, which are active students of Brawijaya University Malang. The used method is nonlinear principal component analysis. There are nine indicators specified in this study, namely gender, regional origin, father's occupation, mother's occupation, type of place of residence, father's last education, mother's last education, parents' income per month, and students' allowance per month. The result of this study shows that the latent demographic variable with samples of a student at Brawijaya University can be obtained by calculating its component scores. The nine indicators formed in PC1 or X_{1} were able to store diversity or information by 19.49%, while the other 80.51% of diversity or other information was not saved in this PC. From these indicators, the strongest indicator in forming latent demographic variables with samples of a student of Brawijaya University is the origin of the region (I_{2}) and type of residence (I_{5}).

Abdeslam Serroukh and Khudhayr A. Rashedi

The aim of this paper is to address the problem of variance break detection in time series in wavelet domain. The maximal overlapped discrete wavelet transform (MODWT) decomposes the series variance across scales into components known as the wavelet variances. We introduce all scale wavelet coefficients based test statistic that allows detecting a break in the homogeneity of the variance of a series through changes in the mean of wavelet variances. The statistic makes use of the traditional CUSUM (cumulative sum) based test designed to test for a break in the mean and constructed using cumulative sums of the square of wavelet coefficients. Under moments and mixing conditions, the test statistic satisfies the functional central limit theorem (FCLT) for a broad class of time series models. The overall performance of our test statistic is compared to the traditional Inclan [8] test statistic. The effectiveness of our statistic is supported by good performances reported in simulations and is as reliable as the traditional statistic. Our method provides a nonparametric test procedure that can be applied to a large class of linear and non linear models. We illustrate the practical use of our test procedure with the quarterly percentage changes in the Americans personal savings data set over the period 1970-2016. Both statistics detect a break in the variance in the second quarter of 2001.

]]>Iryna Halushchak Zoriana Novosad Yurii Tsizhma and Andriy Zagorodnyuk

In this paper, we extend complex polynomial dynamics to a set of multisets endowed with some ring operations (the metric ring of multisets associated with supersymmetric polynomials of infinitely many variables). Some new properties of the ring of multisets are established and a homomorphism to a function ring is constructed. Using complex homomorphisms on the ring of multisets, we proposed a method of investigations of polynomial dynamics over this ring by reducing them to a finite number of scalarvalued polynomial dynamics. An estimation of the number of such scalar-valued polynomial dynamics is established. As an important example, we considered an analogue of the logistic map, defined on a subring of multisets consisting of positive numbers in the interval [0; 1]: Some possible application to study the natural market development process in a competitive environment is proposed. In particular, it is shown that using the multiset approach, we can have a model that takes into account credit debt and reinvestments. Some numerical examples of logistic maps for different growth rate multiset [r] are considered. Note that the growth rate [r] may contain both "positive" and "negative" components and the examples demonstrate the influences of these components on the dynamics.

]]>Girija K. P. Devadas Nayak C Sabitha D’Souza and Pradeep G. Bhat

Graph labeling is an assignment of integers to the vertices or the edges, or both, subject to certain conditions. In literature we find several labelings such as graceful, harmonious, binary, friendly, cordial, ternary and many more. A friendly labeling is a binary mapping such that where and represents number of vertices labeled by 1 and 0 respectively. For each edge assign the label , then the function f is cordial labeling of G if and , where and are the number of edges labeled 1 and 0 respectively. A friendly index set of a graph is { runs over all f riendly labeling f of G} and it is denoted by FI(G). A mapping is called ternary vertex labeling and represents the vertex label for . In this article, we extend the concept of ternary vertex labeling to 3-vertex friendly labeling and define 3-vertex friendly index set of graphs. The set runs over all 3 ? vertex f riendly labeling f f or all is referred as 3-vertex friendly index set. In order to achieve , number of vertices are partitioned into such that for all with and la- bel the edge by where . In this paper, we study the 3-vertex friendly index sets of some standard graphs such as complete graph K_{n}, path P_{n}, wheel graph W_{n}, complete bipartite graph K_{m,n} and cycle with parallel chords PC_{n}.

Mahmoud M. El-Borai and Khairia El-Said El-Nadi

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

]]>Z. R. Rakhmonov A. Khaydarov and J. E. Urunbaev

Mathematical models of nonlinear cross diffusion are described by a system of nonlinear partial parabolic equations associated with nonlinear boundary conditions. Explicit analytical solutions of such nonlinearly coupled systems of partial differential equations are rarely existed and thus, several numerical methods have been applied to obtain approximate solutions. In this paper, based on a self-similar analysis and the method of standard equations, the qualitative properties of a nonlinear cross-diffusion system with nonlocal boundary conditions are studied. We are constructed various self-similar solutions to the cross diffusion problem for the case of slow diffusion. It is proved that for certain values of the numerical parameters of the nonlinear cross-diffusion system of parabolic equations coupled via nonlinear boundary conditions, they may not have global solutions in time. Based on a self-similar analysis and the comparison principle, the critical exponent of the Fujita type and the critical exponent of global solvability are established. Using the comparison theorem, upper bounds for global solutions and lower bounds for blow-up solutions are obtained.

]]>Viliam Ďuriš and Timotej Šumný

The accuracy of geometric construction is one of the important characteristics of mathematics and mathematical skills. However, in geometrical constructions, there is often a problem of accuracy. On the other hand, so-called 'Optical accuracy' appears, which means that the construction is accurate with respect to the drawing pad used. These "optically accurate" constructions are called approximative constructions because they do not achieve exact accuracy, but the best possible approximation occurs. Geometric problems correspond to algebraic equations in two ways. The first method is based on the construction of algebraic expressions, which are transformed into an equation. The second method is based on analytical geometry methods, where geometric objects and points are expressed directly using equations that describe their properties in a coordinate system. In any case, we obtain an equation whose solution in the algebraic sense corresponds to the geometric solution. The paper provides the methodology for solving some specific tasks in geometry by means of algebraic geometry, which is related to cubic and biquadratic equations. It is thus focusing on the approximate geometrical structures, which has a significant historical impact on the development of mathematics precisely because these tasks are not solvable using a compass and ruler. This type of geometric problems has a strong position and practical justification in the area of technology. The contribution of our work is so in approaching solutions of geometrical problems leading to higher degrees of algebraic equations, whose importance is undeniable for the development of mathematics. Since approximate constructions and methods of solution resulting from approximate constructions are not common, the content of the paper is significant.

]]>R. Sivaraman

Huge amount of literature has been written and published about Golden Ratio, but not many had heard about its generalized version called Metallic Ratios, which are introduced in this paper. The methods of deriving them were also discussed in detail. This will help to explore further in the search of universe of real numbers. In mathematics, sequences play a vital role in understanding of the complexities of any given problem which consist of some patterns. For example, the population growth, radioactive decay of a substance, lifetime of an object all follow a sequence called "Geometric Progression". In fact, the rate at which the recent novel corona virus (COVID – 19) is said to follow a Geometric Progression with common ratio approximately between 2 and 3. Almost all branches of science use sequences, for instance, genetic engineers use DNA sequence, Electrical Engineers use Morse-Thue Sequence and this list goes on and on. Among the vast number of sequences used for scientific investigations, one of the most famous and familiar is the Fibonacci Sequence named after the Italian mathematician Leonard Fibonacci through his book "Liber Abaci" published in 1202. In this paper, I shall try to introduce sequences resembling the Fibonacci sequence and try to generalize it to identify general class of numbers called "Metallic Ratios".

]]>Savita Rathee and Priyanka Gupta

In late sixties, Furi and Vignoli proved fixed point results for α-condensing mappings on bounded complete metric spaces. Bugajewski generalized the results to "weakly F-contractive mappings" on topological spaces(TS). Bugajeski and Kasprzak proved several fixed point results for "weakly F-contractive mapping" using the approach of lower(upper) semi-continuous functions. After that, by modifying the concept of "weakly F-contractive mappings", the coupled fixed point results were proved by Cho, Shah and Hussain on topological space. On different spaces, common coupled fixed point results were discussed by Liu, Zhou and damjanovic, Nashine and Shatanawi and many other authors. In this work, we prove the common coupled fixed point theorems by adopting the modified definition of weakly F-contractive mapping r : T→T; where T is a topological space. After that, we extend the result of Cho, Shah and Hussain for Banach spaces to common coupled quasi solutions enriched with a relevant transitive binary relation. Also, we give an example in the support of proved result. Our results extend and generalize several existing results in the literature.

]]>Waego Hadi Nugroho Ni Wayan Surya Wardhani Adji Achmad Rinaldo Fernandes and Solimun

Robust regression analysis is an analysis that is used if there is an outlier in a regression model. Outliers cause data to be abnormal. The most commonly used parameter estimation method is Ordinary Least Squares (OLS). However, outliers in models cause the estimator of the least-squares in the model to be biased, so handling of outliers is required. One of the regressions used for outliers is robust regression. Robust regression method that can be used is M-Estimation. By using Tukey's Bisquare weighted function, a robust M-estimation method can estimate parameters in a model, for example in malnutrition data in East Java Province 2017 to 2012. This study aims to compare the robust method of M-estimation and OLS method on data with several different levels of significance, which is 1%, 5%, and 10%. The predictor variables used in this study were the percentage of poor society, population density, and some health facilities. R^{2} is used to compare the OLS method and the robust method of M-estimation. The results obtained that robust regression is the best method to handle the model if there are outliers in the data. It was supported by almost all results of the value of R^2 on each data that M-estimation has a higher value than the OLS method.

Gwang Hui Kim

The present work continues the study for the superstability and solution of the Pexider type functional equation , which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d'Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation , which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G; +) of functions is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type's functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions.

]]>Yousef Al-Qudah Faisal Yousafzai Mohammed M. Khalaf and Mohammad Almousa

The main motivation behind this paper is to study some structural properties of a non-associative structure as it hasn't attracted much attention compared to associative structures. In this paper, we introduce the concept of an ordered A^{*}G^{**}-groupoid and provide that this class is more generalized than an ordered AG-groupoid with left identity. We also define the generated left (right) ideals in an ordered A^{*}G^{**}-groupoid and characterize a (2; 2)-regular ordered A^{*}G^{**}-groupoid in terms of these ideals. We then study the structural properties of an ordered A^{*}G^{**}-groupoid in terms of its semilattices, (2; 2)-regular class and generated commutative monoids. Subsequently, compare -fuzzy left/right ideals of an ordered AG-groupoid and respective examples are provided. Relations between an -fuzzy idempotent subsets of an ordered A^{*}G^{**}-groupoid and its -fuzzybi-ideals are discussed. As an application of our results, we get characterizations of (2; 2)-regular ordered A^{*}G^{**}-groupoid in terms of semilattices and -fuzzy left (right) ideals. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.

Abdishukurova Guzal Narmanov Abdigappar and Sharipov Xurshid

The concept of differential invariant, along with the concept of invariant differentiation, is the key in modern geometry [1]-[10]. In the Erlangen program [3] Felix Klein proposed a unified approach to the description of various geometries. According to this program, one of the main problems of geometry is to construct invariants of geometric objects with respect to the action of the group defining this geometry. This approach is largely based on the ideas of Sophus Lee, who introduced continuous geometry groups of transformations, now known as Lie groups, into geometry. In particular, when considering classification problems and equivalence problems in differential geometry, differential invariants with respect to the action of Lie groups should be considered. In this case, the equivalence problem of geometric objects is reduced to finding a complete system of scalar differential invariants. The interpretation of the k- order differential invariant as a function on the space of k- jets of sections of the corresponding bundle made it possible to operate with them efficiently, and using invariant differentiation, new differential invariants can be obtained. Differential invariants with respect to a certain Lie group generate differential equations for which this group is a symmetry group. This allows one to apply the well-known integration methods to such equations, and, in particular, the Li- Bianchi theorem [4]. Depending on the type of geometry, the orders of the first nontrivial differential invariants can be different. For example, in the space R^{3} equipped with the Euclidean metric, the complete system of differential invariants of a curve is its curvature and torsion, which are second and third order invariants, respectively. Note that scalar differential invariants are the only type of invariants whose components do not change when changing coordinates. For this reason, scalar differential invariants are effectively used in solving equivalence problems. In this paper differential invariants of Lie group of one parametric transformations of the space of two independent and three dependent variables are studied. It is shown method of construction of invariant differential operator. Obtained results applied for finding differential invariants of surfaces.

V. I. Struchenkov and D. A. Karpov

The article discusses the solution of applied problems, for which the dynamic programming method developed by R. Bellman in the middle of the last century was previously proposed. Currently, dynamic programming algorithms are successfully used to solve applied problems, but with an increase in the dimension of the task, the reduction of the counting time remains relevant. This is especially important when designing systems in which dynamic programming is embedded in a computational cycle that is repeated many times. Therefore, the article analyzes various possibilities of increasing the speed of the dynamic programming algorithm. For some problems, using the Bellman optimality principle, recurrence formulas were obtained for calculating the optimal trajectory without any analysis of the set of options for its construction step by step. It is shown that many applied problems when using dynamic programming, in addition to rejecting unpromising paths lead to a specific state, also allow rejecting hopeless states. The article proposes a new algorithm for implementing the R. Bellman principle for solving such problems and establishes the conditions for its applicability. The results of solving two-parameter problems of various dimensions presented in the article showed that the exclusion of hopeless states can reduce the counting time by 10 or more times.

]]>Nik Muhammad Farhan Hakim Nik Badrul Alam Ajab Bai Akbarally and Silvestru Sever Dragomir

Hermite-Hadamard type inequalities related to convex functions are widely being studied in functional analysis. Researchers have refined the convex functions as quasi-convex, h-convex, log-convex, m-convex, (a,m)-convex and many more. Subsequently, the Hermite-Hadamard type inequalities have been obtained for these refined convex functions. In this paper, we firstly review the Hermite-Hadamard type inequality for both convex functions and log-convex functions. Then, the definition of composite convex function and the Hermite-Hadamard type inequalities for composite convex functions are also reviewed. Motivated by these works, we then make some refinement to obtain the definition of composite log-convex functions, namely composite-^{-1} log-convex function. Some examples related to this definition such as GG-convexity and HG-convexity are given. We also define k-composite log-convexity and k-composite-^{-1} log-convexity. We then prove a lemma and obtain some Hermite-Hadamard type inequalities for composite log-convex functions. Two corollaries are also proved using the theorem obtained; the first one by applying the exponential function and the second one by applying the properties of k-composite log-convexity. Also, an application for GG-convex functions is given. In this application, we compare the inequalities obtained from this paper with the inequalities obtained in the previous studies. The inequalities can be applied in calculating geometric means in statistics and other fields.

Leonid N. Yasnitsky and Sergey L. Gladkiy

One of the main problems in modern mathematical modeling is to obtain high-precision solutions of boundary value problems. This study proposes a new approach that combines the methods of artificial intelligence and a classical analytical method. The use of the analytical method of fictitious canonic regions is proposed as the basis for obtaining reliable solutions of boundary value problems. The novelty of the approach is in the application of artificial intelligence methods, namely, genetic algorithms, to select the optimal location of fictitious canonic regions, ensuring maximum accuracy. A general genetic algorithm has been developed to solve the problem of determining the global minimum for the choice and location of fictitious canonic regions. For this genetic algorithm, several variants of the function of crossing individuals and mutations are proposed. The approach is applied to solve two test boundary value problems: the stationary heat conduction problem and the elasticity theory problem. The results of solving problems showed the effectiveness of the proposed approach. It took no more than a hundred generations to achieve high precision solutions in the work of the genetic algorithm. Moreover, the error in solving the stationary heat conduction problem was so insignificant that this solution can be considered as precise. Thus, the study showed that the proposed approach, combining the analytical method of fictitious canonic regions and the use of genetic optimization algorithms, allows solving complex boundary-value problems with high accuracy. This approach can be used in mathematical modeling of structures for responsible purposes, where the accuracy and reliability of the results is the main criterion for evaluating the solution. Further development of this approach will make it possible to solve with high accuracy of more complicated 3D problems, as well as problems of other types, for example, thermal elasticity, which are of great importance in the design of engineering structures.

]]>Ni Wayan Surya Wardhani Waego Hadi Nugroho Adji Achmad Rinaldo Fernandes and Solimun

WANT-E is a tool created to purify methane gas from organic waste intended as a substitute for renewable gas fuel. The WANT-E product is new because it is necessary to do research related to the public interest in WANT-E products. This study uses primary data obtained from questionnaires with variables based on Theory of Planned Behavior (TPB), namely behavior attitudes, subjective norms, perceived behavior control, and behavior interests that are spread to the community of Cibeber Village, Cikalong Subdistrict, Tasikmalaya Regency that uses LPG gas cylinders or stove using sampling techniques in the form of the judgment sampling method. The analysis used is SEM with the WarpPLS approach, which is to determine the effect of relationships between variables. The results of the analysis obtained the effect of a positive relationship between behavior attitudes variables on subjective norms, behavior attitudes toward perceived behavior control, subjective norms of behavior interests, and perceived behavior control of behavior interests. Then the influence of indirect relations on subjective norms and perceived behavior control was obtained as mediation between behavior attitudes toward behavior interests.

]]>Artykbaev Abdullaaziz and Nurbayev Abdurashid Ravshanovich

This article discusses geometric quantities associated with the concept of surfaces and the indicatrix of a surface in four-dimensional Galileo space. In this case, the second orderly line in the plane is presented as a surface indicatrix. It is shown that with the help of the Galileo space movement, the second orderly line can be brought to the canonical form. The movement in the Galileo space is radically different from the movement in the Euclidean space. Galileo movements include parallel movement, axis rotation, and sliding. Sliding gives deformation in the Euclidean space. The surface indicatrix is deformed by the Galileo movement. When the indicatrix is deformed, the surface will be deformed. In the classification of three-dimensional surface points in the four-dimensional Galileo phase, the classification of the indicatrix of the surface at this point was used. This shows the cyclic state of surface points other than Euclidean geometry. The geometric characteristics of surface curves were determined using the indicatrix test. It is determined what kind of geometrical meaning the identified properties have in the Euclidean phase. It is shown that the Galilean movement gives surface deformation in the Euclidean sense. Deformation of the surface is indicated by the fact that the Gaussian curvature remains unchanged.

]]>M. Khalifa Saad R. A. Abdel-Baky F. Alharbi and A. Aloufi

In a theory of space curves, especially, a helix is the most elementary and interesting topic. A helix, moreover, pays attention to natural scientists as well as mathematicians because of its various applications, for example, DNA, carbon nanotube, screws, springs and so on. Also there are many applications of helix curve or helical structures in Science such as fractal geometry, in the fields of computer aided design and computer graphics. Helices can be used for the tool path description, the simulation of kinematic motion or the design of highways, etc. The problem of the determination of parametric representation of the position vector of an arbitrary space curve according to the intrinsic equations is still open in the Euclidean space E^{3} and in the Minkowski space . In this paper, we introduce some characterizations of a non-null slant helix which has a spacelike or timelike axis in . We use vector differential equations established by means of Frenet equations in Minkowski space . Also, we investigate some differential geometric properties of these curves according to these vector differential equations. Besides, we illustrate some examples to confirm our findings.

Narmanov Abdigappar and Parmonov Hamid

The problem of integrating equations of mechanics is the most important task of mathematics and mechanics. Before Poincare's book "Curves Defined by Differential Equations", integration tasks were considered as analytical problems of finding formulas for solutions of the equation of motion. After the appearance of this book, it became clear that the integration problems are related to the behavior of the trajectories as a whole. This, of course, stimulated methods of qualitative theory of differential equations. Present time, the main method in this problem has become the symmetry method. Newton used the ideas of symmetry for the problem of central motion. Further, Lagrange revealed that the classical integrals of the problem of gravitation bodies are associated with invariant equations of motion with respect to the Galileo group. Emmy Noether showed that each integral of the equation of motion corresponds to a group of transformations preserving the action. The phase flow of the Hamilton system of equations, in which the first integral serves as the Hamiltonian, translates the solutions of the original equations into solutions. The Liouville theorem on the integrability of Hamilton equations was created on this idea. The Liouville theorem states that phase flows of involutive integrals generate an Abelian group of symmetries Hamiltonian methods have become increasingly important in the study of the equations of continuum mechanics, including fluids, plasmas and elastic media. In this paper it is considered the problem on the Hamiltonian system which describes of motion of a particle which is attracted to a fixed point with a force varying as the inverse cube of the distance from the point. We are concerned with just one aspect of this problem, namely the questions on the symmetry groups and Hamiltonian symmetries. It is found Hamiltonian symmetries of this Hamiltonian system and it is proven that Hamiltonian symmetry group of the considered problem contains two dimensional Abelian Lie group. Also it is constructed the singular foliation which is generated by infinitesimal symmetries which invariant under phase flow of the system. In the present paper, smoothness is understood as smoothness of the class C^{∞}.

Jae Won Lee Dae Ho Jin and Chul Woo Lee

Jin [1] defined an ()-type connection on semi-Riemannian manifolds. Semi-symmetric nonmetric connection and non-metric ∅-symmetric connection are two important examples of this connection such that () = (1; 0) and () = (0; 1), respectively. In semi-Riemannian geometry, there are few literatures for the lightlike geometry, so we expose new theories for non-degenerate submanifolds in semi-Riemannian geometry. The goal of this paper is to study a characterization of a (Lie) recurrent lightlike hypersurface M of an indefinite Kaehler manifold with an ()-type connection when the charateristic vector field is tangnet to M. In the special case that an indefinite Kaehler manifold of constant holomorphic sectional curvature is an indefinite complex space form, we investigate a lightlike hypersurface of an indefinite complex space form with an ()-type connection when the charateristic vector field is tangnet to M. Moreover, we show that the total space, the complex space form, is characterized by the screen conformal lightlike hypersurface with an ()-type connection. With a semi-symmetric non-metric connection, we show that an indefinite complex space form is flat.

]]>Mohammad Almousa

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations. Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems. Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields. The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs). In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively. We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind. We introduced efficient method, constructed on modified Bernstein polynomials. The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind. This method is tested for some examples from nonlinear integral equations. Maple software was used to obtain the solutions of these examples. The results demonstrate reliability of the proposed method. Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

]]>Moustafa Omar Ahmed Abu-Shawiesh Muhammad Riaz and Qurat-Ul-Ain Khaliq

In this study, a robust control chart as an alternative to the Tukey's control chart (TCC) based on the modified trimmed standard deviation (MTSD), namely MTSD-TCC, is proposed. The performance of the proposed and the competing Tukey's control chart (TCC) is measured using different length properties such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MDRL). Also, the study covered normal and contaminated cases. We have observed that the proposed robust control chart (MTSD-TCC) is quite efficient at detecting process shifts. Also, it is evident from the simulation results that the proposed robust control chart (MTSD-TCC) offers superior detection ability for different trimming levels as compared to the Tukey's control chart (TCC) under the contaminated process setups. As a result, it is recommended to use the proposed robust control chart (MTSD-TCC) for process monitoring. An application numerical example using real-life data is provided to illustrate the implementation of the proposed robust control chart (MTSD-TCC) which also supported the results of the simulation study to some extent.

]]>Anuradha and SeemaMehra

In 2016, Muralisankar and Jeyabal introduced the concept of ε-Compatible maps and studied the set of common fixed points. They generalized the Banach contraction, Kannan contraction, Reich contraction and Bianchini type contraction to obtain some common fixed point theorems for ε-Compatible mappings which don't involve the suitable containment of the ranges for the given mappings in the setting of metric spaces. Motivated by this new concept of mappings, we establish a new approach for some common fixed point theorems via ϵ -compatible maps in context of complete partial metric space including a directed graph G=(V,E). By the remarkable work of Jachymski in 2008, we extend the results obtained by Muralisankar and Jeyabal in 2016. In 2008, Jachymski obtained some important fixed point results introduced by Ran and Reurings (2004) using the languages of graph theory instead of partial order and gave an interesting approach in this direction. After that, his work is considered as a reference in this domain. Sometimes, there are some mappings which do not satisfy the contractive nature on whole set M(say) but these can be made contractive on some subset of M and this can be done by including graph as shown in our Example 2.6 which is provided to substantiate the validity of our results.

]]>Zahari Md Rodzi and Abd Ghafur Ahmad

In this paper, by combining hesitant fuzzy soft sets (HFSSs) and fuzzy parameterized, we introduce the idea of a new hybrid model, fuzzy parameterized hesitant fuzzy soft sets (FPHFSSs). The benefit of this theory is that the degree of importance of parameters is being provided to HFSSs directly from decision makers. In addition, all the information is represented in a single set in the decision making process. Then, we likewise ponder its basic operations such as AND, OR, complement, union and intersection. The basic properties such as associative, distributive and de Morgan's law of FPHFSSs are proven. Next, in order to resolve the multi-criteria decision making problem (MCDM), we present arithmetic mean score and geometry mean score incorporated with hesitant degree of FPHFSSs in TOPSIS. This algorithm can cater some existing approach that suggested to add such elements to a shorter hesitant fuzzy element, rendering it equivalent to another hesitant fuzzy element, or to duplicate its elements to obtain two sequence of the same length. Such approaches would break the original data structure and modify the data. Finally, to demonstrate the efficacy and viability of our process, we equate our algorithm with existing methods.

]]>Solimun and Adji Achmad Rinaldo Fernandes

The use of regression analysis has not been able to deal with the problems of complex relationships with several response variables and the presence of intervening endogenous variables in a relationship. Analysis that is able to handle these problems is path analysis. In path analysis there are several assumptions, one of which is the assumption of residual normality. If the normality residual assumptions are not met, then estimating the parameters can produce a biased estimator, a large and not consistent range of estimators. Unmet residual normality problems can be overcome by using resampling. Therefore in this study, a simulation study was conducted to apply resampling with the blindfold method to the condition that the normality assumption is not met with various levels of resampling in the path analysis. Based on the simulation results, different levels of closeness occur consistently at different resampling quantities. At a low level of closeness, it is consistent with the resampling magnitude of 1000. At a moderate level, a consistent level of resampling of 500 occurs. At a high level of closeness, it is consistent with the amount of resampling 1400.

]]>Yona Eka Pratiwi Kusbudiono Abduh Riski and Alfian Futuhul Hadi

The development of an increasingly rapid industrial development resulted in increasingly intense competition between industries. Companies are required to maximize performance in various fields, especially by meeting customer demand with agreed timeliness. Scheduling is the allocation of resources to the time to produce a collection of jobs. PT. Bella Agung Citra Mandiri is a manufacturing company engaged in making spring beds. The work stations in the company consist of 5 stages consisting of ram per with three machines, clamps per 1 machine, firing mattresses with two machines, sewing mattresses three machines and packing with one machine. The model problem that was solved in this study was Hybrid Flowshop Scheduling. The optimization method for solving problems is to use the metaheuristic method Migrating Birds Optimization. To avoid problems faced by the company, scheduling is needed to minimize makespan by paying attention to the number of parallel machines. The results of this study are scheduling for 16 jobs and 46 jobs. Decreasing makespan value for 16 jobs minimizes the time for 26 minutes 39 seconds, while for 46 jobs can minimize the time for 3 hours 31 minutes 39 seconds.

]]>Muhammad Asim Khan and Norhashidah Hj. Mohd Ali

The fractional diffusion equation is an important mathematical model for describing phenomena of anomalous diffusion in transport processes. A high-order compact iterative scheme is formulated in solving the two-dimensional time fractional sub-diffusion equation. The spatial derivative is evaluated using Crank-Nicolson scheme with a fourth-order compact approximation and the Caputo derivative is used for the time fractional derivative to obtain a discrete implicit scheme. The order of convergence for the proposed method will be shown to be of . Numerical examples are provided to verify the high-order accuracy solutions of the proposed scheme.

]]>RaziraAniza Roslan Chin Su Na and Darmesah Gabda

The standard method of the maximum likelihood has poor performance in GEV parameter estimates for small sample data. This study aims to explore the Generalized Extreme Value (GEV) parameter estimation using several methods focusing on small sample size of an extreme event. We conducted simulation study to illustrate the performance of different methods such as the Maximum Likelihood (MLE), probability weighted moment (PWM) and the penalized likelihood method (PMLE) in estimating the GEV parameters. Based on the simulation results, we then applied the superior method in modelling the annual maximum stream flow in Sabah. The result of the simulation study shows that the PMLE gives better estimate compared to MLE and PMW as it has small bias and root mean square errors, RMSE. For an application, we can then compute the estimate of return level of river flow in Sabah.

]]>Khadizah Ghazali Jumat Sulaiman Yosza Dasril and Darmesah Gabda

In this paper, we proposed an alternative way to find the Newton direction in solving large-scale unconstrained optimization problems where the Hessian of the Newton direction is an arrowhead matrix. The alternative approach is a two-point Explicit Group Gauss-Seidel (2EGGS) block iterative method. To check the validity of our proposed Newton’s direction, we combined the Newton method with 2EGGS iteration for solving unconstrained optimization problems and compared it with a combination of the Newton method with Gauss-Seidel (GS) point iteration and the Newton method with Jacobi point iteration. The numerical experiments are carried out using three different artificial test problems with its Hessian in the form of an arrowhead matrix. In conclusion, the numerical results showed that our proposed method is more superior than the reference method in term of the number of inner iterations and the execution time.

]]>Mohd Saifullah Rusiman Siti Nasuha Md Nor Suparman and Siti Noor Asyikin Mohd Razali

This paper is focusing on the application of robust method in multiple linear regression (MLR) model towards diabetes data. The objectives of this study are to identify the significant variables that affect diabetes by using MLR model and using MLR model with robust method, and to measure the performance of MLR model with/without robust method. Robust method is used in order to overcome the outlier problem of the data. There are three robust methods used in this study which are least quartile difference (LQD), median absolute deviation (MAD) and least-trimmed squares (LTS) estimator. The result shows that multiple linear regression with application of LTS estimator is the best model since it has the lowest value of mean square error (MSE) and mean absolute error (MAE). In conclusion, plasma glucose concentration in an oral glucose tolerance test is positively affected by body mass index, diastolic blood pressure, triceps skin fold thickness, diabetes pedigree function, age and yes/no for diabetes according to WHO criteria while negatively affected by the number of pregnancies. This finding can be used as a guideline for medical doctors as an early prevention of stage 2 of diabetes.

]]>Nur Hanim Mohd Salleh Husna Hasan and Fariza Yunus

Extreme temperature has been carried out around the world to provide awareness and proper opportunity for the societies to prepare necessary arrangements. In this present paper, the first order Markov chain model was applied to estimate the probability of extreme temperature based on the heat wave scales provided by the Malaysian Meteorological Department. In this study, the 24-year period (1994-2017) daily maximum temperature data for 17 meteorological stations in Malaysia was assigned to the four heat wave scales which are monitoring, alert level, heat wave and emergency. The analysis result indicated that most of the stations had three categories of heat wave scales. Only Chuping station had four categories while Bayan Lepas, Kuala Terengganu, Kota Bharu and Kota Kinabalu stations had two categories. The limiting probabilities obtained at each station showed a similar trend which the highest proportion of daily maximum temperature occurred in the scale of monitoring and followed by the alert level. This trend is apparent when the daily maximum temperature data revealed that Malaysia is experiencing two consecutive days of temperature below 35℃.

]]>Puguh Wahyu Prasetyo Indah Emilia Wijayanti Halina France-Jackson and Joe Repka

In the development of Theory Radical of Rings, there are two kinds of radical constructions. The first radical construction is the lower radical construction and the second one is the upper radical construction. In fact, the class π of all prime rings forms a special class and the upper radical class of forms a radical class which is called the prime radical. An upper radical class which is generated by a special class of rings is called a special radical class. On the other hand, we also have the class of all semiprime rings which is weakly special class of rings. Moreover, we can construct a special class of modules by using a given special class of rings. This condition motivates the existence of the question how to construct weakly special class modules by using a given weakly special class of rings. This research is a qualitative research. The results of this research are derived from fundamental axioms and properties of radical class of rings especially on special and weakly special radical classes. In this paper, we introduce the notion of a weakly special class of modules, a generalization of the notion on a special class of modules based on the definition of semiprime modules. Furthermore, some properties and examples of weakly special classes of modules are given. The main results of this work are the definition of a weakly special class of modules and their properties.

]]>Suparman

A piecewise constant model is often applied to model data in many fields. Several noises can be added in the piecewise constant model. This paper proposes the piecewise constant model with a gamma multiplicative noise and a method to estimate a parameter of the model. The estimation is done in a Bayesian framework. A prior distribution for the model parameter is chosen. The prior distribution for the parameter model is multiplied with a likelihood function for the data to build a posterior distribution for the parameter. Because a number of models are also parameters, a form of the posterior distribution for the parameter is too complex. A Bayes estimator cannot be calculated easily. A reversible jump Monte Carlo Markov Chain (MCMC) is used to find the Bayes estimator of the model parameter. A result of this paper is the development of the piecewise constant model and the method to estimate the model parameter. An advantage of this method can simultaneously estimate the constant piecewise model parameter.

]]>Che Haziqah Che Hussin Ahmad Izani Md Ismail Adem Kilicman and Amirah Azmi

This paper aims to propose and investigate the application of Multistep Modified Reduced Differential Transform Method (MMRDTM) for solving the nonlinear Korteweg-de Vries (KdV) equation. The proposed technique has the advantage of producing an analytical approximation in a fast converging sequence with a reduced number of calculated terms. MMRDTM is presented with some modification of the reduced differential transformation method (RDTM) which is the nonlinear term is replaced by related Adomian polynomials and then adopting a multistep approach. Consequently, the obtained approximation results do not only involve smaller number of calculated terms for the nonlinear KdV equation, but also converge rapidly in a broad time frame. We provided three examples to illustrates the advantages of the proposed method in obtaining the approximation solutions of the KdV equation. To depict the solution and show the validity and precision of the MMRDTM, graphical inputs are included.

]]>Bahtiar Jamili Zaini and Shamshuritawati Sharif

Bivariate data consist of 2 random variables that are obtained from the same population. The relationship between 2 bivariate data can be measured by correlation coefficient. A correlation coefficient computed from the sample data is used to measure the strength and direction of a linear relationship between 2 variables. However, the classical correlation coefficient results are inadequate in the presence of outliers. Therefore, this study focuses on the performance of different correlation coefficient under contaminated bivariate data to determine the strength of their relationships. We compared the performance of 5 types of correlation, which are classical correlations such as Pearson correlation, Spearman correlation and Kendall’s Tau correlation with other robust correlations, such as median correlation and median absolute deviation correlation. Results show that when there is no contamination in data, all 5 correlation methods show a strong relationship between 2 random variables. However, under the condition of data contamination, median absolute deviation correlation denotes a strong relationship compared to other methods.

]]>Gautam Choudhury Akhil Goswami Anjana Begum and Hemanta Kumar Sarmah

The single server queue with two types of heterogeneous services with generalized vacation for unreliable server have been extended to include several types of generalizations to which attentions has been paid by several researchers. One of the most important results which deals with such types of models is the “Stochastic Decomposition Result”, which allows the system behaviour to be analyzed by considering separately distribution of system (queue) size with no vacation and additional system (queue) size due to vacation. Our intention is to look into some sort of united approach to establish stochastic decomposition result for two types of general heterogeneous service queues with generalized vacations for unreliable server with delayed repair to include several types of generalizations. Our results are based on embedded Markov Chain technique which is considerably a most powerful and popular method wisely used in applied probability, specially in queueing theory. The fundamental idea behind this method is to simplify the description of state from two dimensional states to one dimensinal state space. Finally, the results that are derived is shown to include several types of generalizations of some existing well- known results for vacation models, that may lead to remarkable simpliﬁcation while solving similar types of complex models.

]]>Inessa I. Pavlyuk and Sergey V. Sudoplatov

Approximations of syntactic and semantic objects play an important role in various ﬁelds of mathematics. They can create theories and structures in one given class by means of others, usually simpler. For instance, in certain situations, inﬁnite objects can be approximated by ﬁnite or strongly minimal ones. Thus, complicated objects can be collected using simpliﬁed ones. Among these objects, Abelian groups, their ﬁrst order theories, connections and dynamics are of interests. Theories of Abelian groups are characterized by Szmielew invariants leading to the study and descriptions of approximations in terms of these invariants. In the paper we apply a general approach for approximating theories to the class of theories of Abelian groups which characterizes the approximability of a theory of Abelian groups by a given family of theories of Abelian groups in terms of Szmielew invariants and their limits. We describe some forms of approximations for theories of Abelian groups. In particular, approximations of theories of Abelian groups by theories of ﬁnite ones are characterized. In addition, we describe approximations by quasi-cyclic and torsion-free Abelian groups and their combinations with respect to given families of prime numbers. Approximations and closures of families of theories with respect to standard Abelian groups for various sets of prime numbers are also described.

]]>Supawan Yena and Nopparat Pochai

Nitrogen is emitted extensively by industrial companies, increasing nitrogen compounds such as ammonia, nitrate, and nitrite in soil and water as a result of nitrogen cycle reactions. Groundwater contamination with nitrates and nitrites impacts human health. Mathematical models can explain groundwater contamination with nitrates and nitrites. Hydraulic head model provides the hydraulic head of groundwater. Groundwater velocity model provided x- and y- direction vector in groundwater. Groundwater contamination distribution model provides nitrogen, nitrate and nitrite concentration. Finite difference techniques are approximate the models solution. Alternating direction explicit method was used to clarify hydraulic head model. Centered space explained groundwater velocity model. Forward time central space was used to predict groundwater transportation of contamination models. We simulate different circumstances to explain the pollution in leachate water underground, paying attention to the toxic nitrogen, ammonia, nitrate, nitrite blended in the water.

]]>Mohammed M. B. Adam M. B. Zulkafli H. S. and Ali N.

This paper proposes three different statistics to be used to represent the magnitude observations in each class when estimating the statistical measures from the frequency table for continuous data. The existing frequency tables use the midpoint as the magnitude of observations in each class, which results in an error called grouping error. Using the midpoint is due to the assumption that the observations in each class are uniformly distributed and concentrated around their midpoint, which is not always valid. In this research, construction of the frequency tables using the three proposed statistics, the arithmetic mean, median, and midrange and midpoint are respectively named, Method 1, Method 2, Method 3, and the Existing method. The four methods are compared using root-mean-squared error (RMSE) by performing simulation studies using three distributions, normal, uniform, exponential distributions. The simulation results are validated using real data, Glasgow weather data. The ﬁndings indicated that using the arithmetic mean to represent the magnitude of observations in each class of the frequency table leads to minimal error relative to other statistics. It is followed by using the median, for data simulated from normal and exponential distributions, and using midrange for data simulated from uniform distribution. Meanwhile, in choosing the appropriate number of classes used in constructing the frequency tables, among seven different rules used, the freedman and Diaconis rule is the recommended rule.

]]>Ludwik Byszewski Denis Blackmore Alexander A. Balinsky Anatolij K. Prykarpatski and Mirosław Lu´styk

As a ﬁrst step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful deﬁnitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk–Ulam type ﬁxed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the inﬁnite-dimensional problem to a sequence of solvable ﬁnite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach–Steinhaus theorem is brought to bear in the estimation process.

]]>Fausto Galetto

Pooling p-values arises both in practical (in any science and engineering applications) and theoretical (statistical) issues. The p-value (sometimes p value) is a probability used as a statistical decision quantity: in practical applications, it is used to decide if an experimenter has to believe that his/her collected data confirm or disconfirm his/her hypothesis about the “reality” of a phenomenon. It is a real number, determination of a Random Variable, uniformly distributed, related to the data provided by the measurement of a phenomenon. Almost all statistical software provides p-values when statistical hypotheses are considered, e.g. in Analysis of Variance and regression methods. Combining the p-values from various samples is crucial, because the number of degrees of freedom (df) of the samples we want to combine is influencing our decision: forgetting this can have dangerous consequences. One way of pooling p-values is provided by a formula of Fisher; unfortunately, this method does not consider the number of degrees of freedom. We will show other ways of doing that and we will prove that theory is more important than any formula which does not consider the phenomenon on which we have to decide: the distribution of the Random Variables is fundamental in order to pool data from various samples. Manager, professors and scholars should remember Deming’s profound knowledge and Juran’s ideas; profound knowledge means “understanding variation (type of variation)” in any process, production or managerial; not understanding variation causes cost of poor quality (more than 80% of sales value) and do not permits a real improvement.

]]>Anton Epifanov

Paper contains the results of the analysis of the laws of functioning of discrete dynamical systems, as mathematical models of which, using the apparatus of geometric images of automatons, are used numerical sequences which interpreted as sequences of second coordinates of points of geometric images of automatons. The geometric images of the laws of the functioning of the automaton are reduced to numerical sequences and numerical graphs. The problem of constructing an estimate of the complexity of the structures of such sequences is considered. To analyze the structure of sequences, recurrence forms are used that give characteristics of the relative positions of elements in the sequence. The parameters of recurrent forms are considered, which characterize the lengths of the initial segments of sequences determined by recurrence forms of fixed orders, the number of changes of recurrent forms required to determine the entire sequence, the place of change of recurrence forms, etc. All these parameters are systematized into the special spectrum of dynamic parameters used for the recurrent determination of sequences, which is used as a means of constructing estimates of the complexity of sequences. In this paper, it also analyzes return sequences (for example, Fibonacci numbers), for the analysis of the properties of which characteristic sequences are used. The properties of sequences defining approximations of fundamental mathematical constants (number e, pi number, golden ratio, Euler constant, Catalan constant, values of Riemann zeta function, etc.) are studied. Complexity estimates are constructed for characteristic sequences that distinguish numbers with specific properties in a natural series, as well as for characteristic sequences that reflect combinations of the properties of numbers.

]]>Leontiev V. L.

The problem of approximating of a surface given by the values of a function of two arguments in a finite number of points of a certain region in the classical formulation is reduced to solving a system of algebraic equations with tightly filled matrixes or with band matrixes. In the case of complex surfaces, such a problem requires a significant number of arithmetic operations and significant computer time spent on such calculations. The curvilinear boundary of the domain of general type does not allow using classical orthogonal polynomials or trigonometric functions to solve this problem. This paper is devoted to an application of orthogonal splines for creation of approximations of functions in form of finite Fourier series. The orthogonal functions with compact supports give possibilities for creation of such approximations of functions in regions with arbitrary geometry of a boundary in multidimensional cases. A comparison of the fields of application of classical orthogonal polynomials, trigonometric functions and orthogonal splines in approximation problems is carried out. The advantages of orthogonal splines in multidimensional problems are shown. The formulation of function approximation problem in variational form is given, a system of equations for coefficients of linear approximation with a diagonal matrix is formed, expressions for Fourier coefficients and approximations in the form of a finite Fourier series are written. Examples of approximations are considered. The efficiency of orthogonal splines is shown. The development of this direction associated with the use of other orthogonal splines is discussed.

]]>Supawan Yena and Nopparat Pochai

Leachate contamination in a landfill causes pollution that flowing down to the groundwater. There are many methods to measure the groundwater quality. Mathematical models are often used to describe the groundwater flow. In this research, the affection of landfill construction to groundwater-quality around rural area is focused. Three mathematical models are combined. The first model is a two-dimensional groundwater flow model. It provides the hydraulic head of the groundwater. The second model is the velocity potential model. It provides the groundwater flow velocity. The third model is a two-dimensional vertically averaged groundwater pollution dispersion model. The groundwater pollutant concentration is provided. The forward time centered technique with the centered in space, the forward in space and the backward in space with all boundaries are used to obtain approximate hydraulic head, the flow velocity in x- and y- directions, respectively. The approximated groundwater flow velocity is used to input into a two-dimensional vertically averaged groundwater pollution dispersion model. The forward time centered space technique with the centered in space, the forward in space and the backward in space with all boundaries are used to obtain approximate the groundwater pollutant concentration. The proposed explicit forward time centered spaced finite difference techniques to the groundwater flow model the velocity potential model and the groundwater pollution dispersion model give good agreement approximated solutions.

]]>Jindrich Klufa

The entrance examinations tests were shorted from 50 questions to 40 questions at the Faculty of International Relations at University of Economics in Prague due to time reasons. These tests are the multiple choice question tests. The multiple choice question tests are suitable for entrance examinations at University of Economics in Prague - the tests are objective and results can be evaluated quite easily and quickly for large number of students. On the other hand, a student can obtain certain number of points in the test purely by guessing the right answers. This shortening of the tests from 50 questions to 40 questions has negative influence on the probability distributions of number of points in the tests (under assumption of the random choice of answers). Therefore, this paper is suggested a solution of this problem. The comparison of these three ways of acceptance of applicants to study the Faculty of International Relations at University of Economics from probability point of view is performed in present paper. The results of this paper show that there has been a significant improvement of the probability distributions of number of points in the tests. The obtained conclusions can be used for admission process at the Faculty of International Relations in coming years.

]]>GeorgiaIrina Oros and Alina Alb Lupas

In this paper, we define the operator I^{m} : differential-integral operator, where S^{m} is S˘al˘agean differential operator and Lm is Libera integral operator. By using the operator I^{m} the class of univalent functions denoted by is defined and several differential subordinations are studied. Even if the use of linear operators and introduction of new classes of functions where subordinations are studied is a well-known process, the results are new and could be of interest for young researchers because of the new approach derived from mixing a differential operator and an integral one. By using this differential–integral operator, we have obtained new sufficient conditions for the functions from some classes to be univalent. For the newly introduced class of functions, we show that is it a class of convex functions and we prove some inclusion relations depending on the parameters of the class. Also, we show that this class has as subclass the class of functions with bounded rotation, a class studied earlier by many authors cited in the paper. Using the method of the subordination chains, some differential subordinations in their special Briot-Bouquet form are obtained regarding the differential–integral operator introduced in the paper. The best dominant of the Briot-Bouquet differential subordination is also given. As a consequence, sufficient conditions for univalence are stated in two criteria. An example is also illustrated, showing how the operator is used in obtaining Briot–Bouquet differential subordinations and the best dominant.

Mostafa Ftouhi Mohammed Barmaki and Driss Gretete

The class of amenable groups plays an important role in many areas of mathematics such as ergodic theory, harmonic analysis, representation theory, dynamical systems, geometric group theory, probability theory and statistics. The class of amenable groups contains in particular all finite groups, all abelian groups and, more generally, all solvable groups. It is closed under the operations of taking subgroups, taking quotients, taking extensions, and taking inductive limits. In 1959, Harry Kesten proved that there is a relation between the amenability and the estimates of symmetric random walk on finitely generated groups. In this article we study the classification of locally compact compactly generated groups according to return probability to the origin. Our aim is to compare several geometric classes of groups. The central tool in this comparison is the return probability on locally compact groups. we introduce several classes of groups in order to characterize the geometry of locally compact groups compactly generated. Our aim is to compare these classes in order to better understand the geometry of such groups by referring to the behavior of random walks on these groups. As results, we have found inclusion relationships between these defined classes and we have given counterexamples for reciprocal inclusions.

]]>Zainidin Eshkuvatov Massamdi Kommuji Rakhmatullo Aloev Nik Mohd Asri Nik Long and Mirzoali Khudoyberganov

A hypersingular integral equations (HSIEs) of the first kind on the interval [ 1 ; 1 ] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights.

]]>Nurazlina Abdul Rashid Norashikin Nasaruddin Kartini Kassim and Amirah Hazwani Abdul Rahim

Classification studies are widely applied in many areas of research. In our study, we are using classification analysis to explore approaches for tackling the classification problem for a large number of measures using partial least square discriminant analysis (PLS-DA) and decision trees (DT). The performance for both methods was compared using a sample data of breast tissues from the University of Wisconsin Hospital. A partial least square discriminant analysis (PLS-DA) and decision trees (DT) predict the diagnosis of breast tissues (M = malignant, B = benign). A total of 699 patients diagnose (458 benign and 241 malignant) are used in this study. The performance of PLS-DA and DT has been evaluated based on the misclassification error and accuracy rate. The results show PLS-DA can be considered as a good and reliable technique to be used when dealing with a large dataset for the classification task and have good prediction accuracy.

]]>Nurul Shazwani Mohamed Sharifah Kartini Said Husain and Faridah Yunos

Given two algebras and , if lies in the Zariski closure of the orbit , we say that is a degeneration of . We denote this by . Degenerations (or contractions) were widely applied to a range of physical and mathematical point of view. The most well-known example oriented to the application on degenerations is limiting process from quantum mechanics to classical mechanics under that corresponds to the contraction of the Heisenberg algebras to the abelian ones of the same dimension. Research on degenerations of Lie, Leibniz and other classes of algebras are very active. Throughout the paper we are dealing with mathematical background with abstract algebraic structures. The present paper is devoted to the degenerations of low-dimensional nilpotent Leibniz algebras over the field of complex numbers. Particularly, we focus on the classification of three-dimensional nilpotent Leibniz algebras. List of invariance arguments are provided and its dimensions are calculated in order to find the possible degenerations between each pair of algebras. We show that for each possible degenerations, there exists construction of parameterized basis on parameter We proof the non-degeneration case for mentioned classes of algebras by providing some reasons to reject the degenerations. As a result, we give complete list of degenerations and non-degenerations of low-dimensional complex nilpotent Leibniz algebras. In future research, from this result we can find its rigidity and irreducible components.

]]>Busyra Latif Mat Salim Selamat Ainnur Nasreen Rosli Alifah Ilyana Yusoff and Nur Munirah Hasan

Newell-Whitehead-Segel (NWS) equation is a nonlinear partial differential equation used in modeling various phenomena arising in fluid mechanics. In recent years, various methods have been used to solve the NWS equation such as Adomian Decomposition method (ADM), Homotopy Perturbation method (HPM), New Iterative method (NIM), Laplace Adomian Decomposition method (LADM) and Reduced Differential Transform method (RDTM). In this study, the NWS equation is solved approximately using the Semi Analytical Iterative method (SAIM) to determine the accuracy and effectiveness of this method. Comparisons of the results obtained by SAIM with the exact solution and other existing results obtained by other methods such as ADM, LADM, NIM and RDTM reveal the accuracy and effectiveness of the method. The solution obtained by SAIM is close to the exact solution and the error function is close to zero compared to the other methods mentioned above. The results have been executed using Maple 17. For future use, SAIM is accurate, reliable, and easier in solving the nonlinear problems since this method is simple, straightforward, and derivative free and does not require calculating multiple integrals and demands less computational work.

]]>Patricia Abelairas-Etxebarria and Inma Astorkiza

The Exploratory Data Analysis raised by Tuckey [19] has been used in multiple research in many areas but, especially, in the area of the social sciences. This technique searches behavioral patterns of the variables of the study, establishing a hypothesis with the least possible structure. However, in recent times, the inclusion of the spatial perspective in this type of analysis has been revealed as essential because, in many analyses, the observations are spatially autocorrelated and/or they present spatial heterogeneity. The presence of these spatial effects makes necessary to include spatial statistics and spatial tools in the Exploratory Data Analysis. Exploratory Spatial Data Analysis includes a set of techniques that describe and visualize those spatial effects: spatial dependence and spatial heterogeneity. It describes and visualizes spatial distributions, identifies outliers, finds distribution patterns, clusters and hot spots and suggests spatial regimes or other forms of spatial heterogeneity and, it is being increasingly used. With the objective of reviewing the last applications of this technique, this paper, firstly, shows the tools used in Exploratory Spatial Data Analysis and, secondly, reviews the latest Exploratory Spatial Data Analysis applications focused on different areas in the social sciences particularly. As conclusion, it should be noted the growing interest in the use of this spatial technique to analyze different aspects of the social sciences including the spatial dimension.

]]>Agung Prabowo Agus Sugandha Agustini Tripena Mustafa Mamat Sukono and Ruly Budiono

Linear regression is widely used in various fields. Research on linear regression uses the OLS and ML method in estimating its parameters. OLS and ML method require many assumptions to complete. It is frequently found there is an unconditional assumption that both methods are not successfully used. This paper proposes a new method which does not require any assumption with a condition. The new method is called SAM (Simple Averaging Method) to estimate parameters in the simple linear regression model. The method may be used without fulfilling assumptions in the regression model. Three new theorems are formulated to simplify the estimation of parameters in the simple linear regression model with SAM. By using the same data, the simple linear regression model parameter estimation is conducted using SAM. The result shows that the obtained regression parameter is not quite far different. However, to measure the accuracy of both methods, a comparison of errors made by each method is conducted using Root Mean Square Error (RMSE) and Mean Averaged Error (MAE). By comparing the values of RMSE and MAE for both methods, SAM method may be used to estimate parameters in the regression equation. The advantage of SAM is free from all assumptions required by regression, such as error normality assumption while the data should be from the normal distribution.

]]>Jirapud Limthanakul and Nopparat Pochai

A source of contaminated groundwater is governed by the disposal of waste material on a land fill. There are many people in rural areas where the primary source of drinking water is well water. This well water may be contaminated with groundwater from landfills. In this research, a two-dimensional mathematical model for long-term contaminated groundwater pollution measurement around a land fill is proposed. The model is governed by a combination of two models. The first model is a transient two-dimensional groundwater flow model that provides the hydraulic head of the groundwater. The second model is a transient twodimensional advection-diffusion equation that provides the groundwater pollutant concentration. The proposed explicit finite difference techniques are used to approximate the hydraulic head and the groundwater pollutant concentration. The simulations can be used to indicate when each simulated zone becomes a hazardous zone or a protection zone.

]]>Nor Asmaa Alyaa Nor Azlan Effendi Mohamad Mohd Rizal Salleh Oyong Novareza Dani Yuniawan Muhamad Arfauz A Rahman Adi Saptari and Mohd Amri Sulaiman

The purpose of this review paper is to set an augmentation approach and exemplify distribution of augmentation works in Simplex method. The augmentation approach is classified into three forms whereby it comprises addition, substitution and integration. From the diversity study, the result shows that substitution approach appeared to be the highest usage frequency, which is about 45.2% from the total of percentage. This is then followed by addition approach which makes up 32.3% of usage frequency and integration approach for about 22.6% of usage frequency which makes it the least percentage of the overall usage frequency approach. Since it is being the least usage percentage, the paper is then interested to foresee a future study of integration approach that can be performed from the executed distribution of the augmentation works according to Simplex's computation stages. A theme screening is then conducted with a set of criteria and themes to come out with a proposal of new integration approach of augmentation of Simplex method.

]]>Arif Rahman Oke Oktavianty Ratih Ardia Sari Wifqi Azlia and Lavestya Dina Anggreni

Some researches need data homogeneity. The dispersion of data causes research towards an absurd direction. The outlier makes unrealistic homogeneity. The research can reject the extreme data as outlier to estimate trimmed arithmetic mean. Because of the wide data dispersion, it will fail to identify the outliers. The study will evaluate the confidence interval and compare it with the acceptance tolerance. There are three types of invalidity of data gathering: outliers, too wide dispersion, distracted central tendency.

]]>Zahari Md Rodzi and Abd Ghafur Ahmad

The purpose of this work is to present a new theory namely fuzzy parameterized dual hesitant fuzzy soft sets (FPDHFSSs). This theory is an extension of the existing dual hesitant fuzzy soft set whereby the set of parameters have been assigned with respective weightage accordingly. We also introduced the basic operation functions for instance intersection, union, addition and product operations of FPDHFSSs. Then, we proposed the concept of score function of FPDHFSSs of which these scores function were determined based on average mean, geometry mean and fractional score. The said scores function then were divided into the membership and non-membership elements where the distance of FPDHFSSs was introduced. The proposed distance of FPDHFSSs has been applied in TOPSIS which will be able to solve the problem of fuzzy dual hesitant fuzzy soft set environment.

]]>Alec John Villamar Marionne Gayagoy Flerida Matalang and Karen Joy Catacutan

This study aimed to determine the usefulness of Mathematics subjects in the accounting courses for Bachelor of Science in Accountancy. Mathematics subjects, which include College Algebra, Mathematics of Investment, Business Calculus and Quantitative Techniques, were evaluated through its Course Learning Objectives, while its usefulness for accounting courses which include Financial Accounting, Advance Accounting, Cost Accounting, Management Advisory Services, Auditing and Taxation, was evaluated by the students. Descriptive research was employed among all students in their 5^{th}-year in BS-Accountancy who were done with all the Accounting Subjects in the Accountancy Program and they all passed the different Mathematics subjects prerequisite to their courses. A survey questionnaire was used to gather data. Using descriptive statistics, results showed that Mathematics of Investment is the most useful subject in the different accounting courses particularly in Financial Accounting, Advance Accounting and Auditing. Further, by using Mean, the results showed that several skills that can be acquired in the Mathematics subjects are found to be useful in accounting courses and the use of the fundamental operations is the most useful skill in all accounting subjects.

Rafid S. A. Alshkaki

Differential equations are used in modelling many disciplines, in engineering, chemistry, physics, biology, economics, and other fields of sciences, hence can be used to understand and to determine the underlying probabilistic behavior of phenomena through their probability distributions. This paper came to use a simple form of differential equations, namely, the linear form, to determine the probabilistic distributions of some of the most important and popular sub class of discrete distributions used in real-life, the Poisson, the binomial, the negative binomial, and the logarithmic series distributions. A class of finite number of inflated points power series distributions, that contains the Poisson, the binomial, the negative binomial, and the logarithmic series distributions as some of its members, was defined and some of its characteristics properties, along with characterization of the 3-points inflated of these four distributions, through a linear differential equation for their probability generating functions were given. Further, some previous known results were shown to be special cases of our results.

]]>Hasibun Naher Humayra Shafia Md. Emran Ali and Gour Chandra Paul

In this article, the nonlinear partial fractional differential equation, namely the KdV equation is renewed with the help of modified Riemann- Liouville fractional derivative. The equation is transformed into the nonlinear ordinary differential equation by using the fractional complex transformation. The goal of this paper is to construct new analytical solutions of the space and time fractional nonlinear KdV equation through the extended -expansion method. The work produces abundant exact solutions in terms of hyperbolic, trigonometric, rational, exponential, and complex forms, which are new and more general than existing results in literature. The newly generated solutions show that the executed method is a well-organized and competent mathematical tool to investigate a class of nonlinear evolution fractional order equations.

]]>Llesh Lleshaj and Alban Korbi

In this study analyzed 20 different countries that are the origin state of foreign investors, which have invested in Albania (this sample represents 95% of FDI (Foreign Direct Investments) stocks, 2007 - 2014). The analysis technic used is the gravity model of FDI stocks in Albania. The main independent variables in this analysis are GDP, the level of business taxes, the difference of GDP per capita, the similarity economies, etc. The result of this study: The level of FDI stocks in Albania is lower than its potential compare with FDI stock average in the States of the Balkan Region.

]]>Anuradha Seema Mehra and Said Broumi

Motivated by the concepts of fuzzy metric and m-metric spaces, we introduced the notion of Non- Archimedean fuzzy m-metric space which is an extension of partial fuzzy metric space. We present some examples in support of this new notion. Regarding this notion, its topological structure and some properties are specified simultaneously. At the end, some fixed point results are also provided.

]]>Igor Sinitsyn and Vladimir Sinitsyn

Analytical methods of the mathematical statistics of random vectors and matrices based on the parametrization of the distributions are widely used. These methods permit to design practically simple software when it is possible to have the definite information about analytical properties of the distributions under research. The main difficulty in practical applications of the methods based on the parametrization of the distributions is the rapid increase of the number of equations for the moments, the semiinvariants or the coefficients of the truncated orthogonal expansions of the dimension or the state vector (extended in the general case) and the maximal order of the moments involved. The number of equations for the parameters becomes exceedingly large in such cases. For structural parametrization and/or approximation of the probability densities of the random vectors we shall apply the ellipsoidal densities, i.e. the densities whose planes of the levels of equal probability are similar concentric ellipsoids (the ellipses for two-dimensional vectors, the ellipsoids for three-dimensional vectors, the hyperellipsoids for the vectors of the dimension more than three). In particular, a normal distribution in any finite-dimensional space has an ellipsoidal structure. The distinctive characteristics of such distributions consists in the fact that their probability densities are the functions of positively determined quadratic form where is an expectation of the random vector is some positively determined matrix. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters till () where being the number of probabilistic moments. While using ellipsoidal linearization method (ELM) we get Basic EAM and ELM foundations and applications to problems of mathematical statistics and ellipsoidal distributions with invariant measure in populational Volterra differential stochastic nonlinear systems are considered.

]]>Aripov M. Mukimov A. and Mirzayev B.

We study the asymptotic behavior (for ) of solutions of the Cauchy problem for a nonlinear parabolic equation with a double nonlinearity, describing the diffusion of heat with nonlinear heat absorption at the critical value of the parameter ᵝ. For numerical computations as an initial approximation we used founded the long time asymptotic of the solution. Numerical experiments and visualization were carried for one and two dimensional case.

]]>Emil V. Veitsman

This paper is aimed to find a connection between i-dimensional spaces (i=0,…, ‘n') and the long-range j-dimensional attractive forces (j=0,…, ‘m') creating these spaces. The connection is fundamental and unrelated to any processes going in the spaces being studied. A theorem is formulated and strictly proved showing in which cases the long-ranged attractive forces can form real spaces of different dimensions ( i=0,…,n). The existence of the attraction between masses is defined by divergence of the vector of interaction between masses. Weak anisotropic real spaces are studied by rotating an ellipsoid for (3ζ)D-cases when its eccentricity ε<<1. Such spaces cannot be in equilibrium, the time of their existence is substantially limited. The greater is anisotropy, the shorter is the lifetime of such substance. The latter cannot be in equilibrium, the time of their existence is substantially limited.

]]>Taehan Bae and Maral Mazjini

Recent studies on correlated Poisson p