On the Dynamic Flows in Networks

Abstract Considered dynamic network flows identically zero on the negative time axis. The concept of the frontal flow is introduced for such flow. The article contains the theorem about the decomposition of the dynamic flow on an arbitrary time interval ] ; 0 [ T on the sum of two flows, the first of which is equivalent to the original flow on the interval and equals to zero outside the interval, the second flow is equivalent to zero on the interval and coincides with the initial flow outside of the interval. The first is minimum flow which equals the original flow at a given time interval.


Introduction
The subject in the article is the dynamic flows in oriented networks which identically equals to zero for 0 < t ) (x U + -the set of arcs exiting out the vertex x . Now we recall the definition of a stationary flow in the network (see [1]).
where s -the source; r -the drain.
Magnitude of a stationary flow is called flow values sum through any arcs belongs some cut of the network, and in particular, When we talk about the dynamic flow (see [2 -5]), the discrete time Z t ∈ is added to consideration. It is believed that each transition along the arc is made during the unit of discrete time (i.e., per clock).
Here is the definition of dynamic flow:

Definition 2: Dynamic flow
Dynamic flow in the network ) , , ϕ , for a which the next conditions satisfy: where s -the source; r -the drain; t -the time. (3)

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On the Dynamic Flows in Networks I.e. the total flow entering the drain at the time t .

Definition 4: Magnitude of the flow on interval
The magnitude of the flow ϕ on the interval

Claims
It is clear that we have the equality Dynamic flow definition implies: We write down the equality that follows from (5) and (6): Consider the flow is identically equal to zero for 0 < t , and let Z T T ∈ > , 0 . For this flow we have (for 1 , , (  III  III  II  III  II  I III  III  II  III  II  I III  III  II  III  II  I T  III  T  III  T  II  T  III  T  II  T  I Add to these equalities the following identity T  III  T  II  T  I  T  III  T  II  T  I Sum the resulting set of equalities: From the fact that the flow ϕ is identically equal to zero . We have obtained the equality: The expression on the left-hand side of equality (8) It is obvious that for any dynamic flow ϕ and arbitrary we have the following balance relation:  T  III  T  II  T  I  T  V   t  I  T  III  T  II  T  I   T  T Z  T  T  T  T  Z  T  T  ∈ holds.

Example 1.
Consider the network shown in Fig. 1. We assume that the capacity of all arcs is equals 2. Consider two flows 1 ϕ and 2 ϕ shown in Fig. 2 and Fig. 3, respectively, which will be assumed to be constant at the Theorem 2▲ For any dynamic flow ϕ that is identically equal to zero when 0 < t there is frontal flow F ϕ .
Proof: Frontal flow F ϕ is the partial flow of the flow ϕ . From (11) implies that it is identically equal to zero for 0 < t . Its value when 0 = t is determined under definition 6 (c). Next we argue by induction. Let us assume that the frontal flow is already defined for all 0 t t ≤ . We show that it is defined and when 1 0 Take w -arbitrary arc of the network ) , , , We numerate arcs of the set It is clear that Consider the equation: The inductive step is proved.▼ Now we consider the flow . It is clear that it is identically equal to zero for 1 < t . Similarly, the definition 7, we can determine its frontal flow It is clear that it is identically equal to zero for 2 < t . Thus, the following theorem holds.

Theorem 3
For any dynamic flow ϕ on oriented network that is identically equal to zero for 0 < t , and Z T T ∈ > , 0 equality holds: Equation (16) It is clear that for conveying path next equalities satisfies Note that the flow 2 ϕ is equal to the flow ϕ 1 for all T t > .
Proof: ▲In fact, it suffices to prove this theorem for the frontal flow.