Dialectical Logic K-model: On Multidimensional Discrete Dynamical Sampling System and Further Properties of Kirchhoff Matrices

In order to solve the problem of multidimensional logic variable true value function remarked in the paper(Yaozhi Jiang., 2017), now author has used discrete multiple Fourier transform to deal with the problem remarked above, and obtained an theoretical formulations of discrete multidimensional Fourier transform for that multidimensional logic variable true value function is unknown or we need the frequency properties of multidimensional logic variable true value function. Another problem is about further and deeper properties of Kirchhoff matrices defined in author ’s paper(Yaozhi Jiang., 2017), author has established a series of matrix expression for Kirchhoff laws and some new properties of Kirchhoff matrices. These results are all compute-able and complicated.


Introduction
In the dialectical logic K-model, the case about multidimensional logic variable is so important that we have to solve the sort of problem.As shown in author's papers (Yaozhi Jiang, 2017.;Yaozhi Jiang, 2018.), author has solved single-dimensional of these problems.To the case that multidimensional logic variable true value function is unknown or needs analysis properties in frequency domain of the multidimensional logic variable true value function, now author used the multidimensional discrete time and discrete space dynamical sampling system and discrete multiple Fourier transform to the sort of problem of multidimensional logic variable true value function and obtained some results in the section 2. Author also expanded the Kirchhoff graph and Kirchhoff matrix to the multidimensional logic variable true value function in the section 3.These results are never heard of before.

Some Definitions, Symbols about Discrete Multidimensional Dynamical Sampling System and Discrete Multiple Fourier Transform
Some definitions and symbols about discrete multidimensional dynamical sampling system and discrete multiple Fourier transform must be reviewed and recalled as below.
all are truncated interval and   all are sampling number.What shown above means that the sampling is an equal interval sampling and every time pulse is corresponding to z y x N N N P  space pulses, with another word, this is a lattice structure..


if the the sampling number is increasing and the sampling number is Especially when the sampling number is Follow the section 2. 1.2.,2.1.3., 2.1.4.,2.1.5., for the 2-dimensiontruevaluefunction we can established the formulas below If the sampling number is increasing and the sampling number is w N and t N ,then the recurrences formula of discrete 2-dimensional Fourier transform is as below.
The recurrences formula of discrete 2-dimensional Fourier transform and its inverse transform are as below In the last paper by author (Yaozhi Jiang, 2017), we have a series of definitions and concepts about the Kirchhoff matrix initially.In this section, we will explain the properties of Kirchhoff matrix carefully and deeply.(Yaozhi Jiang, 2017;Yaozhi Jiang, 2018)

 
In the formula above, the power function of head vertex minus the power function of tail vertex, otherwise the direction of edge is opposite and need make the direction of the edge redirected by another direction or sign a negative sign before the   t p ij ; B. In the graph, there is a pair of vertex, one vertex is called as source vertex 1  v , its power function is constant +1, and another vertex is called as sink vertex 1  v , its power function is constant -1, and the power function of the edge C. In the graph, for  k dimensional logic variable there is a complete sub-graph , its vertex number is k 2 , of cause its edge number is   . This complete sub-graph can be called as primary sub-graph; D. In the graph, for n -contradiction factors there are   v is connected to every positive vertex, and the sink vertex 1  v is connected to every negative vertex; F. To the every cycle m H in the graph, the formula as below must be satisfied: , to the total edges conjugated with vertex i v , the formula as below is satisfied: In which incoming edges is positive and outgoing edges is negative; I.The formula below is defined as a contradiction function , then the definite integral above is defined as that an energy source produced on edge ij E ; if the 0  ij w , then the definite integral above is defined as that work done on the edge ij E .Thus we have the energy conversation law as below: i.e. the algebraic sum of energy produced on every edge and work done on every edge is zero in the graph.Now we have a series of definitions to a sort of graph, the sort of graph can be called as Kirchhoff graph.Because of the graph is satisfied by Kirchhoff laws.

Kirchhoff Matrix
From definition of Kirchhoff graph above, we can obtain the Kirchhoff matrices via special adjacent matrix for Kirchhoff graph.
A. The first row of Kirchhoff power function matrix is the power function sequence below:  its first column is the transposed sequence of first row; B. Its other elements of power function matrix is the power function The first row of Kirchhoff flow function matrix is the vertex sequence : The first row and first column of Kirchhoff power capacity function matrix are as same as the definition in A., its other elements are that power function E. The first row and first column of Kirchhoff work and energy infinite integral matrix are as same as the definition in A., its other elements are that definite integral So we have the further properties about Kirchhoff matrix below.

Flow Function Matrix
In the Kirchhoff flow function matrix above, the element of the matrix is every flow function .The time t in vertex sequence symbols of the first row and the first column is deleted to save page space.

Contradiction Function Matrix
In the Kirchhoff contradiction function matrix above, the element of the matrix is every contradiction function . The time t in vertex sequence symbols of the first row and the first column is deleted to save page space.

Power Capacity Function Matrix
In the Kirchhoff power capacity matrix above, the element of the matrix is every power capacity function . The time t in vertex sequence symbols of the first row and the first column is deleted to save page space.

Work And Energy Definite Integral Matrix
In the Kirchhoff work and energy definite integral matrix above, the element of the matrix is is the matrix expression for Kirchhoff work and energy definite integral.

The Basic Properties of Kirchhoff Matrices
A. These Kirchhoff matrices above are all "every element on main diagonal is zero"; B. These Kirchhoff matrices above are all axis-symmetrical on the main diagonal taken as symmetrical axis, therefor the Kirchhoff matrices above are all triangular matrix; C. When a graph is named a inverse graph 1  G about graph G , if and only if the 1  G is isomorphic-equality to the graph G , but its direction of every edge is inverse.The Kirchhoff matrix above has been divided into two graphs by the main diagonal, in which one is the Kirchhoff graph and another is the inverse graph of the Kirchhoff graph.With another word, as a boundary by the main diagonal, the matrix can be divided into two parts: the right-upper and the left-lower.The corresponding graph of right-upper and left-lower are inverse graph each other; D. The blocking of Kirchhoff matrices Therefor the flow function Kirchhoff blocked matrix is as below: space such as river channel, highway, railway, etc. Author has explained the situation depended only on time t (Yaozhi Jiang, 2017), now author will explain the situations in which true value function need discrete dynamical sampling system to determine their initial true value function or need multidimensional discrete dynamical sampling system for discrete multiple Fourier transform to analyse true value function properties in frequency domain.In this paper, the true value function the Dirichlet conditions, i.e. the Fourier transform of both the true value functions as above are all existent.
Graph and Kirchhoff Matrices of Multidimensional Logic Variable (Maden, A. D., Cevik, A. S., Cangul, I. N. et al. 2013) is the transposed sequence of first row, and other elements of flow function matrix is the flow function contradiction function matrix is depended on the definition in A., its first row and first column are as same as the definition in A., its other elements are contradiction function . The time t in vertex sequence symbols of the first row and the first column is deleted to save page space.
sub-graph shown in the paper(Yaozhi Jiang., 2018).The blocked matrix's sub-matrix those are axis-symmetrical on main diagonal is transposed matrix each other.We have blocked only on the Kirchhoff flow function matrix, actually we can block on all Kirchhoff matrices defined above, if the blocking is necessary to our calculation; E. The calculating for every vertex flow function in Kirchhoff matrix First, we define a Square-ruler when we move from the first place element of flow function matrix one by one follow the row (or column), if we meet the element zero in main diagonal then we turn from the element zero to follow the column (or row), this looks like a light reflected by the main diagonal, that the every row element or every column element in the Kirchhoff flow function matrix is the adjacent relationship from first place vertex of row or column to the other vertexes on the same row or column.Therefor we can obtain the below:In the Kirchhoff flow function matrix, on the the 1  v row or column and 1 ,  n k v row or column, algebraic sum of the flow function on every row or every column is zero.Or in the Kirchhoff matrix, on the other row or column, the algebraic sum of flow function on every Square-ruler is zero.This is the matrix expression of Kirchhoff flow function law above;F.The calculating for cycle power function in Kirchhoff matrixAt first, we show an example of Kirchhoff power function matrix and its corresponding Kirchhoff graph below.
Fig.1 the example for correspondence Kirchhoff graph the vertexes set and , if the edge exists.A graph can be called as a Kirchhoff graph, if and only if:The graph is a connected, directed, no self-loop, no multiple edge and Kirchhoff-weighted graph ;A.In the graph every vertex ij E is denoted the edge conjunction between vertex i v and vertex j v i v is weighted a power function   t v i , in which t is time variable; in the graph every edge ij E is weighted a power function   t p ij produced by Define the energy produced on the edge ij E or the work done on the edge ij E as below: exists).The time t in vertex sequence symbols of the first row and the first column is deleted to save page space.
ij E