New Criteria for Judging Generalized Strictly Diagonally Dominant Matrix

Generalized strictly diagonally dominant matrices play a wide and important role in computational mathematics, mathematical physics, theory of dynamical systems, etc.But it is difficult to judge a matrix is or not generalized strictly diagonally dominant matrix.In this paper, by using the properties of α-chain diagonally dominant matrix, we obtain new criteria for judging generalized strictly diagonally dominant matrix, which enlarge the identification range.

Criteria Conditions for Generalized Diagonally Dominant Matrices

In this paper, some new sufficient conditions for generalized dominant matrices are given, and some results in [1] are generalized and improved.

Criteria Conditions for Generalized Diagonally Dominant Matrices

In this paper, we provide some new criteria conditions for generalized strictly diagonally dominant matrices, such that the corresponding results in [1] are generalized and improved.

Criteria Condition for Generalized Diagonally Dominant Matrices

In this paper, we provide some new necessary and sufficient conditions for generalized diagonally dominant matrices and also obtain some criteria for nongeneralized dominant matrices.

The Tatonnement of Leon Walras and Diagonal Dominant Hypothesis

This essay poses Walras＇s theory of price mechanism in its merits and limitations. Walras proposed two laws as conditions for general equilibrium, namely： （1） the law of the variation of equilibrium prices, a subjective condition; and （2） the law of the establishment of equilibrium prices, an objective condition. Walras jointed both laws in order to develop his law of supply and demand. This paper offers a formal Walrasian approximation in terms of the Lyapounov＇s function, taking the diagonal dominant hypothesis as departure point, rediscovered almost a century after it was originally proposed by Walras. The paper concludes with critical reflection concerning the idea of equilibrium economics as medium of social cohesion.

非奇异H-张量的新判定

H-tensor plays an important role in identifying positive definiteness of even order real symmetric tensors.In this paper,some definitions and theorems related to H-tensors are introducedfirstly.Secondly,some new criteria for identifying nonsingular H-tensors are proposed,moreover,a new theorem for identifying positive definiteness of even order real symmetric tensors is obtained.Finally,some numerical examples are given to illustrate our results.

The Distribution of Eigenvance of Product Diagonal Dominance Matrix

Let DD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(jj)|≥A_iA_j,i≠j,i,j∈N}.PD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(kk)|≥A_iA_jA_k,i≠j≠k,i,j,k∈N}. In this paper,we show DD_0(R)PD_0(R),and the conditions under which the numbers of eigen vance of A∈PD_0(R)\DD_0(R)are equal to the numbers of a_(ii),i∈N in positive and negative real part respectively.Some couter examples are given which present the condnions can not be omitted.

A New Approximation to the Linear Matrix Equation AX = B by Modification of He’s Homotopy Perturbation Method

It is well known that the matrix equations play a significant role in engineering and applicable sciences. In this research article, a new modification of the homotopy perturbation method (HPM) will be proposed to obtain the approximated solution of the matrix equation in the form AX = B. Moreover, the conditions are deduced to check the convergence of the homotopy series. Numerical implementations are adapted to illustrate the properties of the modified method.

Sufficient Conditions of Nonsingular H-matrices

From the concept of α diagonally dominant matrix, two sufficient conditions of nonsingular H-matrices were obtained in this paper. An example was given to show that these results improve the known results.

Some Criteria for H-Tensors

The H-tensor is a new developed concept in tensor analysis and it is an extension of the M-tensor.In this paper,we present some criteria for identifying nonsingular H-tensors and give two numerical examples.

Criteria of the Nonsingular H-Matrices

The nonsingular H-matrices play an important role in the study of the matrix theory and the iterative method of systems of linear equations, etc. It has always been searched how to verify nonsingular H-matrices. In this paper, nonsingular H-matrices is studies by applying diagonally dominant matrices, irreducible diagonally dominant matrices and comparison matrices and several practical criteria for identifying nonsingular H-matrices are obtained.

Recent developments on applications of sequential loop closing and diagonal dominance control schemes to industrial multivariable system

With a focus on an industrial multivariable system, two subsystems including the flow and the level outputs are analysed and controlled, which have applicability in both real and academic environments. In such a case, at first, each subsystem is distinctively represented by its model, since the outcomes point out that the chosen models have the same behavior as corresponding ones. Then, the industrial multivariable system and its presentation are achieved in line with the integration of these subsystems, since the interaction between them can not actually be ignored. To analyze the interaction presented, the Gershgorin bands need to be acquired, where the results are used to modify the system parameters to appropriate values. Subsequently, in the view of modeling results, the control concept in two different techniques including sequential loop closing control(SLCC) scheme and diagonal dominance control(DDC) schemes is proposed to implement on the system through the Profibus network, as long as the OPC(OLE for process control) server is utilized to communicate between the control schemes presented and the multivariable system. The real test scenarios are carried out and the corresponding outcomes in their present forms are acquired. In the same way, the proposed control schemes results are compared with each other, where the real consequences verify the validity of them in the field of the presented industrial multivariable system control.

Necessary and Sufficient Condition for Generalized Diagonal Dominance Matrices

In the paper,a necessary and sufficeent condition for generalized diagonal domiance matrices is given.Further, the relations among all generalized positive definite matrices are shown, also,some flaws and mistakes in the references are corrected.

Criteria for Stability of Linear Control Systems

As it is well known, it is significant to know whether a matrix is an H-matrix or not in stability analysis of linear control systems. However, to distinguish H-matrices is difficult in real applications. In this paper, a practical extension of the sufficient conditions for H-matrices is investigated under some conditions. A larger scale of H-matrices which can be judged by the proposed method is shown by the numerical examples.

H-tensors and nonsingular H-tensors

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order H-tensors. In this paper, we establish important properties of diagonally dominant tensors and H-tensors. Distributions of eigenvalues of nonsingular symmetricH-tensors are given. An J（t%-tensor is semi-positive, which enlarges the area of semi-positive tensor from H-tensor to H＋-tensor. The spectral radius of Jacobi tensor of a nonsingular （resp. singular） H-tensor is less than （resp. equal to） one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular H-tensor if and only if all of its principal sub-tensors are nonsingular H-tensors. An irreducible tensor H is an H-tensor if and only if it is quasi-diagonally dominant.

Bi-block positive semidefiniteness of bi-block symmetric tensors

The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.

Criteria for strong H-tensors

H-tensor is a new developed concept which plays an important role in tensor analysis and computing. In this paper, we explore the properties of H-tensors and establish some new criteria for strong H-tensors. In particular, based on the principal subtensor, we provide a new necessary and sufficient condition of strong H-tensors, and based on a type of generalized diagonal product dominance, we establish some new criteria for identifying strong H-tensors. The results obtained in this paper extend the corresponding conclusions for strong H-matrices and improve the existing results for strong H-tensors.