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Register a new userBasis for the linear space of matrices under equivalence
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Kuize Zhang
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The semi-tensor product (STP) of matrices which was proposed by Daizhan Cheng in 2001 [2], is a natural generalization of the standard matrix product and well defined at every two finite-dimensional matrices. In 2016, Cheng proposed a new concept of semi-tensor addition (STA) which is a natural generalization of the standard matrix addition and well defined at every two finite-dimensional matrices with the same ratio between the numbers of rows and columns [1]. In addition, an identify equivalence relation between matrices was defined in [1], STP and STA were proved valid for the corresponding identify equivalence classes, and the corresponding quotient space was endowed with an algebraic structure and a manifold structure. In this follow-up paper, we give a new concise basis for the quotient space, which also shows that the Lie algebra corresponding to the quotient space is of countably infinite dimension.
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Horacio Leyva
, Francisco A. Carrillo
, Baltazar Aguirre-Hernandez andn Jesus F. Espinoza
2018
The importance of the Hurwitz Metzler matrices and the Hurwitz symmetric matrices can be appreciated in different applications: communication networks, biology and economics are some of them. In this paper, we use an approach of differential topology for studying such matrices. Our results are as follows: the space of the $ntimes n$ Hurwitz symmetric matrices has a product manifold structure given by the space of the $(n-1) times (n-1)$ Hurwitz symmetric matrices and the euclidean space. Additionally we study the space of Hurwitz Metzler matrices and these ideas let us do an analysis of robustness of Hurwitz Metzler matrices. In particular, we study the Insulin Model as application.
In this paper, a Groebner-Shirshov basis for the Chinese monoid is obtained and an algorithm for the normal form of the Chinese monoid is given.
The well-known Deficiency One Theorem gives structural conditions on a chemical reaction network under which, for any set of parameter values, the steady states of the corresponding mass action system may be easily characterized. It is also known, however, that mass action systems are not uniquely associated with reaction networks and that some representations may satisfy the Deficiency One Theorem while others may not. In this paper we present a mixed-integer linear programming framework capable of determining whether a given mass action system has a dynamically equivalent or linearly conjugate representation which has an underlying network satisfying the Deficiency One Theorem. This extends recent computational work determining linearly conjugate systems which are weakly reversible and have a deficiency of zero.
We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerdens conjecture for HPSD matrices, where the polynomial optimization problem is interpreted as a relaxation of the permanent.
The paper continues the authors study of the linearizability problem for nonlinear control systems. In the recent work [K. Sklyar, Systems Control Lett. 134 (2019), 104572], conditions on mappability of a nonlinear control system to a preassigned linear system with analytic matrices were obtained. In the present paper we solve more general problem on linearizability conditions without indicating a target linear system. To this end, we give a description of invariants for linear non-autonomous single-input controllable systems with analytic matrices, which allow classifying such systems up to transformations of coordinates. This study leads to one problem from the theory of linear ordinary differential equations with meromorphic coefficients. As a result, we obtain a criterion for mappability of nonlinear control systems to linear control systems with analytic matrices.
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