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Register a new userGlobal product structure for a space of special matrices
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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.
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An equivalence of matrices via semi-tensor product (STP) is proposed. Using this equivalence, the quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices leads to the product topology and quotient topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the quotient space. This metric leads to a metric topology. A comparison for these three topologies is presented. Some topological properties are revealed.
Hegyvari and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $Bcdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.
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.
We study a bilinear multiplication rule on 2x2 matrices which is intermediate between the ordinary matrix product and the Hadamard matrix product, and we relate this to the hyperbolic motion group of the plane.
In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(mathbb{F}_p)$ with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyv{a}ri and Hennecart.
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