Monomorphism categories associated with symmetric groups and parity in finite groups

Monomorphism categories of the symmetric and alternating groups are studied via Cayley’s Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley’s theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.     

COMPLEX INTERPOLATION OF Lp（C,H）SPACES WITH RESPECT TO CULLEN-REGULAR

The aim of this paper is to prove a new version of the Riesz-Thorin interpolation theorem on LP(C,H).In the sense of Cullen-regular,we show Hadamard’s three-lines theorem by means of the Maximum modulus principle on a symmetric slice domain.In addition,two applications of the Riesz-Thorin theorem are presented.Finally,we investigate two kinds of Calderón’s complex interpolation methods in LP(C,H).     

Groups with the same order and degree pattern

The degree pattern of a finite group has been introduced in [18].A group M is called k-fold OD- characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M .In particular,a 1-fold OD-characterizable group is simply called OD-characterizable.It is shown that the alternating groups A m and A m+1 ,for m = 27,35,51,57,65,77,87,93 and 95,are OD-characterizable,while their automorphism groups are 3-fold OD-characterizable.It is also shown that the symmetric groups S m+2 ,for m = 7,13,19,23,31,37,43,47,53,61,67,73,79,83,89 and 97,are 3-fold OD-characterizable.From this,the following theorem is derived.Let m be a natural number such that m 100.Then one of the following holds: (a) if m = 10,then the alternating groups A m are OD-characterizable,while the symmetric groups S m are OD- characterizable or 3-fold OD-characterizable;(b) the alternating group A 10 is 2-fold OD-characterizable;(c) the symmetric group S 10 is 8-fold OD-characterizable.This theorem completes the study of OD-characterizability of the alternating and symmetric groups A m and S m of degree m 100.     

SYMMETRIC INTEGRAL AND THE APPROXIMATION THEOREM OF STOCHASTIC INTEGRAL IN THE PLANE

In this paper, we consider the approximation problem of stochastic integral with respect to two-parameter Wiener process. We first introduce a kind of symmetric integral and prove it obeys the chain rule. Then we apply an integral formula of bounded variation functions with two variables to show the approximation theorem of stochastic integral in the plane. In particular, we prove that the symmetric stochastic integral is stable when the limit is taken in the sense of L~2convergence.     

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.     

Inequalities on Positive Semidefinite Hermitian Matrix

1.Introduction. We use A≥0 to denote that A is a positive semidefinite hermitian matrix. S_n denotes the symmetric group of order n, e is the identity of S_n. Two inequalities are given in the follwoing theorem Theorem If A=(α_(ij))≥0,A is of order n, we have     

一阶共形对称空间

The present paper concerns with conformally symmetric spaces of class one, and establish the following theorem.Theorem. If an n-dimensional (n≥4) conformally symmetric space is of class one, then the space is conformally flat or locally symmetric.     

A Vanishing Theorem on a Class of Hartogs Domain

In this paper, we consider the d-boundedness of the Bergman metric and a vanishing theorem of L²-cohomology on a class of Hartogs domain, whose base domain is the production of two irreducible bounded symmetric domains of the first type, by using the Bergman kernel function,invariant function, holomorphic automorphism group and so on.     

Deleting Vertices and Interlacing Laplacian Eigenvalues

The authors obtain an interlacing relation between the Laplacian spectra of a graph G and its subgraph G - U, which is obtained from G by deleting all the vertices in the vertex subset U together with their incident edges. Also, some applications of this interlacing property are explored and this interlacing property is extended to the edge weighted graphs.     

SOLUTIONS OF EULER-POISSON EQUATIONS IN R^n

In this article, the authors study the structure of the solutions for the EuierPoisson equations in a bounded domain of Rn with the given angular velocity and n is an odd number. For a ball domain and a constant angular velocity, both existence and nonexistence theorem are obtained depending on the adiabatic gas constant 7. In addition, they obtain the monotonicity of the radius of the star with both angular velocity and center density. They also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is different to the case of the non-rotating star.     

The fan-pall imbedding theorem over formally real fields

The converse of the Cauchy interlacing theorem, relating eigenvalues of a symmetric real matrix and eigenvalues of a principal submatrix, first proved by Fan and Pall, is extended to the case of symmetric matrices with entries in an arbitrary formally real field.     

The fan-pall imbedding theorem over formally real fields

The converse of the Cauchy interlacing theorem, relating eigenvalues of a symmetric real matrix and eigenvalues of a principal submatrix, first proved by Fan and Pall, is extended to the case of symmetric matrices with entries in an arbitrary formally real field.     

关于Z－矩阵的弱正则分裂

本文主要考虑一类Ｚ－矩阵弱正则分裂的迭代矩阵的谱半径和特征值１的指数问题，推广了Ｒｏｓｅ在１９８４年得到的一个定理．     

Some interlacing theorems on the schur complement

We give a short proof of the interlacing theorem on the eigenvalues of Schur complements of Hermitian matrices due to Hu and Smith, and extend the technique to prove an interlacing theorem on the singular values of Schur complements of rectangular complex matrices.     

The precise consistency consensus matrix in a local AHP-group decision making context

In matrix theory, Fu and Markham showed using majorization technique that if a Hermitian matrix satisfies certain conditions, then the matrix must be block-diagonal. In this paper, we extend this result to the setting of simple Euclidean Jordan algebras by using the Cauchy interlacing theorem and the Schur complement Cauchy interlacing theorem.     

Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains—an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hitting-time distributions for finite-state Markov chains started in stationarity. The proof, and corresponding construction, for one of the two theorems elucidates an intriguing connection between hitting-time distributions and the interlacing eigenvalues theorem for bordered symmetric matrices.     

Matrices with prescribed Ritz values

On the way to establishing a commutative analog to the Gelfand-Kirillov theorem in Lie theory, Kostant and Wallach produced a decomposition of M(n) which we will describe in the language of linear algebra. The “Ritz values” of a matrix are the eigenvalues of its leading principal submatrices of order m=1,2,…,n. There is a unique unit upper Hessenberg matrix H with those eigenvalues. For real symmetric matrices with interlacing Ritz values, we extend their analysis to allow eigenvalues at successive levels to be equal. We also decide whether given Ritz values can come from a tridiagonal matrix.     

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem

We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.

     

The graph with spectrum 141 240 (?4)10?(?6)9

We show that there is a unique graph with spectrum as in the title. It is a subgraph of the McLaughlin graph. The proof uses a strong form of the eigenvalue interlacing theorem to reduce the problem to one about root lattices.     

ON THE CONVERGENCE OFSPLITTINGS FOR A Z-MATRIX

In this paper, first we present a characterization of semiconvergence for nonnegative splittings of a singular Z-mattix, which generalizes the corresponding result of . Second, a characterization of convergence for L1-regular splittings of a singular E-matrix is given, which im-prove the resuh of [3]. Third, convergence of weak nonnegative splittings and regular splittings isdiscussed, and we obtain some necessary and sufficient conditions such that the splittings of a Z-matrix converge,