Symmetric Hermitian decomposability criterion,decomposition, and its applications

The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.     

Compact Hermitian surfaces of constant antiholomorphic sectional curvatures

Compact Hermitian surfaces of constant antiholomorphic sectional curvatures with respect to the Riemannian curvature tensor and with respect to the Hermitian curvature tensor are considered. It is proved: a compact Hermitian surface of constant antiholomorphic Riemannian sectional curvatures is a self-dual Kaehler surface; a compact Hermitian surface of constant antiholomorphic Hermitian sectional curvatures is either a Kaehler surface of constant (non-zero) holomorphic sectional curvatures or a conformally flat Hermitian surface.

     

Twistorial examples of almost Hermitian manifolds with Hermitian Ricci tensor

We construct new twistorial examples of non-Kähler almost Hermitian manifolds with Hermitian Ricci tensor by means of a natural almost Hermitian structures on the twistor space of an almost Hermitian four manifold.     

The Cauchy-Kovalevskaya extension theorem in Hermitian Clifford analysis

Hermitian Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitian monogenic functions, of two Hermitian conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitian monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitian monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitian monogenics, i.e. homogeneous Hermitian monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.     

Differential operators on Hermitian Jacobi forms

We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms.     

Hermitian codes as generalized Reed-Solomon codes

Hermitian codes obtained from Hermitian curves are shown to be concatenated generalized Reed-Solomon codes. This interpretation of Hermitian codes is used to investigate their structure. An efficient encoding algorithm is given for Hermitian codes. A new general decoding algorithm is given and applied to Hermitian codes to give a decoding algorithm capable of decoding up to the full error correcting capability of the code.This work is supported by a Natural Science and Engineering Research Council Grant A7382.     

关于非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解及其扰动估计

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^＊X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.     

Hermitian Groups and K-Theory

This article provides the fundamental constructions and results for Hermitian groups which are necessary for an algebraic understanding of the functors K₁ and K₂ of Hermitian forms. These include a definition of the elementary subgroup EH of a general Hermitian group GH, the Hermitian Whitehead lemma for stabilized GH, a definition of the Hermitian Steinberg group, and the theorem that a stabilized Steinberg group is the universal central extension of its associated stabilized elementary group.     

1-Universal binary and ternary Hermitian lattices over imaginary quadratic fields

The Ramanujan Journal - A positive definite Hermitian lattice is said to be 1-universal if it represents all positive definite unary Hermitian lattices, including both free and non-free Hermitian...     

On Hermitian LCD codes and their Gray image

We provide methods and algorithms to construct Hermitian linear complementary dual (LCD) codes over finite fields. We study existence of self-dual basis with respect to Hermitian inner product, and as an application, we construct Euclidean LCD codes by projecting the Hermitian codes over such a basis. Many optimal quaternary Hermitian and ternary Euclidean LCD codes are obtained. Comparisons with classical constructions are made.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Some aspects of Hermitian Jacobi forms

We introduce a certain differential (heat) operator on the space of Hermitian Jacobi forms of degree 1, show its commutation with certain Hecke operators and use it to construct a map from elliptic cusp forms to Hermitian Jacobi cusp forms. We construct Hermitian Jacobi forms as the image of the tensor product of two copies of Jacobi forms and also from the differentiation of the variables. We determine the number of Fourier coefficients that determine a Hermitian Jacobi form and use the differential operator to embed a certain subspace of Hermitian Jacobi forms into a direct sum of modular forms for the full modular group.     

2-universal Hermitian lattices over imaginary quadratic fields

A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields.     

Hermitian流形上的Bochner公式及其应用

在Hermitian流形上,将Bochner公式推广到了复向量丛上,并以此得到了Hermitian流形之间的调和映射的解析性质.     

矩阵的酉不变范数不等式

利用凹函数和半正定矩阵的性质,讨论并且得到了一些矩阵Rotfel型范数不等式.另外,通过研究Hermitian矩阵和斜Hermitian矩阵和的特征值的模行列式的不等式,得到一些关于Hermitian矩阵和斜Hermitian矩阵和的范数不等式.推广了文献中的相关结果.     

Harmonic maps between compact Hermitian manifolds

In this paper, we generalize the Bochner-Kodaira formulas to the case of Hermitian com- plex (possibly non-holomorphic) vector bundles over compact Hermitian (possibly non-K¨ahler) mani- folds. As applications, we get the complex analyticity of harmonic maps between compact Hermitian manifolds.     

Some curvature properties of complex surfaces

Summary In this paper, we are investigating curvature properties of complex two-dimensional Hermitian manifolds, particularly in the compact case. To do this, we start with the remark that the fundamental form of such a manifold is integrable, and we use the analogy with the locally conformal KÄhler manifolds, which follows from this remark. Among the obtained results, we have the following: a compact Hermitian surface for which either the Riemannian curvature tensor satisfies the KÄhler symmetries or the Hermitian curvature tensor satisfies the Riemannian Bianchi identity is KÄhler; a compact Hermitian surface of constant sectional curvature is a flat KÄhler surface; a compact Hermitian surface M with nonnegative nonidentical zero holomorphie Hermitian bisectional curvature has vanishing plurigenera, c₁(M) 0, and no exceptional curves; a compact Hermitian surface with distinguished metric, and positive integral Riemannian scalar curvature has vanishing plurigenera, etc.     

Several variants of the Hermitian and skew‐Hermitian splitting method for a class of complex symmetric linear systems

This paper is concerned with several variants of the Hermitian and skew‐Hermitian splitting iteration method to solve a class of complex symmetric linear systems. Theoretical analysis shows that several Hermitian and skew‐Hermitian splitting based iteration methods are unconditionally convergent. Numerical experiments from an n‐degree‐of‐freedom linear system are reported to illustrate the efficiency of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.