Discriminant of symmetric matrices as a sum of squares and the orthogonal group

It is proved that the discriminant of n × n real symmetric matrices can be written as a sum of squares, where the number of summands equals the dimension of the space of n‐variable spherical harmonics of degree n. The representation theory of the orthogonal group is applied to express the discriminant of 3 × 3 real symmetric matrices as a sum of five squares and to show that it cannot be written as the sum of less than five squares. It is proved that the discriminant of 4 × 4 real symmetric matrices can be written as a sum of seven squares. These improve results of Kummer from 1843 and Borchardt from 1846. © 2010 Wiley Periodicals, Inc.     

On a class of hyperbolic polynomials

In the present paper, we investigate whether the roots of a biquadratic equation determined by a pair of real symmetric positive definite matrices of order 3 and a three-dimensional vector of parameters are real. We obtain the explicit representation of the discriminant of such a polynomial as the sum of at most two squares.     

The entropic discriminant

The entropic discriminant is a non-negative polynomial associated to a matrix. It arises in contexts ranging from statistics and linear programming to singularity theory and algebraic geometry. It describes the complex branch locus of the polar map of a real hyperplane arrangement, and it vanishes when the equations defining the analytic center of a linear program have a complex double root. We study the geometry of the entropic discriminant, and we express its degree in terms of the characteristic polynomial of the underlying matroid. Singularities of reciprocal linear spaces play a key role. In the corank-one case, the entropic discriminant admits a sum of squares representation derived from the discriminant of a characteristic polynomial of a symmetric matrix.     

Invariant theoretic characterization of subdiscriminants of matrices

An invariant theoretic characterization of subdiscriminants of matrices is given. The structure as a module over the special orthogonal group of the minimal degree non-zero homogeneous component of the vanishing ideal of the variety of real symmetric matrices with a bounded number of different eigenvalues is investigated. These results are applied to the study of sum of squares presentations of subdiscriminants of real symmetric matrices.     

Generation of symmetric exponential sums

In this paper, a new method for generation of infinite series of symmetric identities written for exponential sums in real numbers is proposed. Such systems have numerous applications in theory of numbers, chaos theory, algorithmic complexity, dynamic systems, etc. Properties of generated identities are studied. Relations of the introduced method for generation of symmetric exponential sums to the Morse-Hedlund sequence and to the theory of magic squares are established.     

Characteristic polynomials of symmetric matrices III: some counterexamples

When is a monic polynomial the characteristic polynomial of a symmetric matrix over an integral domain D? Known necessary conditions are shown to be insufficient when D is the field of 2-adic numbers and when D is the rational integers. The latter counterexamples lead to totally real cubic extensions of the rationals whose difierents are not narrowly equivalent to squares. Furthermorex³-4x+1 is the characteristic polynomial of a rational symmetric matrix and is the characteristic polynomial of an integral symmetric p-adic matrix for every prime p, but is not the characteristic polynomial of a rational integral symmetric matrix.     

On Parallel Fast Jacobi Algorithm for the Eigenproblem of Real Symmetric Matrices

Jacobi algorithm has been developed for the eigenproblem of real symmetric matrices, singular value decomposition of matrices and least squares of the overdetermined system on a parallel computer. In this paper, the parallel schemes and fast algorithm are discussed, and the error analysis and a new bound are presented.     

On the Hilbert 2-class field tower of some abelian 2-extensions over the field of rational numbers

It is well known by results of Golod and Shafarevich that the Hilbert 2-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian 2-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian 2-extension over ? in which eight primes ramify and one of theses primes ≡ ?1 (mod 4), the Hilbert 2-class field tower is infinite.     

Completely J —positive linear systems of finite order

Completely J — positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J — positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J — symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J — symmetric realizations are discussed.     

The generalized centro‐symmetric and least squares generalized centro‐symmetric solutions of the matrix equation AYB + CYTD = E

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P^?1 = P^T. An n × n real matrix Y is called a generalized centro‐symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro‐symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro‐symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro‐symmetric matrix, a generalized centro‐symmetric solution (least squares generalized centro‐symmetric solution) can be obtained within a finite number of iterations in the absence of round‐off errors, and the least Frobenius norm generalized centro‐symmetric solution (the minimal Frobenius norm least squares generalized centro‐symmetric solution) can be derived by choosing a special kind of initial generalized centro‐symmetric matrices. We also obtain the optimal approximation generalized centro‐symmetric solution to a given generalized centro‐symmetric matrix Y₀ in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

矩阵方程AXB=C最小二乘D反对称解

首先将对称矩阵推广到D反对称矩阵,然后研究了方程AXB=C的D反对称最小二乘解,利用矩阵对的广义奇异分解、标准相关分解及子空间上的投影定理,得到了最小二乘解的通式．     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.     

On the negative squares of indefinite Sturm-Liouville operators

The number of negative squares of all self-adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a Krein space is characterized in terms of the behaviour of an abstract Titchmarsh-Weyl function near 0 and ∞. These results are applied to a large class of symmetric and self-adjoint indefinite Sturm-Liouville operators with indefinite weight functions.     

Numerical solution of bed load transport equations using discrete least squares meshless (DLSM) method

The discrete least squares meshless (DLSM) method is extended in this paper for solving coupled bedload sediment transport equations. The mathematical formulation of this model consists of shallow water equations for the hydrodynamical component and an Exner equation expressing sediment continuity for the bedload transport. This method uses the moving least squares (MLS) function approximation to construct the shape functions and the minimizing least squares functional method to discretize the system of equations. The method can be viewed as a truly meshless method as it does not need any mesh for both field variable approximation and the construction of system matrices; it also provides the symmetric coefficient matrix. In the present work, several benchmark problems are studied and compared with the work of other researchers; the proposed method shows good accuracy, high convergence rate, and high efficiency, even for irregularly distributed nodes. At the end, a real test problem is performed to show and verify the main benefit and applicability of the proposed method to cope with complex geometry in practical problems.     

An MM Algorithm for Multicategory Vertex Discriminant Analysis

This article introduces a new method of supervised learning based on linear discrimination among the vertices of a regular simplex in Euclidean space. Each vertex represents a different category. Discrimination is phrased as a regression problem involving ?-insensitive residuals and a quadratic penalty on the coefficients of the linear predictors. The objective function can by minimized by a primal MM (majorization–minimization) algorithm that (a) relies on quadratic majorization and iteratively re-weighted least squares, (b) is simpler to program than algorithms that pass to the dual of the original optimization problem, and (c) can be accelerated by step doubling. Limited comparisons on real and simulated data suggest that the MM algorithm is competitive in statistical accuracy and computational speed with the best currently available algorithms for discriminant analysis.     

Composite quantile regression for single-index models with asymmetric errors

For the single-index model, a composite quantile regression technique is proposed in this paper to construct robust and efficient estimation. Theoretical analysis reveals that the proposed estimate of the single-index vector is highly efficient relative to its corresponding least squares estimate. For the single-index vector, the proposed method is always valid across a wide spectrum of error distributions; even in the worst case scenario, the asymptotic relative efficiency has a lower bound 86.4 %. Meanwhile, we employ weighted local composite quantile regression to obtain a consistent and robust estimate for the nonparametric component in the single-index model, which is adapted to both symmetric and asymmetric distributions. Numerical study and a real data analysis can further illustrate our theoretical findings.     

奇数阶泛对角线雪花幻方的构作

The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m l, 6m 5 and 6m 3, m is an odd positive integer and m is an even positive integer 9|6m 3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to the matrices with symmetric partial difference in each direction for orders 6m 1 , 6m 5; to the trisection matrices with symmetric partial difference in each direction for order 6m 3.     

A splitting preconditioner for saddle point problems

For large sparse systems of linear equations iterative techniques are attractive. In this paper, we study a splitting method for an important class of symmetric and indefinite system. Theoretical analyses show that this method converges to the unique solution of the system of linear equations for all t>0 (t is the parameter). Moreover, all the eigenvalues of the iteration matrix are real and nonnegative and the spectral radius of the iteration matrix is decreasing with respect to the parameter t. Besides, a preconditioning strategy based on the splitting of the symmetric and indefinite coefficient matrices is proposed. The eigensolution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomials for the preconditioned matrix is obtained. Numerical experiments of a model Stokes problem and a least‐squares problem with linear constraints presented to illustrate the effectiveness of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

The recognition of symmetric latin squares

A latin square S is isotopic to another latin square S′ if S′ can be obtained from S by permuting the row indices, the column indices and the symbols in S. Because the three permutations used above may all be different, a latin square which is isotopic to a symmetric latin square need not be symmetric. We call the problem of determining whether a latin square is isotopic to a symmetric latin square the symmetry recognition problem. It is the purpose of this article to give a solution to this problem. For this purpose we will introduce a cocycle corresponding to a latin square which transforms very simply under isotopy, and we show this cocycle contains all the information needed to determine whether a latin square is isotopic to a symmetric latin square. Our results relate to 1‐factorizations of the complete graph on n + 1 vertices, K_{n + 1}. There is a well known construction which can be used to make an n × n latin square from a 1‐factorization on n + 1 vertices. The symmetric idempotent latin squares are exactly the latin squares that result from this construction. The idempotent recognition problem is simple for symmetric latin squares, so our results enable us to recognize exactly which latin squares arise from 1‐factorizations of K_{n + 1}. As an example we show that the patterned starter 1‐factorization for the group G gives rise to a latin square which is in the main class of the Cayley latin square for G if and only if G is abelian. Hence, every non‐abelian group gives rise to two latin squares in different main classes. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 291–300, 2008     

Oppenheim不等式的一种类似

证明了实对称正定矩阵或实对称半正定矩阵与 M-矩阵的 Hadamard乘积满足实对称正定矩阵 Hadamard乘积的 Oppenheim不等式 .