强正定二次型与Weyl-Heisenberg 框架的构造

本文研究了一类具有特殊结构的无限维二次型, 得到这类二次型的对称矩阵是符号为多项式的模的平方的Laurent 矩阵, 进一步得到了这类二次型是强正定的判断标准以及一类Weyl-Heisenberg 框架的构造. 本文还研究了这类二次型的矩阵的所有有限维主对角子矩阵的强正定性, 并由此得到一类子空间Weyl-Heisenberg 框架的构造. 最后举例说明本文的主要结果及其应用. 本文建立了两个看似不相关的领域间的联系.     

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it? First, we show that the convexity of the product is determined intrinsically by the condition number of so-called ‘scaled matrices’ associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer’s fixed point of a mapping) with a special structure. Thus, a broader question than the open “Question 11” in Hiriart-Urruty (SIAM Rev. 49, 225–273, 2007) is addressed in this paper.     

Two problems of the theory of quadratic maps

Given quadratic forms q ₁, …, q _k , two questions are studied: Under what conditions does the set of common zeros of these quadratic forms consist of the only point x = 0? When is the maximum of these quadratic forms nonnegative or positive for any x ≠ 0? Criteria for each of these conditions to hold are obtained. These criteria are stated in terms of matrices determining the quadratic forms under consideration.     

Spatial design matrices and associated quadratic forms: structure and properties

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes. We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.     

Sur les génératrices extrémales de certains cônes de formes quadratiques doublement positives

We consider two classes of doubly positive quadratic forms on the space S of real symmetric matrices of order d ≥ 2, and prove that, for d = 2, these forms are the only extremal ones. This result fails for d ≥ 3.     

Composition of binary integral quadratic forms via integral 2 × 2 matrices and composition of matrix classes

A treatment by integral matrices is given for composition of pairs of integral quadratic forms with the same discriminant. The forms are associated with a pair of similar 2 × 2 matrices AB with irrational eigen values which generate the maximal order. The most general integral similarity between AB is given by a matrix whose entries are linear forms in two indeterminates with integral coefficients. This matrix is a "compositum" of two factors of the same nature. By equating determinants a composition of two quadratic forms results. The method can be generalized to n × n matrices.     

On linear transformations preserving rank one matrices over commutative rings

We show that over any cummutative ring R,the combinations, of 2 × 2 minors are the only quadratic forms vanishing on the matrices of rank 1. Hence any invertible linear transformation on matrices that preserves the rank-1 set over R will automatically do the same over all extensions of R. Similarly, the linear combinations of 4 × 4 Paffians are the only quadratic forms vanishing on the alternating matrices of rank 2. Hence again any invertible transformation preserving that set over R will do so formally. This fact allows us to determine the collection of such transformations     

On Copositive Programming and Standard Quadratic Optimization Problems

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FF ^T where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.     

On bilinear forms in normal variables

Bilinear forms in normal variables when the matrices of the forms are rectangular are considered. Explicit expressions for the cumulants, joint cumulants and joint cumulants of bilinear and quadratic forms are given. Necessary and sufficient conditions are established for the independence of two bilinear forms as well as a bilinear and a quadratic form. Special cases are shown to agree with known results.     

Symmetric Bilinear Forms over Valuation Rings

本文讨论赋值环上的对称线性型、二次型和对称矩阵的合同标准形。     

On the quadratic estimation of covariance matrices in multivariate linear models

This paper investigates the estimation of covariance matrices in multivariate mixed models. Some sufficient conditions are derived for a multivariate quadratic form and a linear combination of multivariate quadratic forms to be the BQUE (quadratic unbiased and severally minimum varianced) estimators of its expectations.     

Representations of an even zeta function on theta series

Representation matrices are computed for the Hecke operators corresponding to the coefficients of an even zeta function of the symplectic group on the theta series of positive integral quadratic forms with an even number of variables belonging to a fixed genus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 5–14, 1984.     

Quadratic factorization heuristics for copositive programming

Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem, where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems. In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results on three different classes of test problems are quite promising.     

A class of bivariate exponential distributions

We introduce a class of absolutely continuous bivariate exponential distributions, generated from quadratic forms of standard multivariate normal variates.This class is quite flexible and tractable, since it is regulated by two parameters only, derived from the matrices of the quadratic forms: the correlation and the correlation of the squares of marginal components. A simple representation of the whole class is given in terms of 4-dimensional matrices. Integral forms allow evaluating the distribution function and the density function in most of the cases.The class is introduced as a subclass of bivariate distributions with chi-square marginals; bounds for the dimension of the generating normal variable are underlined in the general case.Finally, we sketch the extension to the multivariate case.     

Quadratic fermionic dynamics with dissipation

Gaussian solutions of the Cauchy problem for the GKS-L equation (in the Schrödinger picture) with quadratic fermionic generators are obtained. These Gaussian solutions are represented both as exponentials of quadratic forms in fermionic creation-annihilation operators and by their normal symbols. The coefficients of these forms are represented as algebraic functions of matrices.

     

2-Universal Positive Definite Integral Quinary Diagonal Quadratic Forms

As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms.     

Finding Symmetry Groups of Some Quadratic Programming Problems

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help decrease the problem dimension, reduce the size of the search space by means of linear cuts. While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space, the present paper considers a larger group of invertible linear transformations. We study a special case of the quadratic programming problem, where the objective function and constraints are given by quadratic forms. We formulate conditions, which allow us to transform the original problem to a new system of coordinates, such that the symmetries may be sought only among orthogonal transformations. In particular, these conditions are satisfied if the sum of all matrices of quadratic forms, involved in the constraints, is a positive definite matrix. We describe the structure and some useful properties of the group of symmetries of the problem. Besides that, the methods of detection of such symmetries are outlined for different special cases as well as for the general case.     

Trace forms, kronecker sums, and the shuffle matrix

The trace map on the ring of square matrices with entries in a field can be used to define various quadratic forms on this ring. This paper makes a study of some of these forms and in particular the "scaled trace forms" are shown to have a symmetric matrix representation involving both Kronecker sums and the shuffle matrix.     

A decomposition for a stochastic matrix with an application to MANOVA

The aim of this paper is to propose a simple method in order to evaluate the (approximate) distribution of matrix quadratic forms when Wishartness conditions do not hold. The method is based upon a factorization of a general Gaussian stochastic matrix as a special linear combination of nonstochastic matrices with the standard Gaussian matrix. An application of previous result is proposed for matrix quadratic forms arising in MANOVA for a multivariate split-plot design with circular dependence structure.