A note on biquadratic forms

It is shown that semidefinite quadratic forms in two by n variables are sums of squares of bilinear forms.     

Tests of Parameters of Several Gamma Distributions with Inequality Restrictions

Gamma distribution is one of the most used methods of modeling lifetime data. However, testing homogeneity of parameters of m 3 gamma distributions against order restrictions is almost non-existent in the current literature. We propose two methods to this end: one uses quadratic forms involving ratios of cumulants as test statistic and the other is a stepwise procedure which uses Fisher's method of combining p-values when shape parameters are equal but unknown. Both procedures allow use of arbitrary sample sizes of m populations. Test of the inequality restrictions as a null hypothesis against unrestricted alternatives is also considered. A Monte Carlo study of power at various alternatives shows that both methods are competitive when they are applicable.     

Quadratic and Symmetric Bilinear Compositions of Quadratic Forms Over Commutative Rings

Over commutative rings in which 2 is a zero-divisor, to compose a quadratic form with symmetric bilinear forms or with quadratic forms is not quite the same. In this article, the relation between the two classes of compositions is clarified and the results applied to find the ranks of minimal compositions.     

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。     

A conjugate directions method for subjective assessment of normal distributions

A procedure is developed that enables the encoding of a subjectiven-dimensional joint normal probability density function through the assessment of its marginal means and variances andn(n–1)/2 conditional means. The new method is based on the theory of conjugate directions for quadratic forms, and it exploits the fact that normal distributions have quadratic equal-likelihood surfaces. Unlike previous approaches, this new method enables easy detection and resolution of inconsistencies in the assessments that could lead to an indefinite estimate of the covariance matrix.     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.     

用正交变换化实二次型为标准形的同步求解问题

作为《关于矩阵的特征值与特征向量同步求解问题》的续篇,利用其给出的方法,证明了新的定理.通过对实对称矩阵进行行列互逆变换,同步求出二次型的标准形及正交变换阵,简化了复杂的施密特正交化法,较好地解决了二次型标准形与正交变换阵同步求解问题.     

Finiteness theorems for positive definite -regular quadratic forms

An integral quadratic form of variables is said to be -regular if globally represents all quadratic forms of variables that are represented by the genus of . For any , it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of variables that are -regular. We also investigate similar finiteness results for almost -regular and spinor -regular quadratic forms. It is shown that for any , there are only finitely many equivalence classes of primitive positive definite spinor or almost -regular quadratic forms of variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).

     

Splitting Patterns and Invariants of Quadratic Forms

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices i_W(φ_k) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ Iⁿ F and give some bounds on m depending on n which assure that they have the same splitting pattern.     

The Probability Content of Cones in Isotropic Random Fields

This paper provides computable representations for the evaluation of the probability content of cones in isotropic random fields. A decomposition of quadratic forms in spherically symmetric random vectors is obtained and a representation of their moments is derived in terms of finite sums. These results are combined to obtain the distribution function of quadratic forms in spherically symmetric or central elliptically contoured random vectors. Some numerical examples involving the sample serial covariance are provided. Ratios of quadratic forms are also discussed.     

Quadratic forms that represent almost the same primes

Jagy and Kaplansky exhibited a table of pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that their list is complete outside of ``trivial' pairs. In this article, we confirm their conjecture, and in fact find all pairs of such forms that represent the same primes outside of a finite set.

     

Quadratic forms in skew normal variates

In this paper first a characterization of the multivariate skew normal distribution is given. Then the joint moment generating functions of two quadratic forms, and a linear compound and a quadratic form in skew normal variates, have been derived and conditions for their independence are given. Distribution of the ratios of quadratic forms in skew normal variates has also been studied.     

A Note on Zinbiel Algebras

In this paper, we consider the decomposition of Zinbiel algebras with a special bilinear form which are called quadratic Zinbiel algebras. We obtain that the decomposition into irreducible ideals of a ...     

On the definity of quadratic forms subject to linear constraints

A short proof is given of the necessary and sufficient conditions for the positivity and nonnegativity of a quadratic form subject to linear constraints.     

On splitting of totally singular quadratic forms

In this note we completely study the standard splitting of quasi-Pfister forms and their neighbors, and we include some general results on standard splitting towers of totally singular quadratic forms. The author was supported by the European research network HPRN-CT-2002-00287 “AlgebraicK-Theory, Linear Algebraic Groups and Related Structures”.     

Cost Cumulant Control: State-Feedback,Finite-Horizon Paradigm with Application to Seismic Protection

The paper is concerned with the study of the optimal control which minimizes a finite linear combination of the first k cost cumulants of a finite-horizon integral quadratic form associated with a linear stochastic system, when the controller measures the states. The solution is investigated by adapting dynamic programming techniques to the nontraditional forms evidenced by cumulant representations. The performance of this k-cost cumulant (kCC) controller is compared to that of the best control paradigms published for the American Society of Civil Engineers first-generation structure benchmark for seismically excited buildings; the simulation results indicate that the newly developed control paradigm makes better use of the available control limits and achieves uniform improvement in the officially defined performance statistics for floor vibrations and accelerations.     

Rankin-Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.