Quadratic forms on completely symmetric spaces

Let V be an n-dimensional Euclidean vector space, and let V(^m) be the corresponding m-th completely symmetric space over V equipped with the induced inner product. The purpose of this paper is to prove the following conjecture of H.A. Robinson: if T is a linear operator on V(^m) and (Tz, z) = 0 for every decomposable element z of V(^m), then T is skew-symmetric.     

Quadratic forms on symmetry classes of tensors

Let V:₁,…,V_m be inner product spaces, and let L be a linear transformation on V₁ ?…?V_m which satisfies (Lz,z)=0 for every decomposable tensor z. It is known that if the field is the complex numbers, then (Lz,z)=0 for every z. This paper contains a short proof of this result, an extension of it to arbitrary symmetry classes of tensors, and an analysis of its failure when the field is the real numbers.     

Quadratic descent of hermitian forms

Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined result is also obtained for hermitian (resp. skew hermitian) forms over a quaternion algebra with symplectic (resp. orthogonal) involution.     

Quadratic forms between euclidean spheres

We study the relationship between the dimensions of euclidean spheres which admit a nonconstant homogeneous quadratic map between them. Givenm (respectivelyn), we determine the least (respectively greatest) possible value ofn (respectivelym) for which there exists a nonconstant homogeneous quadratic mapS ^m →S ⁿ . Research partially supported by NSF DMS-9201204     

Quadratic forms of closed manifolds

In this paper it is shown that for any integral-valued unimodular quadratic form and any number n of the form 8k + 4 (where k1), there exists a smooth closed n-dimensional manifold with this quadratic form. The proof is based on the construction (with the help of the plumbing construction) of smooth closed three-connected eight-dimensional manifolds with given form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 104–108, 1982.     

Quadratic polynomials represented by norm forms

Let ${P(t) \in \mathbb{Q}[t]}$ be an irreducible quadratic polynomial and suppose that K is a quartic extension of ${\mathbb{Q}}$ containing the roots of P(t). Let ${{\bf N}_{K/\mathbb{Q}}({\rm x})}$ be a full norm form for the extension ${K/\mathbb{Q}}$ . We show that the variety $$\begin{array}{ll}P(t)={\bf N}_{K/\mathbb{Q}}({\rm x})\neq 0\end{array}$$ satisfies the Hasse principle and weak approximation. The proof uses analytic methods.     

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.     

Quadratic forms of a matric-t variate

Summary Crowther [2] studied the distribution of a quadratic form in a matrix normal variate. This, in some sense, is extended by De Waal [4]. They represented the density function of this quadratic form in terms of generalized Hayakawa polynomials. Application of some specific results of these authors facilitates the derivation of distributions of quadratic forms of the matric-t variate. Attention is also given to the distributions of the characteristic roots and the trace of this quadratic matrix. Special cases are considered and some useful integrals are formulated. Financially supported by the CSIR and the University of the Orange Free State     

Quadratic forms connected with Fourier coefficients of Maass cusp forms

For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of ∑m12+m22+m32≤xλ(m12+m22+m32)Λ(m12+m22+m32) and ∑m12+m22+m32≤xa(m12+m22+m32)Λ(m12+m22+m32).