Quadratic mappings of spheres

In this paper we obtain a basis-free method for determining the general form of quadratic maps over R between spheres. We show that all quadratic maps (over certain R-lattices) between spheres are Hopf maps, and that the classical Hopf fibrations, S^2m?1→S^m, for m=2, 4, 8, are the unique nontrivial maps over Z, up to action by the orthogonal group.     

On combinatorial properties of spheres in euclidean spaces

For λ>√2 there exists a triangle-free graphG such that for nod canG be imbedded into thed-dimensional unit sphere with two points joined if and only if their distance is >λ.     

Quadratic forms on graphs

We introduce a new graph parameter, called the Grothendieck constant of a graph G=(V,E), which is defined as the least constant K such that for every A:E→ℝ,

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 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 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 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 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 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