Definite quadratic form

About: Definite quadratic form is a research topic. Over the lifetime, 1348 publications have been published within this topic receiving 26989 citations.

Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

Abstract: We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.

Computing the distribution of quadratic forms in normal variables

Abstract: In this paper exact and approximate methods are given for computing the distribution of quadratic forms in normal variables. In statistical applications the interest centres in general, for a quadratic form Q and a given value x, around the probability P{Q > x}. Methods of computation have previously been given e.g. by Box (1954), Gurland (1955) and by Grad & Solomon (1955). None of these methods is very easily applicable except, when it can be used, the finite series of Box. Furthermore, all the methods are valid only for quadratic forms in central variables. Situations occur where quadratic forms in non-central variables must be considered as well. Let x = (x1, ..., xx)' be a column random vector which follows a multidimensional normal law with mean vector 0 and covariance matrix E. Let s = (,t, . . ., ,,7)' be a constant vector, and consider the quadratic form Q = (x + ,)' A(x + ,u). If E is non-singular, one can by means of a non-singular linear transformation (Scheff6 (1959), p. 418) express Q in the form rn 2 Q =E ArXhr; (1 r=1

Introduction to quadratic forms

Abstract: Prerequisites ad Notation Part One: Arithmetic Theory of Fields I Valuated Fields Valuations Archimedean Valuations Non-Archimedean valuations Prolongation of a complete valuation to a finite extension Prolongation of any valuation to a finite separable extension Discrete valuations II Dedekind Theory of Ideals Dedekind axioms for S Ideal theory Extension fields III Fields of Number Theory Rational global fields Local fields Global fields Part Two: Abstract Theory of Quadratic Forms VI Quadratic Forms and the Orthogonal Group Forms, matrices and spaces Quadratic spaces Special subgroups of On(V) V The Algebras of Quadratic Forms Tensor products Wedderburn's theorem on central simple algebras Extending the field of scalars The clifford algebra The spinor norm Special subgroups of On(V) Quaternion algebras The Hasse algebra VI The Equivalence of Quadratic Forms Complete archimedean fields Finite fields Local fields Global notation Squares and norms in global fields Quadratic forms over global fields VII Hilbert's Reciprocity Law Proof of the reciprocity law Existence of forms with prescribed local behavior The quadratic reciprocity law Part Four: Arithmetic Theory of Quadratic Forms over Rings VIII Quadratic Forms over Dedekind Domains Abstract lattices Lattices in quadratic spaces IX Integral Theory of Quadratic Forms over Local Fields Generalities Classification of lattices over non-dyadic fields Classification of Lattices over dyadic fields Effective determination of the invariants Special subgroups of On(V) X Integral Theory of Quadratic Forms over Global Fields Elementary properties of the orthogonal group over arithmetic fields The genus and the spinor genus Finiteness of class number The class and the spinor genus in the indefinite case The indecomposable splitting of a definite lattice Definite unimodular lattices over the rational integers Bibliography Index Bibliography Index

Integral Symmetric Bilinear Forms and Some of Their Applications

Abstract: We set up the technique of discriminant-forms, which allows us to transfer many results for unimodular symmetric bilinear forms to the nonunimodular case and is convenient in calculations. Further, these results are applied to Milnor's quadratic forms for singularities of holomorphic functions and also to algebraic geometry over the reals. Bibliography: 57 titles.