On an oppenheim-type conjecture for systems of quadratic forms

Hermitian forms and locally toroidal regular polytopes

CHAPTER III - PARTICULAR TYPES OF GENERALIZED FUNCTIONS

The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions

In the passage from fields to rings of coefficients quadratic forms with invertible matrices lose their decisive role. It turns out that if all quadratic forms over a ring are diagonalizable, then in effect this is always a local principal ideal ring with . The problem of the construction of a `normal' diagonal form of a quadratic form over a ring faces obstacles in the case of indices greater than 1. In the case of index 2 this problem has a solution given in Theorem 2.1 for (an extension of the law of inertia for real quadratic forms) and in Theorem 2.2 for containing an invertible non-square. Under the same conditions on a ring with nilpotent maximal ideal the number of classes of projectively congruent quadratic forms of the projective space associated with a free -module of rank is explicitly calculated (Proposition 3.2). Up to projectivities, the list of forms is presented for the projective plane over and also (Theorem 3.3) over the local ring with non-principal maximal ideal, where is a field with an invertible non-square in and . In the latter case the number of classes of non-diagonalizable quadratic forms of rank 0 depends on one's choice of the field and is not even always finite; all the other forms make up 21 classes.

For a field F of characteristic not 2, let \({\widehat{F}}\) denote the maximal dimension of anisotropic, weakly isotropic, non-degenerate quadratic forms over F. In this paper, we investigate the behavior of this invariant under field extensions.

SUMMARY This paper is concerned with the characterization of the variance covariance matrix for a normal family in which certain quadratic forms are independent and follow cχ 2 distributions. It is also proved that the necessary and sufficient condition for any quadratic or linear form to be distributed independently of a sum of a finite number of non-negative definite quadratic forms is that it is distributed independently of each one of them.

Heilbronn (6) proved that for every ε ≥ 0 and N ≥ 1 and every real θ there is an integer x such that,where C(ε) depends only on ε and ∥α∥ is the difference between α and the nearest integer, taken positively. Danicic(1) obtained an analogous result for the fractional parts of nkθ, the proof of this is more readily accessible in Davenport(4). Danicic(2) also obtained an estimate for the fractional parts of a real quadratic form in n variables, and in order to extend this result to forms of higher degree it is desirable to first obtain results for additive forms.

In this paper we consider various stability properties of real invariant lagrangian subspaces for real matrices which are either symmetric or skew-symmetric in a real quadratic form which may be symmetric or skew-symmetric itself. In particular, apart from ordinary stability we shall consider strong stability, which seems to be more desirable from a numerical point of view. For the classes of matrices we consider here stable subspaces are not always strongly stable, in contrast with the previous work. We shall completely characterize strongly stable invariant lagrangian subspaces, and in many cases also the stable ones. Invariant lagrangian subspaces with other stability properties, such as Lipschitz stability, are characterized as well.KeywordsInvariant SubspaceReal EigenvalueReal MatriceStrong StabilityStable ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let denote the kth successive inhomogeneous minima for positive values of real indefinite ternary quadratic forms of type (2, 1). Here it is proved that for the class of zero forms, All the critical forms have also been obtained. is already known. For non-zero forms it is proved that .

Chapter 6 - Quadratic forms and Eigenvalue problems

For n > 2, let G(n) denote the two-fold covering group of SO.(l,n).Incase n > 3, G(n) is isomorphic to Spin(l, n) and is simply connected.In a previous paper we determined all the irreducible quasi-simple representations of these groups, up to infinitesimal equivalence.The main purpose of the present paper is to determine which of these representations are unitarizable.Thus, with the aid of some results of Harish-Chandra and Nelson we determine all the irreducible unitary representations of G(n), up to unitary equivalence.One by-product of our analysis is the explicit construction of the infinitesimal equivalences, which are known to exist from our previous work, between the various subquotient representations and certain subrepresentations in the nonirreducible cases of the nonunitary principal series representations of G(ri).1. Introduction.For n>2, let G(n) denote a two-fold covering group of SOe(l, k).The latter group is the identity component of the orthogonal group of a real nondegenerate quadratic form of signature (+, --) and of dimension + 1. Incase > 3, G(ri) is isomorphic to Spin(l,n) and is simply connected.For = 2, G() is isomorphic to SL(2, R).In a previous paper we determined, up to infinitesimal equivalence, all the irreducible quasi-simple representations of G(n).The main purpose of the present paper is to determine which of these representations are unitarizable.Thus, with the aid of some results of b] and Nelson [6], we determine all the unitary representations of G(n), up to unitary equivalence.Before explaining the results and methods of this paper in more detail, we make some remarks concerning the notation.For any unexplained notation we refer the reader to [10].One major change, however, is that C7() refers here to the group Spin(l, ), rather than to the group SOe(l, ).A similar change in Presented to the Society, in part, January 14, 1974 under the title On the explicit determination of Kunze-Stein intertwining operators with applications; received by the editors November 10, 1972.

Unital quadratic Jordan algebras J(q, I) determined by nondegenerate quadratic forms with basepoints over a field axe called full Jordan Clifford algebras. In characteristic 2 they have ample outer ideals which are also simple; they come in 3 sizes, tiny, small, and large, where the large are full Clifford algebras but the tiny and small algebras are lacking some of their parts. The simple algebras played a role in Zelmanov's solution of the Burnside Problem. In this paper we will analyze these in more detail, determining their centroids and their local algebras; this is important in the classification of prime Jordan triples of Clifford type in arbitrary char-acterstics. In addition we make a careful charcterization of the tiny, small, and large Clifford algebras. We use this to straighten one or two missteps in a proof from the classification of simple algebras. An important role in our characterization is played by commutators, and we describe the Jor­dan commutator products and Bergmann formulas for Clifford algebras in general.

Quadratic quaternion forms, introduced by Seip-Hornix (1965), are special cases of generalized quadratic forms over algebras with involutions. We apply the formalism of these generalized qua- dratic forms to give a characteristic free version of difierent results related to hermitian forms over quaternions: 1) An exact sequence of Lewis 2) Involutions of central simple algebras of exponent 2. 3) Triality for 4-dimensional quadratic quaternion forms. 1991 Mathematics Subject Classiflcation: 11E39, 11E88 Keywords and Phrases: Quadratic quaternion forms, Involutions, Tri- ality

Let q q be a power of an odd prime, and F q [ T ] \mathbb {F}_q[T] be the ring of polynomials over a finite field F q \mathbb {F}_q of q q elements. A quadratic form f f over F q [ T ] \mathbb {F}_q[T] is said to be regular if f f globally represents all polynomials that are represented by the genus of f f . In this paper, we study definite regular quadratic forms over F q [ T ] \mathbb {F}_q[T] . It is shown that for a fixed q q , there are only finitely many equivalence classes of regular definite primitive quadratic forms over F q [ T ] \mathbb {F}_q[T] , regardless of the number of variables. Characterizations of those which are universal are also given.

This paper is concerned mostly with the distribution of the ratio of two quadratic forms in normal variables. The probability density of the ratio is obtained as an integral and closed-form expressions are given for several special cases. However, in the general case numerical methods must be used to evaluate the density. A method of numerical integration is presented and illustrated by an example. The behavior of the density around the points at which it is irregular is examined. The general case of the distribution of the ratio of two composite random variables is discussed briefly.