Discriminant Bounds for Spinor Regular Ternary Quadratic Lattices

The main goals of the paper are to establish a priori boundsfor the prime power divisors of the discriminants of spinorregular positive definite primitive integral ternary quadraticlattices, and to describe a procedure for determining all suchlattices.     

On the Number of Representations of Integers by Quadratic Forms in Twelve Variables

A way of finding exact explicit formulas for the number of representations of positive integers by quadratic forms in 12 variables with integral coefficients is suggested.     

Summation of Singular Series Corresponding to Representations of Numbers by Some Quadratic Forms in Twelve Variables

Formulas for calculating the sum of singular series corresponding to the number of representations of integers by some quadratic forms in 12 variables with integral coefficient are derived.     

特征2的有限域上二次型间的表示个数

本文利用特征2有限域Fq上奇异正交几何理论给出Fq上二次型的型与型间具有秩k的表示的个数,并由此得到一般计数公式及其q超几何级数表达式.     

Bi-Element Representations of Ternary Groups#

General properties of ternary semigroups and groups are consi-dered. The bi-element representation theory in which every representation matrix corresponds to a pair of elements is built up. Connections with the standard theory are considered and several concrete examples are constructed. For the sake of clarity and completeness the shortened versions of classical Gluskin-Hosszú and Post theorems are also presented.     

Representation of Integers by Ternary Quadric Forms

In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number. Supported by NSFC and RFDP.     

On Integral Representations and Boundary Properties of Spinor Fields

A new approach to the boundary value problem for the classic Dirac equation is proposed. This approach is based on a recent version of the metaharmonic quaternionic analysis developed in [14–16]. In particular, the following problem is studied: when and how a given function on a surface can be extended to a time-harmonic spinor field.     

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.     

Representation of Integers by Positive Quaternary Quadratic Forms

Let $f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2$ , where $D >1$ is an integer such that $D \ne d^2$ and ${{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \right. \kern-0em} {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}$ . Let $rf(n)$ be the number of representations of n by f. It is proved that $r_f (n) = \pi ^2 \frac{n}{{\sqrt D }}\sigma _f (n) + O\left( {\frac{{n^{1 + \varepsilon - c(\theta )} }}{{\sqrt D }}} \right),$ where $\sigma _f (n)$ is the singular series, $c(\theta ) >0$ , and ε is an arbitrarily small positive constant. Bibliography: 14 titles.     

Comparison of Normal Linear Experiments by Quadratic Forms

Let X and Y be observation vectors in normal linear experiments =N(A, V) and F = N(B, W). We write > Fif for any quadratic form YGY there exists a quadratic formXHX such that E(XHX) = E(Y'GY) and var(X'HX) var(Y'GY).The relation > is characterized by the matrices A, B, V and W. Moreoversome connections with known orderings of linear experiments are given.     

On an Asymptotics of the Number of Representations of a Pair of Integers by Quadratic and Linear Forms with Congruential Condition

Mathematical Notes - Asymptotic formulas with remainder for the number of representations of a pair of integers by quadratic and linear forms with a congruential condition are proved.     

On Primitive Representations of Minimax Nilpotent Groups

We show, in particular, that in the class of minimax two-step nilpotent groups only finitely generated groups can admit exact irreducible primitive representations over a finitely generated field of characteristic zero. We also suggest some approaches to studying induced representations of nilpotent groups of finite rank.     

Number of Representations of p-One-Dimensional Forms by a Genus

We study branching of representations of a locally p-one-dimensional form by a genus of positive definite integral quadratic forms. We give a complete list of minimal representations by a genus for forms of square level. GaussMinkowski formulas are obtained for heights of representations over the ring of integers. As an application, we obtain formulas for heights of primitive representations by genera for specific forms constructed by the method of orthogonal complement. Bibliography: 6 titles.     

Locally Quadratic Modules and Minuscule Representations

We give a new, geometric proof of a theorem by Timmesfeld showing that for simple Chevalley groups, abstract modules where all roots act quadratically are direct sums of minuscule representations. Our proof is uniform, treats finite and infinite fields on an equal footing, and includes Lie rings.     

Representation of Binary Quadratic Forms by a Quaternary Form

The paper presents formulas for finding the number of representations r(A; Q) of a form A with odd square-free level by the form Q defined by the identity matrix of order 4. The case where the difference of the dimensions n — m is even (the codimension is even), which was not considered previously, is analyzed. Bibliography: 8 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 68–80.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).