Composition of binary quadratic forms

Composition of binary quadratic forms over an arbitrary commutative base ring is shown to be closely related to homomorphisms (and in particular isomorphisms) of the corresponding even Clifford algebras.     

Invertibility of 2 × 2 operator matrices

Properties of right invertible row operators, i.e., of 1 × 2 surjective operator matrices are studied. This investigation is based on a specific space decomposition. Using this decomposition, we characterize the invertibility of a 2 × 2 operator matrix. As an application, the invertibility of Hamiltonian operator matrices is investigated.     

Weylness of 2 × 2 operator matrices

Let and be complex separable infinite‐dimensional Hilbert spaces. Given the operators and , we define where is an unknown element. In this paper, a necessary and sufficient condition is given for to be a right Weyl (left Weyl, or Weyl) operator for some . Moreover, some relevant properties and illustrating examples are also given.     

Represented value sets for integral binary quadratic forms and lattices

A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over a Dedekind domain, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property observed by V. Arnold in the case of integral binary quadratic forms.

     

Geometry of 2×2 Hermitian matrices II

Let Z be a field of characteristic ≠2, D be a quaternion division algebra over Z and have a nonstandard involution of the first kind. The fundamental theorem of geometry of 2× 2 Hermitian matrices over D are proved. Thus, if D is a quaternion division algebra over Z with an involution of the first kind, then the fundamental theorem of geometry of 2× 2 Hermitian matrices over D are obtained.     

Representation sets for integral binary quadratic forms over the rationals

Let f be an integral binary form of discriminant d which represents n integrally. Two rational representations (r, s) and (r′, s′), with denominators prime to n, of n by f are called semiequivalent with respect to f if there is a rational automorph of f with determinant 1 and denominator m which takes (r, s) into (r′, s′) where (m, n) = 1 and m contains no factors p of d such that $\text{d}\text{p}{}^2$ is a discriminant. The number of such equivalence classes for a given f and n is sometimes finite. This number is obtained for forms with negative discriminants which have one class in each primitive genus.     

Lorentzian geometry for 2 × 2 real matrices

We give an interpretation and a natural proof of the lemma of Herglotz on 2 × 2 symmetric matrices from the point of view of 3-dimensional Lorentz geometry and 2-dimensional hyperbolic geometry. This interpretation leads naturally to an analogous result on 2 × 2 matrices with trace 0.     

Symplectic 2 × 2 matrices over free algebras

P.M. Cohn has proved the remarkable theorem, that every invertible n × n matrix over a free algebra is the product of elementary n × n matrices, see [C1], [C2]. In this note we prove the analogue for symplectic 2 × 2 matrices over free algebras relative to a homogeneous involution: every symplectic 2 × 2 matrix is the product of elementary symplectic 2 × 2 matrices.In Section 1 we define the group Sp₂(R) of symplectic 2 × 2 matrices over an involutive ring R. The group ESp₂(R) generated by elementary symplectic matrices is introduced in Section 3.In Section 2 we prove a reducibility criterion for homogeneous polynomials in a free algebra KX over a commutative field K. It leads to a special form in the factorization of symmetric homogeneous polynomials, see Corollary to Proposition 2.2.We prove in Section 4 that ESp₂(KX) = Sp₂(KX), if the involution on KX is homogeneous.In a subsequent article we will show that the main result is also true for 2g × 2g symplectic matrices over free algebras relative to homogeneous involutions, g ≥ 1. It seems that a proof of this result will be much more complicated than the case g = 1.     

Analysis of an algorithm for composition of binary quadratic forms

This note presents an algorithm which composes two reduced properly primitive binary quadratic forms of the same nonquadratic determinant D in O(M(log∥D∥)log log∥D∥) elementary operations.     

Hankel matrices and quadratic forms

A characterization of finite Hankei matrices is given and it is shown that such matrices arise naturally as matrix representations of scaled trace forms of field extensions and etale algebras. An algorithm is given for calculating the signature and the Hasse invariant of these scaled trace forms.     

Poincaré series of semi-invariants of 2×2 matrices

The Poincaré series of the algebra of -invariants of m-tuples of 2×2 matrices is presented both as a rational function and as a series of Schur functions. We show that this algebra of invariants is generated by the determinants, the mixed discriminants and the discriminants of 2×2 matrices. Consequences on invariants of three-dimensional matrices of the shape 2×2×m are discussed. For arbitrary n2, we prove an explicit functional equation for the Poincaré series of the -invariants of m-tuples of n×n matrices.     

Some classes of integral matrices

Integral matrices A for which the system Ax=b will have an integral basic solution are considered. Results for these matrices are presented which parallel results concerning other matrices for which the system has integral solutions.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Cubic residues and binary quadratic forms

Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u²−dv² and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.     

Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

Ergodic properties of flows of classes of positive binary quadratic forms in Gaussian genera

The present paper is devoted to further development and refinement of previous results due to A. V. Malyshev and the author concerning the so-called discrete ergodic method of Yu. V. Linnik. An ergodic theorem and a mixing theorem for flows of positive binary quadratic forms are proved; these theorems describe the asymptotic distribution of the coefficients of these forms over the residue classes and over the corresponding surface. Bibliography: 12titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 149–161.     

Commutativity for systems of (2 × 2) selfadjoint matrices

A purely analytic criterion is presented which characterises the commutativity of a finite-collection of (2x2) selfadjoint matrices. Indeed, it is always possible to associate with such matrices their so called Weyl Calculus which is a matirx-valued distribution of finite order. It is shown that the matrices common if and only if their associated Weyl Calculus is a distribution of order zero.