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 .     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

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.     

Fibonacci numbers, alternating parity sequences and faces of the tridiagonal Birkhoff polytope

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.     

On visualization scaling, subeigenvectors and Kleene stars in max algebra

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^-1AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.     

Homogeneous multiplicative polynomial laws are determinants

Let k be a commutative ring. Let R,B be k-algebras with B commutative. Let p:R→B be a homogeneous multiplicative polynomial law of degree n. We show that p is obtained by left and right composing a determinant with some homomorphisms of k-algebras.     

Maximal Linear Subspaces of Strong Self-Dual 2-forms and the Bonan 4-form

The notion of self-duality of 2-forms in 4-dimensions plays an eminent role in many areas of mathematics and physics, but although the 2-forms have a genuine meaning related to curvature and gauge-field-strength in higher dimensions also, their “self-duality” is something which is almost avoided above 4-dimensions. We show that self-duality of 2-forms is a very natural notion in higher (even) dimensions also and we prove the equivalence of some scattered and rarely used definitions in the literature. We demonstrate the usefulness of this higher self-duality by studying it in 8-dimensions and we derive a natural expression for the Bonan form in terms of self-dual 2-forms and we give an explicit expression of the local action of SO(8) on the Bonan form.     

Natural density distribution of Hermite normal forms of integer matrices

The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain. Over the integers, the distribution of the HNFs of randomly looking matrices is far from uniform. The aim of this article is to present an explicit computation of this distribution together with some applications. More precisely, for integer matrices whose entries are upper bounded in absolute value by a large bound, we compute the asymptotic number of such matrices whose HNF has a prescribed diagonal structure. We apply these results to the analysis of some procedures and algorithms whose dynamics depend on the HNF of randomly looking integer matrices.     

On the space of quadruples of quinary alternating forms

We discuss some aspects of the invariant theory and arithmetic of the prehomogeneous vector space of quadruples of quinary alternating forms. In particular, we complete the explicit construction of all prehomogeneous covariants of this space and give the rational classification of maximal singular orbits.     

Comtrans algebras, Thomas sums, and bilinear forms

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.Received: 30 March 2004     

On Hyers-Ulam stability for a class of functional equations

Summary In this paper we prove some stability theorems for functional equations of the formg[F(x, y)]=H[g(x), g(y), x, y]. As special cases we obtain well known results for Cauchy and Jensen equations and for functional equations in a single variable. Work supported by M.U.R.S.T. Research funds (60%).     

The automorphism group of certain higher degree forms

We consider symmetric indecomposable d-linear (d>2) spaces of dimension n over an algebraically closed field k of characteristic 0, whose center (the analog of the space of symmetric matrices of a bilinear form) is cyclic, as introduced by Reichstein [B. Reichstein, On Waring’s problem for cubic forms, Linear Algebra Appl. 160 (1992) 1-61]. The automorphism group of these spaces is determined through the action on the center and through the determination of the Lie algebra. Furthermore, we relate the Lie algebra to the Witt algebra.     

Indecomposable forms of higher degree

All nondegenerate indecomposable forms of higher degree over a perfect field k can be realized as traces of nondegenerate absolutely indecomposable forms of higher degree over a suitable algebraic field extension of k. With the help of trace forms of certain nonassociative algebras we construct classes of indecomposable forms of degree d≥3.     

Singular systems, state feedback problem

In this paper the problem of Kronecker invariants assignment by state feedback in singular linear systems is studied and resolved. This result presents a generalization of the previous results of state feedback action on singular systems.     

Generalized derivations and additive theory II

We return to the theme of generalized derivations related to symmetric functions to correct the hypothesis of one of the main theorems of our first paper, so that all cases are now properly covered.     

Groups acting on circulant matrices

Summary If a groupG permutes a setI, andM is a multiplicative abelian group, a representation ofG onM ^I is given by permutation of coordinates. TheG-module homomorphisms intoM ^I arise from exponential maps. This framework encompasses those systems of functional equations that characterize generalized hyperbolic functions.     

On the continuity of residuals of triangular norms

In this work a long-standing problem related to the continuity of R-implications, i.e., implications obtained as the residuum of t-norms, has been solved. A complete characterization of the class of continuous R-implications obtained from any arbitrary t-norm is given. In particular, it is shown that an R-implication I_T is continuous if and only if T is a nilpotent t-norm. Using this result, the exact intersection between the continuous subsets of R-implications and (S,N)-implications has been determined, by showing that the only continuous (S,N)-implication that is also an R-implication obtained from any t-norm, not necessarily left-continuous, is the ?ukasiewicz implication up to an isomorphism.     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

Quantum Clifford Hopf gebra for quantum field theory

We give arguments for the necessity to employ quantum Clifford Hopf gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design of convolution Hopf gebras is given as a program for future research.