A Second Descent Problem for Quadratic Forms

Let F be a field of characteristic different from 2. We discuss a new descent problem for quadratic forms, complementing the one studied by Kahn and Laghribi. More precisely, we conjecture that for any quadratic form q over F and any Im(W(F) W(F(q))), there exists a quadratic form W(F) such that dim 2 dim and _F(q), where F(q) is the function field of the projective quadric defined by q = 0. We prove this conjecture for dim 3 and any q, and get partial results for dim {4, 5,6}. We also give other related results.     

Erratum: On the Contractions of Vectorial Lattice Representations

An isometric sublattice on a six-dimensional vector space lattice is constructed having a congruence relation such that the factor lattice does not admit any representation via contraction.     

Partial Flocks of Quadratic Cones with a Point Vertex in Pg(n,q), n Odd

We generalise the definition and many properties of flocks ofquadratic cones in PG(3,q) to partial flocks of quadratic coneswith vertex a point in PG(p,q), for n 3 odd.     

Representation of Cubic Lattices by Symmetric Implication Algebras

In this paper a cubic lattice L(S) is endowed with a symmetric implication structure and it is proved that L(S) \ {0} is a power of the three-element simple symmetric implication algebra. The Metropolis–Rota’s symmetries are obtained as partial terms in the language of symmetric implication algebras.     

二次曲面所围封闭图形的体积

讨论三元二次曲面是封闭图形的条件及封闭图形所围图形体积的计算,并将结果的形式推广到二元和一元二次方程所围封闭图形的判定上和面积或长度的计算上.     

Division Algebras, Galois Fields, Quadratic Residues

Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.     

Galois connexion of a fuzzy subset

By their representation theorem, Negoita and Ralescu ‘identify’ a fuzzy set with a family of ordinary sets (levels). Here we ‘identify’ a fuzzy set with a Galois connexion between the boolean of a set and the valuation set. The two representations are closely connected.     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

Holder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for H?lder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ?²ⁿ , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the H?lder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the H?lder exponents, the diameter of Γ and a specific d-sum (d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.     

Algebraic results and bottom-up algorithm for policies generalization in reinforcement learning using concept lattices

The generalization of policies in reinforcement learning is a main issue, both from the theoretical model point of view and for their applicability. However, generalizing from a set of examples or searching for regularities is a problem which has already been intensively studied in machine learning. Thus, existing domains such as Inductive Logic Programming have already been linked with reinforcement learning. Our work uses techniques in which generalizations are constrained by a language bias, in order to regroup similar states. Such generalizations are principally based on the properties of concept lattices. To guide the possible groupings of similar states of the environment, we propose a general algebraic framework, considering the generalization of policies through a partition of the set of states and using a language bias as an a priori knowledge. We give a practical application as an example of our theoretical approach by proposing and experimenting a bottom-up algorithm.     

Global Optimality Conditions in Maximizing a Convex Quadratic Function under Convex Quadratic Constraints

For the problem of maximizing a convex quadratic function under convex quadratic constraints, we derive conditions characterizing a globally optimal solution. The method consists in exploiting the global optimality conditions, expressed in terms of -subdifferentials of convex functions and -normal directions, to convex sets. By specializing the problem of maximizing a convex function over a convex set, we find explicit conditions for optimality.     

Lattices with Complemented Tolerance Lattice

We characterize lattices with a complemented tolerance lattice. As an application of our results we give a characterization of bounded weakly atomic modular lattices with a Boolean tolerance lattice.     

Characterizations of classical orthogonal polynomials on quadratic lattices

This paper is devoted to characterizations of classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal polynomials on quadratic lattices. Moreover a new matrix characterization of classical ortho-gonal polynomials in quadratic lattices is presented. From the Bochner type characterization we derive the three-term recurrence relation coefficients for these polynomials.     

Massless Elementary Particles in a Quantum Theory over a Galois Field

We consider massless elementary particles in a quantum theory based on a Galois field (GFQT). We previously showed that the theory has a new symmetry between particles and antiparticles, which has no analogue in the standard approach. We now prove that the symmetry is compatible with all operators describing massless particles. Consequently, massless elementary particles can have only half-integer spin (in conventional units), and the existence of massless neutral elementary particles is incompatible with the spin–statistics theorem. In particular, this implies that the photon and the graviton in the GFQT can only be composite particles.     

Proof of a conjecture of Metsch

In this paper we prove a conjecture of Metsch about the maximum number of lines intersecting a pointset in PG(2,q), presented at the conference “Combinatorics 2002”. As a consequence, we give a short proof of the famous Jamison, Brouwer and Schrijver bound on the size of the smallest affine blocking set in AG(2,q).     

Characterizations of Complete Sublattices of a Given Complete Lattice

To characterize all complete sublattices of a given complete lattice is an important problem. In this paper we will give three different characterizations of complete sublattices of a complete lattice by using closure operators, kernel operators, and by using Galoisclosed subrelations.The research was supported by a grant of the faculty of Science ChiangMai University Thailand.     

二次系统存在一类四次曲线分界线环的充要条件（续）

本义给出了二次系统存在一类四次曲线分界线环的充要条件，此类分界线环的方程为（ｙ＋ｃｘ￣２）－ｘ￣２（ｘ－ａ）（ｘ－ｂ）＝０。     

实二次域的一个同余式

本文证明了关于实二次域的类数和某类特征和的同余式,同时给出某类实二次域的类数可除性的一个判别法则．     

Global Optimization of a Quadratic Functional with Quadratic Equality Constraints, Part 2

In this paper, we investigate a constrained optimization problem with a quadratic cost functional and two quadratic equality constraints. It is assumed that the cost functional is positive definite and that the constraints are both feasible and regular (but otherwise they are unrestricted quadratic functions). Thus, the existence of a global constrained minimum is assured. We develop a necessary and sufficient condition that completely characterizes the global minimum cost. Such a condition is of essential importance in iterative numerical methods for solving the constrained minimization problem, because it readily distinguishes between local minima and global minima and thus provides a stopping criterion for the computation. The result is similar to one obtained previously by the authors. In the previous result, we gave a characterization of the global minimum of a constrained quadratic minimization problem in which the cost functional was an arbitrary quadratic functional (as opposed to positive-definite here) and the constraints were at least positive-semidefinite quadratic functions (as opposed to essentially unrestricted here).     

Analytic continuation of overconvergent eigenforms

Let be an overconvergent -adic eigenform of level , , with non-zero -eigenvalue. We show how may be analytically continued to a subset of containing, for example, all the supersingular locus. Using these results we extend the main theorem of our earlier work with R. Taylor to many ramified cases.