Spinor equivalence of quadratic forms

Let f be an integral quadratic form in three or more variables and g any form in the genus of f. There exist an effectively determinable prime p and a form g′, belonging to the proper spinor genus of g, such that g′ is a p-neighbor of f in the graph of f. Using this, an alternative decision procedure for the spinor equivalence of quadratic forms is given.     

Spinor genera of binary quadratic forms

Spinor genera are defined for binary quadratic forms with integer coefficients in such a way that the theory fits in with the Gaussian theory of genera. It is shown that spinor generic characters exist which distinguish the various spinor genera in the principal genus, and how they can be determined. It is known that each ambiguous class contains exactly two forms of the type [a, 0, c] or [a, a, c], each with its associate [c, 0, a], [4c ? a, 4c ? a, c]. Since the principal class contains such a form with a = 1, it is an interesting question whether one can predict the second form (not counting associates). This question includes that of Dirichlet about the representability of ?1 by the principal class. Methods are given for evaluating the spinor-generic characters of ambiguous forms in the principal genus for variable discriminants d, and are carried through in the eleven cases where d is fundamental, there are two or four genera, and two spinor genera in the principal genus. The problem of determining the “second form” is thus completely solved except when there is more than one ambiguous class in the principal spinor genus.     

Smooth lattices over quadratic integers

We construct lattices with quadratic structure over the integers in quadratic number fields having the property that the rank of the quadratic structure is constant and equal to the rank of the lattice in all reductions modulo maximal ideals. We characterize the case in which such lattices are free. The construction gives a representative of the genus of such lattices as an orthogonal sum of “standard” pieces of ranks 1–4 and covers the case of the discriminant of the real quadratic number field congruent to 1 modulo 8 for which a general construction was not known.      

On Banach lattices with Levi norms

Schmidt proved that an operator from a Banach lattice into a Banach lattice with property is order bounded if and only if its adjoint is order bounded, and in this case satisfies . In the present paper the result is generalized to Banach lattices with Levi-Fatou norm serving as range, and some characterizations of Banach lattices with a Levi norm are given. Moreover, some characterizations of Riesz spaces having property are also obtained.

     

On extensions of triangular norms on bounded lattices

Smallest and largest possible extensions of triangular norms on bounded lattices are discussed. As such ordinal and horizontal sum like constructions for t-norms on bounded lattices are investigated. Necessary and sufficient conditions for the lattice guaranteeing that the extension is again a t-norm are revealed.     

On certain generalized matrix norms

Two methods of generating classes of generalized matrix norms are presented. Particular norms obtained in this manner are investigated in some detail.     

Orthomodular lattices from 3-dimensional quadratic spaces

If E is a vector space over a field K, then any regular symmetric bilinear form on E induces a polarity on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, ). We show that if K is a proper subfield of K, with K F₂, and E a 3-dimensional K -subspace of E such that the restriction of to E × E is, up to multiplicative constant, a bilinear form on the K -space E , then T(E , ) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, ). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, ) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.Received June 4, 2003; accepted in final form May 18, 2004.     

Random walks on generalized lattices

A generalized lattice is a graph on which the groupZ ^d acts almost transitively. The relations among various features of random walks on generalized lattices are studied. In particular we relate the mean displacement, the drift-freeness of the random walk and the existence of linear harmonic functions. Applications to recurrence criteria are given.     

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.     

Dichotomic lattices and local discretization for Galois lattices

The present paper deals with supervised classification methods based on Galois lattices and decision trees. Such ordered structures require attributes discretization and it is known that, for decision trees, local discretization improves the classification performance compared with global discretization. While most literature on discretization for Galois lattices relies on global discretization, the presented work introduces a new local discretization algorithm for Galois lattices which hinges on a property of some specific lattices that we introduce as dichotomic lattices. Their properties, co-atomicity and \(\vee \)-complementarity are proved along with their links with decision trees. Finally, some quantitative and qualitative evaluations of the local discretization are proposed.     

On p-adic norms and quadratic extensions, II

We investigate the graph theoretical aspects and combinatorics of the action of a unitary group in n variables over a local non-archimedian field on the Bruhat-Tits building of an orthogonal group in 2n?+?1 variables.