Robert F. Lax

Consecutive Weierstrass gaps and minimum distance of Goppa codes

Abstract We prove that if there are consecutive gaps at a rational point on a smooth curve defined over a finite field, then one can improve the usual lower bound on the minimum distance of certain algebraic-geometric codes defined using a multiple of the point.

Graph-theoretic method for merging security system specifications

Computer security policies specify conditions for permissions to access various computer resources and information. Merging two security policies is needed when two organizations, together with their computer systems, merge into one entity as in corporate business acquisition. We propose a graph-theoretic method for merging the role/object hierarchies of two security policies. The formulation of merged hierachies is based on the graph minor relation in graph theory. Ideally, the merged role hierarchy should contain both the participating role hierarchies as graph minors, and similarly for the object hierarchy. We show that one can decide in polynomial time whether this ideal case is possible when the participating hierarchies are trees. We also show that in case the merged hierarchy exists, it can be constructed in polynomial time. Algorithms for detecting the feasibility of an ideal merged tree and for constructing the merged tree are presented. Our hierarchy/tree merge method is also applicable to the integration of heterogeneous databases with generalization hierarchies.

Gap Sequences and Moduli in Genus 4

Let X denote a compact Riemann surface of genus g > 1. At each point P~X, there is a sequence of g integers, l=71(P) g. Every compact Riemann surface of genus > 1 has a finite number of Weierstrass points. (See [6] for details.) The question of describing the moduli of compact Riemann surfaces of genus g which have a Weierstrass point with a specified first nongap has been answered by Rauch [14] and Arbarello [1]. The problem of describing the moduli of curves of genus g which have a Weierstrass point with a specified gap sequence has been considered by Hensel-Landsberg [7] and Rauch [15] and recently results on this question have been obtained by Pinkham [13] and RimVitulli [16]. In [9], we defined complex spaces of Weierstrass points of the universal curve and obtained a theorem concerning the smoothness and dimension of these spaces. From this, one can recover the moduli results of Rauch [14]. In order to study the higher Weierstrass nongaps and the entire gap sequence, we must extend our definition in [9]. We do this in w and then prove a theorem concerning the smoothness and dimension of these spaces which is analogous to the main result in [9]. F rom this, we can recover a moduli result of Pinkham [13]. In w 3, we study the geometry of these complex spaces of Weierstrass points of the universal curve of genus 4. We obtain a description of the moduli of Teichmiiller surfaces of genus 4 which have a Weierstrass point with a specified gap sequence. In the following, a complex space is not necessarily reduced. If Y is a complex space, then IYI will denote the underlying set of points. If Y, Z1 . . . . . Z , are closed complex subspaces of a complex space, then by Y Z 1 Z 2 . . . Z ,

Formulas for approximating pseudo-Boolean random variables

We consider {0,1}^n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. We then derive explicit formulas for approximating a pseudo-Boolean random variable by a linear function if the measure is permutation-invariant, and by a function of degree at most k if the measure is a product measure. These formulas generalize results due to Hammer-Holzman and Grabisch-Marichal-Roubens. We also derive a formula for the best faithful linear approximation that extends a result due to Charnes-Golany-Keane-Rousseau concerning generalized Shapley values. We show that a theorem of Hammer-Holzman that states that a pseudo-Boolean function and its best approximation of degree at most k have the same derivatives up to order k does not generalize to this setting for arbitrary probability measures, but does generalize if the probability measure is a product measure.

Weierstrass weight of Gorenstein singularities with one or two branches

We obtain formulas for the Weierstrass weight of singularities with either one or two branches on a Gorenstein curve in characteristic zero. These formulas generalize results of C. Widland in the cases of simple cusps and ordinary nodes. The formulas arise from a study of the semigroup of values of such a singularity and the relation between this semigroup and a basis for the dualizing differentials on the curve adapted to the singular point.

Generic interpolation polynomial for list decoding

Abstract We extend results of K. Lee and M.E. OʼSullivan by showing how to use Grobner bases to find the interpolation polynomial for list decoding a one-point AG code C = C L ( r P , D ) on any curve X , where P is an F q -rational point on X and D = P 1 + P 2 + ⋯ + P n is the sum of other F q -rational points on X . We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial.

Transforms of pseudo-Boolean random variables

As in earlier works, we consider {0,1}^n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions.

Weierstrass points on rational nodal curves

C. Widland [14] has defined Weierstrass points on integral, projective Gorenstein curves. We show here that the Weierstrass points on a generic integral rational nodal curve have the minimal possible weights or, equivalently, that such a curve has the maximum possible number of distinct nonsingular Weierstrass points. Rational curves with g nodes arise in degeneration arguments involving smooth curves of genus g and they have also recently arisen in connection with g -soliton solutions to certain nonlinear partial differential equations [11], [13].

Efficient learning of pseudo-boolean functions from limited training data

Pseudo-Boolean functions are generalizations of Boolean functions. We present a new method for learning pseudo-Boolean functions from limited training data. The objective of learning is to obtain a function f which is a good approximation of the target function f*. We define suitable criteria for the “goodness” of an approximating function. One criterion is to choose a function f that minimizes the “expected distance” with respect to a distance function d (over pairs of pseudo-Boolean functions) and the uniform distribution over all feasible pseudo-Boolean functions. We define two alternative “distance measures” over pairs of pseudo-Boolean functions, and show that they are are actually equivalent with respect to the criterion of minimal expected distance. We outline efficient algorithms for learning pseudo-Boolean functions according to these criteria. Other reasonable distance measures and “goodness” criteria are also discussed.

Local soft belief updating for relational classification

We introduce local soft belief updating, a new heuristic for taking into account relations that exist between entities in a database. Our idea applies Pearls belief updating, but only in the first-order neighborhood of each node, thus avoiding any problems with loops. We apply our method to a classification problem using a subset of the Cora database of computer science articles, with Coras citation graph giving the relations between entities.