Small Complete Arcs in Projective Planes

In the late 1950s, B. Segre introduced the fundamental notion of arcs and complete arcs [48,49]. An arc in a nite projective plane is a set of points with no three on a line and it is complete if cannot be extended without violating this property. Given a projective plane , determining , the size of its smallest complete arc, has been a major open question in finite geometry for several decades. Assume that has order q, it was shown by Lunelli and Sce [41], more than 40 years ago, that . Apart from this bound, practically nothing was known about , except for the case is the Galois plane. For this case, the best upper bound, prior to this paper, was O(q ^3/4) obtained by Sznyi using the properties of the Galois field GF(q).In this paper, we prove that for any projective plane of order q, where c is a universal constant. Together with Lunelli-Sces lower bound, our result determines up to a polylogarithmic factor. Our proof uses a probabilistic method known as the dynamic random construction or Rödls nibble. The proof also gives a quick randomized algorithm which produces a small complete arc with high probability.The key ingredient of our proof is a new concentration result, which applies for non-Lipschitz functions and is of independent interest.* Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.Part of this work was done at AT&T Bell Labs and Microsoft Research     

Inequalities for <Emphasis Type="Bold">cd</Emphasis>-Indices of Joins and Products of Polytopes

The cd-index is a polynomial which encodes the flag f-vector of a convex polytope. For polytopes U and V, we determine explicit recurrences for computing the cd-index of the free join and the cd-index of the Cartesian product U x V. As an application of these recurrences, we prove the inequality involving the cd-indices of three polytopes.     

Square roots of semihyponormal operators have scalar extensions

In this paper, we study some properties of , i.e., square roots of semihyponormal operators. In particular we show that an operator has a scalar extension, i.e., is similar to the restriction to an invariant subspace of a (generalized) scalar operator (in the sense of Colojoar?-Foia?). As a corollary, we obtain that an operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

Cubic and quartic congruences modulo a prime

Let p>3 be a prime, and denote the number of solutions of the congruence . In this paper, using the third-order recurring sequences we determine the values of N_p(x³+a₁x²+a₂x+a₃) and N_p(x⁴+ax²+bx+c), and construct the solutions of the corresponding congruences, where a₁,a₂,a₃,a,b,c are integers.     

Flux Homomorphisms and Principal Bundles over Infinite Dimensional Manifolds

Flux homomorphisms for closed vector-valued differential forms on infinite dimensional manifolds are defined. We extend the relation between the kernel of the flux for a closed 2-form and Kostants exact sequence associated to a principal bundle with curvature to the context of infinite-dimensional fiber and base space. We then use these results to construct central extensions of infinite dimensional Lie groups.Received April 19, 2002; in revised form October 15, 2002 Published online June 23, 2003     

An Edge Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let f be a function periodic in its two variables x, y with period 2 and having at the origin an edge point jump discontinuity (its graph may be roughly described by a cliff with a smooth but possibly inclined rim). The Fourier series of f exhibits a Gibbs phenomenon as in the one-dimensional case but with an affine horizontal deformation depending on the compass direction of the rim at the origin.Received February 7, 2002; in revised form May 24, 2002 Published online April 4, 2003     

Additive Functions with Respect to Numeration Systems on Regular Languages

Asymptotic formulae for the summatory function of additive arithmetic functions related to numeration systems given by regular languages are derived.This author is supported by the START project Y96-MAT of the Austrian Science Fund.This author warmly thanks P. Grabner for his kind invitation to Graz and his hospitality.Received February 18, 2002; in revised form September 20, 2002 Published online May 15, 2003     

A New Inertia Theorem for Stein Equations,Inertia of Invertible Hermitian Block Toeplitz Matrices and Matrix Orthogonal Polynomials

In this paper we present an inertia result for Stein equations with an indefinite right hand side. This result is applied to establish connnections between the inertia of invertible hermitian block Toeplitz matrices and associated orthogonal polynomials.     

Toeplitz Operators Associated to Unimodular Algebras

We introduce a class of function algebras, that we call unimodular, and study Toeplitz operators on the Hardy spaces associated to representing measures on these algebras.We show that our class of function algebras is very extensive and that a number of important results for Toeplitz operators and their associated C*-algebras extend to the very general setting we consider. Submitted: July 1. 2001.     

Periodic points of positive linear operators and Perron-Frobenius operators

LetC(S) denote the Banach space of continuous, real-valued mapsf:S and letA denote a positive linear map ofC(S) into itself. We give necessary conditions that the operatorA have a strictly positive periodic point of minimal periodm. Under mild compactness conditions on the operatorA, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal periodm. We study a class of Perron-Frobenius operators defined by