Completeness and compactness for varieties over a local field

This paper shows that for a local field K, a subfield k ⊂ K and a variety X over k, X is complete if and only if for every finite field extension Kʹ | K, the set X(Kʹ) is compact in its strong topology. The author likes to thank Florian Pop, Jakob Stix, Stefan Wewers, Gunther Cornelissen and his own parents for their support. Received: 13 April 2006     

A Note on Log-Convexity of q-Catalan Numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14,…, a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted. Received January 2, 2007     

Support problem for the intermediate Jacobians of l-adic representations

We consider the support problem of Erdös in the context of l-adic representations of the absolute Galois group of a number field. Main applications of the results of the paper concern Galois cohomology of the Tate module of abelian varieties with real and complex multiplications, the algebraic K-theory groups of number fields and the integral homology of the general linear group of rings of integers. We answer the question of Corrales-Rodrigáñez and Schoof concerning the support problem for higher dimensional abelian varieties.     

Symmetric incidence groupoids

In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.This paper was sponsored by Vigoni Program 1999.     

On Romanoff's constant

In 1934, Romanoff proved that there are a positive proportion natural numbers which can be expressed as the sum of a prime and a power of 2. In this paper, a quantitative version of this theorem is given. We show that the proportion is larger than 0.0868 and for a positive proportion of odd integers the number of such representations is between 1 and 16.     

Multiple harmonic series related to multiple Euler numbers

In this paper, we define the multiple Euler numbers and consider some multiple harmonic series of Mordell-Tornheim's type, which is a partial sum of the Mordell-Tornheim zeta series defined by Matsumoto. Indeed, we prove a certain reducibility of these series as well as the multiple zeta values.     

Random Walks And The Colored Jones Function

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones function: one in terms of the permanent of a matrix associated to a chord diagram, and another in terms of counting paths of intersecting chords. Electronic supplementary material to this article is available at and is accessible to authorized users. * S. G. was partially supported by an NSF and by an Israel-US BSF grant. † M. L. was partly supported by GAUK 158 grant and by the Project LN00A056 of the Czech Ministry of Education.     

A gas-kinetic scheme for shallow-water equations with source terms

In this paper, the Kinetic Flux Vector Splitting (KFVS) scheme is extended to solving the shallow water equations with source terms. To develop a well-balanced scheme between the source term and the flow convection, the source term effect is accounted in the flux evaluation across cell interfaces. This leads to a modified gas-kinetic scheme with particular application to the shallow water equations with bottom topography. Numerical experiments show better resolution of the unsteady solution than conventional finite difference method and KFVS method with little additional cost. Moreover, some positivity properties of the gas-kinetic scheme is established.