On the products arising from the Kummer conjecture

Let q be a sufficiently large integer and χ be a Dirichlet character modulo q.In this paper,we extend the product ∏χ(-1)=-1L(1,χ) with prime q,arising from the Kummer conjecture,to the products of some general Dirichlet series,and give some meaningful estimates for them.     

Algebraic string operations

We first discuss how open/closed chord diagrams, both with and without marked points, act on appropriate Hochschild complexes possibly coupled with the two-sided cobar complex. Then, in the main part of the paper, we introduce the notion of a V _k -algebra and obtain suitable homotopy versions.     

Chromatic numbers and products

Let f(n) be the smallest number so that there are two n chromatic graphs whose product has chromatic number f(n). Under the assumption that a certain sharper result than one obtained by Duffus et al. [On the chromatic number of the product of graphs, J. Graph Theory 9 (1985) 487–495], and Welzl [Symmetric graphs and interpretations, J. Combin. Theory, Sci. B 37 (1984) 235–244], holds we will prove that f(n)n/2.     

An elliptic K3 surface associated to Heron triangles

A rational triangle is a triangle with rational sides and rational area. A Heron triangle is a triangle with integral sides and integral area. In this article we will show that there exist infinitely many rational parametrizations, in terms of s, of rational triangles with perimeter 2s(s+1) and area s(s²−1). As a corollary, there exist arbitrarily many Heron triangles with all the same area and the same perimeter. The proof uses an elliptic K3 surface Y. Its Picard number is computed to be 18 after we prove that the Néron-Severi group of Y injects naturally into the Néron-Severi group of the reduction of Y at a prime of good reduction. We also give some constructions of elliptic surfaces and prove that under mild conditions a cubic surface in P³ can be given the structure of an elliptic surface by cutting it with the family of hyperplanes through a given line L. Some of these constructions were already known, but appear to have lacked proof in the literature until now.     

The Lax conjecture is true

In 1958 Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov.

     

Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

Metric sparsification and operator norm localization

We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. In particular, we introduce a sufficient geometric condition (called metric sparsification) for the operator norm localization property. This is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. We also show that a sequence of expanding graphs does not possess the operator norm localization property.     

Compact indefinite almost Kähler Einstein manifolds

In the present article, we extend the integral formula on a compact almost Kähler manifold with positive-definite metric to the one on a compact indefinite almost Kähler manifold and give its applications for some special indefinite almost Kähler Einstein manifolds taking the related problems to the indefinite analogy of Goldberg conjecture into consideration.     

The Rademacher conjecture and <Emphasis Type="Italic">q</Emphasis>-partial fractions

In his classic book, Topics in Analytic Number Theory, H. Rademacher posed a natural conjecture concerning the generating function for p(n), the number of partitions of n. In this paper we undertake a systematic study of an expansion technique that has its genesis in the work of Cayley. We apply this to the Rademacher conjecture, and obtain the first positive result providing theoretical evidence for the conjecture.      

Lower bounds for Seshadri constants

One of Demailly's characterization of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note this is translated into algebraic terms by using sections of multiples of the line bundle. The resulting formula for Seshadri constants is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)