Bases of minimal vectors in lattices,I

We prove that a Euclidean lattice of dimension n ≤ 8 which is generated by its minimal vectors possesses a basis of minimal vectors. Received: 7 December 2006     

Contributions to the Hypergeometric Function Theory of Heckman and Opdam: Sharp Estimates, Schwartz Space, Heat Kernel

Under the assumption of positive multiplicity, we obtain basic estimates of the hypergeometric functions F _λ and G _λ of Heckman and Opdam, and sharp estimates of the particular functions F ₀ and G ₀. Next we prove the Paley–Wiener theorem for the Schwartz class, solve the heat equation and estimate the heat kernel. Received: June 2006 Revision: December 2006 Accepted: March 2007     

Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere

We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk. Received: December 2006, Revision: August 2007, Accepted: October 2007     

Bases of minimal vectors in lattices,II

We prove that a Euclidean lattice of dimension n = 5 (resp. 6; resp. 7) having at least 6 (resp. 10; resp. 18) pairs of minimal vectors has a basis of minimal vectors. Received: 7 December 2006     

Kreĭn–Saakyan and Trace Formulas for a Pair of Canonical Differential Operators

An analog of the Kreĭn–Saakyan formula is derived for any pair of relatively prime self-adjoint extensions of a minimal symmetric canonical differential operator. This allows us to deduce a trace formula in the matrix case. I am grateful to Sh. Saakyan for his interest in this work and lively discussion. Received: December 8, 2006. Accepted: December 30, 2006.     

Trace estimates and invariance of the essential spectrum

We provide sufficient conditions under which the difference of the resolvents of two higher-order operators acting in belongs to trace classes . We provide explicit estimates on the norm of the resolvent difference in terms of L^p norms of the difference of the coefficients. Such inequalities are useful in estimating the effect of localized perturbations of the coefficients. Received: 5 September 2005; revised: 15 December 2005     

Non-Wrapping of Hyperbolic Interval Bundles

We demonstrate a condition on the boundary at infinity of a hyperbolic interval bundle N that guarantees that, for any associated geometric limit, there is a compact core for N which embeds under the covering map. The proof involves an analysis of the geometry of torus cusps in a hyperbolic manifold, and techniques of Anderson, Canary and McCullough [AnCM]. Together with results of Holt–Souto [HS] this shows that the locus of non-local-connectivity of the space of once-punctured torus groups is not dense, and describes a relatively open subset of the boundary of the space of once-punctured torus groups consisting of points of non-self-bumping. Received: April 2006, Revision: May 2007, Accepted: December 2007     

On fractional moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions,II

We obtain upper and lower bounds for fractional moments of Dirichlet L-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 606–621, October–December, 2006.     

Path-Factors in the Square of a Tree

In this paper, we give a necessary condition for the existence of P ₃-factor in the square of a tree. Furthermore, we show that the square of any tree on at least 6 vertices has a {P ₃,P ₄}-factor. As a consequence, every connected graph on at least 6 vertices has a {P ₃,P ₄}-factor. Zhao Zhang: The research is supported by NSFC (60603003) and XJEDU. Received: November 15, 2006. Final version received: December 12, 2007.     

On the support of tempered distributions

We show that, given a tempered distribution S whose Fourier transform is a function of polynomial growth, a point x in is outside the support of S if and only if the Fourier integral of S is summable in Bochner-Riesz means to zero uniformly on a neighbourhood of x. Received: 29 December 2005     

On the category Q-Mod

In this paper we consider the category Q-Mod of modules over a given quantale Q. The paper is motivated by constructions and results from the category of modules over a ring. We show that the category Q-Mod is monadic, consider its relation to the category Q-Top of Q-topological spaces and generalize a method of completion of partially ordered sets. Received December 20, 2005; accepted in final form December 4, 2006.     

Elementary properties of categories of acts over monoids

We explore connections between elementary equivalence of categories of acts over monoids and second-order equivalence of monoids. __________ Translated from Algebrai Logika, Vol. 45, No. 6, pp. 687–709, November–December, 2006.     

A Note on the Game Chromatic Index of Graphs

In this note we prove that the game chromatic index χ _g ^′(G) of a graph G of arboricity k is at most Δ + 3k − 1. This improves a bound obtained by Cai and Zhu [J. Graph Theory 36 (2001), 144–155] for k-degenerate graphs. Tomasz Bartnicki: Research of the first author is supported by a PhD grant from Polish Ministry of Science and Higher Education N201 2128 33. Received: November 1, 2006. Final version received: December 22, 2007.     

Subfamilies of special elements of complete numberings

Any subfamily of a given at most countable non-empty family can be converted into a set of all special elements of a suitable numbering. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 758–764, November–December, 2006.     

Zassenhaus conjecture for A 6

For the alternating group A ₆ of degree 6, Zassenhaus’ conjecture about rational conjugacy of torsion units in integral group rings is confirmed. Dedicated to the memory of I. S. Luthar Professor Luthar died at the age of 74 in December 2006.     

The Bipolaron in the Strong Coupling Limit

The bipolaron are two electrons coupled to the elastic deformations of an ionic crystal. We study this system in the Fr?hlich approximation. If the Coulomb repulsion dominates, the lowest energy states are two well separated polarons. Otherwise the electrons form a bound pair. We prove the validity of the Pekar–Tomasevich energy functional in the strong coupling limit, yielding estimates on the coupling parameters for which the binding energy is strictly positive. Under the condition of a strictly positive binding energy we prove the existence of a ground state at fixed total momentum P, provided P is not too large. Tadahiro Miyao: This work was supported by Japan Society for the Promotion of Science (JSPS). Permanent address: The graduate school of natural science and technology, Okayama university, Okayama 700-8530, Japan. Submitted: December 14, 2006. Accepted: March 20, 2007.     

On the intrinsic normalizations of the seminonholonomic complexesSN Gr(m, n, (m+1)(n−m)−ϱ). I

Some intrinsic normalizations of seminonholonomic complexes of then-dimensional projective space are considered. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 517–538, October–December, 1999. Translated by R. Lapinskas     

Spectral Convergence of Quasi-One-Dimensional Spaces

We consider a family of non-compact manifolds X_ε (“graph-like manifolds”) approaching a metric graph X₀ and establish convergence results of the related natural operators, namely the (Neumann) Laplacian and the generalized Neumann (Kirchhoff) Laplacian on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations. Communicated by Claude Alain Pillet Submitted: December 21, 2005 Accepted: January 30, 2006