Journal für die reine und angewandte Mathematik

Published by De Gruyter

Volume 2005 Issue 583

Issue of Journal für die reine und angewandte Mathematik

Contents
Journal Overview

November 7, 2005

CR invariant powers of the sub-Laplacian

A. Rod Gover, C. Robin Graham

Page range: 1-27

Abstract

CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ‘‘conformally invariant powers of the Laplacian’’ via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. A second family is derived from a CR tractor calculus which is developed here; this family includes operators for every positive power of the sub-Laplacian. This result together with work of Čap, Slovák and Souček imply in three dimensions the existence of a curved analogue of each such operator in ﬂat space.

November 7, 2005

The Dirichlet boundary value problem for real solutions of the ﬁrst Painlevé equation on segments in non-positive semi-axis

N. Joshi, A. V. Kitaev

Page range: 29-86

Abstract

We develop a qualitative theory for real solutions of the equation y” = 6 y 2 − x . In this work a restriction x ≦ 0 is assumed. An important ingredient of our theory is the introduction of several new transcendental functions of one, two, and three variables that describe different properties of the solutions. In particular, the results obtained allow us to completely analyse the Dirichlet boundary value problem y ( a ) = y 0 , y ( b ) = y 0 for a < b ≦ 0.

November 7, 2005

Inverse problem and estimates for periodic Zakharov-Shabat systems

Evgeny Korotyaev

Page range: 87-115

Abstract

Consider the Zakharov-Shabat operator T ZS on L 2 (ℝ) ⊕ L 2 (ℝ) with real periodic vector potential q = ( q 1 , q 2 ) ∈ H = L 2 () ⊕ L 2 (). The spectrum of T ZS is absolutely continuous and consists of intervals separated by gaps ( z n − , z n + ), n ∈ ℤ. From the Dirichlet eigenvalues m n , n ∈ ℤ of the Zakharov-Shabat equation with Dirichlet boundary conditions at 0, 1, the center of the gap and the square of the gap length we construct the gap length mapping g : H → ℓ 2 ⊕ ℓ 2 . Using nonlinear functional analysis in Hilbert spaces, we show that this mapping is a real analytic isomorphism. Our proof relies on new identities and a priori estimates contained in the second part of the paper. In order to get these estimates we obtain new results in conformal mapping theory.

November 7, 2005

Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function

Alexander I. Bobenko, Christian Mercat, Yuri B. Suris

Page range: 117-161

Abstract

Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Bäcklund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to ℤ d , where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d -dimensional discrete logarithmic function which is a generalization of Kenyon’s discrete Green’s function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.

November 7, 2005

Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley

Christos A. Athanasiadis

Page range: 163-174

Abstract

It is proved that for a certain class of integer polytopes P the polynomial h ( t ) which appears as the numerator in the Ehrhart series of P , when written as a rational function of t , is equal to the h -polynomial of a simplicial polytope and hence that its co-efficients satisfy the conditions of the g -theorem. This class includes the order polytopes of graded posets, previously studied by Reiner and Welker, and the Birkhoff polytope of doubly stochastic n × n matrices. In the latter case the unimodality of the coefficients of h ( t ), which follows, was conjectured by Stanley in 1983.

November 7, 2005

A new class of integer-valued entire functions

Michael Welter

Page range: 175-192

Abstract

In 1968 Fridman [ G. A. Fridman , Entire integer-valued functions, Mat. Sb. (N.S.) 75 (117) (1968), 417–431] found a lower bound for the growth of transcendental entire functions that together with all their derivatives map ℕ into ℤ. In this paper we study entire functions that satisfy f (σ ) ( n ) ∈ ℤ for all positive integers n and σ = 0, … , s n , where ( s n ) is a sequence of positive integers of exponential growth. As a corollary to our results we get an improvement of Fridman’s lower bound. In the ﬁnal section we brieﬂy report on our analogous results in imaginary quadratic number ﬁelds and for functions that take integer values on a geometric progression.

November 7, 2005

Characteristic elements in noncommutative Iwasawa theory

Otmar Venjakob

Page range: 193-236

About this journal

August Leopold Crelle (1780 - 1855)

The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of "Crelle's Journal". In the 190 years of its existence, "Crelle's Journal" has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals. It belongs to the very top mathematics periodicals, as listed in ISI's Journal Citation Report.