Volume 2005, Issue 583

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.

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.

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.

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.

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.

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.