The influence of the boundary behavior on isometric immersions in the hyperbolic space

This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infinity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curvature vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurface with mean curvature satisfying sup _{p∈Σ ||H(p)|| < 1} has no isolated points in its asymptotic boundary. Our main tool is a Tangency principle for isometric immersions of arbitrary codimension. This work is partially supported by CAPES, Brazil.     

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H _i } of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H _i have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union generates G can be replaced by the condition that it is equal to G). The work presented here is partially supported by NSF/DMS-0805932.     

Sets with more differences than sums

We show that a random set of integers with density 0 has almost always more differences than sums.      

Counting solutions of polynomial systems via iterated fibrations

The paper introduces a new polynomial to count the solutions of a system of polynomial equations and inequations over an algebraically closed field of characteristic zero based on the triangular decomposition algorithm by J. M. Thomas of the nineteen-thirties. In the special case of projective varieties examples indicate that it is a finer invariant than the Hilbert polynomial. Received: 8 March 2008; Revised: 12 August 2008     

Inverse positivity for general Robin problems on Lipschitz domains

It is proved that elliptic boundary value problems in divergence form can be written in many equivalent forms. This is used to prove regularity properties and maximum principles for problems with Robin boundary conditions with negative or indefinite boundary coefficient on Lipschitz domains by rewriting them as a problem with positive coefficient. It is also shown that such methods cannot be applied to domains with an outward pointing cusp. Applications to the regularity of the harmonic Steklov eigenfunctions on Lipschitz domains are given. Received: 26 June 2008; Revised: 12 September 2008     

Value distribution of the difference operator

We investigate the value distribution of difference operators for entire functions, and establish a difference analogue of the Brück conjecture for entire functions of order less than 1. This work was supported by the NNSF of China (No. 10671109).     

Equivalent complete norms and positivity

In the first part of the article we characterize automatic continuity of positive operators. As a corollary we consider complete norms for which a given cone E₊ in an infinite dimensional Banach space E is closed and we obtain the following result: every two such norms are equivalent if and only if has finite codimension. Without preservation of an order structure, on an infinite dimensional Banach space one can always construct infinitely many mutually non-equivalent complete norms. We use different techniques to prove this. The most striking is a set theoretic approach which allows us to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic. Received: 28 January 2009     

A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds

This article provides an Omega-result for the remainder term in Weyl’s law for the spectral counting function of certain rational (2ℓ + 1)-dimensional Heisenberg manifolds. Received: 22 September 2008     

An Improved Cauchy Formula for Hypermonogenic Functions

A new Cauchy-type formula for hypermonogenic functions is derived. Hypermonogenic functions, introduced in [6], are a generalization of holomorphic functions to several dimensions. The power function x^m is hypermonogenic. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš     

Invariant Operators Between Spaces of h-Monogenic Polynomials

In this paper we study properties of invariant differential operators acting between spaces of h-monogenic functions. In this way one obtains differential operators acting between invariant (m)-modules which can be seen as the Hermitean analogues of the classical Rarita-Schwinger operators. Received: October, 2007. Accepted: February, 2008.