Collapsing monoids consisting of permutations and constants

In this paper we determine all collapsing transformation monoids that contain at least one unary constant operation and whose nonconstant operations are permutations. Furthermore, we find an infinite family of transformation monoids that consist of at least three unary constant operations and some permutations for which the corresponding monoidal intervals are 2-element chains. This research is supported by Hungarian National Foundation for Scientific Research grant nos. T 37877 and K 60148.     

Estimation on curves of Dirichlet series with a convex growth majorant

For a class of Dirichlet series defined by a certain convex growth majorant we establish conditions for a sequence of indices which provide the implementation of precise estimates for their increase and decrease on curves that tend to infinity in a special way.     

Sobolev Mappings: Lipschitz Density is not a Bi-Lipschitz Invariant of the Target

We study a question of density of Lipschitz mappings in the Sobolev class of mappings from a closed manifold into a singular space. The main result of the paper, Theorem 1.7, shows that if we change the metric in the target space to a bi-Lipschitz equivalent one, then the property of the density of Lipschitz mappings may be lost. Other main results in the paper are Theorems 1.2, 1.3, 1.6, 1.8. This work was supported by the NSF grant DMS-0500966. Received: June 2005 Revision: December 2005 Accepted: January 2006     

Continuous functions taking every value a given number of times

We give necessary and sufficient conditions for a function f: [0, 1] → {1,2,...,w, c} under which there exists a continuous function F: [0, 1] → [0, 1] such that for every y ɛ [0, 1], |F ⁻¹ (y)| = f(y).      

Topologies on function spaces

Function spaces play an important role in complex analysis, in the theory of differential equations, in functional analysis and in almost every other branch of modern mathematics. In this paper we give and study the notion of clopen convergence. Also, we study the notion of clopen continuity and define new topologies on function spaces. These results generalize basic results of R. Arens, J. Dugundji and A. Di Concilio (see [1], [4], [2] and [3]).     

Generalized bisymmetry on a restricted domain

Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | x ≧ y}. We give the solution of the equation

Some remarks on trigonometric sums

Let

Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources

The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type under nearly optimal conditions on . It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of , obtaining a nontrivial estimate provided . This is a further improvement on [BGK]. Received: May 2007, Revision: October 2007, Accepted: October 2007     

Estimates for Mixed Character Sums

We give sharp estimates for certain families of exponential sums in several variables over finite fields. Received: May 2007, Accepted: September 2007     

The operad of finite labeled tournaments

We study the operad of finite labeled tournaments. We describe the structure of suboperads of this operad generated by simple tournaments. We prove that a suboperad generated by a tournament with two vertices (i.e., the operad of finite linearly ordered sets) is isomorphic to the operad of symmetric groups, and a suboperad generated by a simple tournament with more that two vertices is isomorphic to the quotient operad of the free operad with respect to a certain congruence. We obtain this congruence explicitly.     

Lectures on topology of words

We discuss a topological approach to words introduced by the author in [Tu2]–[Tu4]. Words on an arbitrary alphabet are approximated by Gauss words and then studied up to natural modifications inspired by the Reidemeister moves on knot diagrams. This leads us to a notion of homotopy for words. We introduce several homotopy invariants of words and give a homotopy classification of words of length five. Based on notes by Eri Hatakenaka, Daniel Moskovich, and Tadayuki Watanabe     

Reconstruction of Function Fields

For k an algebraic closure of the finite field , ℓ prime distinct from p and X a surface over k, we prove that the field of rational functions k(X) can be recovered from the maximal pro-ℓ-quotient of its absolute Galois group – in fact already from the second central descending series quotient of . Submitted: July 2004, Revision: October 2005, Final revision: February 2008, Accepted: February 2008     

Method of Variations of Potential of Quasi-Periodic Schrödinger Equations

We study the one-dimensional discrete quasi-periodic Schr?dinger equation

On continuation of a topology from a system of generators of a group to a topology on the group

In this paper we establish conditions under which a topology is continued from a subset of a group to a topology on the group coordinated with the group operation.     

Polytopes passing through circles

Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question.     

On a class of slowly changing functions

We study the slowly changing functions and use them in the study of entire functions. An erratum to this article is available at .     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

Measurable orthogonally additive functions modulo a discrete subgroup

Under appropriate conditions on Abelian topological groups G and H, an orthogonality ⊥ ⊂ G ² and a σ-algebra of subsets of G we decompose an -measurable function f: G → H which is orthogonally additive modulo a discrete subgroup K of H into its continuous additive and continuous quadratic part (modulo K). Research supported by the Silesian University Mathematics Department (Functional Equations on Abstract Structures program — the first author, and Iterative Functional Equations and Real Analysis program — the second author).     

On the Classification of Cartan Actions

We study higher-rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by with k ≥ 3 whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic -actions for k ≥ 3. The first author is supported in part by NSF grants DMS-0140513. The second author is supported in part by NSF grant DMS-0203735. Received: July 2005 Revision: March 2006 Accepted: March 2006