Transitive partial actions of groups

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y _G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y _G such that the global group action of G on Y _G is continuous and the injection Y into Y _G is an open dense equivariant embedding.      

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     

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 .     

Abelian groups small with respect to reduced groups

In this paperwe study

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-small abelian groups, where

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is the class of all reduced groups.

     

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     

On the dual Stiefel-Whitney classes of some Grassmann manifolds

We present some non-vanishing dual Stiefel-Whitney classes of the Grassmann manifolds O(n)/O(4) × O(n − 4) for n = 2 ^s + 2 and n = 2 ^s + 3 (s ≧ 3), providing a supplement to results of Hiller, Stong, and Oproiu. Some applications are also mentioned. Part of this research was carried out while the first author was a member of three research teams supported in part by the grant agencies VEGA and APVV, and the second author was a member of a research team supported in part by VEGA.     

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.     

On the relationships of seven numerical sequences

We analyze the relationships of four classical and three recently defined classes of numerical sequences. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # T042462 and TS44782.     

Some remarks on trigonometric sums

Let

The Nielsen number for free fundamental groups and maps without remnant

For Y any space that has the homotopy type of a wedge of finitely many circles, and for g : Y → Y a map, the Nielsen number of g, N(g), is a homotopy invariant lower bound for the size of the fixed point set of any map homotopic to g. Such a map g has k-remnant if, roughly, there is limited cancellation in any product g _♯(u)g _♯(v) where g _♯ is the induced homomorphism and u, v ∈ π₁(Y) and |u| = |v| = k. We prove that such maps are (k + 1)-characteristic, meaning that in order to determine the Nielsen classes of fixed points, we need only test whether a limited, specified, set of elements z ∈ π₁(Y) are solutions to the equation z = W ⁻¹ _x f _♯(z)W _y , with x and y fixed points that are represented in the fundamental group by W _x and W _y , respectively. The number of elements to be tested is profoundly decreased by using abelianization as well. This work is a significant extension of Wagner’s results involving maps with remnant and Wagner’s algorithm. Our proofs involve new concepts and techniques. We present an algorithm for N(g) for any map g with k-remnant, and we provide examples for which no algebraic techniques previously known would work. One example shows that for any k there is a map that does not have (k − 1)-remnant but does have k-remnant. Dedicated to Edward Fadell for inspirational teaching and guidance as the thesis advisor of the first author     

On the maximum number of points at least one unit away from each other in the unit <Emphasis Type="Italic">n</Emphasis>-cube

We present a very short survey of known results and many new estimates and results on the maximum number of points that can be chosen in the n-dimensional unit cube so that every distance between them is at least 1. Research was supported by Slovak national grant VEGA 1/3839/06.     

On edge-antipodal <Emphasis Type="Italic">d</Emphasis>-polytopes

A convex d-polytope in ℝ ^d is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.      

On some diophantine results related to Euler polynomials

In this paper we prove that there is at most one complex number b for which the shifted Euler polynomial E _n (x) + b has at most two zeros of odd multiplicity. Supported in part, by Grants T48791 and F68872 form HNFSR, and by the Hungarian Academy of Sciences.     

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     

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.     

On the universal A.S. central limit theorem

Let (X _k ) be a sequence of independent r.v.’s such that for some measurable functions gk : R ^k → R a weak limit theorem of the form

On a pythagorean-type functional equation with exponent 3 involving finite fields

We solve the functional equationf(x ³ +y ³) =f(x)³ +f(y)³ for maps of a finite field into itself.     

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem. The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S² V S² V ... →N. The main result is that the images of these renormalized maps converge in L^1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class.