Large regular simplices contained in a hypercube

We prove that the n-dimensional unit hypercube contains an n-dimensional regular simplex of edge length c√n, where c > 0 is a constant independent of n. Supported by MEXT Grant-in-Aid for Scientific Research (B) 20340022.     

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.     

Some remarks on trigonometric sums

Let

Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces

In this paper we extend the Darboux theory of integrability in ℝ ⁿ to the regular algebraic hypersurfaces. Then we apply the extended theory first to the 3-dimensional generalized cylinders ×ℝ^3−r of ℝ⁴ forr=0, 1, 2, 3; and after to then-dimensional sphere of ℝ ⁿ⁺¹.     

Ortho and Causal Closure Operations in Ordered Vector Spaces

On a non-trivial partially ordered real vector space V the orthogonality relation is defined by incomparability and is a complete lattice of double orthoclosed sets. In an earlier paper we defined an integrally open ordered vector space V and proved orthomodularity of . We shall say that is an orthogonal set when for all with , we have . We consider two different closure operations and (ortho and causal closure) and prove: V is integrally open iff for every orthogonal set . Hence follows: if V is integrally open, then . Received July 6, 2007; accepted in final form July 31, 2007.     

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

The ortho-diameters of Nikol’skii and Besov classes in the Lorentz spaces

In this paper we estimate the order of approximation of S. M. Nikol’skii and O. V. Besov classes in the norm of the anisotropic Lorentz space. We also obtain bounds for ortho-diameters of these classes.     

Equations in finite fields with restricted solution sets. II (Algebraic equations)

Generalizing earlier results, it is shown that if are “large” subsets of a finite field F _q , then the equations a + b = cd, resp. ab + 1 = cd can be solved with . Other algebraic equations with solutions restricted to “large” subsets of F _q are also studied. The proofs are based on character sum estimates proved in Part I of the paper. Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No. T 043623, T 043631 and T 049693.     

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.     

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.     

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.     

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²[–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.     

A note on generalized connectedness

This note is devoted to study the preservation of connectedness under the basic operators in generalized topological spaces. Some characterizations of generalized connectedness are given. As an application, we generalize some results in topological spaces. Supported in part by the NSFC (No. 10571151, 10671173).     

A note on holomorphic approximation in Banach spaces

Given a complex Banach space X and a holomorphic function f on its unit ball B, we discuss the problem whether f can be approximated, uniformly on smaller balls, by functions g holomorphic on all of X. Research partially supported by NSF grant DMS0700281.     

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]).     

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 .     

Automorphisms of spine spaces

Automorphisms of non-trivial spine spaces are either, type-preserving, or type-exchanging, that is, they map stars onto stars or exchange them with tops. In the first case they are given by semi-linear bijections, in the latter, by sesqui-linear forms.     

Construction of pseudorandom binary sequences using additive characters over GF(2 k )

In a series of papers Mauduit and Sárközy (partly with coauthors) studied finite pseudorandom binary sequences and they constructed sequences with strong pseudorandom properties. In these constructions fields with prime order were used. In this paper a new construction is presented, which is based on finite fields of order 2 ^k .     

Remarks on extremal solutions of differential equations with advanced argument

We show the existence of absolutely continuous extremal solutions to the problemx′(t)=f(t, x)h(t)))+g(t)),x(0)=x ₀, whereh is an arbitrary continuous deviated argument. Conditions for the uniqueness of solutions are given. Research partialy supported by grant UG BW 5100 - 5 - 0143 - 4     

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