Local Geometry of Fractals Given by Tangent Measure Distributions

 Let μ be a self-similar-measure and ν an ergodic shift-invariant measure on a self-similar set A. We show that under weak conditions ν-almost all points x in A show the same local structure, that is, the same tangent measure distribution of μ. (Received 10 October 2000, in revised form 8 March 2001)     

Local Geometry of Fractals Given by Tangent Measure Distributions

 Let μ be a self-similar-measure and ν an ergodic shift-invariant measure on a self-similar set A. We show that under weak conditions ν-almost all points x in A show the same local structure, that is, the same tangent measure distribution of μ.     

Geodesics in Graphs,an Extremal Set Problem,and Perfect Hash Families

 A set U of vertices of a graph G is called a geodetic set if the union of all the geodesics joining pairs of points of U is the whole graph G. One result in this paper is a tight lower bound on the minimum number of vertices in a geodetic set. In order to obtain that result, the following extremal set problem is solved. Find the minimum cardinality of a collection 𝒮 of subsets of [n]={1,2,…,n} such that, for any two distinct elements x,y∈[n], there exists disjoint subsets A _x ,A _y ∈𝒮 such that x∈A _x and y∈A _y . This separating set problem can be generalized, and some bounds can be obtained from known results on families of hash functions. Received: May 19, 2000 Final version received: July 5, 2001     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

Generating primitive positive clones

Suppose is a set of operations on a finite set A. Define PPC() to be the smallest primitive positive clone on A containing . For any finite algebra A, let PPC#(A) be the smallest number n for which PPC(CloA) = PPC(Clo _n A). S. Burris and R. Willard [2] conjectured that PPC#(A) ≤|A| when CloA is a primitive positive clone and |A| > 2. In this paper, we look at how large PPC#(A) can be when special conditions are placed on the finite algebra A. We show that PPC#(A) ≤|A| holds when the variety generated by A is congruence distributive, Abelian, or decidable. We also show that PPC#(A) ≤|A| + 2 if A generates a congruence permutable variety and every subalgebra of A is the product of a congruence neutral algebra and an Abelian algebra. Furthermore, we give an example in which PPC#(A) ≥|A| - 1)² so that these results are not vacuous. Received August 30, 1999; accepted in final form April 4, 2000.     

The rate of convergence of Fourier expansions in the plane: a geometric viewpoint

Let A be an appropriate planar domain and let f be a piecewise smooth function on . We discuss the rate of convergence of in terms of the interaction between the geometry of A and the geometry of the singularities of f. The most subtle case is when x belongs to the singular set of f. Received: 21 December 2000; in final form: 4 September 2001 / Published online: 4 April 2002     

Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces

Let C be a closed and convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself, A be an α-inverse strongly-monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. We introduce an iteration scheme of finding a point of F (T)∩(A+B)⁻¹0, where F (T) is the set of fixed points of T and (A+B)⁻¹0 is the set of zero points of A+B. Then, we prove a strong convergence theorem, which is different from the results of Halpern’s type. Using this result, we get a strong convergence theorem for finding a common fixed point of two nonexpansive mappings in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping.     

One class of simultaneous pursuit games

Let A and B be given convex closed bounded nonempty subsets in a Hilbert space H; let the first player choose points in the set A and let the second one do those in the set B. We understand the payoff function as the mean value of the distance between these points. The goal of the first player is to minimize the mean value, while that of the second player is to maximize it. We study the structure of optimal mixed strategies and calculate the game value.     

Box and Packing Dimensions of Typical Compact Sets

 Let (X,ρ) be a complete metric space and let dim A be the upper box dimension of the set . We show that packing dimension of the typical (in the sense of Baire category) compact set is at least . (Received 27 March 2000; in revised form 5 June 2000)     

Small transversals in hypergraphs

For each positive integerk, we consider the setA _k of all ordered pairs [a, b] such that in everyk-graph withn vertices andm edges some set of at mostam+bn vertices meets all the edges. We show that eachA _k withk2 has infinitely many extreme points and conjecture that, for every positive , it has only finitely many extreme points [a, b] witha. With the extreme points ordered by the first coordinate, we identify the last two extreme points of everyA _k , identify the last three extreme points ofA ₃, and describeA ₂ completely. A by-product of our arguments is a new algorithmic proof of Turán's theorem.     

Finite sets which contain their Radon points

A point r is a Radon point of a finite set S if r=conv Aconv B, where A and B are disjoint subsets of S. Two characterizations are given for those finite sets in R ^d which contain their Radon points, and the convex hull of such a set is described.     

Zum satz von ljapunov in vektorverbänden

Let K be a closed convex order-bounded set of an order-complete vector lattice and let A be a him continuous linear operator. Then the equality AK = A(ext K) is proved, where ext K is the set of all extremal points of K. It is shown that various generalizations of the Ljapunov-theorem on the range of vector-measures are special cases of this general statement.     

Relative Chebyshev centers in normed linear spaces, part II

Let E be a normed linear space, A a bounded set in E, and G an arbitrary set in E. The relative Chebyshev center of A in G is the set of points in G best approximating A. We have obtained elsewhere general results characterizing the spaces in which the center reduces to a singleton in terms of structural properties related to uniform and strict convexity. In this paper, an analysis of the Chebyshev norm case, which falls outside the scope of the previous analysis, is presented.     

Conditions on Normal Spaces of Finite Rank That Guarantee C(X) is SV

An f-ring A is an SV f-ring if for every minimal prime ?-ideal P of A, A/P is a valuation domain. A topological space X is an SV space if C(X) is an SV f-ring. For normal spaces, several conditions are shown to guarantee the space is an SV space. For example, a normal space of finite rank for which the closure of the set of points of rank greater than 1 is an F-subspace, is an SV space. For normal spaces of rank 2, a characterization of SV spaces is given.     

The generation of formulas held in common knowledge

Common knowledge of a finite set of formulas implies a special relationship between syntactic and semantic common knowledge. If S, a set of formulas held in common knowledge, is implied by the common knowledge of some finite subset of S, and A is a non-redundant semantic model where exactly S is held in common knowledge, then the following are equivalent: (a) S is maximal among the sets of formulas that can be held in common knowledge, (b) A is finite, and (c) the set S determines A uniquely; otherwise there are uncountably many such A. Even if the knowledge of the agents are defined by their knowledge of formulas, 1) there is a continuum of distinct semantic models where only the tautologies are held in common knowledge and, 2) not assuming that S is finitely generated (a) does not imply (c), (c) does not imply (a), and (a) and (c) together do not imply (b). Received November 1999/Revised version January 2000     

Estimation for the region of attraction of nonlinear difference equations

Summary. We consider nonlinear systems of difference equations with and , where A is any matrix, is a continuous vector-function, and is a numeral parameter. The spectrum of A belongs to the unit circle . We give the estimations for the region of attraction and the speed of convergence solutions to the zero solution of the systems. We indicate a set M such that for solutions of the system with parameter the limit is true. Received April 7, 2000 / Revised version received January 8, 2001 / Published online November 15, 2001     

<Emphasis Type="Italic">A</Emphasis><Subscript>2</Subscript>-continued fraction representation of real numbers and its geometry

We study the geometry of representations of numbers by continued fractions whose elements belong to the set A ₂ = {α₁, α₂} (A ₂-continued fraction representation). It is shown that, for α₁α₂ ≤ 1/2 , every point of a certain segment admits an A ₂-continued fraction representation. Moreover, for α₁α₂ = 1/2, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose A ₂-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its A ₂-continued fraction representation form a homogeneous Markov chain are also investigated.     

Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem

In a set without linear structure equipped with a preorder, we give a general existence result for efficient points. In a topological vector space equipped with a partial order induced by a closed convex cone with a bounded base, we prove another kind of existence result for efficient points; this result does not depend on the Zorn lemma. As applications, we study a solution problem in vector optimization and generalize the Bishop–Phelps theorem to a topological vector space setting by showing that the B-support points of any sequentially complete closed subset A of a topological vector space E is dense in A, where B is any bounded convex subset of E.