A Minimal Candidate for the Existence of a Weakly-Ruinous Set

We show that Richard Thompson's group F is nonamenable if and only if the three element set A ₂ = {x ₀ x ₀ x ₀, x ₀ x ₀ x ₁, x ₀ x ₀ x ₂} is weakly-ruinous. We then show that no two element set is weakly-ruinous, thereby showing that A ₂ is minimal in both the number of elements that a weakly-ruinous set must contain and also in the number of carets that each tree in a weakly-ruinous set must have.     

Remotality of Closed Bounded Convex Sets in Reflexive Spaces

Let X be a Banach space and E be a closed bounded subset of X. For x ? X, we define D(x, E) = sup{‖ x ? e‖:e ? E}. The set E is said to be remotal (in X) if, for every x ? X, there exists e ? E such that D(x, E) = ‖x ? e‖. The object of this paper is to characterize those reflexive Banach spaces in which every closed bounded convex set is remotal. Such a result enabled us to produce a convex closed and bounded set in a uniformly convex Banach space that is not remotal. Further, we characterize Banach spaces in which every bounded closed set is remotal.     

Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in ⁿ which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on ^s and ⁿ , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map and a constant C such that ||ϕ (x) - (x)|| for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | η; and if X is a Meyer set, then all algebraic conjugates | η ' | 1. Received May 9, 1997, and in revised form March 5, 1998.     

On the Structure of the Solution Set of u″ = f(t,u), u(0) = u(1) = 0

The solution set of a Dirichlet problem x″ = f(t, x), x(0) = x(1) = 0, on a Banach space E and with f satisfying a Lipschitz condition, is homeomorphic to a closed subset of E. We prove that to an closed subset C of E there is a function f with Lipschitz constant arbitrarily close to π², such that the solution set of the corresponding Dirichlet problem is homeomorphic to C.     

On coding uncountable sets by reals

If A ? ω₁, then there exists a cardinal preserving generic extension ??[A ][x ] of ??[A ] by a real x such that 1) A ∈ ??[x ] and A is Δ₁^HC (x) in ??[x ]; 2) x is minimal over ??[A ], that is, if a set Y belongs to ??[x ], then either x ∈ ??[A, Y ] or Y ∈ ??[A ]. The forcing we use implicitly provides reshaping of the given set A (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semiconductor Equations for variable Mobilities Based on Boltzmann Statistics or Fermi-Dirac Statistics

The first aim of this paper is to characterize those limit spaces (X, τ) which can be valuated, i.e. for which a set E of valuations exists such that for each x ∈ X, τ(x) equals the set of filters on X which converge to x relative to E. It is shown further that a separated pretopological space is a URYSOHN -space iff it can be valuated by a set of subadditive valuations. In the second part of the paper a completion is constructed for the separated subadditively valuated limit spaces which can be considered as a generalization of the usual completion of a uniform space.     

Rewarding Maps: On Greedy Optimization of Set Functions

Given a set function, that is, a map ƒ: (E) → {−∞} from the set (E) of subsets of a finite set E into the reals including −∞, the standard greedy algorithm (GA) for optimizing ƒ starts with the empty set and then proceeds by enlarging this set greedily, element by element. A set function ƒ is said to be tractable if in this way a sequence x₀ , x₁, . . ., x_N E (N #E) of subsets with max(ƒ) {ƒ(x₀), ƒ(x₁), . . ., ƒ(x_N)} will always be found. In this note, we will reinterpret and transcend the traditions of classical GA-theory (cf., e.g., [KLS]) by establishing necessary and sufficient conditions for a set function ƒ not just to be tractable as it stands, but to give rise to a whole family of tractable set functions ƒ(η) : (E) → : x ƒ(x) + Σ_{e x}η(e), where η runs through all real valued weighting schemes η : E → , in which case ƒ will be called rewarding. In addition, we will characterize two important subclasses of rewarding maps, viz. truncatably rewarding (or well-layered) maps, that is, set functions ƒ such that [formula] is rewarding for every i = 1, . . ., N, and matroidal maps, that is, set functions ƒ such that for every η : E → and every ƒ_eta-greedy sequence x₀, x₁, . . ., x_N as above, one has max(ƒ_η) = ƒ_η(x_i) for the unique i {0, . . ., N} with ƒ_η(x₀) < ƒ_η(x₁) < ··· < ƒ_η(x_i) ≥ ƒ_η(x_{i + 1}).     

One-dimensional Tilings Using Tiles with Two Gap Lengths

Let n,p,k,q,l be positive integers with n=k+l+1. Let x₁,x₂, . . . ,x_n be a sequence of positive integers with x₁<x₂<···<x_n. A set {x₁,x₂, . . . ,x_n} is called a set of type (p,k;q,l) if the set of differences {x₂–x₁,x₃–x₂, . . . ,x_n–x_n–1} equals {p, . . . ,p,q, . . . ,q} as a multiset, where p and q appear k and l times, respectively. Among other results, it is shown that for any p,k,q, there exists a finite interval I in the set of integers such that I is partitioned into sets of type (p,k;q,1).     

The Cendian of a Tree

The eccentricity e(x) and the distance sum s(x) of a vertex x of a connected graph G are well-known functions which measure the centrality of the vertex x in G. The set of vertices which minimize e(x) is called the center of G and the set of vertices which minimize s(x) is known as the median. In this paper we introduce the idea of the so-called cendian of a graph, which unifies the concepts of center and median, and study its structure in trees.     

Cycles with a given number of vertices from each partite set in regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d ⁺(x) and d ⁻(x) the outdegree and the indegree of x, respectively. A digraph D is called regular, if there is a number p ∈ ℕ such that d ⁺(x) = d ⁻(x) = p for all vertices x of D. A c-partite tournament is an orientation of a complete c-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether c-partite tournaments with r vertices in each partite set contain a cycle with exactly r − 1 vertices of every partite set. In 1982, Beineke and Little [2] solved this problem for the regular case if c = 2. If c ⩾ 3, then we will show that a regular c-partite tournament with r ⩾ 2 vertices in each partite set contains a cycle with exactly r − 1 vertices from each partite set, with the exception of the case that c = 4 and r = 2.     

Local bootstrap

The local bootstrap method draws bootstrap samples from a neighborhood of each data point. Unlike the common bootstrap, it is especially useful for heteroscedastic data. This paper considers local bootstrap of y for a data set {(x _i , y _i )} with the neighborhood defined by the window in a kernel regression model. It shows that for each x, the bootstrap distribution and moments converge almost surely to the true distribution and moments of y respectively, and this convergence is also uniform in x within the data set.     

Measures of affinity of a sequence for a number

We define strong and weak affinities of a number a for a sequence (x_k ) denoted by L (a,(x_k )) and U (a, (x_k )) respectively. We show U (a,(x_k )) > 0 if and only if the number a is a statistical limit point of the sequence (x_k ). We consider the distribution of sequences with positive weak and strong measures of affinity within the space l ^∞ of bounded sequences. The main result is that the set of bounded sequences with U (a,(x_k )) > 0, that is, the set of sequences with statistical limit points, is a dense subset in l ^∞ of the first category. We also show the set of sequences with positive strong affinities is a nowhere dense subset of l ^∞.     

Angle orders and zeros

An angle order is a partially ordered set whose points can be mapped into unbounded angular regions in the plane such that x is less than y in the partial order if and only if x's angular region is properly included in y's. The zero augmentation of a partially ordered set adds one point to the set that is less than all original points. We prove that there are finite angle orders whose augmentations are not angle orders. The proof makes extensive use of Ramsey theory.     

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 spectral curve for functional-difference Schrödinger equation

We suggest a method for constructing a set of finite-gap solutions for a functional-difference deformation of the Schr?dinger equation v(x)f(x +2h)+ f(x)= λf(x + h). It is shown that the edges of gaps of the corresponding spectral curve depend on x. Examples are given. Bibliography: 7 titles.     

Not all quadrative norms are strongly stable

A norm N on an algebra A is called quadrative if N(x²) ≤ N(x)² for all x A, and strongly stable if N(x^k) ≤ N(x)^k for all x A and all k = 2, 3, 4…. Our main purpose in this note is to show that not all quadrative norms are strongly stable.     

Continuum cardinality of the set of solutions of one class of equations that contain the function of frequency of ternary digits of a number

We study the equation ν ₁(x) = x, where ν ₁(x) is the function of frequency of the digit 1 in the ternary expansion of x. We prove that this equation has a unique rational root and a continuum set of irrational solutions. An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions. Some additional facts about the equations ν _i (x) = x, i = 0, 2, are given. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1414–1421, October, 2008.     

Cohen-macaulay bipartite graphs

Let G be a graph on the vertex set V={x ₁, ..., x _n}. Let k be a field and let R be the polynomial ring k[x ₁, ..., x _n]. The graph ideal I(G), associated to G, is the ideal of R generated by the set of square-free monomials x _ix_j so that x _i, is adjacent to x _j. The graph G is Cohen-Macaulay over k if R/I(G) is a Cohen-Macaulay ring. Let G be a Cohen-Macaulay bipartite graph. The main result of this paper shows that G{v} is Cohen-Macaulay for some vertex v in G. Then as a consequence it is shown that the Reisner-Stanley simplicial complex of I(G) is shellable. An example of N. Terai is presented showing these results fail for Cohen-Macaulay non bipartite graphs. Partially supported by COFAA-IPN, CONACyT and SNI, México.     

Projection Theorems for Harmonic Measure in NTA Domains

Suppose D is an NTA domain, E D is any closed set, and P ^x ₀(E) is the projection with respect to a point x ₀D of the set E onto the boundary of D. The projection P ^x ₀ satisfies certain geometric properties so that it is a generalization of the notion of radial projection with respect to a point x ₀ onto a boundary of a domain. It is shown that the harmonic measure of E with respect to the domain DE evaluated at the point x ₀ is bounded below by a constant times the harmonic measure of the set P ^x ₀(E) with respect to the domain D evaluated at the point x ₀. The constant is independent of the set E but it may depend upon x ₀.     

Generic existence of a nonempty compact set of fixed points

Let X be a complete metric space, a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset contained in such that for every mapping T in the set {x ε X: Tx = x} is nonempty, it is proved that most mappings (in the Baire category sense) in do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived.