Irregular abelian semigroups with weakly amenable semigroup algebra

In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra ℓ ¹(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but ℓ ¹(S) is weakly amenable. Examples are then given.     

Approximate amenability of Banach category algebras with application to semigroup algebras

Let C be a small category. Then we consider ℓ ¹(C) as the ℓ ¹ algebra over the morphisms of C, with convolution product and also consider as the ℓ ¹ algebra over the objects of C, with pointwise multiplication. The main purpose of this paper is to show that approximate amenability of ℓ ¹(C) implies of and clearly this implies that C has only finitely many objects. Some applications are given, the main one is the characterization of approximate amenability for ℓ ¹(S), where S is a Brandt semigroup, which corrects a result of Lashkarizadeh Bami and Samea (Semigroup Forum 71:312–322, 2005).     

Ring semigroups whose subsemigroups form a chain

A multiplicative semigroup S is called a ring semigroup if an addition may be defined on S so that (S,+,⋅) is a ring. Such semigroups have been well-studied in the literature (see Bell in Words, Languages and Combinatorics, pp. 24–31, World Scientific, Singapore, 1994; Jones in Semigroup Forum 47(1):1–6, 1993; Jones and Ligh in Semigroup Forum 17(2):163–173, 1979). In this note, we use Mihăilescu’s Theorem (formerly Catalan’s Conjecture) to characterize the ring semigroups whose subsemigroups containing 0 form a chain with respect to set inclusion.     

Bounds on the Total Restrained Domination Number of a Graph

Let G = (V, E) be a graph. A set S í V{S \subseteq V} is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ _tr (G) ≤ n − δ − 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r − regular, where 4 ≤ r ≤ n − 3, then γ _tr (G) ≤ n − diam(G) − r + 1.     

On maximal weakly separated set-systems

For a permutation ω∈S _n , Leclerc and Zelevinsky in Am. Math. Soc. Transl., Ser. 2 181, 85–108 (1998) introduced the concept of an ω-chamber weakly separated collection of subsets of {1,2,…,n} and conjectured that all inclusionwise maximal collections of this sort have the same cardinality ℓ(ω)+n+1, where ℓ(ω) is the length of ω. We answer this conjecture affirmatively and present a generalization and additional results.     

A Generalisation of Tverberg’s Theorem

We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝ ^d of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A ₁,A ₂,…,A _k such that for any C⊂S of at most r points, the convex hulls of A ₁\C,A ₂\C,…,A _k \C are intersecting. This was conjectured first by Natalia García-Colín (Ph.D. thesis, University College of London, 2007).     

A remark on presentations of certain Chevalley groups

Let Φ be a root system of typeA _ℓ, ℓ ≧ 2,D _ℓ, ℓ ≧ 4 orE _ℓ, 6 ≧ ℓ ≧ 8 andG a group generated by nonidentity abelian subgroupsA _r,r∈Φ, satisfying:

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].     

Loss of skills in coordination games

This paper deals with 2-player coordination games with vanishing actions, which are repeated games where all diagonal payoffs are strictly positive and all non-diagonal payoffs are zero with the following additional property: At any stage beyond r, if a player has not played a certain action for the last r stages, then he unlearns this action and it disappears from his action set. Such a game is called an r-restricted game. To evaluate the stream of payoffs we use the average reward. For r = 1 the game strategically reduces to a one-shot game and for r ≥ 3 in Schoenmakers (Int Game Theory Rev 4:119–126, 2002) it is shown that all payoffs in the convex hull of the diagonal payoffs are equilibrium rewards. In this paper for the case r = 2 we provide a characterization of the set of equilibrium rewards for 2 × 2 games of this type and a technique to find the equilibrium rewards in m × m games. We also discuss subgame perfection.     

<Emphasis Type="BoldItalic">C</Emphasis>(<Emphasis Type="BoldItalic">X</Emphasis>)-objects in the Category of Semi-affine Lattices

In our previous paper (Hušek and Pulgarín, Topol Appl, doi:, 2009) we characterized the set C(X) of real-valued continuous functions on a topological space X as a real ℓ-group. The present paper weakens the situation to the level of semi-affine lattices.     

Exact simulation of the stationary distribution of the FIFO M/G/c queue: the general case for <Emphasis Type="Italic">ρ</Emphasis><<Emphasis Type="Italic">c</Emphasis>

We present an exact simulation algorithm for the stationary distribution of customer delay for FIFO M/G/c queues in which ρ=λ/μ<c. In Sigman (J. Appl. Probab. 48A:209–216, 2011) an exact simulation algorithm was presented but only under the strong condition that ρ<1 (super stable case). We only assume that the service-time distribution G(x)=P(S≤x), x≥0, with mean 0<E(S)=1/μ<∞, and its corresponding equilibrium distribution $G_{e}(x)=\mu\int_{0}^{x} P(S>y)\,dy$G_{e}(x)=\mu\int_{0}^{x} P(S>y)\,dy are such that samples of them can be simulated. Unlike the methods used in Sigman (J. Appl. Probab. 48A:209–216, 2011) involving coupling from the past, here we use different methods involving discrete-time processes and basic regenerative simulation, in which, as regeneration points, we use return visits to state 0 of a corresponding random assignment (RA) model which serves as a sample-path upper bound.     

Large almost monochromatic subsets in hypergraphs

We show that for all ℓ and ε > 0 there is a constant c = c(ℓ, ε) > 0 such that every ℓ-coloring of the triples of an N-element set contains a subset S of size $ c\sqrt {\log N} $ c\sqrt {\log N} such that at least 1 − ε fraction of the triples of S have the same color. This result is tight up to the constant c and answers an open question of Erdős and Hajnal from 1989 on discrepancy in hypergraphs. For ℓ ≥ 4 colors, it is known that there is an ℓ-coloring of the triples of an N-element set whose largest monochromatic subset has cardinality only Θ(log log N). Thus, our result demonstrates that the maximum almost monochromatic subset that an ℓ-coloring of the triples must contain is much larger than the corresponding monochromatic subset. This is in striking contrast with graphs, where these two quantities have the same order of magnitude. To prove our result, we obtain a new upper bound on the ℓ-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erdős and Hajnal.     

Entering and leaving j-facets

Let S be a set of n points in d -space, no i+1 points on a common (i-1) -flat for 1 ≤ i ≤ d . An oriented (d-1) -simplex spanned by d points in S is called a j-facet of S if there are exactly j points from S on the positive side of its affine hull. We show: (*) {\em For j ≤ n/2 - 2 , the total number of (≤ j) -facets (i.e. the number of i -facets with 0 ≤ i ≤ j ) in 3-space is maximized in convex position (where these numbers are known).} A large part of this presentation is a preparatory review of some basic properties of the collection of j -facets—some with their proofs—and of relations to well-established concepts and results from the theory of convex polytopes (h -vector, Dehn—Sommerville relations, Upper Bound Theorem, Generalized Lower Bound Theorem). The relations are established via a duality closely related to the Gale transform—similar to previous works by Lee, by Clarkson, and by Mulmuley. A central definition is as follows. Given a directed line ℓ and a j -facet F of S , we say that {\it ℓ enters F } if ℓ intersects the relative interior of F in a single point, and if ℓ is directed from the positive to the negative side of F . One of the results reviewed is a tight upper bound of j+d-1 \choose d-1 on the maximum number of j -facets entered by a directed line. Based on these considerations, we also introduce a vector for a point relative to a point set, which—intuitively speaking—expresses ``how interior' the point is relative to the point set. This concept allows us to show that statement (*) above is equivalent to the Generalized Lower Bound Theorem for d -polytopes with at most d+4 vertices. Received May 21, 1999, and in revised form July 6, 2000. Online publication January 17, 2001.     

The embracing Voronoi diagram and closest embracing number

A point q is embraced by a set of points S if q is interior to the convex hull of S [8]. In some illumination applications where points of S are lights and q is a point to be illuminated, the embracing concept is related to a good illumination [4, 6], also known as the ∆-guarding [12] and the well-covering [10]. In this paper, we are not only interested in convex dependency (which is actually the embracing notion) but also in proximity. Suppose that the sites of S are lights or antennas with limited range; due to their limited power, they cover a disk of a given radius r centered at the sites of S. Only the points lying in such disks are illuminated. If we want to embrace the point q with the minimum range r, we need to know which is the closest light s _q to q such that q lies in the convex hull formed by s _q and the lights of S closer to q than s _q . This subset of S related to point q is called the closest embracing set for q in relation to S and its cardinality is the closest embracing number of q. By assigning every point q in the convex hull of S to its closest embracing site s _q , we obtain a partition of the convex hull that we call the embracing Voronoi diagram of S. This paper proves some properties of the embracing Voronoi diagrams and describes algorithms to compute such diagrams, as well as the levels in which the convex hull is decomposed regarding the closest embracing number.     

Trees with Equal Domination and Restrained Domination Numbers

Let G = (V,E) be a graph and let S V. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a RDS of G. The graph G is γ-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with γ(T)=γ_r(T); (ii) T is a γ-excellent tree and T ≠ K₂; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with ℓ leaves, then γ_r(T) ≤ (n + ℓ + 1)/2, and we characterize those trees achieving equality.     

Optimal quality of exceptional points for the Lebesgue density theorem

In spite of the Lebesgue density theorem, there is a positive δ such that, for every non-trivial measurable set S⊂ℝ, there is a point at which both the lower densities of S and of ℝ∖S are at least δ. The problem of determining the supremum of possible values of this δ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.     

Hypergraphs in which all disjoint pairs have distinct unions

Let ℓ be a set-system ofr-element subsets on ann-element set,r≧3. It is proved that if |ℓ|>3.5 then ℓ contains four distinct membersA, B, C, D such thatA∪B=C∪D andA∩B=C∩D=0.     

On Order Types of Systems of Segments in the Plane

Let r(n) denote the largest integer such that every family C\mathcal{C} of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Pach and Tóth gave a construction that shows r(n) < n ^log8/log9 (Pach and Tóth 2009). They also stated that one can apply the Erdős–Szekeres theorem for convex sets in Pach and Tóth (Discrete Comput Geom 19:437–445, 1998) to obtain r(n) > log₁₆ n. In this note, we will show that r(n) > cn ^1/4 for some absolute constant c.     

Linear codes with covering radius 3

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by ℓ _q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on ℓ _q (r, 3). General constructions are given and upper bounds on ℓ _q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.     

On the number of Galois points for a plane curve in positive characteristic,III

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(p ^e  + 1) of degree p ^e  + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \mathbbF₂{\mathbb{F}_{2}} -rational points in \mathbbP²{\mathbb{P}^{2}}.