On αβ-sets

A closed setE⊂T is an αβ-set, where α and β are elements of infinite order inT ifE⊂E−α). ∪ (E−β). We give two constructions of “thin” αβ-sets.     

Codes from the incidence matrices of graphs on 3-sets

We examine the p-ary linear codes from incidence matrices of the three uniform subset graphs with vertex set the set of subsets of size 3 of a set of size n, with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. All the main parameters of the codes and the nature of the minimum words are obtained, and it is shown that the codes can be used for full error-correction by permutation decoding. We examine also the binary codes of the line graphs of these graphs.     

On Banach spaces whose norm-open sets are -sets in the weak topology

We investigate non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is σ-discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces C(K) with a σ-discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina concerning Borel structures and Kadec renormings in Banach spaces.     

Completely Regular Designs of Strength One

We study a class of highly regular t-designs. These are the subsets of vertices of the Johnson graph which are completely regular in the sense of Delsarte [2]. In [9], Meyerowitz classified the completely regular designs having strength zero. In this paper, we determine the completely regular designs having strength one and minimum distance at least two. The approach taken here utilizes the incidence matrix of (t+1)-sets versus k-sets and is related to the representation theory of distance-regular graphs [1, 5].     

Partitions and sums of (m,p, c)-sets

The (m, p, c)-sets characterize those sets which contain solutions to every partition-regular system of homogeneous linear equations. We show that if N is partitioned into finitely many classes there is one class and for each (m, p, c) ϵ N × N × N, an (m, p, c) set B(m, p, c) such that all finite sums choosing at most one term from each B(m, p, c) lie in the given class.     

There do not exist minimal algebras between C*(X) and C(X) with prescribed real maximal ideal spaces

Let C(X) be the algebra of all real-valued continuous functions on a completely regular Hausdorff space X, and C*(X) the subalgebra of bounded functions. We prove that for any intermediate algebra A between C*(X) and C(X), other than C*(X), there exists a smaller intermediate algebra with the same real maximal ideals as in A. The space X is called A-compact if any real maximal ideal in A corresponds to a point in X. It follows that, for a noncompact space X, there does not exist any minimal intermediate algebra A for which A is A-compact. This completes the answer to a question raised by Redlin and Watson in 1987.     

New coverings of t-sets with (t + 1)-sets

The minimum number of k-subsets out of a v-set such that each t-set is contained in at least one k-set is denoted by C(v, k, t). In this article, a computer search for finding good such covering designs, leading to new upper bounds on C(v, k, t), is considered. The search is facilitated by predetermining automorphisms of desired covering designs. A stochastic heuristic search (embedded in the general framework of tabu search) is then used to find appropriate sets of orbits. A table of upper bounds on C(v, t + 1, t) for v 28 and t 8 is given, and the new covering designs are listed. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 217–226, 1999     

Continuous Selections and Finite C-Spaces

Characterization of paracompact finite C-spaces via continuous selections avoiding Z -sets are given. We apply these results to obtain some properties of finite C-spaces. Factorization theorems and a completion theorem for finite C-spaces are also proved.     

Some properties of P-sets of finite-automaton functions

Classes of finite-automation functions are considered in the paper and each state of those functions realizes a function from some closed class D of the k-valued logic (P-sets). It is proved that there exists continuum of precomplete classes C containing an arbitrary P-set. The problem of existence of a completeness criterion for systems containing P-sets is also considered.     

Fuzzy subobjects in a category and the theory of C-sets

In this paper we shall define the concept of a fuzzy subobject of an object in arbitrary categories. This concept is generated by the representation theorem of fuzzy sets. By using fuzzy subobjects one can include most of the fuzzy concepts defined in the literature, such as: fuzzy groups, fuzzy relations and fuzzy convex sets. In the second part of the paper we shall define a new concept; that of a $\text{C}$-set. This concept will generalize that of a fuzzy set and we shall also prove that $\text{C}$-sets can be represented by some sets of functors. More precisely, $\text{C}$-sets form a category which can be represented by a category of functors. The utility of $\text{C}$-sets resides in the fact that one can replace “ordering” by the more general concept of a morphism in category. The new representation of $\text{C}$-sets is weaker than that of fuzzy sets.     

n2-sets in a projective plane which determine exactly n2+n lines

In this paper n²-sets, in a projective plane, determining exactly n²+n lines are completely characterized.Research partially supported by G.N.S.A.G.A. (C.N.R.).Paper written while the second author was at the University of Rome supported by a grant within the Italian Hungarian cultural exchange program.     

A note on δ-I-sets

We characterize δ-I-sets, discuss the relation between δ-sets, δ-I-sets and I _δ-sets and generalize some of the results.     

On n2 -sets of type (0,1,n) in projective planes

In this paper n²-sets of type (0,1,n), in projective planes of order greater than 3, are completely characterized.This paper was written while the author was studying at the University of Rome, under the direction of Prof. G. Tallini, supported by a grant from the Institute for Italian Hungarian Cultural Exchange.     

Riesz spaces of real continuous functions

An Archimedean Riesz space E is isomorphic to C(X) for some completely regular Hausdorff space X if and only if there exists a weak order unit e > 0 for which E is e-uniformly complete, e-semisimple, e-separating and 2-universally e-complete.     

On Λb-sets and the associated topologyτΛ b

We define the concept of Λ_b-sets (resp. V_b-sets) of a topological space, i.e., the intersection of b-open (resp. the union of b-closed) sets. We study the fundamental property of Λ_b-sets (resp. V_b-sets) and investigate the topologies defined by these families of sets.     

[1, 2]-sets in graphs

A subset S⊆V$S\subseteq V$ in a graph G=(V,E)$G=(V,E)$ is a [j,k]$[j,k]$-set if, for every vertex v∈V?S$v\in V?S$, j≤|N(v)∩S|≤k$j\le |N\left(v\right)\cap S|\le k$ for non-negative integers j$j$ and k$k$, that is, every vertex v∈V?S$v\in V?S$ is adjacent to at least j$j$ but not more than k$k$ vertices in S$S$. In this paper, we focus on small j$j$ and k$k$, and relate the concept of [j,k]$[j,k]$-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k$k$-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G$G$, the restrained domination number is equal to the domination number of G$G$.     

Irregular 1-sets on the graphs of continuous functions

One can define in a natural way irregular 1-sets on the graphs of several fractal functions, like Takagi’s function, Weierstrass-Cellerier type functions and the typical continuous function. These irregular 1-sets can be useful during the investigation of level-sets and occupation measures of these functions. For example, we see that for Takagi’s function and for certain Weierstrass-Cellerier functions the occupation measure is singular with respect to the Lebesgue measure and for almost every level the level set is finite.     

A note on <Emphasis Type="Italic">f</Emphasis><Subscript><Emphasis Type="Italic">I</Emphasis></Subscript>-sets and <Emphasis Type="Italic">f</Emphasis><Subscript><Emphasis Type="Italic">I</Emphasis></Subscript> -continuous functions

Summary In [6] we introduced and investigated the notions of f_I -sets and f_I -continuous functions in ideal topological spaces. In this paper, we investigate their further important properties.     

Covering and guarding polygons using L k -sets

The Art Gallery Problem is the problem of finding a minimum number of points (called guards) in a given polygon such that every point in the polygon is visible to at least one of the guards. Chvátal [5] was the first to show that, in the worst case, [n/3] such points will suffice for any polygon of n sides. O'Rourke [15] later showed that only [n/4] guards were needed if line segments, rather than points, were allowed as guards. In this paper, we unify these results, and extend them to many other classes of guards, while using a generalization of visibility known as link-visibility. In particular, we present the following theorems:

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.