Between closed sets andg-closed sets

Quite recently, by using semi-open (resp.α-open, preopen,β-open) sets in a topological space, the notions ofsg*-closed (resp.αg*-closed,pg*-closedβg*-closed) sets are indroduced and investigated in [8]. These subsets place between closed sets andg-closed sets due to Levine [5]. In this paper, we introduce the notion ofmg*-closed sets and obtain the unified theory for collections of subsets between closed sets andg-closed sets.     

On IP* sets and central sets

IP^* sets and central sets are subsets of which are known to have rich combinatorial structure. We establish here that structure is significantly richer that was previously known. We also establish that multiplicatively central sets have rich additive structure. The relationship among IP^* sets, central sets, and corresponding dynamical notions are also investigated.The authors gratefully acknowledge support from the National Science Foundation (USA) via grants DMS-9103056 and DMS-9025025 respectively.     

Multi-fuzzy sets: An extension of fuzzy sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory.     

Crisp sets as classes of discontinuous fuzzy sets

This paper aims to show how, by using a threshold-based approach, a path from imprecise information to a crisp ‘decision’ can be developed. It deals with the problem of the logical transformation of a fuzzy set into a crisp set. Such threshold arises from the ideas of contradiction and separation, and allows us to prove that crisp sets can be structurally considered as classes of discontinuous fuzzy sets. It is also shown that continuous fuzzy sets are computationally indistinguishable from some kind of discontinuous fuzzy sets.     

An extension of building sets and relative difference sets

Building sets are a successful tool for constructing semi‐regular divisible difference sets and, in particular, semi‐regular relative difference sets. In this paper, we present an extension theorem for building sets under simple conditions. Some of the semi‐regular relative difference sets obtained using the extension theorem are new in the sense that their ambient groups have smaller ranks than previously known. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 50–57, 2000     

The element sets and value sets of two-dimensional continued fractions

For two-dimensional continued fractions we prove the existence and uniqueness of an optimal sequence of value sets corresponding to an arbitrarily given sequence of element sets. We compute the element set for a given sequence of disk value sets and as a corollary, give the element sets and value sets that are used in convergence criteria for two-dimensional continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 55–61.     

Levels in the toposes of simplicial sets and cubical sets

The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide with dimensions and give rise to the usual notions of n-skeletal and n-coskeletal simplicial sets. In addition to the obvious ordering, the levels provide a stricter means of comparing the complexity of objects, which is determined by the answer to the following question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal? This paper, which subsumes earlier unpublished work of some of the authors, answers this question for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical sets, and reflexive globular sets. For the latter, n-skeletal implies (n+1)-coskeletal but for the other two examples the situation is considerably more complicated: n-skeletal implies (2n−1)-coskeletal for simplicial sets and 2n-coskeletal for cubical sets, but nothing stronger. In a discussion of further applications, we prove that n-skeletal cyclic sets are necessarily (2n+1)-coskeletal.     

Special sets of the Hermitian surface and indicator sets

An interesting connection between special sets of the Hermitian surface of PG(3,q²), q odd, (after Shult 13 ) and indicator sets of line‐spreads of the three‐dimensional projective space is provided. Also, the CP‐type special sets are characterized. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 18–24, 2008     

Level sets of the Takagi function: local level sets

The Takagi function ??: [0,1] ?? [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y)?=?{x : ??(x)?=?y} of the Takagi function ??(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a ??generic?? full Lebesgue measure set of ordinates y, the level sets are finite sets. In contrast, here it is shown for a ??generic?? full Lebesgue measure set of abscissas x, the level set L(??(x)) is uncountable. An interesting singular monotone function is constructed associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly ${\frac{3}{2}}$ .     

On*-operfect sets and α-*-closed sets

The main purpose of this paper is to introduce *-operfect, τ*-clopen, α-*-closed, strongly α-*-closed and pre-*-closed sets. We compare them and obtain a diagram to show their relationships among these sets and related sets.     

Finite-tight sets

We introduce two notions of tightness for a set of measurable functions — the finite-tightness and the Jordan finite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune-Valadier’s theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness locates the great growths of a set of measurable mappings on a finite family of sets of small measure. In the Euclidean case, the Jordan finite-tight sets form a subclass of finite-tight sets for which the finite family of sets of small measure is composed by d-dimensional intervals. The main result affirms that each tight set H ⊆ W ^1,1 for which the set of the gradients ∇H is a Jordan finite-tight set is relatively compact in measure. This result offers very good conditions to use fiber product lemma for obtaining a relaxed lower semicontinuity condition.      

Substitution-like minimal sets

Substitution-like minimal sets are a class of symbolic minimal sets on two symbols which includes the discrete substitution minimal sets on two symbols. They are almost automorphic extensions of then-adic integers and they are constructed by using special subsets of then-adics from which the almost automorphic points are determined by following orbits in then-adics. Through their study a complete classification is obtained for substitution minimal sets of constant length on two symbols. Moreover, the classification scheme is such that for specific substitutions the existence or non-existence of an isomorphism can be determined in a finite number of steps.     

Difference sets and sum-free sets in groups of order 16

Transversals for sum-free sets in the nine nonabelian order 16 groups are given. It is shown that exactly eight of the order 16 groups have difference sets Dwithλ = 2and D = -D. It is proven that the only (υ, k, λ) difference sets with υ = 2^t and λ = 2, have parameters (16, 6, 2). It is shown that exactly four of the order 16 groups can be partitioned into three sum-free sets.     

On polynomial sets of classS k,and sets of rankk

Summary Al-Salam and Verma introduced the class S^k of polynomial sets, and characterized such sets by their generating functions. Here we define a class V^k of sets of rank k by means of recurrence relations, and show that S^k ⊂ V^k; and we characterize those sets of S^k, and of V^k, that are orthogonal. Entrata in Redazione il 24 giugno 1977.     

Supremum metric and relatively compact sets of fuzzy sets

The supremum metric D between fuzzy subsets of a metric space is the supremum of the Hausdorff distances of the corresponding level sets. In this paper some new criteria of compactness with respect to the distance D are given; they concern arbitrary fuzzy sets (see Theorem 7), fuzzy sets having no proper local maximum points (see Theorem 12) and, finally, fuzzy sets with convex sendograph (see Theorem 13). In order to compare results with a previous characterization of compactness of Diamond–Kloeden, the criteria will be expressed by equi-(left/right)-continuity. In the proofs a first author's purely topological criterion of D  -compactness and a variational convergence (called Γ$\Gamma$-convergence) which was introduced by De Giorgi and Franzoni, are fundamental.     

Simploidals sets: Definitions, operations and comparison with simplicial sets

The combinatorial structure of simploidal sets generalizes both simplicial complexes and cubical complexes. More precisely, cells of simploidal sets are cartesian product of simplices. This structure can be useful for geometric modeling (e.g. for handling hybrid meshes) or image analysis (e.g. for computing topological properties of parts of n-dimensional images). In this paper, definitions and basic constructions are detailed. The homology of simploidal sets is defined and it is shown to be equivalent to the classical homology. It is also shown that products of Bézier simplicial patches are well suited for the embedding of simploidal sets.     

p-Sidon sets

Let G be a compact Abelian group with character group X. Bo?ejko and Pytlik [Colloq. Math.25 (1972), 117–124] introduced and studied several special types of lacunary subsets of X. This paper is based upon a hitherto unpublished detailed study of those types that most resemble Sidon sets, which the present authors had independently introduced and studied under the name of p-Sidon sets. Some, but not all, aspects of the theory of Sidon (= 1-Sidon) sets carry over to the more general setting. In Section 1 some properties of sets equivalent to p-Sidonicity are given. Section 2 contains several useful consequences of p-Sidonicity; see Theorems 2.1 and 2.4 and Corollaries 2.6 and 2.7. In Section 3, it is shown that certain Λ_q sets also satisfy some of the consequences listed in Section 2. Nevertheless, Λ_q sets need not be p-Sidon sets; see Theorem 3.1. Examples of ($\text{4}\text{3}$)-Sidon sets that are not Sidon sets are given in Section 5. The proof that these sets are ($\text{4}\text{3}$)-Sidon sets requires a brief study of 4-norms in Varopoulos algebras; see Section 4. In Section 6, some special results for the circle group are deduced. Many of these results appear to be new even for p = 1.     

Maps of Borel sets

Questions concerning the structure of Borel sets were raised in special cases by Luzin, Aleksandrov, and Uryson as the problems of distinguishing the sets with certain homogeneous properties in Borel classes and determining the number of such pairwise nonhomemorphic sets. The universal homogeneity, i.e., the property to contain an everywhere closed copy of any Borel set of the same or smaller class, was considered by L.V. Keldysh. She called the sets of classes Π _α ⁰ , α > 2, of first category in themselves that possess this homogeneity property canonical and proved their uniqueness. Thus she revealed the central role of the universality property when describing homeomorphic Borel sets. These investigations led her to the problem of universality of Borel sets and to the problem of finding conditions under which there exists an open map between Borel sets. In this paper, such conditions are presented and similar questions are considered for closed, compact-covering, harmonious, and other stable maps.     

Partitionable sets,almost partitionable sets,and their applications

This paper introduces almost partitionable sets (APSs) to generalize the known concept of partitionable sets. These notions provide a unified frame to construct $\mathbb{Z}$‐cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and APSs are investigated. As an application, a large number of optical orthogonal codes achieving the Johnson bound or the Johnson bound minus one are constructed.