Equivalence systems with finitely many relations

We consider a setX with a finite totally ordered setE of equivalence relations onX. We describe the automorphism group of this system, that is, the group of all those permutations ofX that leave each relation inE invariant.     

On regular linear relations

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.     

Conjugative relations and applications

In this paper, the concepts of conjugative relations and dual conjugative relations are introduced and generalized. Intrinsic characterizations of them are obtained. In addition, we give algebraic and topological characterizations of posets for which the relation ≰ is conjugative/dual conjugative.     

Tolerance relations on eventually regular semigroups

A semigroup is called eventually regular if a power of each element is regular. Regular and group-bound semigroups are each eventually regular. A tolerance relation on an eventually regular semigroup is introduced, and Lallement's result for regular semigroups is generalized to eventually regular semigroups. Weakly compatible tolerances on semigroups are studied. This research was partially supported by the National Science Foundation of Qufu Normal University. The author wishes to thank T. E. Hall for his comments and help in preparing this paper for publication.     

Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.     

Weak stability of almost regular contact foliations

We prove that on a compact manifold, a contact foliation obtained by a smallC ¹ perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC ¹-close to almost regular contact forms.Supported in part by NSF Grant DMS 90-01861     

Every finitely generated regular field extension has a stable transcendence base

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex ₁, …,x _n with the following properties: The field extensionF/K(x ₁,…,x _n ) is separable and the Galois hull ofF/K(x ₁,…,x _n ) remains regular overK, i.e.K is algebraically closed in . We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.     

Minimal regular 2-graphs and applications

A 2-graph is a hypergraph with edge sizes of at most two. A regular 2-graph is said to be minimal if it does not contain a proper regular factor. Let f2(n) be the maximum value of degrees over all minimal regular 2-graphs of n vertices. In this paper, we provide a structure property of minimal regular 2-graphs, and consequently, prove that f2(n) = n 3-i/3, where 1 ≤ i ≤ 6, i ≡ n (mod 6) and n ≥ 7, which solves a conjecture posed by Fan, Liu, Wu and Wong. As applications in graph theory, we are able to characterize unfactorable regular graphs and provide the best possible factor existence theorem on degree conditions. Moreover, fa(n) and the minimal 2-graphs can be used in the universal switch box designs, which originally motivated this study.     

Regular relations and strictly completely regular ordered spaces

The classical theorem of Zareckiı̆ about regular relations is slightly extended and an intrinsic characterization of regularity is given. Based on the extended Zareckiı̆ theorem and the intrinsic characterization of regularity, we give a characterization of the strict complete regularity of ordered spaces by means of a certain regular relation between the closed and the open upper sets. As an application, it is shown that a quasicontinuous domain endowed with the Lawson topology is strictly completely regular, provided that the Lawson-open lower sets are contained in the lower topology. By means of regular relations we present a new proof of the strict Tychonoff embedding theorem for strictly completely regular ordered spaces.     

Valuations dominating regular local rings and proximity relations

We study zero-dimensional valuations dominating a regular local ring of dimension n≥2. For this we introduce the proximity matrix and the multiplicity sequence (extending classical definitions of the case n=2) that are associated with the sequence of the successive quadratic transforms of the ring along the valuation. We describe the precise relations between these invariants and study their properties.     

Strong convergence theorems for finitely many nonexpansive mappings and applications

Let E$E$ be a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Recently, Takahashi and Shimoji [W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000) 1463–1471] introduced an iterative scheme given by finitely many nonexpansive mappings in E$E$ and proved weak convergence theorems which are connected with the problem of image recovery. In this paper we introduce a new iterative scheme which includes their iterative scheme as a special case. Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gâteaux differentiable and which has a weakly continuous duality mapping, we prove strong convergence theorems which are connected with the problem of image recovery. Using the established results, we consider the problem of finding a common fixed point of finitely many nonexpansive mappings.     

Homomorphisms close to regular and their applications

This paper contains new and known results on homomorphisms that are close to regular. The main results are presented with proofs.     

Optimal cocircuits in regular matroids and applications

Let M = (E, C) be a regular matroid with circuit set H and cocircuit set H and let $\text{(}\text{H}\text{, +, ?)}$ be an ordered group. To given partitions D = D⁺ ∈ D^? for all D ?H and weighting functions ?,k: E → H optimale ê-cocircuits are defined by having a minimal value $\text{K}\text{(}\text{D}{}^{\mathrm{+}}\text{) ? ol(}\text{D}{}^{\mathrm{?}}\text{ø 1^e1}$.It is shown that P as well as NP problems can be formulated by means of optimal cocircuits. We discuss a class of optimal cocircuit problems which can be solved using flow theory in regular matroids. Applications of the derived theory and algorithms to graphic and cographic matroids and to special ordered groups yield polynomial algorithms for some new minimal cut and shortest path problems which are useful in combinatorial and integer optimization problems.     

Refined discrete regular variation and its applications

We introduce a new class of the so‐called regularly varying sequences with respect to τ and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to τ. We reveal also a connection with generalized regularly varying functions. As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, in some way, a new view on the tests for convergence and divergence of series; we establish the statement that generalizes Raabe test and Bertrand test.     

Weak Morrey spaces with applications

The main object of this investigation is to study weak Morrey spaces. Block spaces, which are preduals of weak Morrey spaces, are characterized. Besides, the Fatou property of block spaces is proved. Finally, as an application, we study the boundedness of singular integral operators in weak Morrey spaces.     

Weak convergence of partial sums of absolutely regular sequences

We show that weak convergence results for partial sums of absolutely regular sequences can easily be derived from the corresponding convergence results for independent triangular arrays. The link to be used is a simple lemma on the total variation norm.