Co-RS-compact Topologies

A topoogy R(r)is constructed from a given topology r on a set X.R(r)iscoarser than r,and the following are some results based on this topology:1.Continuity and RS-continuity are equivalent if the codomain is retopologized by R(τ).2.Each topological spase has a coarser extremely disconnected topology3.The class of semi-open sets with respect to R(r)is a topology4.T_2 and semi-T_2 properties are equivalent on a space whose topology is R(r).5.Minboal R_0-spaces are RS-compact.6.MaxAnalα-compact spaces are RS-compact.     

The Measure-Preserving Shift on the Cantor Set

ＴｈｅＭｅａｓｕｒｅ┐ＰｒｅｓｅｒｖｉｎｇＳｈｉｆｔｏｎｔｈｅＣａｎｔｏｒＳｅｔ＊）ＺｅｎｇＦａｎＰｉｎｇ＊＊）（曾凡平）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮａｎｊｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｎａｎｊｉｎｇ，２１００９３）Ａｂｓｔｒａ...     

Fixed Point Theorems on a Non-convex Set

ＦｉｘｅｄＰｏｉｎｔＴｈｅｏｒｅｍｓｏｎａＮｏｎ－ｃｏｎｖｅｘＳｅｔ￥ＺｈａｏＸｉａｏｑｕａｎ（赵晓全）ａｎｄＳｕｎＸｕｅｓｉ（孙学思）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃａ，ＨａｒｂｉｎＩｎｓｔｉｔｕｔｅｏｆＥｌｅｃｔｒｉｃａｌＴｅｃｈｎ...     

A Generalization about Congruent Pairs on a Set

ＡＧｅｎｅｒａｌｉｚａｔｉｏｎａｂｏｕｔＣｏｎｇｒｕｅｎｔＰａｉｒｓｏｎａＳｅｔＨｕａｎｇＪｕｎｍｉｎ（黄骏敏）（ＳｈａｎｇｈａｉＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｃａｎｃｅｌｔｈｅｃｏｎｄｉｔｉｏｎ...     

On the Behaviors of Hyperspace Topologies for Subspaces

For a topological space X we denote by CL(X) the collection of all nonempty closed subsets of X. Suppose we have a map T which assigns in some coherent way to every topological space X some topology T(X) on CL(X). In this paper we study continuity and inverse continuity of the map iA,X :(CL(A),T{A)) → (CL(X),T(X)) defined by iA,x(F) = F whenever F ∈CL(A), for various assignment T; in particular, for locally finite topology, upper Kuratowski topology, and Attouch-Wets topology, etc.     

Note on the Blow-up Set for A Semilinear Parabolic Differential System

ＮｏｔｅｏｎｔｈｅＢｌｏｗ－ｕｐＳｅｔｆｏｒＡＳｅｍｉｌｉｎｅａｒＰａｒａｂｏｌｉｃＤｉｆｆｅｒｅｎｔｉａｌＳｙｓｔｅｍ￥ＣｈｅｎＦｕｊｕｎ（陈富均）（ＳｈａｎｇＱｉｕＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ４７６０００）Ａｂｓｔｒａｃｔ：Ｔｈｉｓｎｏｔｅｄ...     

On Cluster Set

Ginsbury and Saks have proved that if the power X~2 is countably compact,X is a Hausdorff space,then every power of X is countably compact([1]).Many natural questions were raised,for example,Is the cardinal 2° optimal inthe conclusion? Saks raised a conjecture in [2]. Conjecture. ω~*(=βω\ω)1s not the union of <2°cluster sets.The conjec-ture is very useful to answer those questions.For example,if the conjecture istrue,then the cardinal 2°can not decrease in the conclusion.The conjecture isconsistent with ZFC and Saks guessed that it is also true in ZFC.We claim thatthe conjecture is equivalent to the following proposition.     

The Largest Cardinals of Isomorphic Class of Topology-set-lattices on an Infinite Set

Let X be an infinite set,K={τ:τ is a topology on X},defineτ≌σ iff (f)(f is a lattice-isomorphism from τ to σ),for a given τ∈K we define τ={σ:σ∈K ＆σ≌τ}K(K)is an imcomparable class iff (τ∈K)(σ∈K)(τ≠→τ and σ are incomparable),M={K:K is an incomparable class}. Theorem 1 sup{|τ|:τ∈K}=max{|τ|:τ∈k}=|K|=2~2~[X]=exp(exp(|X|)). Theorem 2 sup{sup{|τ|:τ∈K}:KK ＆ K is an incomparable class}=sup     

On the Boundary of the Set of Morse-Smale Vector Fields on Two-dimensional Manifolds

NotatianM a compact,boundaryless,orientable or non—orientable two-dimensional manifold of class C~∞H~r(M) the Banach space of C~r(r≥1)vector fields on M∑~r(M) the set of structurally stable vector fields in H~r(M)MS~r(M) the set of Morse—Smale vector fields in H~r(M)     

Families of Semiopen Sets and Their Accompanied Topologies

Introduction Let (X, U) be a topological space. A set sX is called a semiopen set of U, if there exists u∈U such that usu~-, where u~- expresses the clo sure of u. The family of all semiopen sets of U is called the family of semiopen sets of U, and the topology U an accompanied topology of. The     

Total Dominating Set Games

1 DefinitionsIn this paper we discuss cooperative cost games that arise from total domination problems on graphs. A cooperative cost game (Г) = (N,c) consists of a player set N = {1,2,…,n} and a characteristic function c : 2N→R, where for each subset S(？)N, c(S) represents the cost incurred by the coalition of players in S. The central problem in cooperative game is     

Existence of a Phase Transition for the Potts p-adic Model on the Set ℤ

We consider the Potts model on the set in the field Q _p of p-adic numbers. The range of the spin variables (n), , in this model is . We show that there are some values q=q(p) for which phase transitions occur.     

The Largest Cardinals of Homeomorphic Class and Isomorphic Class of Set-lattices on an Infinite Set

Let X be an infinite set, K={τ:τ is a topology on X}, C={τ:τ∈K&τ is acomplete lattice of sets with respect to operations of arbitrary intersection andunion}. Theorem 1 For a given τ∈K, we define (τ)={σ:σ∈K & σo and τ are homeomorphic}, then sup{|<τ>):τ∈K}=max{|<τ>|:τ∈K}=2~|X|=exp(|X|).     

Equivalence Relation on the Set of Subsets of Z_v and Enumeration of the Equivalence Classes

Motivated by the concept of equivalent cyclic difference sets on Z_v (the residue ring modulo v), we introduce a similar equivalence relation on the set of subsets of Z_v as follows. Difinition Let D_1 and D_2 be two subsets of Z_v. If D_1=tD_2+s for some t, s∈Z_v(ged(t,v)=1), then D_1 and D_2 are said to be equivalent, denoted by D_～D_2.     

Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property

Let μ and ν be probability measures on a group Γ and let G _μ and G _ν denote Green’s function with respect to μ and ν. The group Γ is said to admit instability of Green’s function if there are symmetric, finitely supported measures μ and ν and a sequence {x _n } such that G _μ (e, x _n )/G _ν (e,x _n ) →0, and Γ admits instability of recurrence if there is a set S that is recurrent with respect to ν but transient with respect to μ. We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously known groups with these instabilities did not have the Liouville property.     

Newton’s Method for Minimizing a Convex Twice Differentiable Function on a Preconvex Set

The problem of minimizing a convex twice differentiable function on the set-theoretic difference between a convex set and the union of several convex sets is considered. A generalization of Newton’s method for solving problems with convex constraints is proposed. The convergence of the algorithm is analyzed.     

Congruent Pairs On a Set

ＣｏｎｇｒｕｅｎｔＰａｉｒｓＯｎａＳｅｔＺｈｕＺｕｏｔｏｎｇ；（朱作桐）ＨｕａｎｇＪｕｎｍｉｎ（黄军民）（ＮａｎｊｉｎｇＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ）（ＮａｎｊｉｎｇＵｎｉｖｅｒｓｉｔｙ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｉｎｔｒｏｄ...     

Set inertia-preserving matrices∗ †

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.     

The Range Set of Meromorphic Derivatives

ＴｈｅＲａｎｇｅＳｅｔｏｆＭｅｒｏｍｏｒｐｈｉｃＤｅｒｉｖａｔｉｖｅｓＩｎｄｒａｊｉｔＬａｈｉｒｉ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＪａｄａｖｐｕｒＵｎｉｖｅｒｓｉｔｙ，Ｃａｌｃｕｔａ－７０００３２，Ｉｎｄｉａ）ＡｂｓｔｒａｃｔＩｎｔ...