The number of all isormorphic classes of atomic subfields in the power set of an infinite set

Let X be an infinite set, the power set of X is denoted by P(X). = {G:G and F are isomorphic in the sense of Boolean algebra}, C=(:F is an atomicsubfield in P(X)}. Define C_1(C) is an incomparable class if (_1,_2∈C_1)(F_1≠F_2→F_1 and F_2 are incomparable), then |C| = 2~2~(|x|)= max{|C_1|}. In proof of this result, the AC and GCH are used. The sense of "isomorphism" in [2] is understood as"φ(φ is a bejection on     

A reverse Denjoy theorem Ⅲ

Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ : re iθ∈ D} has measure at most 2α, where 0 α π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2 ∪～D, where Φ(|z|) is a convex, non-decreasing function of |z| and ～D is the complement of D. Let A D (r, u) = inf{u(z) : z ∈ D and |z| = r}. It is shown that if A D (r, u) = O(1) then lim inf r→∞ B(r, u)/r π/(2α) 0.     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

Equivalent Separating Curves on Closed Surface of Genus 2

Let {A,B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to (A, B) if it is disjoint from A∪B and it cuts F into two once-punctured tori X, Y with A(?)X, B(?)Y. Letγbe a simple closed curve on F which is disjoint from A∪B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hnγ(C):n∈Z} is the set containing all the simple closed curves on F which is separating with respect to (A,B), where hγis the Dehn twist alongγon F. This also shows how two simple closed curves on F which are separating with respect to (A,B) are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.     

Constructing super Gabor frames: the rational time-frequency lattice case

For a time-frequency lattice Λ = A Z d B Z d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det( AB) | = 1 L . The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles R d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)>0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N>0,k(s)>0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

On p-hypercenter of Finite Groups

In this paper, we investigate p-hypercenter of the finite group, and obtain some propertiesof p-hypercenter. As its application, we draw some conclusions on the p-nilpotent groups, ourconclusions generalize some well-known results.Throughout this paper, G is a finite group, p is a prime number. Set Zp(G) ={g ∈ G |'[g, G] 5 O.' (G)} and call it the p-center of G. Z;(G)(the p-hypercenter of G) is the largestterm of the chain of subgroupsWe give the characterizations of Z7(G) as follows:Pr…     

两个酉算子的线性组合与框架分解

In this paper, we prove that the norm closure of all linear combinations of two unitary operators is equal to the norm closure of all invertible operators in B(H). We apply the results to frame representations and give some simple and alternative proofs of the propositions in “P. G. Casazza, Every frame is a sum of three (but not two) orthonormal bases-and other frame representations, J. Fourier Anal. Appl., 4(6)(1998), 727-732.”     

关于Hutton定理的一个注

本文证明了存在一个一一对应$\varphi: {\cal J}\cup{\cal J}'\longrightarrow\delta\cup\delta'$,它满足: \ \ (1) $\varphi|{\cal J}: ({\cal J},\subset)\longrightarrow(\delta,\leq)$是frame同构. \ \ (2) $\varphi|{\cal J}': ({\cal J}',\subset)\longrightarrow(\delta',\leq)$是coframe同构.     

Totally Order-disconnected Compact Topologies

Posets with property DINT which are Priestely spaces with respect interval topologies are characterized. Also, posets which are Priestley spaces with respect to bi-Scott topologies are characterized.     

集值映射空间上的自然映射(英文)

In this paper,we discuss the continuities of some natural mappings on pointcompact continuous set-valued mapping spaces with compact-open topology and obtain the properties of set-valued injective mappings,set-valued diagonal mappings,induced mappings,set-valued evaluation mappings,set-valued topological sum mappings and set-valued topological product mappings.     

Selections and hyperspace topologies via special metrics

If (Y, d) is a complete metric space with a non-Archimedean metric d, then there exists a selection on the set of all nonempty closed subsets of Y which is continuous with respect to both the Vietoris and Hausdorff topologies on .     

区间值映射的半连续性与凸性

首先给出了区间数空间子集的有界性及确界等概念,并给出了确界的存在性定理;然后讨论了区间值映射的半连续性问题,给出了区间值映射的半连续性概念及相关性质;最后讨论了半连续区间值映射的凸性问题,给出了半连续区间值映射为凸区间值映射的两个充分条件.     

An open mapping theorem for Young measures

Ditor and Eifler consider the open mapping theorem for the probability spaces. Here we attempt to generalize the theorem to the spaces of Young measures.

     

模糊连续映射和开映射的新特征

给定模糊格Ｌ及ＬＦ拓扑空间（Ｌ＾ｘ,δ）,对Ｌ的素元素ｒ及分子ａ,在（Ｌ＾ｘ,δ）中引入了ｒ－开集及α－闭集的概念。证明了所有的ｒ－开集Ｏｒ（δ）和α－闭集Ｃａ（δ）分别形成一个ＬＦ拓扑和ＬＦ余拓扑,借助于它俩得到了ＬＦ连续映射和ＬＦ开映射的若干新特征。     

On continuous noncomplete lattices

We consider noncomplete continuous and algebraic lattices and prove that finitely generated free lattices are algebraic. We also study the Lawson topology, the second most important topology in the theory of continuous domains, on finitely presented lattices. In particular, we prove that algebraic finitely presented lattices are linked bicontinuous and the Lawson topology on these lattices coincides with the interval topology. Several examples of non-distributive and noncomplete algebraic and continuous lattices are given in the paper. Received April 5, 2000; accepted in final form December 12, 2000.     

Tietze's extension theorem in fuzzy topological spaces

This paper discusses the conceptual difficulties of generalizing standard topological terms of L-fuzzy topological terms. In particular, a theory of relative topologies and relative functions for L-fuzzy topological spaces is developed. The extension of a relative L-fuzzy continuous function into the fuzzy unit interval is defined. The equivalence of L-fuzzy continuous functions and monotone families of open sets is proved, and this equivalence is exploited to establish a fuzzy version of Tietze's Extension Theorem. Finally, a partial converse to the theorem is proved.     

高维空间的拟正则映照

将所有维数的 Beltrami方程组 Dtf· Df =J2 /n G化为一个“Beltrami方程”并利用它研究了 Bel-trami方程组的解的正则性 ,得到一个比文献 [8]更大的正则性区间 .