$L^{\Phi}(I,X)$中的最佳同时逼近

Let X be a Banach space and Ф be an Orlicz function. Denote by L^Ф（I,X） the space of X-valued （I）-integrable functions on the unit interval I equipped with the Luxemburg norm. For f1,f2,... ,fm ∈ L^Ф（I,X）, a distance formula distv（f1,f2,... ,fm,L^Ф（I,G）） is presented, where G is a close subspace of X. Moreover, some existence and characterization results concerning the best simultaneous approximation of L^Ф （I, G） in L^Ф （I, X） axe given.     

图映射的非游荡集

§ 1　IntroductionLet N be the set of all natural numbers.Write Z+=N∪ { 0 } ,Nn={ 1 ,2 ,...,n} andZn={ 0 }∪Nnfor any n∈N.Let X be a topological space and f:X→X be a continuous map.Forx∈X,O(x,f) ={ fk(x) :k∈ Z+} is called the orbit of x.The set of periodic points,the set of recurrentpoints,the set ofω-limit points for some x∈X and the set of non-wandering points of fare denoted by P(f) ,R(f) ,ω(x,f) andΩ(f) ,respectively(for the definitions see[1 ] ) .Let A X,we use int(A) ,A…     

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis R, and(I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I → I be a continuous multi-valued map. Assume that Pn =(x_0, x_1,..., xn) is a return tra jectory of f and that p ∈ [min Pn, max Pn] with p ∈ f(p). In this paper, we show that if there exist k(≥ 1) centripetal point pairs of f(relative to p)in {(x_i; x_i+1) : 0 ≤ i ≤ n-1} and n = sk + r(0 ≤ r ≤ k-1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r 0. Besides,we also study stability of periodic orbits of continuous multi-valued maps from I to I.     

APPLICATIONS OF GRBNER BASIS FOR WEYL ALGEBRA

L et K be a field with zero character and K[x, ]denotes the Wey1 - algebra K[x1 ,x2 ,… ,xn, 1 , 2 ,… , n].I is a left ideal of K[x, ].F={f1 ,f2 ,… ,fm}is a Grobner basis for I.(f1 ,f2 ,… ,fm) denotes the left ideal generated by f1 ,f2 ,… ,fm.T deno     

L(d1, d2,..., dt)-Number λ(Cn; d1, d2,...,dt) of Cycles

An L(d1,d2,...,dt)-labeling of a graph G is a function f from its vertex set V(G) to the set {0, 1,..., k} for some positive integer k such that {f(x) - f(y)| ≥ di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L(d1,d2,...,dt)-number λ(G;d1,d2,... ,dt) of G is the smallest integer number k such that G has an L(d1,d2,... ,dt)labeling with max{f(x)|x ∈ V(G)} = k. In this paper, we obtain the exact values for λ(Cn; 2, 2,1) and λ(Cn; 3, 2, 1), and present lower and upper bounds for λ(Cn; 2,..., 2,1,..., 1)     

DOUBLE φ-INEQUALITIES FOR BANACH-SPACE-VALUED MARTINGALES

Let B be a Banach space, Φ1 , Φ2 be two generalized convex Φ-functions and Ψ 1 , Ψ 2 the Young complementary functions of Φ1 , Φ2 respectively with ∫t t 0 ψ2 (s) s ds ≤ c 0 ψ1 (c 0 t) (t > t 0 ) for some constants c 0 > 0 and t 0 > 0, where ψ1 and ψ2 are the left-continuous derivative functions of Ψ 1 and Ψ 2 , respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f = (f n ) n ≥ 0 , ‖f*‖Φ1 ≤ c‖S (p) (f ) ‖Φ2 (or ‖S (q) (f )‖Φ1 ≤ c‖f*‖Φ2 , respectively), where f and S (p) (f ) are the maximal function and the p-variation function of f respec- tively; (ii) If B is a UMD space, T v f is the martingale transform of f with respect to v = (v n ) n ≥ 0 (v*≤ 1), then ‖(T v f )*‖Φ1 ≤ c ‖f *‖Φ2 .     

Gargano,Lewinter and Malerba问题的解

§1　IntroductionLet G be a graph with vertex-set V(G) ={ v1 ,v2 ,...,vn} .A labeling of G is a bijectionL:V(G)→{ 1,2 ,...,n} ,where L (vi) is the label of a vertex vi.A labeled graph is anordered pair (G,L) consisting of a graph G and its labeling L.Definition1.An increasing nonconsecutive path in a labeled graph(G,L) is a path(u1 ,u2 ,...,uk) in G such thatL(ui) + 1相似文献   

The Infinite—dimensional σ—widths of Besov Classes

Let (X(Rd),|.|X) be a normed space of real functions on Rd.Let >0 and Pα be the operator in X(Rd) defined by Pα f(x)=χα(x) f(x) (where χα(x) is thecharacteristic function of the cube Idα=［-α,α］d). Let L be asubspace of X(Rd). Set PαL=Pα f:f∈L. Suppose L islocally-finite dimensional, I.e., dim(Pα L,X)<+∞ for every >0.Then the following quantity is said to be the average dimension of L in X(in the sense of LED)(see ［1］).     

The Infinite-dimensional σ-widthsof Besov Classes

Let (X(Rd), x) be a normed space of real functions on Rd. Let > 0 and P be theoperator in X(Rd) defined by P f(x) = x(x)f(x) (where (x) is the characteristic functionof the cube Id = [- , ]d). Let L be a subspace of X(Rd). Set P L= {P f: f L}. SupposeL is locally-finite dimensional, i.e., dim(P L,X) < for every > 0. Then the followingquantity is said to be the average dimension of L in X (in the sense of LED)(see [1]).Let > 0, and C be a centrally symmetric subset of X(Rd). The i…     

On the Factorizations of （mg, mf）-graph

Let G be an (mg, mf)-graph, where g and f are integer-valued functions defined on V(G) and such that 0≤g(x)≤f(x) for each x ∈ V(G). It is proved that(1) If Z ≠ , both g and f may be not even, G has a (g, f)-factorization, where Z = {x ∈ V(G): mf(x)-dG(x)≤t(x) or dG(x)-mg(x)≤ t(x), t(x)= f(x)-g(x)>0}.(2) Let G be an m-regular graph with 2n vertices, m≥n. If (P1, P2,..., Pr) is a partition of m, P1 ≡ m (mod 2), Pi ≡ 0 (mod 2), i = 2,..., r, then the edge set E(G) of G can be parted into r parts E1 , E2,...,Er of E(G) such that G[Ei] is a Pi-factor of G.     

Entropy,Periodicity, and Graphs with Zero Euler Characteristic

Let G be a graph which contains exactly one simple closed curve. We prove that a continuous map f : G → G has zero topological entropy if and only if there exist at most k ≤ |(Edg(G) End(G) 3)/2] different odd numbers n1,...,nk such that Per(f) is contained in ∪i=1^k ∪j=0^∞ ni2^j, where Edg(G) is the number of edges of G and End(G) is the number of end points of G.     

Extension of a quantized enveloping algebra by a Hopf algebra

Suppose that H is a Hopf algebra,and g is a generalized Kac-Moody algebra with Cartan matrix A =(aij)I×I,where I is an index set and is equal to either {1,2,...,n} or the natural number set N.Let f,g be two mappings from I to G(H),the set of group-like elements of H,such that the multiplication of elements in the set {f(i),g(i)|i ∈I} is commutative.Then we define a Hopf algebra Hgf Uq(g),where Uq(g) is the quantized enveloping algebra of g.     

Inverse-preserving Linear Maps Between Spaces of Matrices over Fields

Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.     

ON THE EMBEDDING OF TOP IN THE CATEGORY OF STRATIFIED L-TOPOLOGICAL SPACES

§1. Introduction and PreliminariesLet L be a complete lattice. An L-topology on a set X is a subset τ of LX closed withrespect to ?nite meets and arbitrary joins. (X,τ) is called an L-topological space. The L-topology τ is called strati?ed if it contains all the constant maps from X to L, and in this case(X,τ) is called a strati?ed L-topological space. A continuous map between two L-topologicalspaces (X,τX) and (Y,τY ) is a function f : X ?→ Y such that f←(λ) = λ ? f ∈ τX…     

A local version of Hardy spaces associated with operators on metric spaces

Let(X,d,μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ.Let L be a second order self-adjoint positive operator on L2(X).Assume that the semigroup e tL generated by L satisfies the Gaussian upper bounds on L 2(X).In this article we study a local version of Hardy space h1L(X) associated with L in terms of the area function characterization,and prove their atomic characters.Furthermore,we introduce a Moser type local boundedness condition for L,and then we apply this condition to show that the space h1L(X) can be characterized in terms of the Littlewood-Paley function.Finally,a broad class of applications of these results is described.     

Some Dynamical Properties in Set-valued Discrete Systems

1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).     

圈的$L(d_1,d_2,\ldots,d_t)$-数$\lambda(C_n;d_1,d_2,\ldots,d_t)$

An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L（d1,d2,...,dt）-number λ（G;d1,d2,... ,dt） of G is the smallest integer number k such that G has an L（d1,d2,...,dr）- labeling with max{f （x）｜x ∈ V（G）} = k. In this paper, we obtain the exact values for λ（Cn; 2, 2, 1） and λ（Cn; 3, 2, 1）, and present lower and upper bounds for λ（Cn; 2,..., 2, 1,..., 1）     

Non-wandering sets of the powers of dendrite maps

Let(X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f)and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold:(1) If x ∈Ω(f)-Ω(f~n) for some n ≥ 2,then x ∈ EP(f).(2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = ?0(the cardinal number of the set of positive integers).     

The Existence of Invariant Subspaces for Some Weighted Composition Operators

In this bulletin, () denotes a complete, σ-finite measure space so that L2(X,u) is separable. All statments about functions or sets are assumed to hold up to sets of measure zero. For a measurable function f, the support of f is defined as σ(f).Let Y be a nonempty measurable subset of X. A measurable transformation T is a point transformation mappimg Y into X with the properties that T-1Y and uoT-1 u. The function h will always denote du o T-1 /du, and assume that h is a.e. finite value…     

Maps preserving peripheral spectrum of generalized Jordan products of operators

Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X1 and X2, respectively. For k ≥ 2, let(i1, i2,..., im) be a finite sequence such that {i1, i2,..., im} = {1, 2,..., k} and assume that at least one of the terms in(i1,..., im) appears exactly once. Define the generalized Jordan product T1 o T2 o ··· o Tk= Ti1Ti2··· Tim+ Tim··· Ti2Ti1 on elements in Ai. This includes the usual Jordan product A1A2 + A2A1, and the Jordan triple A1 A2 A3 + A3 A2 A1. Let Φ : A1 → A2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σπ(Φ(A1) o ··· o Φ(Ak)) = σπ(A1 o ··· o Ak) for all A1,..., Ak,where σπ(A) stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.