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…     

RESEARCH ANNOUGCEMENTS—Kolmogorov Width of Classes of Smooth Functions on the Sphere s^d—1

Let Rd be the d-dimensional Euclidean space and Sd-1 = {(x1,…,xd): x21 + … + x2d = 1} be the unit sphere of Rd equipped with the normal Lebesgue measure dσ(x).For r ＞ 0,denotc by Brp := Brp(Sd-1) (1 ≤ p ≤∞) the class of functions f on the sphere Sd-1 representable in the form     

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).     

Local generalized empirical estimation of regression

Let f(x) be the density of a design variable X and m(x) = E[Y\X = x] the regression function. Then m(x) - G(x)/f(x), where G(x) = m(x)f(x). The Dirac δ-function is used to define a generalized empirical function Gn (x) for G(x) whose expectation equals G(x). This generalized empirical function exists only in the space of Schwartz distributions, so we introduce a local polynomial of order p approximation to Gn(.) which provides estimators of the function G(x) and its derivatives. The density f(x) can be estimated in a similar manner. The resulting local generalized empirical estimator (LGE) of m(x) is exactly the Nadaraya-Watson estimator at interior points when p = 1, but on the boundary the estimator automatically corrects the boundary effect. Asymptotic normality of the estimator is established. Asymptotic expressions for the mean squared errors are obtained and used in bandwidth selection. Boundary behavior of the estimators is investigated in details. We use Monte Carlo simulations to show that the     

ABSOLUTE CONTINUITIES OF EXIT MEASURES AND TOTAL WEIGHTED OCCUPATION TIME MEASURES FOR SUPER-α-STABLE PROCESSES

Suppose X is a super-α-stable process in Rd, (0<α<2), whose branching rate function is dt, and branching mechanism is of the form ■(z) = z1 β(0 <β≤1). Let Xr and Yr denote the exit measure and the total weighted occupation time measure of X in a bounded smooth domain D, respectively. The absolute continuities of Xτ and Yτ are discussed.     

组合群论中的一个定理

Let F= F(X) be a free group of rand n, A be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of A if x is in the normal closure of A in F(X). Finally, an application of the theorem in Heegaard splitting of 3manifolds is given.     

ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Suppose that X is a right process which is associated with a semi-Dirichlet form(ε,D(ε)) on L~2(E;m).Let J be the jumping measure of(ε,D(ε)) satisfying J(E×E-d) ∞.Let u ∈ D(ε)_b:= D(ε)∩ L~∞(E;m),we have the following Fukushima's decomposition u(X_t)-u(X_0) =M_t~u+N_t~u.Define P_t~uf(x)=E_x[e~(N_t~u)f(X_t)].Let Q~u(f,g) =ε(f,g)+ε(u,fg)for f,g∈ D(ε)_b.In the first part,under some assumptions we show that(Q~u,D(ε)_b) is lower semi-bounded if and only if there exists a constant α_0≥0 such that ‖P_t~u‖2≤e~(α_0~t) for every t0.If one of these assertions holds,then(P_t~u)t≥0 is strongly continuous on L~2(E;m).If X is equipped with a differential structure,then under some other assumptions,these conclusions remain valid without assuming J(E×E-d)∞.Some examples are also given in this part.Let A_t be a local continuous additive functional with zero quadratic variation.In the second part,we get the representation of A_t and give two sufficient conditions for P_t~A f(x) = E_x[e~(A_t) f(X_t)]to be strongly continuous.     

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…     

FRACTIONAL (g, f)-FACTORS OF GRAPHS

1 IntroductionThe graphs considered in this paper will be finite undirected graphs wllicll 11lay llavemultiple edges but no loops. Let G be a grapll with vertex set V(G) and edge set E(G). Fora vertex x of G, the degree of x in G is denoted by dG(z). Let g and f be two integer-valuedfunctions defined o11 V(G) such that 0 < g(z) 5 f(x) fOr all x E V(G). Then a (g, f)-factorof G is a spanning 8ubgraph F of G satisfying g(x) < dG(z) 5 f(x) for all x E V(F). Ifg(x) = f(x) for all x E V(…     

QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

Let f : U(x0) (?) E→F be a. C1 map and f'(X0) be the Prechet derivative of /fat X0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f'(x0) is double splitting and R(f'(x))∩N(T0 ) = {0} near X0. However, in infinite dimensional Banach spaces, f'(x0) is not always double splitting (i.e., the generalized inverse of f'(xo) does not always exist), but its bounded outer inverse of f'(x0) always exists. Only using the C1 map f and the outer inverse T0# of f'(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if X0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.