A NEW CLASS OF LOCALLY CONVEX SPACES AND THE GENERALIZATION OF KALTON'S CLOSED GRAPH THEOREM

By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.

THE PATHWISE SOLUTION FOR A CLASS OF QUASILINEAR STOCHASTIC EQUATIONS OF EVOLUTION IN BANACH SPACE Ⅲ

This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to the stochastic case.

The calculation for C-G coefficients

The eigenfunction method put forward by Chen Jin-quan is illustrated. We apply this theory to the space group D1_ 6h. The selection rules of this space are worked out in the points of higher symmetry A,K,H in the first Brillion Zone. The C-G coefficients are calculated for K.HA.

Gene selection in class space for molecular classification of cancer

Gene selection (feature selection) is generally pertormed in gene space(feature space), where a very serious curse of dimensionality problem always existsbecause the number of genes is much larger than the number of samples in gene space(G-space). This results in difficulty in modeling the data set in this space and the lowconfidence of the result of gene selection. How to find a gene subset in this case is achallenging subject. In this paper, the above G-space is transformed into its dual space,referred to as class space (C-space) such that the number of dimensions is the verynumber of classes of the samples in G-space and the number of samples in C-space isthe number of genes in G-space. it is obvious that the curse of dimensionality in C-spacedoes not exist. A new gene selection method which is based on the principle of separatingdifferent classes as far as possible is presented with the help of Principal ComponentAnalysis (PCA). The experimental results on gene selection for real data set areevaluated with Fisher criterion, weighted Fisher criterion as well as leave-one-out crossvalidation, showing that the method presented here is effective and efficient.

DECOMPOSITION OF A CLASS OF FUNCTIONALS AND THE PREDICTABLE REPRESENTATION THEOREM ON BANACH SPACES

Let B be a separable real Banach space and X(t) be a symmetric conservative diffusionprocess taking values in B. In this paper, we decompose the functional u(X(t),t) into a sumof a square integrable martingale and a regular 0-quadratic variation process. On this basis, weestablish the predictable representation theorem of X(t).

ATOMIC DECOMPOSITION FOR WEIGHTED H^p SPACES ON PRODUCT DOMAINS

By using the weighted versions of Journe's covering lemma and its extension to highetdimensions, this paper contributes an atomic decomposition theorem for the weighted H^p(0<p≤1)spaces on product domains, gets a vector value for the index of the moment conditions whichextends the corresponding result in the case with one--parameter to the case with arbitrarynumber of parameters and solves the problem proposed by S. Y. A. Chang &R. Fefferman.

Carleson measures, BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L^2(R^n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R^n) denote the BMO space associated to the Schrdinger operator L on R^n. In this article, we show that for every f ∈ BMO_L(R^n) with compact support, then there exist g ∈ L~∞(R^n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R^n), where S_(μ,P)=∫(R_+^(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-^(t(L)^(1/2))}t>0 on L^2(R^n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R^n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.