Fuzzy-valued integrals based on a constructive methodology

The procedures for constructing a fuzzy number and a fuzzy-valued function from a family of closed intervals and two families of real-valued functions, respectively, are proposed in this paper. The constructive methodology follows from the form of the well-known “Resolution Identity” (decomposition theorem) in fuzzy sets theory. The fuzzy-valued measure is also proposed by introducing the notion of convergence for a sequence of fuzzy numbers. Under this setting, we develop the fuzzy-valued integral of fuzzy-valued function with respect to fuzzy-valued measure. Finally, we provide a Dominated Convergence Theorem for fuzzy-valued integrals.     

国有商业银行企业不良贷款的主因子分析

论文首先在银行企业不良贷款成因的研究成果进行回顾梳理基础上,从探究现行发展环境下,影响企业不良贷款的主要原因的角度,通过对国有商业银行贷款部的中高层访谈问卷调查,运用探索性因子分析模型,对不良贷款主成因进行萃取和诠释,最后构建银行业不良贷款的成因层次构面,以支持国有商业银行授信决策之依据。     

A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums Σ_{j = 1}ⁿa_jV_j of independent random elements {V_n, n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form (Σ_{j = 1}ⁿa_jV_j − v_n)/b_n →^p 0 is established, where {v_n, n ≥ 1} and b_n → ∞ are suitable sequences. It is assumed that {V_n, n ≥ 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of |V| and the growth behaviors of the constants {a_n, n ≥ 1} and {b_n, n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the {V_n, n >- 1}.     

一类奇异非线性多重调和方程存在正整解的充分必要条件

研究一类奇异非线性多重调和方程?~mu=f(|x|,u,|▽u|)u~(-β),给出了方程存在正的径向对称整体解的充分必要条件和解的性质.     

Facile fabrication of anatase TiO2 microspheres on solid substrates and surface crystal facet transformation from {001} to {101}

Anatase TiO(2) microspheres with controlled surface morphologies and exposed crystal facets were directly synthesized on metal titanium foil substrates by means of a facile, one-pot hydrothermal method without use of any templating reagent. The obtained products were characterized using X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), X-ray photoelecron spectroscopy (XPS), and the focused ion beam (FIB) technique. The sizes of the resultant microspheres ranged from 1.1 to 2.1 μm. The transformation of anatase TiO(2) microspheres with exposed {001} facets surface to nanosheets surface with {101} facets was achieved by simply controlling the hydrothermal reaction time. The anatase TiO(2) microspheres with exposed square-shaped plane {001} facets were obtained by controlling the reaction time at 1 h. The prolonged reaction time transforms the anatase TiO(2) microspheres with exposed square-shaped plane {001} facets to eroded {001} facets then to a nanosheet surface with exposed {101} facets. With hydrothermal synthesis, the surface morphological structure and crystal facets formation are highly dependent on dissolution/deposition processes, which can be strongly influenced by attributes, such as pH of the reaction media, the total concentration of dissolved and suspended titanium species, and the concentration of fluoride in the reaction solution. The changes of these attributes during the hydrothermal process were therefore measured and used to illustrate the morphology and crystal-facet transformation processes of anatase TiO(2) microspheres. The surface morphologies and crystal-facet transformations during hydrothermal processes were found to be governed by the compositional changes of the reaction media, driven by dynamically shifted dissolution/deposition equilibria. The photocatalytic activities of the photoanodes made of anatase TiO(2) microspheres were evaluated. The experimental results demonstrated that the photocatalytic activity of anatase TiO(2) microspheres with exposed {001} facets was found to be 1.5 times higher than that of the anatase TiO(2) microspheres with exposed {101} facets.     

An expectation formula with applications

In this paper, we derive an expectation formula of a random variable having distribution W(x;q). As applications of the expectation formula, we give a transformation formula and an expansion of Sears? transformation formula.     

Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

This paper investigates the asymptotic behavior of tail probability of randomly weighted sums of dependent and real-valued random variables with dominated variation, where the weights form another sequence of nonnegative random variables. The result we obtain extends the corresponding result of Wang and Tang[7].     

A Construction of Sequentially Cohen-Macaulay Graphs

For every simple graph G,a class of multiple clique cluster-whiskered graphs Geπm is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex IndGeπm is sequentially Cohen-Macaulay.The properties of the graphs Geπm and Gπ constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind Gπ and the calculation of Betti numbers of the cover ideal Ic(Geπ").We also prove that the complex △ =IndH is strongly shellable and pure for either a Boolean graph H =Bn or the full clique-whiskered graph H =Gw of G,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△) and the cover ideal Ic(H) have linear quotients.     

The partition problem

In this paper we describe several forms of thek-partition problem and give integer programming formulations of each case. The dimension of the associated polytopes and some basic facets are identified. We also give several valid and facet defining inequalities for each of the polytopes.Partial Support from NSF Grants DMS 8606188 and ECS 8800281 is gratefully acknowledged.