模糊集的基数与连续统假设

本文在模糊映射的基础上给出了模糊集基数的定义，它把普通集的基数作为特款；不但得到有关基数的大部分结论，而且有其自身的特殊性质；特别，对于连续统假设这一世界难题可能有新的启示．     

On the specificity of a possibility distribution

The specificity of a possibility distribution measures the degree to which the distribution allows one and only one element as its manifestation. As such it is a measure of amount of uncertainty or information. We investigate a number of issues related to specificity measures. We discuss the connection between the specificity of a possibility distribution and the entropy of a probability distribution. We describe unifying view for constructing specificity measures. We look at the relationship of the specificity of a distribution and its negation. We consider the case where the base set is continuous.     

WebMTA: a web-interface for ab initio geometry optimization of large molecules using molecular tailoring approach

A web-interface for geometry optimization of large molecules using a linear scaling method, i.e., cardinality guided molecular tailoring approach (CG-MTA), is presented. CG-MTA is a cut-and-stitch, fragmentation-based method developed in our laboratory, for linear scaling of conventional ab initio techniques. This interface provides limited access to CG-MTA-enabled GAMESS. It can be used to obtain fragmentation schemes for a given spatially extended molecule depending on the maximum allowed fragment size and minimum cut radius values provided by the user. Currently, we support submission of single point or geometry optimization jobs at Hartree-Fock and density functional theory levels of theory for systems containing between 80 to 200 first row atoms and comprising up to 1000 basis functions. The graphical user interface is built using HTML and Python at the back end. The back end farms out the jobs on an in-house Linux-based cluster running on Pentium-4 Class or higher machines using an @Home-based parallelization scheme (http://chem.unipune.ernet.in/ approximately tcg/mtaweb/).     

Commutative Group Algebras of Cardinality {\cal N} 1

We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define G_p and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if G_p is Hausdorff and both F and G have cardinalities not exceeding ₁, then S(FG)/G_p is a direct sum of cyclics. Thus G_p is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power ₁ whose p-component G_p is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between G_p and H_p. In particular, when G is of power ₁ and is p-mixed of torsion-free rank 1 whose G_p is torsion-complete, we have G H. If F is in power p and G, with cardinality ₁, is a direct sum of p-local algebraically compact groups such that FG FH as F-algebras for some group H, then G H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.     

Sufficient Conditions for the Existence of Disintegrations

Let be a probability space and a partition of . A necessary and sufficient condition is given for the existence of a -additive and measurable disintegration of P on . It is also shown that P admits a -additive (but not measurable) disintegration on whenever is a standard space and the set (₁, ₂):₁ and ₂ are in the same element of } is coanalytic in ×. Finally, sufficient statistics (in the classical Fisherian sense) are investigated by using -additive disintegrations as conditional probabilities.     

A ZFC Dowker space in : An application of pcf theory to topology

The existence of a Dowker space of cardinality and weight is proved in ZFC using pcf theory.

     

带基数约束的次模+超模(BP)函数最大化问题的流算法

本文研究在基数约束下具有单调性的次模+超模函数最大化问题的流模型。该问题在数据处理、机器学习和人工智能等方面都有广泛应用。借助于目标函数的收益递减率（$\gamma$）,我们设计了单轮读取数据的过滤-流算法,并结合次模、超模函数的全局曲率（$\kappa^{g}$）得到算法的近似比为$\min\left\{\frac{（1-\varepsilon）\gamma}{2^{\gamma}},1-\frac{\gamma}{2^{\gamma}（1-\kappa^{g}）^{2}}\right\}$。数值实验验证了过滤-流算法对BP最大化问题的有效性并且得出：次模函数和超模函数在同量级条件下,能保证在较少的时间内得到与贪婪算法相同的最优值。     

任意基数集上的拟阵之单扩张

对于由Betten和Wenzel于2003年提出的任意基数集上的拟阵其相应的秩公理给予了证明,并将此结果用于研究任意基数集上的拟阵的单扩张问题.     

Two cardinals models with gap one revisited

We succeed to say something on the identities of (μ⁺, μ) when μ > θ > cf(μ) with μ strong limit θ‐compact or even μ is limit of compact cardinals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)