POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES

Abstract

This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology. In particular, those aspects of fuzzy sets, hinging around (crisp) powersets of fuzzy subsets and around powerset operators between such powersets lifted from ordinary functions between the underlying base sets, are examined and characterized using point-set and lattice-theoretic methods. The basic goal is to uniquely derive the powerset operators and not simply stipulate them, and in doing this we explicitly distinguish between the “fixed-basis” case (where the underlying lattice of membership values is fixed for the sets in question) and the “variable-basis” case (where the underlying lattice of membership values is allowed to change). Applications to fuzzy sets/logic include: development and justification/characterization of the Zadeh Extension Principle [36], with applications for fuzzy topology and measure theory; characterizations of ground category isomorphisms; rigorous foundation for fuzzy topology in the poslat sense; and characterization of those fuzzy associative memories in the sense of Kosko [18] which are powerset operators. Some results appeared without proof in [31], some with partial proofs in [32], and some in the fixed-basis case in Johnstone [13] and Manes [22].     

Initial-boundary value problem for a compressible dusty gas model

Initial-boundary value problem describing one-dimensional two-phase flows of a compressible viscous dusty gas in a channel is considered. Global-in-time unique solvability for small initial data and the exponential decay of the energy for a small total mass of the dust are proved.     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.     

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

The classic Cayley identity states that

det(∂)(detX)^s=s(s+1)?(s+n−1)(detX)^s−1$\det (\partial ){(\det X)}^s=s(s+1)?(s+n-1){(\det X)}^{s-1}$

where X=(_xij)$X=(x_{ij})$ is an n×n$n\times n$ matrix of indeterminates and ∂=(∂/∂_xij)$\partial =(\partial /\partial x_{ij})$ is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities.     

方柱微结构表面上FC-72的流动沸腾强化换热实验研究

为了提高电子器件的冷却效率,研究了不导电介质FC-72在表面加工有方柱微结构的模拟芯片上的流动沸腾强化换热性能.采用了两种方柱微结构,其边长均为30μm,但高度分别为60μm和120 μm.方柱微结构芯片与光滑芯片相比显示出较好的强化沸腾换热效果,且增加方柱高度可有效提高流动沸腾强化换热性能.方柱微结构芯片的临界热流密度随着流速和过冷度的增大而增大,且到达临界热流密度(CHF)时芯片的表面温度低于芯片回路正常工作的上限温度85℃.     

C72O3的结构和光谱的理论研究

采用AM 1方法理论研究了C70 五元环酸酐衍生物C72 O3 的 8种可能异构体的结构和稳定性 ;以各异构体稳定构型为基础 ,分别用AM 1和ZINDO/CI方法计算了它们的振动光谱和电子光谱。结果表明 ,酸酐基团—C2 O3 主要加成在CⅠ CⅡ(异构体A)和CⅢ CⅢ(异构体B)键上形成闭环结构 ,异构体B的稳定性与实验已证实存在的异构体A十分相近 ;异构体A的振动光谱理论计算值与实验值符合较好 ,B的振动光谱理论计算值与A相似 ;对C72 O3 各异构体的电子跃迁进行了理论指认 ,讨论了其电子光谱的红移现象 ;其他异构体的振动和电子光谱属于理论预测。     

Magnetic {Mo72Fe30}-embedded hybrid nanocapsules

Magnetic nanocapsules were constructed by fabricating nanometer scaled C₆₀-like “Keplerate” type {Mo₇₂Fe₃₀} with molecular formula [Mo₇₂^VIFe₃₀^IIIO₂₅₂(CH₃COO)₁₂{Mo₂O₇(H₂O)}₂{H₂Mo₂O₈(H₂O)}(H₂O)₉₁]ca.150H₂O into nanocapsule shells using the LbL technique. The morphology of the obtained hybrid nanocapsules was examined by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). Shell thickness of the {Mo₇₂Fe₃₀}-embedded nanocapsules can be tailored at the nanometer level more precisely than other nanoparticle-embedded capsules due to the homogeneous diameter and surface charges of {Mo₇₂Fe₃₀}. Interestingly, the {Mo₇₂Fe₃₀}-embedded nanocapsules could be separated and aligned under a circumstance of magnetic field, though {Mo₇₂Fe₃₀} is a paramagnetic molecule. This is the first time to fabricate hybrid magnetic materials containing {Mo₇₂Fe₃₀} using LbL technique. The obtained nanocapsules can be a good candidate for bioseparation as well as targeted delivery.     

Enumeration of bilaterally symmetric 3-noncrossing partitions

Schützenberger’s theorem for the ordinary RSK correspondence naturally extends to Chen et al.’s correspondence for matchings and partitions. Thus the counting of bilaterally symmetric k-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for 3-noncrossing partitions, we use a different technique to develop a Maple package for 2-dimensional vacillating lattice walk enumeration problems. The package also applies to the hesitating case. As applications, we find several interesting relations for some special bilaterally symmetric partitions.     

A propos du lemme fondamental pondéré tordu

In order to use the trace formula of Arthur–Selberg in the twisted case, we need to prove the “twisted weighted fundamental lemma”, that is a sophisticated version of the fundamental lemma. Here, we prove that this twisted weighted fundamental lemma follows from two others lemmas, where the torsion has disappeared: the weighted fundamental lemma for Lie algebras and a “non-standard weighted fundamental lemma”, concerning Lie algebras too.     

On the irrationality measure for certain series

Let K be either the rational number field or an imaginary quadratic field. We give irrationality results for the number , where q (∣q∣ > 1) is an integer in K, r∈ K ^× (∣r∣ < ∣q∣), and with q ⁿ ≠ r ^l (n ≥ 1). Authors’ addresses: Kenji Amano, NS solutions Corporation, 2-27-1 Shinkawa, Chuo-ku, Tokyo 104-0033, Japan; Yohei Tachiya, Department of Mathematics, Keio University, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan