Synthesis and Reactivity of the First Isolated Hydrogen‐Bridged Silanol–Silanolate Anions

We report on the first examples of isolated silanol–silanolate anions, obtained by utilizing weakly coordinating phosphazenium counterions. The silanolate anions were synthesized from the recently published phosphazenium hydroxide hydrate salt with siloxanes. The silanol–silanolate anions are postulated intermediates in the hydroxide‐mediated polymerization of aryl and alkyl siloxanes. The silanolate anions are strong nucleophiles because of the weakly coordinating character of the phosphazenium cation, which is perceptible in their activity in polysiloxane depolymerization.     

左消半群M与幂单半群B的Schǔzenberger积

本文讨论了每个元都有幂等元作为右单位元的左消半群与幂单半群N的Schuzenberger积M◇N的ρ类，证明了这种半群M与N的Schuzenberger积M◇N的ρ类是右E一半适合半群和弱E-headged半群．     

Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

The universal nilpotent group compactification of a semigroup

The purpose of this paper is to introduce an algebra of functions on a semitopological semigroup and to study these functions from the point of view of universal semigroup compactification. We show that the corresponding semigroup compactification of this algebra is universal with respect to the property of being a nilpotent group.

     

The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

K-非常凸空间

本文引入了一种新的K凸空间K-非常凸空间,及其对偶空间K-非常光滑空间,它们分别是非常凸空间和非常光滑空间的推广但又严格弱于非常凸空间和非常光滑空间,因此它们又有许多独特的性质.本文讨论了它们的一些特性及与其它K凸性和K光滑性的关系,推广了[3]、[6]、[7]、[8]中的一些结果.     

凸度量空间中单值和集值映象的公共不动点定理

§ 1.　Introduction　　Anecessaryandsufficientconditionandasufficientconditiontoensurethataset valuedmappingandasingle valuedmappinginacompletemetricspaceandacompleteconvexmetricspacehavingacommonfixedpointarerespectivelygivenin [1 ] .Meanwhilethemainresultsin[2 ]— [5]areimprovedandextendedin [1 ] .Inthispaper ,theexistenceatcommonfixedpointsoftwoset valuedmappingandasing valuedmappingwerestudied ,andthecorrespond ingresultsin [1 ]— [5]wereextendedandimproved .Let(X ,d)beanon emptymetricspa…     

Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type

We consider two classes of integrable nonlinear hyperbolic systems on Lie algebras. These systems generalize the principal chiral model. Each system is related to a pair of compatible Lie brackets and has a Lax representation, which is determined by the direct sum decomposition of the Lie algebra of Laurent series into the subalgebra of Taylor series and the complementary subalgebra corresponding to the pair. New examples of compatible Lie brackets are given.     

Slutsky定理在样本分位数的极限分布证明上的应用

为了更好地理解和应用样本分位数的极限分布,利用Slutsky定理,推导了样本分位数的极限分布.     

A Class of Cauchy Problem for Quasilinear Hyperbolic Systems with the Weakly Dissipative Terms

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