Reducing Exhausters

The notions of upper and lower exhausters were introduced by Demyanov (Optimization 45:13–29, 1999). Upper and lower exhausters can be employed to study a very wide range of positively homogeneous functions, for example, various directional derivatives of nonsmooth functions. Exhausters are not uniquely defined; hence, the problem of minimality arises naturally. This paper describes some techniques for reducing exhausters, both in size and amount of sets. We define also a modified convertor which provides much more flexibility in converting upper exhausters to lower ones and vice versa, and allows us to obtain much smaller sets.     

Comparison Between Quasidifferentials and Exhausters

Directional derivatives play one of the major roles in optimization. Optimality conditions can be described in terms of these objects. These conditions, however, are not constructive. To overcome this problem, one has to represent the directional derivative in special forms. Two such forms are quasidifferentials and exhausters proposed by V.F. Demyanov. Quasidifferentials were introduced in 1980s. Optimality conditions in terms of these objects were developed by L.N. Polyakova and V.F. Demyanov. It was described how to find directions of steepest descent and ascent when these conditions are not satisfied. This paved a way for constructing new optimization algorithms. Quasidifferentials allow one to treat a wide class of functions. V.F. Demyanov introduced the notion of exhausters in 2000s to expand the class of functions that can be treated. It should be noted that a great contribution to the emergence of this notion was made by B.N. Pshenichny and A.M. Rubinov. In this work it is shown that exhausters not only allow one to treat a wider class of functions than quasidifferentials (since every quasidifferentiable function has exhausters) but is also preferable even for quasidifferentiable functions when solving nonsmooth optimization problems.     

Exhausters af a positively homogeneous function *

Notions of upper exhauster and lower exhauster of a positively homogeneous (of the first degree) function h: ? ⁿ →? are introduced. They are linked to exhaustive families of upper convex and lower concave approximations of the function h. The pair of an upper exhauster and a lower exhauster is called a biexhauster of h. A calculus for biexhausters is described (in particular, a composition theorem is formulated). The problem of minimality of exhausters is stated. Necessary and sufficient conditions for a constrained minimum and a constrained maximum of a directionally differentiable function f: ? ⁿ →? are formulated in terms of exhausters of the directional derivative of f. In general, they are described by means of exhausters of the Hadamard upper and lower directional derivatives of the function f. To formulate conditions for a minimum, an upper exhauster is employed while conditions for a maximum are formulated via a lower exhauster of the respective directional derivative (the Hadamard lower derivative for a minimum and the Hadamard upper derivative for a maximum).

If a point x _o is not stationary then directions of steepest ascent and descent can also be calculated by means of exhausters.     

Exhausters, optimality conditions and related problems

The notions of exhausters were introduced in (Demyanov, Exhauster of a positively homogeneous function, Optimization 45, 13–29 (1999)). These dual tools (upper and lower exhausters) can be employed to describe optimality conditions and to find directions of steepest ascent and descent for a very wide range of nonsmooth functions. What is also important, exhausters enjoy a very good calculus (in the form of equalities). In the present paper we review the constrained and unconstrained optimality conditions in terms of exhausters, introduce necessary and sufficient conditions for the Lipschitzivity and Quasidifferentiability, and also present some new results on relationships between exhausters and other nonsmooth tools (such as the Clarke, Michel-Penot and Fréchet subdifferentials).     

弱横截与弱混沌

在保留混沌基本特征的前提下，该文引进弱横截与弱混沌概念．证明了弱横截的平面微分同胚必产生弱混沌．     

Weak Reflections and Weak Factorization Systems

We describe a one-to-one correspondence between saturated weak factorization systems and weak reflections in categories C\mathcal{C} with finite products. This actually extends to an adjunction between the category of natural weak factorization systems on C\mathcal{C} (in the sense of Grandis and Tholen, Arch Math 42:397–408, 2006, and Garner, arXiv preprint, 2007) and the category of monads on C\mathcal{C}. Explicit comparisons are made with the parallel result of Cassidy et al. (J Aust Math Soc 38:287–329, 1985), linking factorization systems and reflective subcategories.     

Weak Covering Properties of Weak Topologies

We consider covering properties of weak topologies of Banachspaces, especially of weak or point-wise topologies of functionspaces C(K), for compact spaces K. We answer questions posedby A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Ourmain results are the following. A Banach space of density atmost ₁ is hereditarily metaLindel of in its weak topology. Ifthe weight of a compact spaceK is at most ₁, then the spacesC_w(K) and C_p(K) are hereditarily metaLindel of. Let be the one-point compactificationof a treeT. Then the space is hereditarily -metacompact. If T is an infinitely branchingfull tree of uncountable height and of cardinality bigger thanc, then the weak topology of the unit sphere of is not -fragmented by any metric. The space C_p(rß₁)is neither metaLindel of nor -relatively metacompact. The spaceC_p(rß₂) is not -relatively metaLindel of. Under theset-theoretic axiom , there exists a scattered compact spaceK₁ such that the space C_p(K₁) is not -relatively metacompact,and under a related axiom , there exists a scattere compactspace K₂ such that the space C_p(K₂) is not -relatively metaLindelof. 1991 Mathematics Subject Classification: 54C35, 46B20, 54E20,54D30.     

偏序集中的弱理想和弱滤子

在偏序集上利用上集算子和下集算子引入了弱理想,弱滤子,弱素理想,弱素滤子,弱极大理想和弱极大滤子等概念,研究了它们的若干性质,同时给出了偏序集上的(DPI),(BPI),(DMI),(BUF)公理并建立了它们与Zorn引理之问的相互关系.     

WEAK MONOTONE REGRESSION AND WEAK REGRESSION DEPENDENCE

This paper gives the exhaustive relations among several concepts of positive dependencefor two dimensional r.v.,and two weak types of positive dependence are studied in detail     

弱Hopf群T-余代数上的弱Doi-Hopf群模

在弱Hopf群T-余代数情形下,弱量子Yetter-Drinfeld群模的概念被引入,并证明了弱量子Yetter-Drinfeld群模是特殊的弱Doi-Hopf群模.接着建立了弱量子Yetter Drinfeld群模范畴与弱Hopf群双余模代数的余不动点子代数B上模范畴之间的伴随对.最后考虑了弱量子Yetter-Drinfeld群模的积分.     

On Weak Crossed Products of Weak Hopf Algebras

Let H be a weak Hopf algebra in the sense of Böhm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A# _σ H be a weak crossed product with σ invertible. Then in this paper we first give some conditions for A# _σ H to be a weak Hopf algebra. Next the spectral sequence for Ext will be constructed which yields an estimate for the global dimension of A# _σ H in terms of the corresponding data for H and A. Furthermore, we will investigate when A???A# _σ H becomes a separable extension. Finally, we prove that if H and its dual H* are both semisimple, then the finitistic dimension of A# _σ H is equal to that of A.     

关于弱MCP空间与弱层空间

本文引入了与已知的一些空间类非常类似的两个概念,我们分别称它们为弱MCP空间与弱层空间.本文主要讨论了这两空间类的等价命题与一些基本性质,及它们与原有空间类的关系.     

Invertible Weak Entwining Structures and Weak C-cleft Extensions

In this note we improve the definition of invertible weak entwining structure. Also we clarify what invertibility means for weak cleft extensions and weak C-Galois extensions.      

Weak Mp-groups

Abstract

Let G be a finite group and p be a prime. We prove in this note that if every irreducible monolithic p-Brauer character of G is monomial then G is solvable.

Communicated by J. Zhang     

Weak MV-algebras

In a recent paper [CHAJDA, I.—KüHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301–312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property. The financial support by the grant of Czech Government MSM 6198959214 is gratefully acknowledged.     

Weak structures

We define weak structures and show that these structures can replace in many situations generalized topologies or minimal structures.     

弱斜配对双代数和弱相关Long双代数

该文在弱双代数$H$上给出了扭曲积$(H^\sigma,\cdot_\sigma)$成为弱双代数的充分必要条件.设$[B, H, \tau]$是一个弱斜配对, 并且$\tau$可逆,则在某个条件下弱双交叉积$B\bowtie_\tau H$是一个弱双代数. 如果$(B,H, \sigma)$是弱相关Long双代数, 并且$\sigma$可逆,则弱双交叉积$B^{OP}\bowtie_\sigma H$可以被构造. 它的乘法是:$(x\otimes h)(y\otimes g)=\Sigma\sigma(y_1, h_1)y_2x\otimes h_2g\sigma^{-1}(y_3, h_3),$ 特别地, 如果$(B, H,\sigma)$是相关Long双代数, 则$(B^{OP \bowtie_\sigma H,\beta)$是Long双代数当且仅当对任意$b, d\in B^{OP}; g, \ell\in H$,$\Sigma\sigma^{-1}(b, g_2\ell)\sigma(d, g_1)=\Sigma\sigma^{-1}(b,\ell g_1)\sigma(d, g_2),$ 其中$B$为$H$的子Hopf代数,$\beta$定义为$\beta(b\bowtie_\sigma h\otimes c\bowtie_\sigma g)=\varepsilon_H(h)\varepsilon_{B^{OP}}(c)\sigma^{-1}(b, g).$ 对于Sweedler 4维Hopf代数$H$, 作者给出一个例子说明:此弱双交叉积$(B^{OP}\bowtie_\sigma H, \beta)$不仅是一个Long双代数,而且是一个非可换和非余可换的8维Hopf代数. 最后, 设$B,H$都是弱双代数, $\sigma: B\otimes H\rightarrow k$是一个线性映射, 作者给出了$(B,\sigma,\leftharpoonup, \Delta_B)$是弱相关右$(H, B)$ -重模代数的充分必要条件.