Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

Existence of strong minimizers for the Griffith static fracture model in dimension two

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate $SB{D}^p$ functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSB{D}^p$ functions by $SB{V}^p$ ones.     

Molybdenum(VI) Bis(imido) Complexes: From Frustrated Lewis Pairs to Weakly Coordinating Cations

Molybdenum(VI) bis(imido) complexes [Mo(NtBu)₂(L^R)₂] (R=H 1 a ; R=CF₃ 1 b ) combined with B(C₆F₅)₃ ( 1 a /B(C₆F₅)₃, 1 b /B(C₆F₅)₃) exhibit a frustrated Lewis pair (FLP) character that can heterolytically split H−H, Si−H and O−H bonds. Cleavage of H₂ and Et₃SiH affords ion pairs [Mo(NtBu)(NHtBu)(L^R)₂][HB(C₆F₅)₃] (R=H 2 a ; R=CF₃ 2 b ) composed of a Mo(VI) amido imido cation and a hydridoborate anion, while reaction with H₂O leads to [Mo(NtBu)(NHtBu)(L^R)₂][(HO)B(C₆F₅)₃] (R=H 3 a ; R=CF₃ 3 b ). Ion pairs 2 a and 2 b are catalysts for the hydrosilylation of aldehydes with triethylsilane, with 2 b being more active than 2 a . Mechanistic elucidation revealed insertion of the aldehyde into the B−H bond of [HB(C₆F₅)₃]⁻. We were able to isolate and fully characterize, including by single-crystal X-ray diffraction analysis, the inserted products Mo(NtBu)(NHtBu)(L^R)₂][{PhCH₂O}B(C₆F₅)₃] (R=H 4 a ; R=CF₃ 4 b ). Catalysis occurs at [HB(C₆F₅)₃]⁻ while [Mo(NtBu)(NHtBu)(L^R)₂]⁺ (R=H or CF₃) act as the cationic counterions. However, the striking difference in reactivity gives ample evidence that molybdenum cations behave as weakly coordinating cations (WCC).     

A Simple Homoleptic Gallium(I) Olefin Complex: Mimicking Transition‐Metal Chemistry at a Main‐Group Metal?

The earth‐metal olefin complex [Ga ^I (COD)₂]⁺[Al(OR^F)₄]^? (COD=1,5‐cyclooctadiene; R^F=C(CF₃)₃) constitutes the first homoleptic olefin complex of any main‐group metal accessible as a bulk compound. It is straight forward to prepare in good yield and constitutes an olefin complex of a main‐group metal that—similar to many transition‐metals—may adopt the +1 and +3 oxidation states opening potential applications. Crystallographic‐, vibrational‐ and computational investigations give an insight to the atypical bonding between an olefin and a main‐group metal. They are compared to classical transition‐metal relatives.     

基于拟相对内部集值优化问题弱有效元的最优性条件

本文研究了基于拟相对内部的非凸集值优化问题弱有效元的最优性条件.首先,讨论了弱有效元与线性子空间之间的关系,利用涉及拟相对内部的凸集分离定理,获得了弱有效元的最优性条件.其次,给出了基于拟相对内部弱有效元的Lagrange乘子定理.     

Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form × D,where h has infinite order and D is finite.     

旅行商问题的基因整合算法

本文针对旅行商问题提出了基因整合算法。它是通过设置扰动矩阵构造与原商问题等价的近似问题，使用最优罚函数选择回路分枝得到一系列局部最优回路，从中提取频度高的分技——基因进行整合，得到更优的回路。该算法计算量小，对大规模问题计算效果显著。利用该算法对CHN144问题给出了目前最件的结果。     

Global Weak Solutions for the Weakly Dissipative Camassa-Holm Equation

In this paper we prove the existence and uniqueness of global weak solutions to the weakly dissipative Camassa-Holm equation.     

随机环境中的马氏链状态分类

给出了随机环境中马氏链的特征数和状态的定义,讨论了状态间的传递性,自反性,对称性,是经典马氏链相应结果的一般化.并运用位势方法研究了状态间的关系,它们在极限理论的研究中非常有用.     

囚禁弱相互作用玻色气体的势场优化准则

根据Thomas-Fermi近似,在基于最小动量态上玻色-爱因斯坦凝聚的前提下,研究了囚禁弱相互作用玻色气体势场的最优化问题.导出了指数吸引势阱中有效势场和粒子数极限判据,粒子数给定时,可由此判据求出所需势场强度;势场强度给定时,可由此判据求出粒子数极限.根据吸引相互作用系统的稳定性以及求出的排斥相互作用的最大粒子数极限,结合有效势场判据,分别给出了囚禁吸引和排斥相互作用玻色气体时,势场强度的最佳取值范围. 关键词： 玻色-爱因斯坦凝聚 弱相互作用 粒子数极限 势场强度