Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

Convergence Results for Henig Proper Efficient Solution Sets of Vector Optimization Problems

In this article, we discuss the convergence of Henig proper minimal point sets and Henig proper efficient solution sets for (strict) proper quasi-convex vector optimization problems when the data of the perturbed problems converges to the data of the original problem in the sense of Painlevé-Kuratowski. Our main results are new and different from those in the literature.     

Non-periodic finite-element formulation of Kohn-Sham density functional theory

We present a real-space, non-periodic, finite-element formulation for Kohn-Sham density functional theory (KS-DFT). We transform the original variational problem into a local saddle-point problem, and show its well-posedness by proving the existence of minimizers. Further, we prove the convergence of finite-element approximations including numerical quadratures. Based on domain decomposition, we develop a parallel finite-element implementation of this formulation capable of performing both all-electron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature. We also evaluate the numerical performance of the implementation with regard to its scalability and convergence rates. We view this work as a step towards developing a method that can accurately study defects like vacancies, dislocations and crack tips using density functional theory (DFT) at reasonable computational cost by retaining electronic resolution where it is necessary and seamlessly coarse-graining far away.     

Negative derivatives and special functions

The formalism underlying the concept of negative derivatives is proved to be a powerful tool to study new series expansions of special functions defined by integral representations. Its importance is also discussed within the context of multiple integrations and of integral equations.     

再论Γ-环由元素性质确定的根

本文在 Γ-环中继续研究由元素性质确定的根性质 .首先证明了文献 [1]中主要定理的逆定理 , 从而使满足某些条件的元素所具有的性质 P与根性质 R可互相确定 .进而讨论确定的唯一性问题.利用 这些结果可得出 Γ-环的 Baer根是由元素的 m-幂零性所确定的根.     

Pseudo core inverses in rings with involution

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary ?-ring case by Raki?, Din?i? and Djordjevi? in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of ?-ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.     

Derivative of power series attached to Γ-transform of p-adic measures

We extend the result of Anglès (2007) [1], namely for the Iwasawa power series . For the derivative , a numerical polynomial Q on Zp, and a prime π in over p, we show that if and only if i.e. for all x∈Zp. This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in .     

Compactness,Kinetic Formulation,and Entropies for a Problem Related to Micromagnetics

Abstract

We carry on the study of (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294–338.) on the asymptotics of a family of energy-functionals related to micromagnetics. We prove compactness for families of uniformly bounded energies releasing the LBP condition we had previously set. Such families converge to unit-valued divergence-free vector-fields that are tangent to the boundary of the domain, and we found in (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294–338.) that the energy-functionals Γ-converge to a limiting jump-energy of such configurations. We examine the behavior of certain truncated fields which serve to construct “entropies,” and to provide an improved lower bound. We give a kinetic formulation of the problem, and show that the limiting divergence-free problem is supplemented, in the case of minimizers, with a sign condition which can in turn, using the kinetic formulation, be interpreted as an entropy condition that plays a role in uniqueness questions.