Hölder Continuity of the Saddle Point Set for Real-Valued Functions

This paper is concerned with Hölder continuity of the solution to a saddle point problem. Some new su?cient conditions for the uniqueness and Hölder continuity of the solution for a perturbed saddle point problem are established. Applications of the result on Hölder continuity of the solution for perturbed constrained optimization problems are presented under mild conditions. Examples are given to illustrate the obtained results.     

Koszulity of some path categories

We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.     

On the theta completions for maximal subalgebras

Let L be a finite dimensional Lie algebra. Then for a maximal subalgebra M of L, a 𝜃-completion for M is a subalgebra C of L such that CM and M_L?C and C∕M_L contains no non-zero ideal of L∕M_L, properly. And a 𝜃-completion C of M is said to be a strong 𝜃-completion, if C = L or there exists a subalgebra B of L such that C be maximal in B and B is not a 𝜃-completion for M. These are analogous to the concepts of 𝜃-completion and strong 𝜃-completion of a maximal subgroup of a finite group. Now, we consider the influence of these concepts on the structure of a finite dimensional Lie algebra.     

Silica‐supported polymeric monolithic column with a mixed mode of hydrophilic and strong cation‐exchange interactions for microcolumn liquid chromatography

A novel sulfonic acid group containing hydrophilic strong cation‐exchange monolith was prepared by in situ coating 5 μm bare silica particles with the copolymers of glycidyl methacrylate and pentaerythritol triacrylate and further sulfonating the prepared polymer matrix with Na₂SO₃ inside a 150 μm id capillary. The preparation conditions were investigated, and the method was described in detail. The prepared column was characterized by comparing with its counterparts reported previously in terms of matrix morphology, preparation reproducibility, permeability, swelling–shrinking behavior, mechanical stability, hydrophilicity, binding capacity, and column efficiency. The swelling–shrinking behavior of the present column in solvents of different polarities was negligible, the hydrophobicity could be suppressed at the acetonitrile concentrations higher than 40% v/v, and the binding capacities were 256 μequiv/mL and 20.1 mg/mL for Cu²⁺ and lysozyme, respectively. The minimum theoretical plate heights were 8, 10, and 13 μm, and the values of the C term in van Deemter equation were 9, 12, and 35 ms for the test analytes of Na⁺, thiourea, and cytidine 5ʹ‐monophosphate, respectively. This column exhibited an excellent performance in the separations of monovalent inorganic cations, uncharged polar, and charged polar compounds.     

Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.     

Integral representation and estimation of harmonic functions in the quaternionic half space

In this article, we discuss the integral representation of quaternionic harmonic functions in the half space with the general boundary condition. Next, we derive a lower bound from an upper one for quaternionic harmonic functions. These results generalize some of the classic results from the case of plane to the case of noncommutative quaterninionic half space. Copyright © 2017 John Wiley & Sons, Ltd.     

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.     

非齐次树上m重非齐次马氏链的一类强偏差定理

研究给出了非齐次树上m重非齐次马氏链的一类强偏差定理.