Tensor logarithmic norm and its applications

Matrix logarithmic norm is an important quantity, which characterize the stability of linear dynamical systems. We propose the logarithmic norms for tensors and tensor pairs, and extend some classical results from the matrix case. Moreover, the explicit forms of several tensor logarithmic norms and semi‐norms are also derived. Employing the tensor logarithmic norms, we bound the real parts of all the eigenvalues of a complex tensor and study the stability of a class of nonlinear dynamical systems. Copyright © 2016 John Wiley & Sons, Ltd.     

一类函数的对数完全单调性

通过对参数λ,μ的讨论,主要利用函数的单调性理论,已有对数完全单调函数的性质以及幂函数的积分表达式研究了函数Gλ,μ(x)及函数[Gλ,μ(x)]-1的对数完全单调性,并在此基础上得到了一定条件下函数Gλ,μ(x)及[Gλ,μ(x)]-1对数完全单调的充要条件.     

Continuity of the efficient solution mapping for vector optimization problems

This paper aims at investigating the continuity of the efficient solution mapping of perturbed vector optimization problems. First, we introduce the concept of the level mapping. We give sufficient conditions for the upper semicontinuity and the lower semicontinuity of the level mapping. The upper semicontinuity and the lower semicontinuity of the efficient solution mapping are established by using the continuity properties of the level mapping. We establish a corollary about the lower semicontinuity of the minimal point set-valued mapping. Meanwhile, we give some examples to illustrate that the corollary is different from the ones in the literature.     

加权流形上加权p-Laplace特征值问题的第一特征值下界估计

本文研究了加权流形上加权p-Laplacian特征值问题的第一特征值下界估计的问题.利用余面积公式、Cavalieri原理以及Federer-Fleming定理,获得了由Cheeger常数或等周常数确定的第一特征值的下界估计.     

Polymerization of Acrylamide in Aqueous Solution of Poly(N‐isopropylacrylamide) at Lower Critical Solution Temperature

Acrylamide (AAm) was found to polymerize in a solution of poly(N‐isopropylacrylamide) (PNIPAAm) in water at around its lower critical solution temperature (LCST) (32°C) without any initiators. This phenomenon was specifically observed in aqueous solutions of the polymers having LCST such as PNIPAAm and poly(methylvinylether) (PMVE). AAm polymerized only when PNIPAAm and AAm were dissolved in water below LCST of PNIPAAm and then the solution was warmed up to the polymerization temperature (40°C). On the other hand, the polymerization of AAm did not proceed when AAm was added into aqueous PNIPAAm solution during and after the phase separation above 32°C. Furthermore the polymerizability of AAm was remarkably affected by the concentration and molecular weight of the PNIPAAm additives. Under the condition of lower PNIPAAm concentration (0.30 mol/L), the increase in the molecular weight of PNIPAAm considerably increased the molecular weight of the resulting PAAm but decreased the yield of PAAm. Under the condition of higher PNIPAAm concentration (0.60 mol/L) the polymerizability was not so affected by the molecular weight of PNIPAAm, while the molecular weight of PAAm formed by using higher molecular weight PNIPAAm was higher than those of PAAm formed by using lower molecular weight PNIPAAm. Moreover, the molecular weight of PAAm formed by the PNIPAAm induced polymerization of AAm was much higher than that of the polymer obtained by the radical polymerization using AIBN in THF or VA‐ 061 in water.     

Temperature/Light Dual‐Responsive Inclusion Complexes of α‐Cyclodextrins and Azobenzene‐Containing Polymers

Three kinds of photoresponsive copolymers with azobenzene side chains were synthesized by radical polymerization of N‐4‐phenylazophenylacrylamide (PAPA) with N‐isopropylacrylamide (NIPAM), N,N‐diethylacrylamide (DEAM) or N,N‐dimethylacrylamide (DMAM) respectively. Their structures were characterized by FT‐IR, ¹H‐NMR and UV/Vis spectroscopy. Their reversible photoresponses were studied with or without α‐cyclodextrin (α‐CD), which showed that both the copolymers and their inclusion complexes with α‐CD underwent rapid photoisomerization. The lower critical solution temperature (LCST) of the copolymers and their inclusion complexes with α‐CD were investigated by cloud point measurement, which showed that the LCST of three kinds of copolymers increased largely after adding α‐CD. After UV irradiation on the solutions of copolymers and their inclusion complexes, the LCST of the copolymers increased slightly with the absence of α‐CD, while decreased largely with the presence of α‐CD. Furthermore, the LCST reverted to its originality after visible light irradiation. This change of LCST could be reversibly controlled by UV and visible light irradiation alternately. In particular, in the copolymer of PAPA and DMAM, the reversible water solubility of the inclusion complexes could be triggered by alternating UV and visible light irradiation.     

Spherically Symmetric Solvent is Sufficient to Explain the LCST Mechanism in Polymer Solutions

The mechanism of the lower critical solution temperature (LCST) in thermoresponsive polymer solutions has been studied by means of a coarse‐grained single polymer chain simulation and a theoretical approach. The simulation model includes solvent explicitly and thus accounts for solvent interactions and entropy directly. The theoretical model consists of a single chain polymer in an implicit solvent where the effect of solvent is included through the intrapolymer solvophobic potential proposed by Kolomeisky and Widom. The results of this study indicate that the LCST behavior is determined by the competition between the mean energy difference between the bulk and bound solvent, and the entropy loss due to the bound solvent. At low temperatures, solvent molecules are bound to the polymer and the solvophobicity of the polymer is screened, resulting in a coiled state. At high temperatures the entropy loss due to bound solvent offsets the energy gain due to binding which causes the solvent molecules to unbind, leading to the collapse of the polymer chain to a globular state. Furthermore, the coarse‐grained nature of these models indicates that mean interaction energies are sufficient to explain LCST in comparison to specific solvent structural arrangements.

     

Recent approaches for clickable poly(2-oxazoline)-based functional stimuli-responsive polymers and related applications

Abstract

Poly(2-oxazoline)s (POxs) are well-known thermo-responsive polymers that exhibit reversible hydrophilic–hydrophobic phase transitions at the lower critical solution temperature (LCST). Using living cationic ring-opening polymerisation, various functional groups can be introduced into POxs. Several clickable POxs with propargyl or azide end groups have been designed and subsequently reacted with various functional groups to prepare multifunctional POxs that respond to stimuli such as temperature, pH, chemicals and light. In this article, we briefly review recent approaches for clickable POx-based functional stimuli-responsive polymers and related applications.     

A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H¹ norm is achieved when the mesh sizes satisfy h = O(H²). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.     

An unfeasibility view of neural network learning

We define the notion of a continuously differentiable perfect learning algorithm for multilayer neural network architectures and show that such algorithms do not exist provided that the length of the data set exceeds the number of involved parameters and the activation functions are logistic, tanh or sin.