Semilinear wave models with friction and viscoelastic damping

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.     

基于化学学科认知障碍突破的中考复习教学实践* ——以“二氧化碳的性质”为例

分析学生对二氧化碳的3个典型学科认知障碍,以“二氧化碳的捕捉”为情境,通过“尝试捕捉二氧化碳”“谁在捕捉二氧化碳”“捕捉背后的原理”等3个主要环节,实现对学科认知障碍的突破,展示了一种新的中考复习教学思路。     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

Potts models with magnetic field: Arithmetic,geometry, and computation

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.     

Epidemiological models with quadratic equation for endemic equilibria—A bifurcation atlas

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplicity, such an equation can describe an amazing number of dynamical behaviors. In this paper, we shall provide a comprehensive survey of possible bifurcation patterns, deriving explicit conditions on the equation's parameters for the occurrence of each of them, and discuss illustrative examples.     

The effects of resource limitation and consensual incentives on cooperation

Exploring the evolution of cooperation has been one of the most challenging topics and attracted increasing attention in various realms. Hereby, we institute an analytic model where the relationship between the common benefits and contributions is the typical saturation nonlinearity. We investigate the evolutionary dynamics of cooperation with and without consensual incentives respectively. The population always evolves to full defection when the incentives are absent. Cooperation, however, can be promoted by consensual incentives when the degree of cooperators' preferences for incentives is great. There exists an intermediate amount of resources at which the positive effect of consensual incentives on cooperation is minimized. Furthermore, there is a U-shape relationship between the degree of cooperators' preference for an incentive and the evolution of cooperation. A greater basin of full cooperation can be induced when the cooperators' preference for reward is more pronounced than that for punishment.     

Norbornene derivatives from a metal‐free,strain‐promoted cycloaddition reaction: New building blocks for ring‐opening metathesis polymerization reactions

The preparation of new ring opening metathesis polymerization (ROMP) monomers using a 1,3‐dipolar cycloaddition between aryl azides and norbornadiene is described. Various norbornenetriazolines, obtained through a solvent‐and catalyst‐free reaction, can subsequently be incorporated into polymer backbones through ROMP reactions. Furthermore, thermal decomposition of the triazoline moiety can allow for further polymer functionalization. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 2357–2362     

A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.