Entropy sets, weakly mixing sets and entropy capacity
François Blanchard Wen Huang
Entropy sets are defined both topologically and for a measure. The set of topological entropy sets is the union of the sets of entropy sets for all invariant measures. For a topological system $(X,T)$ and an invariant measure $\mu$ on $(X,T)$, let $H(X,T)$ (resp. $H^\mu(X,T)$) be the closure of the set of all entropy sets (resp. $\mu$- entropy sets) in the hyperspace $2^X$. It is shown that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$), the subsystem $(H(X,T),\hat{T})$ (resp. $(H^\mu(X,T),\hat{T}))$ of $(2^X,\hat{T})$ has an invariant measure with full support and infinite topological entropy.
    Weakly mixing sets and partial mixing of dynamical systems are introduced and characterized. It is proved that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$) the set of all weakly mixing entropy sets (resp. $\mu$-entropy sets) is a dense $G_\delta$ in $H(X,T)$ (resp. $H^\mu(X,T)$). A Devaney chaotic but not partly mixing system is constructed.
     Concerning entropy capacities, it is shown that when $\mu$ is ergodic with $h_\mu(T)>0$, the set of all weakly mixing $\mu$-entropy sets $E$ such that the Bowen entropy $h(E)\ge h_\mu(T)$ is residual in $H^\mu(X,T)$. When in addition $(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$ with $h(E)=h_{\text{top}}(T)$ is residual in $H(X,T)$.
keywords: partial mixing Entropy set entropy capacity.
The thermodynamic formalism for sub-additive potentials
Yongluo Cao De-Jun Feng Wen Huang
The topological pressure is defined for sub-additive potentials via separated sets and open covers in general compact dynamical systems. A variational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the topological pressure are discussed. The result will have some potential applications in the multifractal analysis of iterated function systems with overlaps, the distribution of Lyapunov exponents and the dimension theory in dynamical systems.
keywords: variational principle entropy topological pressure products of matrices. Thermodynamical formalism Lyapunov exponents
Reaction, diffusion and chemotaxis in wave propagation
Shangbing Ai Wenzhang Huang Zhi-An Wang
By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion-chemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
keywords: traveling waves Reaction-diffusion-chemotaxis minimal wave speed. cell growth
Dynamics of an SIS reaction-diffusion epidemic model for disease transmission
Wenzhang Huang Maoan Han Kaiyu Liu
Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition has been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many important and interesting properties have been obtained: such as the role of diffusion coefficients in defining the reproductive number; the global stability of disease-free equilibrium; the existence and uniqueness of a positive endemic steady; global stability of endemic steady for some particular cases; and the asymptotical profiles of the endemic steady states as the diffusion coefficient for susceptible individuals is sufficiently small. In this research we will study two modified SIS diffusion models with the Dirichlet boundary condition that reflects a hostile environment in the boundary. The reproductive number is defined which plays an essential role in determining whether the disease will extinct or persist. We have showed that the disease will die out when the reproductive number is less than one and that the endemic equilibrium occurs when the reproductive number is exceeds one. Partial result on the global stability of the endemic equilibrium is also obtained.
keywords: stability. epidemic model reaction-diffusion equations
Traveling wave solutions for a diffusive sis epidemic model
Wei Ding Wenzhang Huang Siroj Kansakar
In this paper, we study the traveling wave solutions of an SIS reaction-diffusion epidemic model. The techniques of qualitative analysis has been developed which enable us to show the existence of traveling wave solutions connecting the disease-free equilibrium point and an endemic equilibrium point. In addition, we also find the precise value of the minimum speed that is significant to study the spreading speed of the population towards to the endemic steady state.
keywords: minimum wave speed. traveling wave solution SIS epidemic model
Increase statistical reliability without losing predictive power by merging classes and adding variables
Wenxue Huang Xiaofeng Li Yuanyi Pan

It is usually true that adding explanatory variables into a probability model increases association degree yet risks losing statistical reliability. In this article, we propose an approach to merge classes within the categorical explanatory variables before the addition so as to keep the statistical reliability while increase the predictive power step by step.

keywords: Association categorical data category merging statistical reliability predictive power
On balancing between optimal and proportional categorical predictions
Wenxue Huang Yuanyi Pan
A bias-variance dilemma in categorical data mining and analysis is the fact that a prediction method can aim at either maximizing the overall point-hit accuracy without constraint or with the constraint of minimizing the distribution bias. However, one can hardly achieve both at the same time. A scheme to balance these two prediction objectives is proposed in this article. An experiment with a real data set is conducted to demonstrate some of the scheme's characteristics. Some basic properties of the scheme are also discussed.
keywords: Bias-variance dilemma point estimation conditional distribution. proportional prediction optimal prediction categorical data
Weakly coupled traveling waves for a model of growth and competition in a flow reactor
Wenzhang Huang
For a reaction-diffusion model of microbial flow reactor with two competing populations, we show the coexistence of weakly coupled traveling wave solutions in the sense that one organism undergoes a population growth while another organism remains in a very low population density in the first half interval of the space line; the population densities then exchange the position in the next half interval. This type of traveling wave can occur only if the input nutrient slightly exceeds the maximum carrying capacity for these two populations. This means, lacking an adequate nutrient, two competing organisms will manage to survive in a more economical way.
keywords: Microbial flow reactor model unstable manifold reaction-diffusion equations weakly coupled traveling waves.
Soliton solutions for a quasilinear Schrödinger equation with critical exponent
Wentao Huang Jianlin Xiang
This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$ with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort laser in matter. By working with a perturbation approach which was initially proposed in [26], we prove that the given problem has a positive ground state solution.
keywords: Ground state quasilinear Schrödinger equation positive solution perturbation method. critical exponent
Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor
Wenzhang Huang
Consider a reaction-diffusion model for a microbial flow reactor with two competing species. Suppose that the amount of nutrient is input in a constant velocity at one end of the flow reactor and is washed out at the other end of the reactor. We study the dynamical behavior of population growth of these two species. In particular we are interested in the problem on the coexistence of traveling waves that best describes the long time dynamical behavior. By developing shooting method and continuation argument with the aid of an appropriately Liapunov function, we obtain the sufficient conditions for the coexistence of traveling waves as well as the minimum wave speed.
keywords: shooting method Microbial flow reactor continuation argument. reaction-diffusion equations traveling waves

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