Divergence points in systems satisfying the specification property
Jinjun Li Min Wu
Let $f$ be a continuous transformation of a compact metric space $(X,d)$ and $\varphi$ any continuous function on $X$. In this paper, under the hypothesis that $f$ satisfies the specification property, we determine the topological entropy of the following sets: $$K_{I}=\Big\{x\in X: A\big(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^{i}(x))\big)=I\Big\},$$ where $I$ is a closed subinterval of $\mathbb{R}$ and $A(a_{n})$ denotes the set of accumulation points of the sequence $\{a_{n}\}_{n}$. Our result generalizes the classical result of Takens and Verbitskiy ( Ergod. Th. Dynam. Sys., 23 (2003), 317-348 ). As an application, we present another concise proof of the fact that the irregular set has full topological entropy if $f$ satisfies the specification property.
keywords: Divergence points topological entropy the specification property.
Topological pressure and topological entropy of a semigroup of maps
Dongkui Ma Min Wu
By using the Carathéodory-Pesin structure(C-P structure), with respect to arbitrary subset, the topological pressure and topological entropy, introduced for a single continuous map, is generalized to the cases of semigroup of continuous maps. Several of their basic properties are provided.
keywords: topological pressure Semigroup of continuous maps C-P structure topological entropy.
Generic property of irregular sets in systems satisfying the specification property
Jinjun Li Min Wu
Let $f$ be a continuous map on a compact metric space. In this paper, under the hypothesis that $f$ satisfies the specification property, we prove that the set consisting of those points for which the Birkhoff ergodic average does not exist is either residual or empty.
keywords: residual. specification property Irregular set
Hausdorff dimension of certain sets arising in Engel continued fractions
Lulu Fang Min Wu
In the present paper, we are concerned with the Hausdorff dimension of certain sets arising in Engel continued fractions. In particular, the Hausdorff dimension of sets
$\big\{x ∈ [0,1): b_n(x) ≥ \phi (n)~i.m.~n ∈ \mathbb{N}\big\}\ \ \text{and}\ \ \big\{x ∈ [0,1): b_n(x) ≥ \phi(n),\ \forall n ≥ 1\big\}$
are completely determined, where
means infinitely many,
$\{b_n(x)\}_{n ≥ 1}$
is the sequence of partial quotients of the Engel continued fraction expansion of
is a positive function defined on natural numbers.
keywords: Engel continued fractions growth rate of partial quotients Hausdorff dimension
Optimal isolation strategies of emerging infectious diseases with limited resources
Yinggao Zhou Jianhong Wu Min Wu
A classical deterministic SIR model is modified to take into account of limited resources for diagnostic confirmation/medical isolation. We show that this modification leads to four different scenarios (instead of three scenarios in comparison with the SIR model) for optimal isolation strategies, and obtain analytic solutions for the optimal control problem that minimize the outbreak size under the assumption of limited resources for isolation. These solutions and their corresponding optimal control policies are derived explicitly in terms of initial conditions, model parameters and resources for isolation (such as the number of intensive care units). With sufficient resources, the optimal control strategy is the normal Bang-Bang control. However, with limited resources the optimal control strategy requires to switch to time-variant isolation at an optimal rate proportional to the ratio of isolated cases over the entire infected population once the maximum capacity is reached.
keywords: limited resource. Optimal control isolation SIR model
Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case
Mingzhu Wu Zuodong Yang
In this paper, we study positive solution of the following system of quasilinear elliptic equations

div$(|\nabla u|^{p-2}\nabla u)=u^{m_1}v^{n_1},$ in $\Omega$

div$(|\nabla v|^{q-2}\nabla v)=u^{m_2}v^{n_2},$ in $\Omega,$ $\qquad\qquad\qquad\qquad$ (0.1)

where $m_1>p-1,n_2>q-1, m_2,n_1>0$, and $\Omega\subset R^N$ is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: $u=\lambda, v=\mu$ or $u=v=+\infty$ or $u=+\infty, v=\mu$ on $\partial\Omega$, where $\lambda, \mu>0$. Under several hypotheses on the parameters $m_1,n_1,m_2,n_2$, we show that the existence of positive solutions. We further provide the asymptotic behaviors of the solutions near $\partial\Omega$. Some more general related problems are also studied.

keywords: Boundary blow-up positive solution quasilinear elliptic system.

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