DCDS-B

In this paper, we study a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. By Schauder's fixed point theorem and Laplace transform, we show that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. Some examples are listed to illustrate the theoretical results. Our results generalize some known results.

DCDS-B

A disease transmission model of SIRS type with latent period and
nonlinear incidence rate is considered. Latent period is assumed
to be a constant $\tau$, and the incidence rate is assumed to be
of a specific nonlinear form, namely,
$\frac{kI(t-\tau)S(t)}{1+\alpha I^{h}(t-\tau)}$,
where $h\ge 1$. Stability of the disease-free equilibrium, and
existence, uniqueness and stability of an endemic equilibrium for
the model, are investigated. It is shown that, there exists the
basic reproduction number $R_0$ which is independent of the form
of the nonlinear incidence rate, if $R_0\le 1$, then the
disease-free equilibrium is globally asymptotically stable,
whereas if $R_0>1$, then the unique endemic equilibrium is
globally asymptotically stable in the interior of the feasible
region for the model in which there is no latency, and periodic
solutions can arise by Hopf bifurcation from the endemic
equilibrium for the model at a critical latency. Some numerical
simulations are provided to support our analytical conclusions.

MBE

In this paper, we study an age-structured virus dynamics model with Beddington-DeAngelis infection function. An explicit formula for the basic reproductive number $\mathcal{R}_{0}$ of the model is obtained. We investigate the global behavior of the model in terms of $\mathcal{R}_{0}$: if $\mathcal{R}_{0}\leq1$, then the infection-free equilibrium is globally asymptotically stable, whereas if $\mathcal{R}_{0}>1$, then the infection equilibrium is globally asymptotically stable. Finally, some special cases, which reduce to some known HIV infection models studied by other researchers, are considered.

DCDS

In this paper we establish a closing
property and a hyperbolic closing property for thin
trapped chain hyperbolic homoclinic classes with one dimensional
center in partial hyperbolicity setting. Taking advantage of theses
properties, we prove that the growth rate of the number of
hyperbolic periodic points is equal to the topological entropy. We also
obtain that the hyperbolic periodic measures are dense in the space of invariant measures.

JIMO

The affine rank minimization problem is to find a low-rank matrix satisfying a set of linear equations, which includes the well-known matrix completion problem as a special case and draws much attention in recent years. In this paper, a new model for affine rank minimization problem is proposed. The new model not only enhances the robustness of affine rank minimization problem, but also leads to high nonconvexity. We show that if the classical projected gradient method is applied to solve our new model, the linear convergence rate can be established under some conditions. Some preliminary experiments have been conducted to show the efficiency and effectiveness of our method.

NACO

In this paper we propose two two-step methods for image zooming
using duality strategies. In the first method, instead of smoothing
the normal vector directly as did in the first step of the classical
LOT model, we reconstruct the unit normal vector by means of
Chambolle's dual formulation. Then, we adopt the split Bregman
iteration to obtain the zoomed image in the second step. The second
method is based on the TV-Stokes model. By smoothing the tangential
vector and imposing the divergence free condition, we propose an
image zooming method based on the TV-Stokes model using the dual
formulation. Furthermore, we give the convergence analysis of the
proposed algorithms. Numerical experiments show the efficiency of
the proposed methods.

IPI

In this paper, we discuss the nonsmooth second-order
regularization, suggested by Lysaker, Lundervold and Tai, and its
application in image denoising. A function space $BV^2$ is given
and the well-posedness of the LLT model is proved in this function
space. By means of the Fisher-Burmeister NCP function, we
reformulate the dual formula of the LLT model in discrete setting
as a system of semismooth equations. Then we propose a semismooth
Newton method for the LLT model to build up a Q-superlinearly
convergent numerical scheme. The computational experiments are
supplied to demonstrate the efficiency of the proposed method.

DCDS

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, and
$H_0^1(\Omega)$ be the standard Sobolev space. Define for any $p>1$,
$\lambda_p(\Omega)=$i n f$_u\in H_0^1(\Omega),$u≠0(for some u)$\|\|\nabla u\|\|_2^2/\|\|u\|\|_p^2,$

where $|\|\cdot\||_p$ denotes $L^p$ norm. We derive in this paper a
sharp form of the following improved Moser-Trudinger inequality
involving the $L^p$-norm using the method of blow-up analysis:

$s u p_{u\in H_0^1(\Omega),\|\|\nabla u\|\|_2=1}\int_{\Omega}
e^{4\pi (1+\alpha\|\|u\|\|_p^2)u^2}dx<+\infty$

for $0\leq \alpha
<\lambda_p(\Omega)$, and the supremum is infinity for all
$\alpha\geq \lambda_p(\Omega)$. We also prove the existence of the
extremal functions for this inequality when $\alpha$ is sufficiently
small.

DCDS

We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$ mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$ and then deduce that
the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+\alpha}$ mapping $f$
is greater than or equal to $h_{\mu}(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.