Convex spacelike hypersurfaces of constant curvature in de Sitter space
Joel Spruck Ling Xiao
We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(\kappa)=\sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity has at least one smooth solution (if $l=1$ or $l=2$ there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l=\sigma,\,1 \leq l < n$ in both de Sitter and Hyperbolic space.
keywords: de Sitter Hyperbolic space fully nonlinear. constant curvature asymptotic Plateau
Second-order weak composed epiderivatives and applications to optimality conditions
Qilin Wang Xiao-Bing Li Guolin Yu
In this paper, one introduces the second-order weak composed contingent epiderivative of set-valued maps, and discusses some of its properties. Then, by virtue of the second-order weak composed contingent epiderivative, necessary optimality conditions and sufficient optimality conditions are obtained for set-valued optimization problems. As consequences, recent existing results are derived. Several examples are provided to show the main results obtained.
keywords: Set-valued optimization second-order optimality conditions. second-order weak composed contingent epiderivatives
Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space
Baoquan Yuan Xiao Li
In this paper, we investigate the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations in three dimensions. We obtain that if $ \nabla_{h}u,\nabla_{h}\omega\in L^{1}(0,T;\dot{B}^{0}_{\infty,\infty})$ or $ \nabla_{h}u,\nabla_{h}\omega\in L^{\frac{8}{3}}(0,T;\dot{B}^{-1}_{\infty,\infty})$ then the solution $(u,\omega)$ can be extended smoothly beyond $t=T$.
keywords: Micropolar fluid equations blow-up criteria regularity criteria smooth solutions Besov space.
Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching
Hongfu Yang Xiaoyue Li George Yin
This work is concerned with permanence and ergodicity of stochastic Gilpin-Ayala models involve continuous states as well as discrete events. A distinct feature is that the Gilpin-Ayala parameter and its corresponding perturbation parameter are allowed to be varying randomly in accordance with a random switching process. Necessary and sufficient conditions of the stochastic permanence and extinction are established, which are much weaker than the previous results. The existence of the unique stationary distribution is also established. Our approach treats much wider class of systems, uses much weaker conditions, and substantially generalizes previous results. It is shown that regime switching can suppress the impermanence. Furthermore, several examples and simulations are given to illustrate our main results.
keywords: positive recurrence Markov chain Stochastic permanence Gilpin-Ayala model stationary distribution.
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
Stability of solution mapping for parametric symmetric vector equilibrium problems
Xiao-Bing Li Xian-Jun Long Zhi Lin
This paper is concerned with the stability for a parametric symmetric vector equilibrium problem. A parametric gap function for the parametric symmetric vector equilibrium problem is introduced and investigated. By virtue of this function, we establish the sufficient and necessary conditions for the Hausdorff lower semicontinuity of solution mapping to a parametric symmetric vector equilibrium problem. The results presented in this paper generalize and improve the corresponding results in the recent literature.
keywords: parametric gap function. solution mapping semicontinuity Symmetric vector equilibrium problem continuity
Well-posedness for the three-dimensional compressible liquid crystal flows
Xiaoli Li Boling Guo
This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.
keywords: existence and uniqueness. strong solution vacuum compressible Liquid crystals
Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation
Xiaoyue Li Xuerong Mao
In this paper, we consider a non-autonomous stochastic Lotka-Volterra competitive system $ dx_i (t) = x_i(t)$[($b_i(t)$-$\sum_{j=1}^{n} a_{ij}(t)x_j(t))$$dt$$+ \sigma_i(t) d B_i(t)]$, where $B_i(t)$($i=1 ,\ 2,\cdots,\ n$) are independent standard Brownian motions. Some dynamical properties are discussed and the sufficient conditions for the existence of global positive solutions, stochastic permanence, extinction as well as global attractivity are obtained. In addition, the limit of the average in time of the sample paths of solutions is estimated.
keywords: Brownian motion Itô's formula stochastic differential equation stochastic permanence global attractivity.
The "hot spots" conjecture on higher dimensional Sierpinski gaskets
Xiao-Hui Li Huo-Jun Ruan
In this paper, using spectral decimation, we prove that the ``hot spots" conjecture holds on higher dimensional Sierpinski gaskets.
keywords: ``hot spots'' conjecture eigenfunction higher dimensional Sierpinski gaskets spectral decimation. eigenvalue Neumann Laplacian
Inverse problems with partial data in a slab
Xiaosheng Li Gunther Uhlmann
In this paper we consider several inverse boundary value problems with partial data on an infinite slab. We prove the unique determination results of the coefficients for the Schrödinger equation and the conductivity equation when the corresponding Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same single hyperplane.
keywords: Inverse problems Incomplete data Slab.

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