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3.0

Jun 30, 2018
06/18

by
Xue Luo; Stephen S. -T. Yau; Mingyi Zhang; Huaiqing Zuo

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This work is a natural continuation of our previous work \cite{yz}. In this paper, we give a complete classification of toric surface codes of dimension less than or equal to 6, except a special pair, $C_{P_6^{(4)}}$ and $C_{P_6^{(5)}}$ over $\mathbb{F}_8$. Also, we give an example, $C_{P_6^{(5)}}$ and $C_{P_6^{(6)}}$ over $\mathbb{F}_7$, to illustrate that two monomially equivalent toric codes can be constructed from two lattice non-equivalent polygons.

Topics: Information Theory, Combinatorics, Computing Research Repository, Mathematics

Source: http://arxiv.org/abs/1402.0060

3
3.0

Jun 30, 2018
06/18

by
Xue Luo

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In this paper, we propose new spectral viscosity methods based on the generalized Hermite functions for the solution of nonlinear scalar conservation laws in the whole line. It is shown rigorously that these schemes converge to the unique entropy solution by using compensated compactness arguments, under some conditions. The numerical experiments of the inviscid Burger's equation support our result, and it verifies the reasonableness of the conditions.

Topics: Mathematics, Numerical Analysis

Source: http://arxiv.org/abs/1402.3775

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31

Sep 22, 2013
09/13

by
Xue Luo; Dong Ye; Feng Zhou

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In this note, we investigate the regularity of extremal solution $u^*$ for semilinear elliptic equation $-\triangle u+c(x)\cdot\nabla u=\lambda f(u)$ on a bounded smooth domain of $\mathbb{R}^n$ with Dirichlet boundary condition. Here $f$ is a positive nondecreasing convex function, exploding at a finite value $a\in (0, \infty)$. We show that the extremal solution is regular in low dimensional case. In particular, we prove that for the radial case, all extremal solution is regular in dimension...

Source: http://arxiv.org/abs/1004.3956v2

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31

Jun 30, 2018
06/18

by
Xue Luo; Roman Shvydkoy

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In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $u = \nabla^\perp \Psi$, $\Psi(r,\theta) = r^{\lambda} \psi(\theta)$, for $\lambda >0$, we show that only trivial solutions exist in the range $0

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.4322

3
3.0

Jun 30, 2018
06/18

by
Xue Luo; Stephen S. -T. Yau

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The singular parabolic problem $u_t-\triangle u=\lambda{\frac{1+\delta|\nabla u|^2}{(1-u)^2}}$ on a bounded domain $\Omega$ of $\mathbb{R}^n$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $\lambda_\delta^*>0$ such that if $0

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1402.0066

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5.0

Jun 29, 2018
06/18

by
Xue Luo; Roman Shvydkoy

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In this addendum note we fill in the gap left in \cite{ls} in the description of 2D homogeneous solutions to the stationary Euler system with the help of the results of \cite{sd}. This gives a complete classification of all solutions. The note includes updated classification tables, and a reverse application to the results of \cite{sd}.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1608.00061

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Sep 21, 2013
09/13

by
Xue Luo; Stephen S. -T. Yau

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In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on...

Source: http://arxiv.org/abs/1301.1403v1

4
4.0

Jun 30, 2018
06/18

by
Xue Luo; Shing-Tung Yau; Stephen S. -T. Yau

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A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theorethical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method...

Topics: Mathematics, Numerical Analysis

Source: http://arxiv.org/abs/1412.0427