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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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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

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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

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Sep 18, 2013
09/13

by
Hing Sun Luk; Stephen S. -T. Yau

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The purpose of this paper is to give a counterexample of Theorem 10.4 in [Ann. of Math. 102 (1975), 223-290]. In the Harvey-Lawson paper, a global result is claimed, but only a local result is proven. This theorem has had a big impact on CR geometry for almost a quarter of a century because one can use the theory of isolated singularities to study the theory of CR manifolds and vice versa.

Source: http://arxiv.org/abs/math/9811188v1

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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

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Sep 19, 2013
09/13

by
Stephen S. -T. Yau; Yung Yu

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In the paper "Algebraic classification of rational CR structures on topological 5-sphere with transversal holomorphic S^1-action in C^4" (Yau and Yu, Math. Nachrichten 246-247(2002), 207-233), we give algebraic classification of rational CR structures on the topological 5-sphere with transversal holomorphic S^1-action in C^4. Here, algebraic classification of compact strongly pseudoconvex CR manifolds X means classification up to algebraic equivalence, i.e. roughly up to isomorphism...

Source: http://arxiv.org/abs/math/0303302v1

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Sep 19, 2013
09/13

by
Tan Jiang; Stephen S. -T. Yau

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Let $\scr A^*=\{l_1,l_2,\cdots,l_n\}$ be a line arrangement in $\Bbb{CP}^2$, i.e., a collection of distinct lines in $\Bbb{CP}^2$. Let $L(\scr A^*)$ be the set of all intersections of elements of $A^*$ partially ordered by $X\leq Y\Leftrightarrow Y\subseteq X$. Let $M(\scr A^*)$ be $\Bbb{CP}^2-\bigcup\scr A^*$ where $\bigcup\scr A^*= \bigcup\{l_i\colon\ 1\leq i\leq n\}$. The central problem of the theory of arrangement of lines in $\Bbb{CP}^2$ is the relationship between $M(\scr A^*)$ and...

Source: http://arxiv.org/abs/math/9307228v1

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Jun 29, 2018
06/18

by
Bingyi Chen; Dan Xie; Shing-Tung Yau; Stephen S. -T. Yau; Huaiqing Zuo

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We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N=2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.

Topics: High Energy Physics - Theory, Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1604.07843

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Jun 29, 2018
06/18

by
Yifan Wang; Dan Xie; Stephen S. -T. Yau; Shing-Tung Yau

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Detailed studies of four dimensional N=2 superconformal field theories (SCFT) defined by isolated complete intersection singularities are performed: we compute the Coulomb branch spectrum, Seiberg-Witten solutions and central charges. Most of our theories have exactly marginal deformations and we identify the weakly coupled gauge theory descriptions for many of them, which involve (affine) D and E shaped quiver gauge theories and theories formed from Argyres-Douglas matters. These...

Topic: High Energy Physics - Theory

Source: http://arxiv.org/abs/1606.06306