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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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Sep 22, 2013
09/13

by
Xianfeng Gu; Feng Luo; Jian Sun; S. -T. Yau

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In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computational geometry is established.

Source: http://arxiv.org/abs/1302.5472v1

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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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Jul 19, 2013
07/13

by
L. M. Lui; T. W. Wong; W. Zeng; X. F. Gu; P. M. Thompson; T. F. Chan; S. T. Yau

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In shape analysis, finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings is of great importance. Such process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making exhaustive search for the best mapping challenging. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs), which are...

Source: http://arxiv.org/abs/1005.3292v1

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Sep 20, 2013
09/13

by
S. Hosono; A. Klemm; S. Theisen; S. -T. Yau

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Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Source: http://arxiv.org/abs/hep-th/9308122v2

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Sep 20, 2013
09/13

by
S. Hosono; B. H. Lian; S. -T. Yau

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In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of \KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called...

Source: http://arxiv.org/abs/alg-geom/9603020v2

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Sep 21, 2013
09/13

by
Yong Lin; Linyuan Lu; S. -T. Yau

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A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges. Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau, Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009]. In this paper, we classified all Ricci-flat connected graphs with girth at least five: they are the infinite path, cycle $C_n$ ($n\geq 6$), the dodecahedral graph, the Petersen graph, and the half-dodecahedral graph. We also construct many Ricci-flat graphs with...

Source: http://arxiv.org/abs/1301.0102v1

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Sep 20, 2013
09/13

by
S. Hosono; B. H. Lian; S. -T. Yau

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Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel'fand-Kapranov-Zelevinsky(GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exists certain special boundary points, which we called maximal degeneracy points, at which all but one solutions become singular.

Source: http://arxiv.org/abs/alg-geom/9603014v2

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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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Sep 21, 2013
09/13

by
R. P. Thomas; S. -T. Yau

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We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-H\"older-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's...

Source: http://arxiv.org/abs/math/0104197v3

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Jul 20, 2013
07/13

by
B. Lian; K. Liu; S. T. Yau

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We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles -- including any direct sum of line bundles -- on $\P^n$. This includes proving the formula of Candelas-de la Ossa-Green-Parkes hence completing the program...

Source: http://arxiv.org/abs/alg-geom/9712011v1

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Sep 19, 2013
09/13

by
B. Lian; K. Liu; S. T. Yau

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We generalize the theorems in {\it Mirror Principle I} and {\it II} to the case of general projective manifolds without the convexity assumption. We also apply the results to balloon manifolds, and generalize to higher genus.

Source: http://arxiv.org/abs/math/9912038v1

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Jul 20, 2013
07/13

by
A. Klemm; B. H. Lian; S. S. Roan; S. -T. Yau

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We give close formulas for the counting functions of rational curves on complete intersection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the instanton corrected Yukawa coupling.

Source: http://arxiv.org/abs/hep-th/9407192v1

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Sep 22, 2013
09/13

by
Yi Hu; S. -T. Yau

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In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.

Source: http://arxiv.org/abs/math/0111089v5

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Sep 23, 2013
09/13

by
S. Hosono; B. H. Lian; S. -T. Yau

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We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Gr\"obner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the...

Source: http://arxiv.org/abs/alg-geom/9511001v1

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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

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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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Sep 21, 2013
09/13

by
Bong H. Lian; Kefeng Liu; S. T. Yau

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We generalize our theorems in "Mirror Principle I" to a class of balloon manifolds. Many of the results are proved for convex projective manifolds. In a subsequent paper, Mirror Principle III, we will extend the results to projective manifolds without the convexity assumption.

Source: http://arxiv.org/abs/math/9905006v1

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49

Sep 19, 2013
09/13

by
I. Smith; R. P. Thomas; S. -T. Yau

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We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman and Tian. We describe several examples which show that there are either many more Calabi-Yau manifolds (e.g. rigid ones) than previously thought or there exist ``symplectic Calabi-Yaus'' -- non-Kaehler symplectic 6-folds with c_1=0. The analogous surgery in...

Source: http://arxiv.org/abs/math/0209319v2

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58

Jul 20, 2013
07/13

by
Edward Witten; S. -T. Yau

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Let $M$ be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary $N$ of positive scalar curvature. We show that under these conditions, $H_n(M;Z)=0$ and in particular $N$ must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence.

Source: http://arxiv.org/abs/hep-th/9910245v1

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3.0

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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47

Sep 18, 2013
09/13

by
T. -M. Chiang; A. Klemm; S. -T. Yau; E. Zaslow

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We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of...

Source: http://arxiv.org/abs/hep-th/9903053v4

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46

Sep 23, 2013
09/13

by
J. Froehlich; G. M. Graf; D. Hasler; J. Hoppe; S. -T. Yau

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We derive the power law decay, and asymptotic form, of SU(2) x Spin(d) invariant wave-functions which are zero-modes of all s_d=2(d-1) supercharges of reduced (d+1)-dimensional supersymmetric SU(2) Yang Mills theory, resp. of the SU(2)-matrix model related to supermembranes in d+2 dimensions.

Source: http://arxiv.org/abs/hep-th/9904182v2

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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

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155

Jul 19, 2013
07/13

by
W. Zeng; L. M. Lui; F. Luo; J. S. Liu T. F. Chan; S. T. Yau; X. F. Gu

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Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a {\it quasi-conformal} map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichm\"uller theory, there is an 1-1 correspondence between the set of Beltrami differentials and the set of...

Source: http://arxiv.org/abs/1005.4648v2