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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 MongeAmpere equation (DMAE). A link between the discrete optimal transport, discrete MongeAmpere equation and the power diagram in computational geometry is established.
Source: http://arxiv.org/abs/1302.5472v1
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Yong Lin; Linyuan Lu; S. T. Yau
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A graph is called Ricciflat if its Riccicurvatures vanish on all edges. Here we use the definition of Riccicruvature on graphs given in [LinLuYau, Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009]. In this paper, we classified all Ricciflat 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 halfdodecahedral graph. We also construct many Ricciflat graphs with...
Source: http://arxiv.org/abs/1301.0102v1
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Xue Luo; ShingTung Yau; Stephen S. T. Yau
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A timedependent HermiteGalerkin spectral method (THGSM) is investigated in this paper for the nonlinear convectiondiffusion equations in the unbounded domains. The timedependent 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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S. Hosono; A. Klemm; S. Theisen; S. T. Yau
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Mirror Symmetry, PicardFuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of CalabiYau 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/hepth/9308122v2
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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 (GKZhypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of CalabiYau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Gr\"obner basis for the socalled toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the...
Source: http://arxiv.org/abs/alggeom/9511001v1
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L. Erdos; H. T. Yau
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We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section.
Source: http://arxiv.org/abs/mathph/9901020v1
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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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S. Hosono; B. H. Lian; S. T. Yau
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In this note, we give a list of CalabiYau hypersurfaces in weighted projective 4spaces 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 3spaces. The first list significantly enlarges the list of K3fibrations of \KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the socalled...
Source: http://arxiv.org/abs/alggeom/9603020v2
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Stephen S. T. Yau; Yung Yu
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In the paper "Algebraic classification of rational CR structures on topological 5sphere with transversal holomorphic S^1action in C^4" (Yau and Yu, Math. Nachrichten 246247(2002), 207233), we give algebraic classification of rational CR structures on the topological 5sphere with transversal holomorphic S^1action 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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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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This paper proposes a problem of maximum.
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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'fandKapranovZelevinsky(GKZ) hypergeometric system. Some of these solutions arise as period integrals for CalabiYau 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/alggeom/9603014v2
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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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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), 223290]. In the HarveyLawson 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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R. P. Thomas; S. T. Yau
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We make a conjecture about mean curvature flow of Lagrangian submanifolds of CalabiYau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a JordanH\"oldertype 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 ShapereVafa's...
Source: http://arxiv.org/abs/math/0104197v3
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C. Landim; J. A. Ramirez; H. T. Yau
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It was proved \cite{EMYa, QY} that stochastic lattice gas dynamics converge to the NavierStokes equations in dimension $d=3$ in the incompressible limits. In particular, the viscosity is finite. We proved that, on the other hand, the viscosity for a two dimensional lattice gas model diverges faster than $\log \log t$. Our argument indicates that the correct divergence rate is $(\log t)^{1/2}$. This problem is closely related to the logarithmic correction of the time decay rate for the velocity...
Source: http://arxiv.org/abs/math/0505090v1
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Xue Luo; Stephen S. T. Yau
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The singular parabolic problem $u_t\triangle u=\lambda{\frac{1+\delta\nabla u^2}{(1u)^2}}$ 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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This note discusses alphanumerics and solutions related to the Smarandache function.
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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 Amodel calculations using the mirror principle (localization). Our results agree with Bmodel computations from solutions of PicardFuchs differential equations constructed form the local geometry near a Fano surface within a CalabiYau manifold. We interpret the GromovWittentype 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/hepth/9903053v4
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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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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 Candelasde la OssaGreenParkes hence completing the program...
Source: http://arxiv.org/abs/alggeom/9712011v1
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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 nonequivalent polygons.
Topics: Information Theory, Combinatorics, Computing Research Repository, Mathematics
Source: http://arxiv.org/abs/1402.0060
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C. Landim; J. Quastel; M. Salmhofer; H. T. Yau
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We prove that the diffusion coefficient for the asymmetric exclusion process diverges at least as fast as $t^{1/4}$ in dimension $d=1$ and $(\log t)^{1/2}$ in $d=2$. The method applies to nearest and nonnearest neighbor asymmetric exclusion processes.
Source: http://arxiv.org/abs/math/0201317v1
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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 CalabiYau 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/hepth/9407192v1
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In order to make students from the American competions to learn and understand better this notion, used in many east  european national mathematical competions, the author: calculates it for some small numbers, establishes a few proprieties of it, and involves it in relations with other famous functions in the number theory.
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Bingyi Chen; Dan Xie; ShingTung 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 miniversal deformation of these singularities, and therefore solve the Coulomb branch spectrum and SeibergWitten solution.
Topics: High Energy Physics  Theory, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1604.07843
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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/hepth/9910245v1
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Yifan Wang; Dan Xie; Stephen S. T. Yau; ShingTung 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, SeibergWitten 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 ArgyresDouglas matters. These...
Topic: High Energy Physics  Theory
Source: http://arxiv.org/abs/1606.06306
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I. Smith; R. P. Thomas; S. T. Yau
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We introduce a symplectic surgery in six dimensions which collapses Lagrangian threespheres and replaces them by symplectic twospheres. Under mirror symmetry it corresponds to an operation on complex 3folds studied by Clemens, Friedman and Tian. We describe several examples which show that there are either many more CalabiYau manifolds (e.g. rigid ones) than previously thought or there exist ``symplectic CalabiYaus''  nonKaehler symplectic 6folds with c_1=0. The analogous surgery in...
Source: http://arxiv.org/abs/math/0209319v2
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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 11 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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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 quasiconformal} map. Many surface maps in our physical world are quasiconformal. The angular distortion of a quasiconformal map can be represented by Beltrami differentials. According to quasiconformal Teichm\"uller theory, there is an 11 correspondence between the set of Beltrami differentials and the set of...
Source: http://arxiv.org/abs/1005.4648v2
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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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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 wavefunctions which are zeromodes of all s_d=2(d1) 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/hepth/9904182v2
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S. Guha; J. D. Rice; Y. T. Yau; C. M. Martin; M. Chandrasekhar; H. R. Chandrasekhar; R. Guentner; P. Scandiucci de Freitas; U. Scherf
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We present photoluminescence studies as a function of temperature from a series of conjugated polymers and a conjugated molecule with distinctly different backbone conformations. The organic materials investigated here are: planar methylated ladder type poly paraphenylene, semiplanar polyfluorene, and nonplanar para hexaphenyl. In the longerchain polymers the photoluminescence transition energies blue shift with increasing temperatures. The conjugated molecules, on the other hand, red shift...
Source: http://arxiv.org/abs/condmat/0206357v1
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We note that the most unsolved problems of the world on the same subject are related to the Smarandache Function in the Analytic Number Theory.
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We ask to the question: For what triplets Smarandache function verifies the Fibonacci relationship?