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NUMERICAL STUDY OF A 2-POINT CORRELATION-FUNCTION AND LIOUVILLE FIELD PROPERTIES IN 2-DIMENSIONAL QUANTUM-GRAVITY

MODERN PHYSICS LETTERS A 1991年第12期6卷 1115-1131页

作者： AGISHTEIN, ME BENAV, R MIGDAL, AA SOLOMON, S Program in Applied and Computational Mathematics Fine Hall Princeton University Princeton NJ 08544–1000 USA Nuclear Physics Department Weizmann Institute of Science Rehovot 76100 Israel Physics Department and Program in Applied and Computational Mathematics Fine Hall Princeton University Princeton NJ 08544–1000 USA Physics Department Hebrew University Jerusalem Israel

Two-point Green's function is measured on the manifolds of a 2-dimensional quantum gravity. The recursive sampling technique is used to generate the triangulations, lattice sizes being up to hundred thousand triangles. The grid Laplacian was inverted by means of the algebraic multi-grid solver. The free field model of the Quantum Gravity assumes the Gaussian behavior of Liouville field and curvature. We measured histograms as well as six momenta of these fields. Our results support the Gaussian assumption.

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Mean field limit of a dynamical model of polymer systems

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Science China mathematics 2013年第12期56卷 2591-2598页

作者： E Weinan SHEN Hao School of Mathematical Sciences and BICMR Peking University Department of Mathematics and PACM Princeton UniversityPrinceton NJ 08544 USA Program in Applied and Computational Mathematics Princeton UniversityPrinceton NJ 08544 USA

This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDEs）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.

关键词： polymers mean field limit stochastic PDEs McKean-Vlasov equation

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Hybrid spectral-element-low-order methods for incompressible flows

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Journal of Scientific Computing 1991年第2期6卷 79-100页

作者： Henderson, Ron Karniadakis, George Em. Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics Princeton University Princeton 08544 New Jersey United States

In this article we present a new formulation for coupling spectral element discretizations to finite difference and finite element discretizations addressing flow problems in very complicated geometries. A general iterative relaxation procedure (Zanolli patching) is employed that enforces C¹ continuity along the patching interface between the two differently discretized subdomains. In fluid flow simulations of transitional and turbulent flows the high-order discretization (spectral element) is used in the outer part of the domain where the Reynolds number is effectively very high. Near "rough" wall boundaries (where the flow is effectively very viscous) the use of low-order discretizations provides sufficient accuracy and allows for efficient treatment of the complex geometry. An analysis of the patching procedure is presented for elliptic problems, and extensions to incompressible Navier-Stokes equations are implemented using an efficient high-order splitting scheme. Several examples are given for elliptic and flow model problems and performance is measured on both serial and parallel processors. © 1991 Plenum Publishing Corporation.

关键词： complex geometries finite differences finite elements Incompressible flows spectral elements

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Multidimensional Hyperbolic Problems and Computations 1

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丛书名： The IMA Volumes in mathematics and its Applications

1000年

作者： James Glimm Andrew J. Majda

ISBN: (数字)9781461391210

ISBN: (纸本)9781461391234

This IMA Volume in mathematics and its Applications MULTIDIMENSIONAL HYPERBOLIC PROBLEMS AND COMPUTATIONS is based on the proceedings of a workshop which was an integral part ofthe 1988-89 IMA program on NONLINEAR WAVES. We are grateful to the Scientific Commit tee: James Glimm, Daniel Joseph, Barbara Keyfitz, Andrew Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for planning and implementing an exciting and stimulating year-long program. We especially thank the Work shop Organizers, Andrew Majda and James Glimm, for bringing together many of the major figures in a variety of research fields connected with multidimensional hyperbolic problems. A vner Friedman Willard Miller PREFACE A primary goal of the IMA workshop on Multidimensional Hyperbolic Problems and Computations from April 3-14, 1989 was to emphasize the interdisciplinary nature of contemporary research in this field involving the combination of ideas from the theory of nonlinear partial differential equations, asymptotic methods, numerical computation, and experiments. The twenty-six papers in this volume span a wide cross-section of this research including some papers on the kinetic theory of gases and vortex sheets for incompressible flow in addition to many papers on systems of hyperbolic conservation laws. This volume includes several papers on asymptotic methods such as nonlinear geometric optics, a number of articles applying numerical algorithms such as higher order Godunov methods and front tracking to physical problems along with comparison to experimental data, and also several interesting papers on the rigorous mathematical theory of shock waves.

关键词： Analysis

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Anomalous suppression of large-scale density fluctuations in classical and quantum spin liquids

arXiv

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arXiv 2025年

作者： Chen, Duyu Samajdar, Rhine Jiao, Yang Torquato, Salvatore Materials Research Laboratory University of California Santa BarbaraCA93106 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Center for Theoretical Science Princeton University PrincetonNJ08544 United States Materials Science and Engineering Arizona State University TempeAZ85287 United States Department of Physics Arizona State University TempeAZ85287 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Materials Institute Princeton University PrincetonNJ08540 United States Program in Applied and Computational Mathematics PrincetonNJ08544 United States

Classical spin liquids (CSLs) are intriguing states of matter that do not exhibit long-range magnetic order and are characterized by an extensive ground-state degeneracy. Adding quantum fluctuations, which induce dynamics between these different classical ground states, can give rise to quantum spin liquids (QSLs). QSLs are highly entangled quantum phases of matter characterized by fascinating emergent properties, such as fractionalized excitations and topological order. One such exotic quantum liquid is the Z2 QSL, which can be regarded as a resonating valence bond (RVB) state formed from superpositions of dimer coverings of an underlying lattice. In this work, we unveil a hidden large-scale structural property of archetypal CSLs and QSLs known as hyperuniformity, i.e., normalized infinite-wavelength density fluctuations are completely suppressed in these systems. In particular, we first demonstrate that classical ensembles of close-packed dimers and their corresponding quantum RVB states are perfectly hyperuniform in general. Subsequently, we focus on a ruby-lattice spin liquid that was recently realized in a Rydberg-atom quantum simulator, and show that the QSL remains effectively hyperuniform even in the presence of a finite density of spinon and vison excitations, as long as the dimer constraint is still largely preserved. Moreover, we demonstrate that metrics based on the framework of hyperuniformity can be used to distinguish the QSL from other proximate quantum phases. These metrics can help identify potential QSL candidates, which can then be further analyzed using more advanced, computationally-intensive quantum numerics to confirm their status as true QSLs. © 2025, CC BY-NC-ND.

关键词： Quantum entanglement

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ISING-MODEL SIMULATIONS ON THE MANIFOLDS OF 2-DIMENSIONAL QUANTUM-GRAVITY

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MODERN PHYSICS LETTERS A 1991年第17期6卷 1615-1628页

作者： AGISHTEIN, ME BAILLIE, CF Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544-1000 USA Physics Department University of Colorado Boulder CO 80309 USA

The Ising model is stimulated on the manifolds of 2-dimensional quantum gravity, which are represented by fixed random triangulations (so-called quenched Ising model). Unlike the case of the Ising model on a dynamical random triangulation, there is no analytical prediction for the quenched case, since these manifolds do not have internal Hausdorff dimension and the problem cannot be formulated in matrix model language. The recursive sampling technique is used to generate the triangulations, lattice sizes being up to ten thousand triangles. The Metropolis algorithm was used for the spin update in order to obtain the initial estimation of the Curie point. After that we used the Wolff cluster algorithm in the critical region. We observed a second order phase transition, similar to that for the Ising model on a regular 2-dimensional lattice, and measured the critical exponents.

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GAUGE METHOD FOR VISCOUS INCOMPRESSIBLE FLOWS

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Communications in Mathematical Sciences 2003年第2期1卷 317-332页

作者： Weinan, E. Liu, Jian-Guo Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton 08544 NJ United States Institute for Physical Science and Technology and Department of Mathematics University of Maryland College Park 20742 MD United States

We present a new formulation of the incompressible Navier-Stokes equation in terms of an auxiliary field that differs from the velocity by a gauge transformation. The gauge freedom allows us to assign simple and specific boundary conditions for both the auxiliary field and the gauge field, thus eliminating the issue of pressure boundary condition in the usual primitive variable formulation. The resulting dynamic and kinematic equations can then be solved by standard methods for heat and Poisson equations. A normal mode analysis suggests that in contrast to the classical projection method, the gauge method does not suffer from the problem of numerical boundary layers. Thus the subtleties in the spatial discretization for the projection method are removed. Consequently, the projection step in the gauge method can be accomplished by standard Poisson solves. We demonstrate the efficiency and accuracy of the gauge method by several numerical examples, including the flow past cylinder © 2003 International Press

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Heterogeneous Multiscale Methods: A Review

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Communications in computational Physics 2007年第3期2卷 367-450页

作者： Weinan E Bjorn Engquist Xiantao Li Weiqing Ren Eric Vanden-Eijnden Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Department of Mathematics The University of Texas at AustinAustinTX 78712USA Department of Mathematics Pennsylvania State UniversityUniversity ParkPA 16802USA Department of Mathematics Courant Institute of Mathematical SciencesNew York UniversityNew YorkNY 10012USA.

This paper gives a systematic introduction to HMM,the heterogeneous multiscale methods,including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular *** is illustrated by examples from several application areas,including complex fluids,micro-fluidics,solids,interface problems,stochastic problems,and statistically self-similar *** is given to the technical tools,such as the various constrained molecular dynamics,that have been developed,in order to apply HMM to these *** of mathematical results on the error analysis of HMM are *** review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

关键词： Multi-scale modeling heterogeneous multi-scale method multi-physics models constrained micro-scale solver data estimation.

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Recovery of frequency-sparse signals from compressive measurements

Recovery of frequency-sparse signals from compressive measur...

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48th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2010

作者： Duarte, Marco F. Baraniuk, Richard G. Program in Applied and Computational Mathematics Princeton University United States Department of Electrical and Computer Engineering Rice University United States

ISBN: (纸本)9781424482146

Compressive sensing (CS) is a new approach to simultaneous sensing and compression for sparse and compressible signals. While the discrete Fourier transform has been widely used for CS of frequency-sparse signals, it provides optimal sparse representations only for signals with components at integral frequencies. There exist redundant frames that provide compressible representations for frequency-sparse signals, but such frames are highly coherent and severely affect the performance of standard CS recovery. In this paper, we show that by modifying standard CS recovery algorithms to prevent coherent frame elements from being present in the signal estimate, it is possible to bypass the shortcomings introduced by the coherent frame. The resulting algorithm comes with theoretical guarantees and is shown to perform significantly better for frequency-sparse signal recovery than its standard counterparts. The algorithm can also be extended to similar settings that use coherent frames. ©2010 IEEE.

关键词： Compressed sensing

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Effective Maxwell Equations from Time-dependent Density Functional Theory

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Acta Mathematica Sinica,English Series 2011年第2期27卷 339-368页

作者： Weinan E Jianfeng LU Xu YANG Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Beijing International Center for Mathematical Research and School of Mathematical Sciences Peking University Beijing 100871 P. R. China Department of Mathematics Courant Institute of Mathematical Sciences New York University New York NY 10012 USA

The behavior of interacting electrons in a perfect crystal under macroscopic external electric and magnetic fields is studied. Effective Maxwell equations for the macroscopic electric and magnetic fields are derived starting from time-dependent density functional theory. Effective permittivity and permeability coefficients are obtained.

关键词： Time-dependent density functional theory Maxwell equations effective permittivity electromagnetic fields

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