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A second moment closure model for MHD turbulence

ZAMP Journal of applied mathematics and Physics 1989年第5期40卷 740-757页

作者： Galperin, Boris Program in Applied and Computational Mathematics Princeton University Princeton 08544 NJ United States

A second moment turbulence closure model of the type used before for flows with density stratification, frame rotation and streamline curvature is augmented to describe MHD flows with small magnetic Reynolds number. It is shown that all three configurations of the constant unidirectional magnetic field (longitudinal, transverse and azimuthal) suppress the components of the Reynolds stress tensor as well as turbulence energy. Criterions are derived for the extinction of 3-D turbulence based on magnetic Richardson numbers. Monin-Obukhov type similarity theory is developed for MHD turbulent flows. The combined effect of longitudinal magnetic field and spanwise rotation is considered. A discussion is presented on the extra strains exerted by the electromagnetic force compared to the effects of buoyancy, frame rotation and streamline curvature. © 1989 Birkhäuser Verlag.

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Simulation of Impurity Diffusion in a Strained Nanowire Using Off-Lattice KMC

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Communications in computational Physics 2007年第1期2卷 164-176页

作者： Weidong Guo Tim P.Schulze Weinan E Department of Mathematics University of TennesseeKnoxvilleTN 37996-1300USA Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544-1000USA.

Kinetic Monte Carlo(KMC)is a stochastic model used to simulate crystal ***,most KMC models rely on a pre-defined lattice that neglects dislocations,lattice mismatch and strain *** this paper,we investigate the use of a 3D off-lattice KMC *** test this method by investigating impurity diffusion in a strained FCC *** faster than a molecular dynamics simulation,the most general implementation of off-lattice KMC is much slower than a lattice-based *** improved procedure is achieved for weakly strained systems by precomputing approximate saddle point locations based on unstrained lattice *** this way,one gives up some of the flexibility of the general method to restore some of the computational speed of lattice-based *** addition to providing an alternative approach to nano-materials simulation,this type of simulation will be useful for testing and calibrating methods that seek to parameterize the variation in the transition rates for lattice-based KMC using continuum modeling.

关键词： Off-lattice KMC nanowire strain rate impurity diffusion.

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A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrdinger Equations with Multiply Components

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Communications in Theoretical Physics 2014年第6期61卷 703-709页

作者：钱旭宋松和李伟斌 Department of Mathematics and Systems Science National University of Defense Technology Program in Applied and Computational Mathematics Princeton University State Key Laboratory of High Performance Computing National University of Defense Technology

Considering the coupled nonlinear Schr¨odinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time ***, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.

关键词： conservative algorithm coupled nonlinear SchrSdinger equation multiply components unstablewave

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Rank-width and Well-quasi-ordering of Skew-symmetric Matrices. (extended abstract)

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Electronic Notes in Discrete mathematics 2005年 22卷 281-285页

作者： Oum, Sang-il Program in Applied and Computational Mathematics Princeton University Princeton NJ 08540 United States

Robertson and Seymour prove that a set of graphs of bounded tree-width is well-quasi-ordered by the graph minor relation. By extending their methods to matroids, Geelen, Gerards, and Whittle prove that a set of matroids representable over a fixed finite field are well-quasi-ordered if it has bounded branch-width. More recently, it is shown that a set of graphs of bounded rank-width (or clique-width) is well-quasi-ordered by the graph vertex-minor relation. The proof of the last one uses isotropic systems defined by A. Bouchet. We obtain a common generalization of the above three theorems in terms of skew-symmetric matrices over a fixed finite field. © 2005 Elsevier B.V. All rights reserved.

关键词： branch-width isotropic system principal pivot rank-width skew-symmetric matrix tree-width well-quasi-ordering

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computational test of the renormalization group theory of turbulence

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Journal of Scientific Computing 1988年第2期3卷 139-147页

作者： Yakhot, Victor Orszag, Steven A. Panda, Raj Program in Applied and Computational Mathematics Princeton University Princeton 08544 New Jersey United States

The results of numerical simulations of random-force-driven Navier-Stokes turbulence designed to test predictions of the renormalization group theory of turbulence are presented. By specially choosing the random force, we generate fully developed turbulence with a relatively long inertial range. The results of these simulations provide direct numerical verification of the correspondence principle [V. Yakhot and S.A. Orszag, Phys. Rev. Lett.57, 1722 (1986)] and agree with the theoretical predictions based on the e{open}-expansion to about 2%-5%. © 1988 Plenum Publishing Corporation.

关键词： numerical simulation renormalization group Turbulence

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MORSE AND MELNIKOV FUNCTIONS FOR NLS PDES

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COMMUNICATIONS IN MATHEMATICAL PHYSICS 1994年第1期162卷 175-214页

作者： LI, Y MCLAUGHLIN, DW 1. Department of Mathematics Princeton University 08544 Princeton NJ USA 2. Department of Mathematics Program in Applied and Computational Mathematics Princeton University 08544 Princeton NJ USA

The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat liner operator L, is developed in sufficient detail for later use in studies of perturbations of the NLS equation. ''Counting lemmas'' for the non-selfadjoint operator L, are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite disc D in the complex eigenvalue plane. The radius of the disc D is controlled by the H-1 norm of the potential q. For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds - ''whiskered tori'' for the NLS pde. The Floquet discriminant DELTA(lambda;q) used to introduce a natural sequence of NLS constants of motion, [F(j)(q) = DELTA(lambda = lambda(j)c(q);q), where lambda(j)c denotes the j(th) critical point of the Floquet discriminant DELTA(lambda)]. A Taylor series expansion of the constants F(j)(q), with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of F(j)(q), which themselves are expressed in terms of quadratic products of eigenfunctions of L. The second variation permits identification, within the disc D, of important bifurcations m the spectral configurations of the operator L. The constant F(j)(q), as the height of the Floquet discriminant over the critical point lambda(j)c, admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, {F(j)(q)}\j\>N, which is associated with critical points outside the disc D. Within this disc, the interpretation is only valid locally, with the s

关键词： 34L05 35Q55 58F05 58F19

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Accurate Deep Potential model for the Al-Cu-Mg alloy in the full concentration space

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Chinese Physics B 2021年第5期30卷 12-19页

作者： Wanrun Jiang Yuzhi Zhang Linfeng Zhang Han Wang Songshan Lake Materials Laboratory Dongguan 523808China Institute of Physics Chinese Academy of SciencesBeijing 100190China Beijing Institute of Big Data Research Beijing 100871China Yuanpei College of Peking University Beijing 100871China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088China

Combining first-principles accuracy and empirical-potential efficiency for the description of the potential energy surface(PES)is the philosopher's stone for unraveling the nature of matter via atomistic *** has been particularly challenging for multi-component alloy systems due to the complex and non-linear nature of the associated *** this work,we develop an accurate PES model for the Al-Cu-Mg system by employing deep potential(DP),a neural network based representation of the PES,and DP generator(DP-GEN),a concurrent-learning scheme that generates a compact set of ab initio data for *** resulting DP model gives predictions consistent with first-principles calculations for various binary and ternary systems on their fundamental energetic and mechanical properties,including formation energy,equilibrium volume,equation of state,interstitial energy,vacancy and surface formation energy,as well as elastic *** benchmark shows that the DP model is ready and will be useful for atomistic modeling of the Al-Cu-Mg system within the full range of concentration.

关键词： potential energy surface deep learning Al-Cu-Mg alloy materials simulation

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mathematics for cryo-electron microscopy

Mathematics for cryo-electron microscopy

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2018 International Congress of Mathematicians, ICM 2018

作者： Singer, Amit Department of Mathematics Program in Applied and Computational Mathematic Princeton University United States

ISBN: (纸本)9789813272934

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM. © ICM *** rights reserved.

关键词： X ray crystallography

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A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics

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Science China mathematics 2020年第7期63卷 1235-1258页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.

关键词： two-layer neural network random feature model Gram matrix generalization error

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Machine learning from a continuous viewpoint,I

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Science China mathematics 2020年第11期63卷 2233-2266页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

We present a continuous formulation of machine learning,as a problem in the calculus of variations and differential-integral equations,in the spirit of classical numerical *** demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous *** also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous *** discuss how the issues of generalization error and implicit regularization can be studied under this framework.

关键词： machine learning continuous formulation flow-based model gradient flow particle approximation

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