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2014 52nd Annual Allerton Conference on Communication, Control, and Computing, Allerton 2014

作者： Lee, Eun Jee Abbe, Emmanuel Program in Applied and Computational Mathematics Princeton University United States

ISBN: (纸本)9781479980093

In secure multi-party computations (SMC), parties wish to compute a function on their private data without revealing more information about their data than what the function reveals. In this paper, we investigate two Shannontype questions on this problem. We first consider the traditional one-shot model for SMC which does not assume a probabilistic prior on the data. In this model, private communication and randomness are the key enablers to secure computing, and we investigate a notion of randomness cost and capacity. We then move to a probabilistic model for the data, and propose a Shannon model for discrete memoryless SMC. In this model, correlations among data are the key enablers for secure computing, and we investigate a notion of dependency which permits the secure computation of a function. While the models and questions are general, this paper focuses on summation functions and relies on polar code constructions. © 2014 IEEE.

关键词： Random processes

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Structural properties of hyperuniform Voronoi networks

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Physical Review E 2025年第3期111卷 034123-034123页

作者： Eli Newby Wenlong Shi Yang Jiao Reka Albert Salvatore Torquato Department of Physics Pennsylvania State University University Park Pennsylvania 16802 USA Materials Science and Engineering Arizona State University Tempe Arizona 85287 USA Department of Physics Arizona State University Tempe Arizona 85287 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by the complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, while lacking conventional long-range order, as in liquids and glasses. In this work, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform Poisson process, nonhyperuniform random sequential addition (RSA) saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks by determining multiple metrics that characterize the distribution, including their average areas and corresponding variances as well as higher-order cumulants (i.e., skewness γ1 and excess kurtosis γ2). We show that the HIP distribution is far from Gaussian, as evidenced by a high skewness (γ1=3.16) and large positive excess kurtosis (γ2=16.2). The Poisson (with γ1=1.07 and γ2=1.79) and non-stealthy hyperuniform (with γ1=0.257 and γ2=0.0217) distributions are Gaussian-like distributions, since they exhibit a small but positive skewness and excess kurtosis. The RSA (with γ1=0.450 and γ2=−0.0384) and the highest stealthy hyperuniform distributions (with γ1=0.0272 and γ2=−0.0626) are also non-Gaussian because of their low skewness and negative excess kurtosis, which is diametrically opposite of the non-Gaussian behavior of the HIP. The fact that the cell-area distributions of large, finite-sized RSA and stealthy hyperuniform networks (e.g., with N≈10,000 nodes) are narrower, have larger peaks, and smaller tails than a Gaussian distribution implies that in the thermodynamic limit th

关键词： Gaussian distribution

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SIMULATIONS OF 4-DIMENSIONAL SIMPLICIAL QUANTUM-GRAVITY AS DYNAMIC TRIANGULATION

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MODERN PHYSICS LETTERS A 1992年第12期7卷 1039-1061页

作者： AGISHTEIN, ME MIGDAL, AA Program in Applied and Computational Mathematics Fine Hall Princeton University Princeton NJ 08544–1000 USA

Four-Dimensional Simplicial Quantum Gravity is simulated using the dynamical triangulation approach. We studied simplicial manifolds of spherical topology and found the critical line for the cosmological constant as a function of the gravitational one, separating the phases of opened and closed Universe. When the bare cosmological constant approaches this line from above, the four-volume grows: we reached about 5 x 10(4) simplexes, which proved to be sufficient for the statistical limit of infinite volume. However, for the genuine continuum theory of gravity, the parameters of the lattice model should be further adjusted to reach the second order phase transition point, where the correlation length grows to infinity. We varied the gravitational constant, and we found the first order phase transition, similar to the one found in three-dimensional model, except in 4D the fluctuations are rather large at the transition point, so that this is close to the second order phase transition. The average curvature in cutoff units is large and positive in one phase (gravity), and small negative in another (antigravity). We studied the fractal geometry of both phases, using the heavy particle propagator to define the geodesic map, as well as with the old approach using the shortest lattice paths. The heavy propagator geodesic appeared to be much smoother, so that the scaling laws were found, corresponding to finite fractal dimensions: D+ approximately 2.3 in the gravity phase and D- approximately 4.6 in the antigravity phase. Similar, but somewhat lower numbers were obtained from the heat kernel singularity. The influence of the alpha-R2 terms in 2, 3 and 4 dimensions is discussed.

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Short- and long-time behavior of eddy-viscosity models

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Theoretical and computational Fluid Dynamics 1993年第5期4卷 197-207页

作者： Smith, Leslie M. Yakhot, Victor Applied and Computational Mathematics Program Princeton University Princeton 08544 NJ United States

The short-time behavior of the turbulent viscosity is inferred from the immediate response of the Reynolds stress deduced by Crow [1] for the problem of isotropic turbulence subjected to a mean strain at time t=0. The turbulent viscosity v is defined for t → 0 by the relation T_ij=-2 vS_ij, where T_ij is the Reynolds stress and S_ij is the mean rate of strain. It follows that the viscosity is v=O(t) for t → 0. Matching the short- and long-time behaviors, we propose an analytic expression for the effective viscosity valid for all time. Using the proposed viscosity, the K - E model for homogeneously sheared turbulence is reformulate to be valid in both the short- and long-time limits. Previously, the K - E model has been used with the long-time form of the effective viscosity for all time. Comparison of theoretical predictions with the results of physical and numerical experiments is presented. Implications of the short-time response for large-eddy simulations and spectral-space closure theories are discussed. © 1993 Springer-Verlag.

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CONVERGENCE OF THE SPECTRAL METHOD FOR STOCHASTIC GINZBURG-LANDAU EQUATION DRIVEN BY SPACE-TIME WHITE NOISE

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Communications in Mathematical Sciences 2003年第2期1卷 361-375页

作者： Liu, Di Program in Applied and Computational Mathematics Princeton University Princeton 08544 NJ United States

In this paper, a spectral method is formulated as a numerical solution for the stochastic Ginzburg-Landau equation driven by space-time white noise. The rates of pathwise convergence and convergence in expectation in Sobolev spaces are given based on the convergence rates of the spectral approximation for the stochastic convolution. The analysis can be generalized to other spectral methods for stochastic PDEs driven by additive noises, provided the regularity condition for the noises © 2003 International Press

关键词： Ginzburg-landau equation Rate of convergence Space-time white noise Spectral method Stochastic partial differential equations

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A second moment closure model for MHD turbulence

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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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Learning algorithms for mean field optimal control

arXiv

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arXiv 2025年

作者： Soner, H. Mete Teichmann, Josef Yan, Qinxin Department of Operations Research and Financial Engineering Princeton University PrincetonNJ08540 United States Department of Mathematics ETH Zürich Switzerland Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

We analyze an algorithm to numerically solve the mean-field optimal control problems by approximating the optimal feedback controls using neural networks with problem specific architectures. We approximate the model by an N-particle system and leverage the exchangeability of the particles to obtain substantial computational efficiency. In addition to several numerical examples, a convergence analysis is provided. We also developed a universal approximation theorem on Wasserstein *** Codes 35Q89, 35D40, 49L25, 60G99 © 2025, CC BY-NC-ND.

关键词： Optimal control systems

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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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