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A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions

Journal of Scientific Computing 1993年第1期8卷 1-40页

作者： Goldhirsch, I. Tan, M.-L. Zanetti, G. Department of Fluid Mechanics and Heat Transfer Faculty of Engineering Tel-Aviv University Ramat Aviv 69978 Tel-Aviv Israel Program in Applied and Computational Mathematics Princeton University Princeton 08544 New Jersey United States

Results of a numerical study of the dynamics of a collection of disks colliding inelastically in a periodic two-dimensional enclosure are presented. The properties of this system, which is perhaps the simplest model for rapidly flowing granular materials, are markedly different from those known for atomic or moleclar gases, in which collisions are of elastic nature. The most prominent feature characterizing granular systems, even in the idealized situation in which no external forcing exists and the initial condition is statistically homogeneous, is their inherent instability to inhomogeneous fluctuations. Granular "gases" are thus generically nonuniform, a fact that suggests extreme caution in pursuing direct analogies with molecular gases. We find that once an inhomogeneous state sets in, the velocity distribution functions differ from the classical Maxwell-Boltzmann distribution. Other characteristics of the system are different from their counterparts in molecular systems as well. For a given value of the coefficient of restitution, e, a granular system forms clusters of typical separation L₀≈l/(1-e²)^1/2, where l is the mean free path in the corresponding homogeneous system. Most of the fluctuating kinetic energy then resides in the relatively dilute regions that surround the clusters. Systems whose linear dimensions are less then L⁰ do not give rise to clusters;still they are inhomogeneous, the scale of the corresponding inhomogeneity being the longest wavelength allowed by the system's size. The present article is devoted to a detailed numerical study of the above-mentioned clustering phenomenon in two dimensions and in the absence of external forcing. A theoretical framework explaining this phenomenon is outlined. Some general implications as well as practical ramifications are discussed. © 1993 Plenum Publishing Corporation.

关键词： Granular flow dynamics molecular dynamics numerical simulation

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GEOMETRIC CHARACTERIZATION OF STATES IN 2-DIMENSIONAL QUANTUM-GRAVITY

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MODERN PHYSICS LETTERS A 1990年第12期5卷 965-977页

作者： AGISHTEIN, ME JACOBS, L MIGDAL, AA RICHARDSON, JL Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544 USA Center for Theoretical Physics 6-403 Massachusetts Institute of Technology Cambridge MA 02139 USA Work supported in part by the U. S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069. Department of Physics and Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544 USA Thinking Machines Corporation 245 First Street Cambridge MA 02142 USA

We report first results of a large-scale simulation of two-dimensional quantum gravity using the dynamical triangulation model for systems of up to sixteen thousand triangles. Our results for the internal geometry show an unexpectedly complicated behavior of the internal volume as function of the internal radius. A simple fractal characterization is inadequate to describe the geometry of the states in the system.

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Featurized bidirectional GAN: Adversarial defense via adversarially learned semantic inference

arXiv

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arXiv 2018年

作者： Bao, Ruying Liang, Sihang Wang, Qingcan Program in Applied and Computational Mathematics Princeton University Department of Physics Princeton University

Deep neural networks have been demonstrated to be vulnerable to adversarial attacks, where small perturbations intentionally added to the original inputs can fool the classifier. In this paper, we propose a defense method, Featurized Bidirectional Generative Adversarial Networks (FBGAN), to extract the semantic features of the input and filter the non-semantic perturbation. FBGAN is pre-trained on the clean dataset in an unsupervised manner, adversarially learning a bidirectional mapping between the high-dimensional data space and the low-dimensional semantic space;also mutual information is applied to disentangle the semantically meaningful features. After the bidirectional mapping, the adversarial data can be reconstructed to denoised data, which could be fed into any pre-trained classifier. We empirically show the quality of reconstruction images and the effectiveness of defense. Copyright © 2018, The Authors. All rights reserved.

关键词： Generative adversarial networks

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A spectral-infinite-element solution of Poisson’s equation: An application to self gravity

arXiv

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arXiv 2017年

作者： Gharti, Hom Nath Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ08544 United States Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

We solve Poisson’s equation by combining a spectral-element method with a mapped infinite-element method. We focus on problems in geostatics and geodynamics, where Earth’s gravitational field is determined by Poisson’s equation inside the Earth and Laplace’s equation in the rest of space. Spectral elements are used to capture the internal field, and infinite elements are used to represent the external field. To solve the weak form of Poisson/Laplace equation, we use Gauss-Legendre-Lobatto quadrature in spectral elements inside the domain of interest. Outside the domain, we use Gauss-Radau quadrature in the infinite direction, and Gauss-Legendre-Lobatto quadrature in the other directions. We illustrate the efficiency and accuracy of the method by comparing the gravitational fields of a homogeneous sphere and the Preliminary Reference Earth Model (PREM) with (semi-)analytical solutions. Copyright © 2017, The Authors. All rights reserved.

关键词： Earth (planet)

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The Weyl-Heisenberg ensemble: Statistical mechanics meets time-frequency analysis

The Weyl-Heisenberg ensemble: Statistical mechanics meets ti...

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International Conference on Sampling Theory and Applications

作者： Luis Daniel Abreu Joao Pereira Jose Luis Romero Salvatore Torquato Austrian Academy of Sciences Acoustics Research Institute Program in Applied and Computational Mathematics Princeton University Department of Chemistry Princeton Materials Institute and Program in Applied and Computational Mathematics Princeton University

ISBN: (纸本)9781538615669

We present our recent work on the Weyl-Heisenberg ensemble and its statistical properties [4]. The WH ensemble is a class of determinantal point processes associated with the Schrodinger representation of the Heisenberg group. As a special example, WH ensembles include a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels. We describe the hyperuniformity of WH ensembles, which characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. Our approach is based on methods from time-frequency analysis. We introduce the main results from [4] highlighting time-frequency techniques and connections to the theory of polyanalytic functions, and also present some small extensions.

关键词： Weyl-Heisenberg ensemble Hyperuniformity Higher Landau levels Disorder Fluctuations

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Differential gene expression in trabecular bone osteocytes is related to the local strain and strain gradient

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Scientific reports 2025年第1期15卷 18501页

作者： Meghana Machireddy Sarah Nano Lucas Debiase Jonathan Lewis Sara Cole Jun Li Glen L Niebur University of Notre Dame Bioengineering Graduate Program Notre Dame IN USA. Department of Aerospace and Mechanical Engineering Notre Dame IN USA. Notre Dame Integrated Imaging Facility Notre Dame IN USA. Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame IN USA. University of Notre Dame Bioengineering Graduate Program Notre Dame IN USA. gniebur@nd.edu. Department of Aerospace and Mechanical Engineering Notre Dame IN USA. gniebur@nd.edu.

Osteocytes regulate the response of osteoclasts and osteoblasts to mechanical loading through signaling molecules, the levels of which are controlled by post-translational modification or degradation and by differential gene transcription and translation. The magnitude and mode of bone tissue deformation that elicits a transcriptional response in individual osteocytes in situ has been difficult to quantify. We measured SOST, Wnt11, TNF, and FRZB gene expression in osteocytes within loaded and unloaded control porcine trabecular bone explants using RNAScope® and compared the local tissue level strain and strain gradient-which we used as an indicator of potential poroelastic fluid flow-in the tissue surrounding osteocytes with high vs. low gene expression. The measured expression of all four genes differed between loaded and unloaded explants, on average, with the mean SOST expression level decreasing by 45%. In the loaded explants, gene expression was altered from baseline in about 30% of the osteocytes, and they were surrounded by tissue with higher strain and strain gradient than the 20 to 25% of osteocytes that remained near baseline expression. Both deviatoric strain and hydrostatic strain gradient were sensitive and specific predictors of the mechanobiological response for individual genes as well as combinations. SOST expression was highly related to elevated strain gradient, providing evidence that osteocytes respond to fluid flow in the lacunar-canalicular system.

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GLOBAL SOLUTIONS TO THE SUPERCOOLED STEFAN PROBLEM WITH BLOW-UPS: REGULARITY AND UNIQUENESS

Probability and Mathematical Physics

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Probability and Mathematical Physics 2022年第1期3卷 171-213页

作者： Delarue, François Nadtochiy, Sergey Shkolnikov, Mykhaylo Laboratoire J.A.Dieudonné Université de Nice Sophia-Antipolis UMR CNRS-UNS No. 7351 Nice France Department of Applied Mathematics Illinois Institute of Technology Rettaliata Engineering Center Chicago IL United States ORFE Department Bendheim Center for Finance Program in Applied and Computational Mathematics Princeton University Princeton NJ United States

We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows us to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Hölder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal 1889 paper for the ordinary Stefan problem. In our second main theorem, we establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present. © 2022 Mathematical Sciences Publishers.

关键词： blow-ups free boundary problem heat equation interacting particle systems mean-field interaction physical solutions probabilistic reformulation self-excitation supercooled Stefan problem zero set

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Extreme-scale AMR 10

Extreme-scale AMR

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Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis

作者： Burstedde, Carsten Ghattas, Omar Gurnis, Michael Isaac, Tobin Stadler, Georg Warburton, Tim Wilcox, Lucas C. Institute for Computational Engineering and Sciences University of Texas Austin United States Jackson School of Geosciences University of Texas Austin United States Department of Mechanical Engineering University of Texas Austin United States Seismological Laboratory California Institute of Technology United States Department of Computational and Applied Mathematics Rice University United States

ISBN: (纸本)9781424475575

Many problems are characterized by dynamics occurring on a wide range of length and time scales. One approach to overcoming the tyranny of scales is adaptive mesh refinement/coarsening (AMR), which dynamically adapts the mesh to resolve features of interest. However, the benefits of AMR are difficult to achieve in practice, particularly on the petascale computers that are essential for difficult problems. Due to the complex dynamic data structures and frequent load balancing, scaling dynamic AMR to hundreds of thousands of cores has long been considered a challenge. Another difficulty is extending parallel AMR techniques to high-order-accurate, complex-geometry-respecting methods that are favored for many classes of problems. Here we present new parallel algorithms for parallel dynamic AMR on forest-of-octrees geometries with arbitrary-order continuous and discontinuous finite/spectral element discretizations. The implementations of these algorithms exhibit excellent weak and strong scaling to over 224,000 Cray XT5 cores for multiscale geophysics problems. © 2010 IEEE.

关键词： Mesh generation

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Time-frequency and time-scale canonical representations of doubly spread channels 12

Time-frequency and time-scale canonical representations of d...

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12th European Signal Processing Conference, EUSIPCO 2004

作者： Balan, Radu Vincent Poor, H. Rickard, Scott Verdú, Sergio Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Siemens Corporate Research PrincetonNJ United States Electronic and Electrical Engineering Department University College Dublin Ireland

ISBN: (纸本)9783200001657

A general technique for the generation of canonical channel models and demonstrate the application of the technique to time-frequency and time-scale integral kernel operators is developed. As an example, the derivation of Sayeed/Aazhang's time-frequency canonical channel characterization that forms the basis for the time-frequency RAKE receiver is shown. Then, a canonical time-scale channel model for wideband communication is developed. © 2004 EUSIPCO.

关键词： Time measurement

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Second-order Sigma-Delta (ΣΔ) quantization of finite frame expansions

Second-order Sigma-Delta (ΣΔ) quantization of finite frame...

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作者： Benedetto, John J. Powell, Alexander M. Yilmaz, Özgür Department of Mathematics University of Maryland College Park MD 20742 United States Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Department of Mathematics University of British Columbia Vancouver BC V6T 1Z2 Canada

The second-order Sigma-Delta (ΣΔ) scheme with linear quantization rule is analyzed for quantizing finite unit-norm tight frame expansions for Rd. Approximation error estimates are derived, and it is shown that for certain choices of frames the quantization error is of order 1/N2, where N is the frame size. However, in contrast to the setting of bandlimited functions there are many situations where the second-order scheme only gives approximation error of order 1/N. For example, this is the case when quantizing harmonic frames of odd length in even dimensions. An important component of the error analysis involves extending existing stability results to yield smaller invariant sets for the linear second-order ΣΔ scheme. © 2005 Elsevier Inc. All rights reserved.

关键词： Estimation

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