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arXiv 2024年

作者： Li, Lianxia Gruninger, Cole Lee, Jae H. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

In the class of immersed boundary (IB) methods, the choice of the regularized delta function plays a crucial role in transferring information between fluid and solid domains through interpolation and spreading operators. Most prior work using the IB method has used isotropic kernels that do not preserve the divergence-free condition of the velocity field, leading to loss of incompressibility of the solid when interpolating the Eulerian velocity to Lagrangian markers. To address this issue, in simulations involving large deformations of incompressible hyperelastic structures immersed in fluid, researchers often use a volumetric stabilization approach such as adding a volumetric energy term and using modified invariants in the constitutive model of the immersed structure. Composite B-spline (CBS) kernels offer an alternative approach by inherently maintaining the discrete divergence-free property. This work evaluates the performance of CBS kernels in terms of their volume conservation and accuracy, comparing them with several traditional isotropic kernel functions using a construction introduced by Peskin (referred to as IB kernels) and B-spline (BS) kernels. Benchmark tests include pressure-loaded and shear-dominated flows, such as an elastic band under differential pressure loads, a pressurized membrane, a compressed block, Cook’s membrane, a slanted channel flow, and a modified Turek-Hron problem. Additionally, we validate our methodology using a complex fluid-structure interaction model of bioprosthetic heart valve dynamics in a pulse duplicator. Results demonstrate that CBS kernels achieve superior volume conservation compared to conventional isotropic kernels, eliminating the need for additional volumetric stabilization techniques typically required to address instabilities arising from volume conservation errors. Further, CBS kernels achieve convergence on coarser fluid grids, while IB and BS kernels need finer grids for comparable accuracy. Unlike IB and BS ke

关键词： Delta functions

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Quality of internal representation shapes learning performance in feedback neural networks

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Physical Review Research 2021年第1期3卷 013176-013176页

作者： Lee Susman Francesca Mastrogiuseppe Naama Brenner Omri Barak Interdisciplinary Program in Applied Mathematics Technion Israel Institute of Technology Haifa 32000 Israel Network Biology Research Laboratories Technion Israel Institute of Technology Haifa 32000 Israel Gatsby Computational Neuroscience Unit University College London London W1T 4JG United Kingdom Department of Chemical Engineering Technion Israel Institute of Technology Haifa 32000 Israel Rappaport Faculty of Medicine Technion Israel Institute of Technology Haifa 32000 Israel

A fundamental feature of complex biological systems is the ability to form feedback interactions with their environment. A prominent model for studying such interactions is reservoir computing, where learning acts on low-dimensional bottlenecks. Despite the simplicity of this learning scheme, the factors contributing to or hindering the success of training in reservoir networks are in general not well understood. In this work, we study nonlinear feedback networks trained to generate a sinusoidal signal, and analyze how learning performance is shaped by the interplay between internal network dynamics and target properties. By performing exact mathematical analysis of linearized networks, we predict that learning performance is maximized when the target is characterized by an optimal, intermediate frequency which monotonically decreases with the strength of the internal reservoir connectivity. At the optimal frequency, the reservoir representation of the target signal is high-dimensional, desynchronized, and thus maximally robust to noise. We show that our predictions successfully capture the qualitative behavior of performance in nonlinear networks. Moreover, we find that the relationship between internal representations and performance can be further exploited in trained nonlinear networks to explain behaviors which do not have a linear counterpart. Our results indicate that a major determinant of learning success is the quality of the internal representation of the target, which in turn is shaped by an interplay between parameters controlling the internal network and those defining the task.

关键词： Learning theory Artificial neural networks Dynamical systems Neuronal network models

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Comparing demographics of signatories to public letters on diversity in the mathematical sciences

arXiv

引用

arXiv 2019年

作者： Topaz, Chad M. Cart, James Eaton, Carrie Diaz Shrout, Anelise Hanson Higdon, Jude A. İnce, Kenan Katz, Brian Lewis, Drew Libertini, Jessica Smith, Christian Michael Department of Mathematics and Statistics Williams College WilliamstownMA United States Institute for the Quantitative Study of Inclusion Diversity and Equity WilliamstownMA United States Office of Institutional Research Williams College WilliamstownMA United States Digital and Computational Studies Program Bates College LewistonME United States Department of Mathematics Westminster College Salt Lake CityUT United States Department of Mathematics and Statistics Smith College NorthamptonMA United States Department of Mathematics and Statistics University of South Alabama MobileAL United States Department of Applied Mathematics Virginia Military Institute LexingtonVA United States Department of Sociology University of Wisconsin-Madison MadisonWI United States

In its December 2019 edition, the Notices of the American Mathematical Society published an essay critical of the use of diversity statements in academic hiring. The publication of this essay prompted many responses, including three public letters circulated within the mathematical sciences community. Each letter was signed by hundreds of people and was published online, also by the American Mathematical Society. We report on a study of the signatories’ demographics, which we infer using a crowdsourcing approach. Letter A highlights diversity and social justice. The pool of signatories contains relatively more individuals inferred to be women and/or members of underrepresented ethnic groups. Moreover, this pool is diverse with respect to the levels of professional security and types of academic institutions represented. Letter B does not comment on diversity, but rather, asks for discussion and debate. This letter was signed by a strong majority of individuals inferred to be white men in professionally secure positions at highly research intensive universities. Letter C speaks out specifically against diversity statements, calling them "a mistake," and claiming that their usage during early stages of faculty hiring "diminishes mathematical achievement." Individuals who signed both Letters B and C, that is, signatories who both privilege debate and oppose diversity statements, are overwhelmingly inferred to be tenured white men at highly research intensive universities. Our empirical results are consistent with theories of power drawn from the social sciences. Copyright © 2019, The Authors. All rights reserved.

关键词： Population statistics

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clotFoam: An Open-Source Framework to Simulate Blood Clot Formation Under Arterial Flow

arXiv

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arXiv 2023年

作者： Montgomery, David Municchi, Federico Leiderman, Karin Department of Applied Mathematics and Statistics Colorado School of Mines 1500 Illinois St GoldenCO80401 United States Department of Mechanical Engineering Colorado School of Mines 1500 Illinois St GoldenCO80401 United States Department of Mathematics University of North Carolina at Chapel Hill 216 Lenoir Dr Chapel HillNC27599 United States Computational Medicine Program University of North Carolina at Chapel Hill 216 Lenoir Dr Chapel HillNC27599 United States

Blood clotting involves the coupled processes of platelet aggregation and coagulation. Simulating clotting under flow in complex geometries is challenging due to multiple temporal and spatial scales and high computational cost. clotFoam is an open-source software developed in OpenFOAM that employs a continuum model of platelet advection, diffusion, and aggregation in a dynamic fluid environment and a simplified coagulation model with proteins that advect, diffuse, and react within the fluid and with wall-bound species through reactive boundary conditions. Our framework provides the foundation on which one can build more complex models and perform reliable simulations in almost any computational domain. © 2023, CC BY.

关键词： Coagulation

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Local-order metric for condensed-phase environments

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Physical Review B 2018年第6期97卷 064105-064105页

作者： Fausto Martelli Hsin-Yu Ko Erdal C. Oğuz Roberto Car Department of Chemistry Princeton University Princeton New Jersey 08544 USA Institut für Theoretische Physik II Heinrich-Heine-Universität D-40225 Düsseldorf Germany Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We introduce a local order metric (LOM) that measures the degree of order in the neighborhood of an atomic or molecular site in a condensed medium. The LOM maximizes the overlap between the spatial distribution of sites belonging to that neighborhood and the corresponding distribution in a suitable reference system. The LOM takes a value tending to zero for completely disordered environments and tending to one for environments that perfectly match the reference. The site-averaged LOM and its standard deviation define two scalar order parameters, S and δS, that characterize with excellent resolution crystals, liquids, and amorphous materials. We show with molecular dynamics simulations that S, δS, and the LOM provide very insightful information in the study of structural transformations, such as those occurring when ice spontaneously nucleates from supercooled water or when a supercooled water sample becomes amorphous upon progressive cooling.

关键词： Glass transition Nucleation Order parameters Phase transitions

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Method for obtaining upper bounds on photonic band gaps

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Physical Review B 2009年第15期80卷 155126-155126页

作者： Mikael C. Rechtsman Salvatore Torquato Courant Institute of Mathematical Sciences 251 Mercer Street New York New York 10012 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08544 USA

We present a method to calculate upper bounds on the photonic band gaps of two-component photonic crystals. The method involves calculating both upper and lower bounds on the frequency bands for a given structure, and then maximizing over all possible two-component structures. We apply this method to a number of examples, including a one-dimensional photonic crystal (or “Bragg grating”) and two-dimensional photonic crystals (in both the TM and TE polarizations) with both four and sixfold rotational symmetries. We compare the bounds to band gaps of numerically optimized structures and find that the bounds are extremely tight. We prove that the bounds are “sharp” in the limit of low dielectric contrast ratio between the two components. This method and the bounds derived here have important implications in the search for optimal photonic band-gap structures.

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A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrödinger Equations with Multiply Components

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Communications in Theoretical Physics 2014年第6期61卷

作者： QIAN Xu SONG Song-He LI Wei-Bin 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?dinger 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 computations. Furthermore, 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 Schrödinger equation multiply components unstable wave

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Generating large disordered stealthy hyperuniform systems with ultra-high accuracy to determine their physical properties

arXiv

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arXiv 2023年

作者： Morse, Peter K. Kim, Jaeuk Steinhardt, Paul J. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute of Materials Princeton University PrincetonNJ08544 United States Princeton Center for Theoretical Science Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Hyperuniform many-particle systems are characterized by a structure factor S(k) that is precisely zero as |k| → 0;and stealthy hyperuniform systems have S(k) = 0 for the finite range 0 30 for systems sizes more than an order of magnitude larger. We further show that this ultra-high accuracy is required to draw conclusions about their corresponding characteristics, such as the nature of the associated energy landscape and the presence or absence of Anderson localization, which might be masked, even when deviations are relatively small. © 2023, CC BY.

关键词： Physical properties

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Dense packings of polyhedra: Platonic and Archimedean solids

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Physical Review E 2009年第4期80卷 041104-041104页

作者： S. Torquato Y. Jiao Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08540 USA Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA

Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. The preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of nonoverlapping, nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the adaptive shrinking cell (ASC) scheme. This optimization problem is solved here (using a variety of multiparticle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3, except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823…, 0.836…, 0.904…, and 0.947…, respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other nontiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to Platonic solids, Archimedean solids, superballs, and ellipsoids. Provided that what we term the “asphericity” (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This can be regarded to be

关键词： Optimization

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Connected coulomb columns: Analysis and numerics

arXiv

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arXiv 2020年

作者： Dondl, P. Novaga, M. Wojtowytsch, S. Wolff-Vorbeck, S. Abteilung für Angewandte Mathematik Albert-Ludwigs-Universität Freiburg Hermann-Herder-Str. 10 Freiburg i. Br.79104 Germany Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo 5 Pisa56127 Italy Program in Applied and Computational Mathematics Princeton University 205 Fine Hall - Washington Road PrincetonNJ08544 United States

We consider a version of Gamow’s liquid drop model with a short range attractive perimeter-penalizing potential and a long-range Coulomb interaction of a uniformly charged mass in R3. Here we constrain ourselves to minimizing among the class of shapes that are columnar, i.e., constant in one spatial direction. Using the standard perimeter in the energy would lead to non-existence for any prescribed cross-sectional area due to the infinite mass in the constant spatial direction. In order to heal this defect we use a connected perimeter instead. We prove existence of minimizers for this connected isoperimetric problem with long-range interaction and study the shapes of minimizers in the small and large cross section regimes. For an intermediate regime we use an Ohta-Kawasaki phase field model with connectedness constraint to study the shapes of minimizers *** Codes 5308, 28A75, 65N99 Copyright © 2020, The Authors. All rights reserved.

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