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arXiv 2020年

作者： Susman, Lee Mastrogiuseppe, Francesca Brenner, Naama Barak, Omri Interdisciplinary Program in Applied Mathematics Technion Israel Institute of Technology Haifa32000 Israel Network Biology Research Laboratories Technion Israel Institute of Technology Haifa32000 Israel Gatsby Computational Neuroscience Unit University College London London United Kingdom Dept. of Chemical Engineering Technion Israel Institute of Technology Haifa32000 Israel Rappaport Faculty of Medicine Technion Israel Institute of Technology Haifa32000 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 non-linear 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, de-synchronized, and thus maximally robust to noise. We show that our predictions successfully capture the qualitative behaviour of performance in non-linear networks. Moreover, we find that the relationship between internal representations and performance can be further exploited in trained non-linear networks to explain behaviours 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. © 2020, CC BY.

关键词： Quality control

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Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem

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Physical Review E 2014年第2期89卷 022721-022721页

作者： Yang Jiao Timothy Lau Haralampos Hatzikirou Michael Meyer-Hermann Joseph C. Corbo Salvatore Torquato Princeton Institute of the Science and Technology of Materials Princeton University Princeton New Jersey 08540 USA Materials Science and Engineering Arizona State University Tempe Arizona 85287 USA Department of Pathology and Immunology Washington University School of Medicine St. Louis Missouri 63110 USA Department of Systems Immunology Helmholtz Centre for Infection Research Braunschweig Germany Center for Advancing Electronics Dresden TU Dresden 01062 Dresden Germany Department of Chemistry Department of Physics Program in Computational and Applied Mathematics Princeton University Princeton New Jersey 08544 USA

Optimal spatial sampling of light rigorously requires that identical photoreceptors be arranged in perfectly regular arrays in two dimensions. Examples of such perfect arrays in nature include the compound eyes of insects and the nearly crystalline photoreceptor patterns of some fish and reptiles. Birds are highly visual animals with five different cone photoreceptor subtypes, yet their photoreceptor patterns are not perfectly regular. By analyzing the chicken cone photoreceptor system consisting of five different cell types using a variety of sensitive microstructural descriptors, we find that the disordered photoreceptor patterns are “hyperuniform” (exhibiting vanishing infinite-wavelength density fluctuations), a property that had heretofore been identified in a unique subset of physical systems, but had never been observed in any living organism. Remarkably, the patterns of both the total population and the individual cell types are simultaneously hyperuniform. We term such patterns “multihyperuniform” because multiple distinct subsets of the overall point pattern are themselves hyperuniform. We have devised a unique multiscale cell packing model in two dimensions that suggests that photoreceptor types interact with both short- and long-ranged repulsive forces and that the resultant competition between the types gives rise to the aforementioned singular spatial features characterizing the system, including multihyperuniformity. These findings suggest that a disordered hyperuniform pattern may represent the most uniform sampling arrangement attainable in the avian system, given intrinsic packing constraints within the photoreceptor epithelium. In addition, they show how fundamental physical constraints can change the course of a biological optimization process. Our results suggest that multihyperuniform disordered structures have implications for the design of materials with novel physical properties and therefore may represent a fruitful area for future research

关键词： PHOTORECEPTORS ARRAY processing BIRDS MULTISCALE modeling SAMPLING (Signal processing) EPITHELIUM

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computational design of anisotropic stealthy hyperuniform composites with engineered directional scattering properties

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Physical Review E 2023年第4期108卷 045306-045306页

作者： Wenlong Shi David Keeney Duyu Chen Yang Jiao Salvatore Torquato Materials Science and Engineering Arizona State University Tempe Arizona 85287 USA Materials Research Laboratory University of California Santa Barbara California 93106 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 materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here we generalize the Fourier-space-based numerical construction procedure for designing and generating digital realizations of isotropic disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. 142, 152 (2018)] to anisotropic microstructures with targeted spectral densities. Our generalized construction procedure explicitly incorporates the vector-dependent spectral density function χ̃V(k) of arbitrary form that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of anisotropic stealthy hyperuniform microstructures with χ̃V(k)=0 for k∈Ω, i.e., complete suppression of scattering in an “exclusion” region Ω around the origin in Fourier space. We show how different exclusion-region shapes with various discrete symmetries, including circular-disk, elliptical-disk, square, rectangular, butterfly-shaped, and lemniscate-shaped regions of varying size, affect the resulting statistically anisotropic microstructures as a function of the phase volume fraction. The latter two cases of Ω lead to directionally hyperuniform composites, which are stealthy hyperuniform only along certain directions and are nonhyperuniform along others. We find that while the circular-disk exclusion regions give rise to isotropic hyperuniform composite microstructures, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotropic exclusion regions

关键词： Amorphous materials Complex materials Simulated annealing

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Erratum to: An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations

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Journal of Mathematical Fluid Mechanics 2014年第3期16卷 481-481页

作者： Dongho Chae Peter Constantin Jiahong Wu Department of Mathematics College of Natural Science Chung-Ang University Seoul South Korea Program in Applied and Computational Mathematics Princeton University Princeton USA Department of Mathematics Oklahoma State University Stillwater USA Department of Mathematics Chung-Ang University Seoul Republic of Korea

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Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

arXiv

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arXiv 2018年

作者： Oguz, Erdal C. Socolar, Joshua E.S. Steinhardt, Paul J. Torquato, Salvatore School of Mechanical Engineering Sackler Center for Computational Molecular and Materials Science Tel Aviv University Tel Aviv Israel Physics Department Duke University DurhamNC27708 United States Department of Physics Princeton Center for Theoretical Science Princeton University PrincetonNJ08544 United States Center for Cosmology and Particle Physics Department of Physics New York University New YorkNY10003 United States Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

We consider the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long wavelength fluctuations in a broad class of one-dimensional substitution tilings. We present a simple argument that predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and confirm with numerical computations that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra, and limit-periodic tilings. Tilings with quasiperiodic or singular continuous spectra can be constructed with α arbitrarily close to any given value between −1 and 3. Limit-periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law. Copyright © 2018, The Authors. All rights reserved.

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Quantifying Gerrymandering in North Carolina

arXiv

引用

arXiv 2018年

作者： Herschlag, Gregory Kang, Han Sung Luo, Justin Graves, Christy Vaughn Bangia, Sachet Ravier, Robert Mattingly, Jonathan C. Department of Mathematics Duke University DurhamNC27708 United States Department of Biomedical Engineering Duke University DurhamNC27708 United States Department of Computer Science Duke University DurhamNC27708 United States Department of Electrical and Computer Engineering Duke University DurhamNC27708 United States Department of Economics Duke University DurhamNC27708 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Statistical Science Duke University DurhamNC27708 United States

Using an ensemble of redistricting plans, we evaluate whether a given political districting faithfully represents the geo-political landscape. Redistricting plans are sampled by a Monte Carlo algorithm from a probability distribution that adheres to realistic and non-partisan criteria. Using the sampled redistricting plans and historical voting data, we produce an ensemble of elections that reveal geo-political structure within the state. We showcase our methods on the two most recent districtings of NC for the U.S. House of Representatives, as well as a plan drawn by a bipartisan redistricting panel. We find the two state enacted plans are highly atypical outliers whereas the bipartisan plan accurately represents the ensemble both in partisan outcome and in the fine scale structure of district-level results. Copyright © 2018, The Authors. All rights reserved.

关键词： Probability distributions

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An Immersed Interface Method for Incompressible Flows and Geometries with Sharp Features

arXiv

引用

arXiv 2024年

作者： Facci, Michael J. Kolahdouz, Ebrahim M. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Flatiron Institute Simons Foundation New YorkNY 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

The immersed interface method (IIM) for models of fluid flow and fluid-structure interaction imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Most prior work using the IIM for fluid dynamic applications has focused on smooth interfaces, but boundaries with sharp features such as corners and edges can appear in practical analyses, particularly on engineered structures. The present study builds on our work to integrate finite element-type representations of interface geometries with the IIM. Initial realizations of this approach used a continuous Galerkin (CG) finite element discretization for the boundary, but as we show herein, these approaches generate large errors near sharp geometrical features. To overcome this difficulty, this study introduces an IIM approach using discontinuous Galerkin (DG) representation of the jump conditions. Numerical examples explore the impacts of different interface representations on accuracy for both smooth and sharp boundaries, particularly flows interacting with fixed interface configurations. We demonstrate that using a DG approach provides accuracy that is comparable to the CG method for smooth cases. Further, we identify a time step size restriction for the CG representation that is directly related to the sharpness of the geometry. In contrast, time step size restrictions imposed by DG representations are demonstrated to be insensitive to the presence of sharp *** Codes 76-10 (Primary) 76D05 (Secondary) © 2024, CC BY-NC-ND.

关键词： Incompressible flow

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Fast principal component analysis for cryo-electron microscopy images

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Biological imaging 2023年 3卷 e2-e2页

作者： Nicholas F Marshall Oscar Mickelin Yunpeng Shi Amit Singer Department of Mathematics Oregon State University Corvallis Oregon 97331 USA. Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA. Department of Mathematics Princeton University Princeton New Jersey 08544 USA.

Principal component analysis (PCA) plays an important role in the analysis of cryo-electron microscopy (cryo-EM) images for various tasks such as classification, denoising, compression, and ab initio modeling. We introduce a fast method for estimating a compressed representation of the 2-D covariance matrix of noisy cryo-EM projection images affected by radial point spread functions that enables fast PCA computation. Our method is based on a new algorithm for expanding images in the Fourier-Bessel basis (the harmonics on the disk), which provides a convenient way to handle the effect of the contrast transfer functions. For images of size × , our method has time complexity ( + ) and space complexity ( + ). In contrast to previous work, these complexities are independent of the number of different contrast transfer functions of the images. We demonstrate our approach on synthetic and experimental data and show acceleration by factors of up to two orders of magnitude.

关键词： Covariance estimation Fourier–Bessel cryo-EM denoising principal component analysis single particle reconstruction

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Three-Dimensional Construction of Hyperuniform, Nonhyperuniform and Antihyperuniform Random Media via Spectral Density Functions and Their Transport Properties

arXiv

引用

arXiv 2024年

作者： Shi, Wenlong Jiao, Yang Torquato, Salvatore Arizona State University TempeAZ85287 United States Department of Physics Arizona State University TempeAZ85287 United States 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 Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function χ˜V(k), which can be obtained experimentally from scattering techniques, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability S(t) and effective dynamic dielectric constant ϵe for electromagnetic wave propagation. Moreover, χ˜V(k) has been employed to derive sharp bounds on the fluid permeability k. Given the importance of χ˜V(k), we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical χ˜V(k) functions that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions . We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by a power-law autocovariance function χV (r) and contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability S(t) and estimate the fluid permeability k associated with all of the constructed materials directly from the corresponding χ˜V(k) functions. Although it is well established that the long-time asymptotic scaling behavior of S(t) only depends on the functional form of χ˜V(k), with the stealthy hyperuniform and antihyperuniform media respectively

关键词： Spectral density

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Oscillator models and collective motion

IEEE Control Systems Magazine

引用

IEEE Control Systems Magazine 2007年第4期27卷 89-105页

作者： Paley, Derek A. Leonard, Naomi Ehrich Sepulchre, Rodolphe Grünbaum, Daniel Parrish, Julia K. Princeton University Department of Mechanical and Aerospace Engineering Princeton NJ 08544 United States Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Department Electrical Engineering and Computer Science University of Liège Liège Belgium School of Oceanography Department of Biology University of Washington Seattle WA United States Biology Department School of Aquatic and Fishery Sciences University of Washington Seattle WA United States

A description of the so called "particles with coupled oscillator dynamics" (PCOD) is presented which is used to model, analyze and synthesize collective motion. An oscillator model with spatial dynamics is presented to help describe how to design steering control laws while it is being used to study biological collectives. Lastly, both engineering and biological analysis were described.

关键词： Oscillators (mechanical)

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