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Journal of Software for Algebra and Geometry 2019年第1期9卷 55-63页

作者： Chen, Justin Kileel, Joe School of Mathematics Georgia Institute of Technology Atlanta GA United States Program in Applied and Computational Mathematics Princeton University Princeton NJ United States

We present the NumericalImplicitization.m2 package for Macaulay2, which allows for user-friendly computation of the invariants of the image of a polynomial map, such as dimension, degree, and Hilbert function values. This package relies on methods of numerical algebraic geometry, including homotopy continuation and monodromy. © 2019, Mathematical Sciences Publishers. All rights reserved.

关键词： homotopy continuation implicitization interpolation monodromy numerical algebraic geometry

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Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions

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Physical Review Letters 2020年第6期125卷 068002-068002页

作者： Zheng Ma Enrique Lomba Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The creation of disordered hyperuniform materials with extraordinary optical properties (e.g., large complete photonic band gaps) requires a capacity to synthesize large samples that are effectively hyperuniform down to the nanoscale. Motivated by this challenge, we propose a feasible equilibrium fabrication protocol using binary paramagnetic colloidal particles confined in a 2D plane. The strong and long-ranged dipolar interaction induced by a tunable magnetic field is free from screening effects that attenuate long-ranged electrostatic interactions in charged colloidal systems. Specifically, we numerically find a family of optimal size ratios that makes the two-phase system effectively hyperuniform. We show that hyperuniformity is a general consequence of low isothermal compressibilities, which makes our protocol suitable to treat more general systems with other long-ranged interactions, dimensionalities, and/or polydispersity. Our methodology paves the way to synthesize large photonic hyperuniform materials that function in the visible to infrared range and hence may accelerate the discovery of novel photonic materials.

关键词： Applications of soft matter Colloids Disordered systems

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Deep neural network for the dielectric response of insulators

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Physical Review B 2020年第4期102卷 041121(R)-041121(R)页

作者： Linfeng Zhang Mohan Chen Xifan Wu Han Wang Weinan E Roberto Car Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We introduce a deep neural network to model in a symmetry preserving way the environmental dependence of the centers of the electronic charge. The model learns from ab initio density functional theory, wherein the electronic centers are uniquely assigned by the maximally localized Wannier functions. When combined with the deep potential model of the atomic potential energy surface, the scheme predicts the dielectric response of insulators for trajectories inaccessible to direct ab initio simulation. The scheme is nonperturbative and can capture the response of a mutating chemical environment. We demonstrate the approach by calculating the infrared spectra of liquid water at standard conditions, and of ice under extreme pressure, when it transforms from a molecular to an ionic crystal.

关键词： Electric polarization Electrical properties Electronic structure Ferroelectrics Liquids Piezoelectrics Density functional theory First-principles calculations Machine learning Molecular dynamics Multiscale modeling Wannier function methods

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Generation and structural characterization of Debye random media

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Physical Review E 2020年第4期102卷 043310-043310页

作者： Zheng Ma Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In their seminal paper on scattering by an inhomogeneous solid, Debye and coworkers proposed a simple exponentially decaying function for the two-point correlation function of an idealized class of two-phase random media. Such Debye random media, which have been shown to be realizable, are singularly distinct from all other models of two-phase media in that they are entirely defined by their one- and two-point correlation functions. To our knowledge, there has been no determination of other microstructural descriptors of Debye random media. In this paper, we generate Debye random media in two dimensions using an accelerated Yeong-Torquato construction algorithm. We then ascertain microstructural descriptors of the constructed media, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Accurate semianalytic and empirical formulas for these descriptors are devised. We compare our results for Debye random media to those of other popular models (overlapping disks and equilibrium hard disks) and find that the former model possesses a wider spectrum of hole sizes, including a substantial fraction of large holes. Our algorithm can be applied to generate other models defined by their two-point correlation functions, and their other microstructural descriptors can be determined and analyzed by the procedures laid out here.

关键词： Patterns in complex systems

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Risk-averse contextual predictive maintenance and operations scheduling with flexible generation under wind energy uncertainty

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European Journal of Operational Research 2025年

作者： Natalie Randall Beste Basciftci Department of Mathematics Applied Mathematical and Computational Sciences Program College of Liberal Arts and Science University of Iowa United States of America Department of Business Analytics Tippie College of Business University of Iowa United States of America

Ensuring resiliency and sustainability of power systems operations under the uncertainty of the intermittent nature of renewables is becoming a critical concern while considering the integration of flexible generation resources that provide additional adjustability during planning. To address this emerging issue, this study proposes a risk-averse contextual predictive generator maintenance and operations scheduling problem with traditional and flexible generation resources under wind energy uncertainty. We formulate this problem as a two-stage risk-averse stochastic mixed-integer program, where the first-stage determines the maintenance and unit commitment related decisions of the traditional generation units, whereas the second-stage determines the corresponding decisions for flexible generators along with the production related plans of all generators. To integrate contextual information and the uncertainty around the wind power, we propose a Gaussian Process Regression approach for predicting wind power generation, which is then leveraged into this stochastic program. Since this problem is computationally challenging to solve with a mixed-integer recourse due to the second-stage decisions involving flexible generation resources, we provide two versions of a progressive hedging based solution algorithm by first utilizing the classical progressive hedging approach and then leveraging the Frank-Wolfe algorithm for improving the solution quality. In both versions, we extend these algorithms to the risk-averse setting and present various computational enhancements. Our results on the IEEE 118-bus instances demonstrate the impact of adopting a risk-averse approach compared to risk-neutral and deterministic alternatives with a better worst-case performance, and highlight the value of integrating flexible generation and contextual information with resilient maintenance and operations schedules leading to cost-effective plans with less component failures. Furthermore, our

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Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability

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Physical Review E 2024年第6期109卷 064108-064108页

作者： Adam Hitin-Bialus Charles Emmett Maher Paul J. Steinhardt Salvatore Torquato Department of Physics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA. Department of Chemistry <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA. Jefferson Physical Laboratory <a href="https://***/03vek6s52">Harvard University</a> Cambridge Massachusetts 02138 USA and Department of Physics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA. Department of Chemistry <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA Department of Physics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA Princeton Institute of Materials <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA and Program in Applied and Computational Mathematics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA.

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|α in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)−S(t) can be fit to a power law of the form t−(d−α)/2, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ̃V(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of

关键词： Quasicrystals

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Flagellum Pumping Efficiency in Shear-Thinning Viscoelastic Fluids

arXiv

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arXiv 2023年

作者： Barrett, Aaron Fogelson, Aaron L. Gregory Forest, M. Gruninger, Cole Lim, Sookkyung Griffith, Boyce E. Department of Mathematics University of Utah Salt Lake CityUT United States Departments of Mathematics and Biomedical Engineering University of Utah Salt Lake CityUT United States Department of Mathematics University of North Carolina Chapel HillNC United States Department of Applied Physical Sciences 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 Department of Mathematical Sciences University of Cincinnati CincinnatiOH 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

Microorganism motility often takes place within complex, viscoelastic fluid environments, e.g., sperm in cervicovaginal mucus and bacteria in biofilms. In such complex fluids, strains and stresses generated by the microorganism are stored and relax across a spectrum of length and time scales and the complex fluid can be driven out of its linear response regime. Phenomena not possible in viscous media thereby arise from feedback between the swimmer and the complex fluid, making swimming efficiency co-dependent on the propulsion mechanism and fluid properties. Here we parameterize a flagellar motor and filament properties together with elastic relaxation and nonlinear shear-thinning properties of the fluid in a computational immersed boundary model. We then explore swimming efficiency, defined as a particular flow rate divided by the torque required to spin the motor, over this parameter space. Our findings indicate that motor efficiency (measured by the volumetric flow rate) can be boosted or degraded by relatively moderate or strong shear-thinning of the viscoelastic environment. Copyright © 2023, The Authors. All rights reserved.

关键词： Shear thinning

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Biwhitening reveals the rank of a count matrix

arXiv

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arXiv 2021年

作者： Landa, Boris Zhang, Thomas T.C.K. Kluger, Yuval Program in Applied Mathematics Yale University United States Department of Electrical and Systems Engineering University of Pennsylvania United States Interdepartmental Program in Computational Biology and Bioinformatics Yale University United States Department of Pathology Yale University School of Medicine United States

Estimating the rank of a corrupted data matrix is an important task in data analysis, most notably for choosing the number of components in PCA. Significant progress on this task was achieved using random matrix theory by characterizing the spectral properties of large noise matrices. However, utilizing such tools is not straightforward when the data matrix consists of count random variables, e.g., Poisson, in which case the noise can be heteroskedastic with an unknown variance in each entry. In this work, we consider a Poisson random matrix with independent entries, and propose a simple procedure termed biwhitening for estimating the rank of the underlying signal matrix (i.e., the Poisson parameter matrix) without any prior knowledge. Our approach is based on the key observation that one can scale the rows and columns of the data matrix simultaneously so that the spectrum of the corresponding noise agrees with the standard Marchenko-Pastur (MP) law, justifying the use of the MP upper edge as a threshold for rank selection. Importantly, the required scaling factors can be estimated directly from the observations by solving a matrix scaling problem via the Sinkhorn-Knopp algorithm. Aside from the Poisson, our approach is extended to families of distributions that satisfy a quadratic relation between the mean and the variance, such as the generalized Poisson, binomial, negative binomial, gamma, and many others. This quadratic relation can also account for missing entries in the data. We conduct numerical experiments that corroborate our theoretical findings, and showcase the advantage of our approach for rank estimation in challenging regimes. Furthermore, we demonstrate the favorable performance of our approach on several real datasets of single-cell RNA sequencing (scRNA-seq), High-Throughput Chromosome Conformation Capture (Hi-C), and document topic *** Codes 62H12, 62H25. Copyright © 2021, The Authors. All rights reserved.

关键词： Random variables

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Realizability of Iso-g2 Processes via Effective Pair Interactions

arXiv

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arXiv 2022年

作者： Wang, Haina Stillinger, Frank H. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Institute of Materials and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Natural Sciences Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by d-dimensional many-body systems. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. In light of this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse methodology that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably numerically realizable up to the packing fraction = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal" packing fraction c = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through c. For near but below c, the large-r behaviors of the effective potentials are given exactly by the functional forms exp[−κ()r] for d = 1, r−1/2 exp[−κ()r] for d = 2, and r−1 exp[−κ()r] (Yukawa form) for d = 3, where κ−1() is a screening length, and for = c, the potentials at large r are given by the pure Coulomb forms in the respective dimensions, as predicted by Torquato and Stillinger [Phys. Rev. E, 68, 041113 1-25 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost" random sequential addition at the saturation density c = 1/8 is much shorter-ranged than that for the 3D unit-step-function g2 at c, and thus does not constrain the realizability

关键词： Inverse problems

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Liquid-Liquid Transition in Water from First Principles

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Physical Review Letters 2022年第25期129卷 255702-255702页

作者： Thomas E. Gartner, III Pablo M. Piaggi Roberto Car Athanassios Z. Panagiotopoulos Pablo G. Debenedetti Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Department of Chemical and Biological Engineering Princeton University Princeton New Jersey 08544 USA

A long-standing question in water research is the possibility that supercooled liquid water can undergo a liquid-liquid phase transition (LLT) into high- and low-density liquids. We used several complementary molecular simulation techniques to evaluate the possibility of an LLT in an ab initio neural network model of water trained on density functional theory calculations with the SCAN exchange correlation functional. We conclusively show the existence of a first-order LLT and an associated critical point in the SCAN description of water, representing the first definitive computational evidence for an LLT in water from first principles.

关键词： Classical statistical mechanics First order phase transitions First-principles calculations Liquid-liquid phase transition Phase transitions Thermodynamics Deep learning Density functional theory Molecular dynamics

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