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Uncovering multiscale order in the prime numbers via scattering

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arXiv 2018年

作者： Torquato, .S. Zhang, .G. De Courcy-Ireland, M. Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Mathematics Princeton University PrincetonNJ08544 United States

The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor S(k), proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes. Copyright © 2018, The Authors. All rights reserved.

关键词：

HIDDEN MULTISCALE ORDER IN THE PRIMES

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arXiv 2018年

作者： Torquato, Salvatore Zhang, Ge Courcy-Ireland, Matthew De Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Mathematics Princeton University PrincetonNJ08544 United States

We study the pair correlations between prime numbers in an interval M ≤ p ≤ M + L with M → ∞, L/M → β > 0. By analyzing the structure factor, we prove, conditionally on the Hardy-Littlewood conjecture on prime pairs, that the primes are characterized by unanticipated multiscale order. Specifically, their limiting structure factor is that of a union of an infinite number of periodic systems and is characterized by dense set of Dirac delta functions. Primes in dyadic intervals are the first examples of what we call effectively limit-periodic point configurations. This behavior implies anomalously suppressed density fluctuations compared to uncorrelated (Poisson) systems at large length scales, which is now known as hyperuniformity. Using a scalar order metric τ calculated from the structure factor, we identify a transition between the order exhibited when L is comparable to M and the uncorrelated behavior when L is only logarithmic in M. Our analysis for the structure factor leads to an algorithm to reconstruct primes in a dyadic interval with high accuracy. Copyright © 2018, The Authors. All rights reserved.

关键词： Delta functions

Model reduction with memory and the machine learning of dynamical systems

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arXiv 2018年

作者： Ma, Chao Wang, Jianchun Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen518055 China Beijing Institute of Big Data Research Beijing100871 China

The well-known Mori-Zwanzig theory tells us that model reduction leads to memory effect. For a long time, modeling the memory effect accurately and efficiently has been an important but nearly impossible task in developing a good reduced model. In this work, we explore a natural analogy between recurrent neural networks and the Mori-Zwanzig formalism to establish a systematic approach for developing reduced models with memory. Two training models-a direct training model and a dynamically coupled training model-are proposed and compared. We apply these methods to the Kuramoto-Sivashinsky equation and the Navier-Stokes equation. Numerical experiments show that the proposed method can produce reduced model with good performance on both short-term prediction and long-term statistical properties. Copyright © 2018, The Authors. All rights reserved.

关键词： Recurrent neural networks

Hyperuniformity of generalized random organization models

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arXiv 2018年

作者： Ma, Zheng Torquato, Salvatore Department of Physics Princeton University and Princeton New Jersey08544 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 PrincetonNJ08544 United States

Studies of random organization models of monodisperse (i.e., identical) spherical particles have shown that a hyperuniform state is achievable when the system goes through an absorbing phase transition to a critical state. Here we investigate to what extent hyperuniformity is preserved when the model is generalized to particles with a size distribution and/or nonspherical shapes. We begin by examining binary disks in two dimensions and demonstrate that their critical states are hyperuniform as two-phase media, but not hyperuniform nor multihyperuniform as point patterns formed by the particle centroids. We further confirm the generality of our findings by studying particles with a continuous size distribution. Finally, to study the effect of rotational degrees of freedom, we extend our model to noncircular particles, namely, hard rectangles with various aspect ratios, including the hard-needle limit. Although these systems exhibit only short-range orientational order, hyperuniformity is still preserved. Our analysis reveals that the redistribution of the "mass" of the particles rather than the particle centroids is central to this dynamical process. The consideration of the "active volume fraction" of generalized random organization models may help to resolve which universality class they belong to and hence may lead to a deeper theoretical understanding of absorbing-state models. Our results suggest that general particle systems subject to random organization can be a robust way to fabricate a wide class of hyperuniform states of matter by tuning the structures via different particle-size and -shape distributions. This in turn potentially enables the creation of multifunctional hyperuniform materials with desirable optical, transport and mechanical properties. Copyright © 2018, The Authors. All rights reserved.

关键词： Critical current density (superconductivity)

Toward single particle reconstruction without particle picking: Breaking the detection limit

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arXiv 2018年

作者： Bendory, Tamir Boumal, Nicolas Leeb, William Levin, Eitan Singer, Amit The School of Electrical Engineering Tel Aviv University Tel Aviv Israel Lausanne Switzerland School of Mathematics University of Minnesota MinneapolisMN United States Department of Computing and Mathematical Sciences California Institute of Technology PasadenaCA United States The Program in Applied and Computational Mathematics Department of Mathematics Princeton University PrincetonNJ United States

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method to resolve biological macromolecules. In a cryo-EM experiment, the microscope produces images called micrographs. Projections of the molecule of interest are embedded in the micrographs at unknown locations, and under unknown viewing directions. Standard imaging techniques first locate these projections (detection) and then reconstruct the 3-D structure from them. Unfortunately, high noise levels hinder detection. When reliable detection is rendered impossible, the standard techniques fail. This is a problem, especially for small molecules. In this paper, we pursue a radically different approach: we contend that the structure could, in principle, be reconstructed directly from the micrographs, without intermediate detection. The aim is to bring small molecules within reach for cryo-EM. To this end, we design an autocorrelation analysis technique that allows to go directly from the micrographs to the sought structures. This involves only one pass over the micrographs, allowing online, streaming processing for large experiments. We show numerical results and discuss challenges that lay ahead to turn this proof-of-concept into a complementary approach to state-of-the-art algorithms. Copyright © 2018, The Authors. All rights reserved.

关键词： X ray crystallography

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

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

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

Computing Optimal Interfacial Structure of Modulated Phases

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Communications in computational Physics 2017年第1期21卷 1-15页

作者： Jie Xu Chu Wang An-Chang Shi Pingwen Zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew Jersey 08544USA Department of Physics and Astronomy McMaster UniversityHamiltonOntario L8S4M1Canada

We propose a general framework of computing interfacial structures between two modulated *** we propose to use a computational box consisting of two half spaces,each occupied by a modulated phase with given position and *** boundary conditions and basis functions are chosen to be commensurate with the bulk *** observe that the ordered nature of modulated structures stabilizes the interface,which enables us to obtain optimal interfacial structures by searching local minima of the free energy *** framework is applied to the Landau-Brazovskii model to investigate interfaces between modulated phases with different relative positions and *** types of novel complex interfacial structures emerge from the calculations.

关键词： Interface modulated phase metastable state compatibility Landau-Brazovskiimodel

3D ab initio modeling in cryo-em by autocorrelation analysis

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arXiv 2017年

作者： Levin, Eitan Bendory, Tamir Boumal, Nicolas Kileel, Joe Singer, Amit Princeton University Program in Applied and Computational Mathematics PrincetonNJ United States

Single-Particle Reconstruction (SPR) in Cryo-Electron Microscopy (cryo-EM) is the task of estimating the 3D structure of a molecule from a set of noisy 2D projections, taken from unknown viewing directions. Many algorithms for SPR start from an initial reference molecule, and alternate between refining the estimated viewing angles given the molecule, and refining the molecule given the viewing angles. This scheme is called iterative refinement. Reliance on an initial, user-chosen reference introduces model bias, and poor initialization can lead to slow convergence. Furthermore, since no ground truth is available for an unsolved molecule, it is difficult to validate the obtained results. This creates the need for high quality ab initio models that can be quickly obtained from experimental data with minimal priors, and which can also be used for validation. We propose a procedure to obtain such an ab initio model directly from raw data using Kam's autocorrelation method. Kam's method has been known since 1980, but it leads to an underdetermined system, with missing orthogonal matrices. Until now, this system has been solved only for special cases, such as highly symmetric molecules or molecules for which a homologous structure was already available. In this paper, we show that knowledge of just two clean projections is sufficient to guarantee a unique solution to the system. This system is solved by an optimization-based heuristic. For the first time, we are then able to obtain a low-resolution ab initio model of an asymmetric molecule directly from raw data, without 2D class averaging and without tilting. Numerical results are presented on both synthetic and experimental data. Copyright © 2017, The Authors. All rights reserved.

关键词： Molecules

Steerable ePCA: Rotationally Invariant Exponential Family PCA

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arXiv 2018年

作者： Zhao, Zhizhen Liu, Lydia T. Singer, Amit Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign UrbanaIL61820 United States Department of Electrical Engineering and Computer Sciences University of California at Berkeley BerkeleyCA94720 United States Department of Mathematics Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

—In photon-limited imaging, the pixel intensities are affected by photon count noise. Many applications, such as 3-D reconstruction using correlation analysis in X-ray free electron laser (XFEL) single molecule imaging, require an accurate estimation of the covariance of the underlying 2-D clean images. Accurate estimation of the covariance from low-photon count images must take into account that pixel intensities are Poisson distributed, hence the classical sample covariance estimator is sub-optimal. Moreover, in single molecule imaging, including in-plane rotated copies of all images could further improve the accuracy of covariance estimation. In this paper we introduce an efficient and accurate algorithm for covariance matrix estimation of count noise 2-D images, including their uniform planar rotations and possibly reflections. Our procedure, steerable ePCA, combines in a novel way two recently introduced innovations. The first is a methodology for principal component analysis (PCA) for Poisson distributions, and more generally, exponential family distributions, called ePCA. The second is steerable PCA, a fast and accurate procedure for including all planar rotations for PCA. The resulting principal components are invariant to the rotation and reflection of the input images. We demonstrate the efficiency and accuracy of steerable ePCA in numerical experiments involving simulated XFEL datasets and rotated Yale B face data. Copyright © 2018, The Authors. All rights reserved.

关键词： Principal component analysis