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Algebraic Constraints and Algorithms for Common Lines in Cryo-EM

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arXiv 2024年

作者： Muller, Tommi Duncan, Adriana L. Verbeke, Eric J. Kileel, Joe Mathematical Institute University of Oxford OxfordOX2 6GG United Kingdom Department of Mathematics University of Texas at Austin AustinTX78712 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States Oden Institute University of Texas at Austin AustinTX78712 United States

We revisit the topic of common lines between projection images in single particle cryo-electron microscopy (cryo-EM). We derive a novel low-rank constraint on a certain 2n × n matrix storing properly-scaled basis vectors for the common lines between n projection images of one molecular conformation. Using this algebraic constraint and others, we give optimization algorithms to denoise common lines and recover the unknown 3D rotations associated to the images. As an application, we develop a clustering algorithm to partition a set of noisy images into homogeneous communities using common lines, in the case of discrete heterogeneity in cryo-EM. We demonstrate the methods on synthetic and experimental *** Codes 90C26, 14Q30 © 2024, CC BY.

关键词： Clustering algorithms

Modeling of Measurement Error in Financial Returns Data

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arXiv 2024年

作者： Jasra, Ajay Maama, Mohamed Mijatović, Aleksandar School of Data Science The Chinese University of Hong Kong Shenzhen China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Statistics University of Warwick United Kingdom

In this paper we consider the modeling of measurement error for fund returns data. In particular, given access to a time-series of discretely observed log-returns and the associated maximum over the observation period, we develop a stochastic model which models the true log-returns and maximum via a Lévy process and the data as a measurement error there-of. The main technical difficulty of trying to infer this model, for instance Bayesian parameter estimation, is that the joint transition density of the return and maximum is seldom known, nor can it be simulated exactly. Based upon the novel stick breaking representation of [12] we provide an approximation of the model. We develop a Markov chain Monte Carlo (MCMC) algorithm to sample from the Bayesian posterior of the approximated posterior and then extend this to a multilevel MCMC method which can reduce the computational cost to approximate posterior expectations, relative to ordinary MCMC. We implement our methodology on several applications including for real data. Copyright © 2024, The Authors. All rights reserved.

关键词： Markov chains

On Time Uniform Wong-Zakai Approximation Theorems

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arXiv 2023年

作者： Del Moral, Pierre Hu, Shulan Jasra, Ajay Ruzayqat, Hamza Wang, Xinyu Institut de Mathematiques de Bordeaux Bordeaux33405 France School of Statistics and Mathematics Zhongnan University of Economics and Law China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Wenlan School of Business Zhongnan University of Economics and Law China

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations associated to random smooth fluctuations;e.g. robust filtering problems. In many examples, the mean error estimate bounds that have been derived in the literature can grow exponentially with respect to the time horizon. We show in a simple example that indeed mean error estimates do explode exponentially in the time parameter, i.e. in that case a Wong-Zakai approximation is only useful for extremely short time intervals. Under spectral conditions, we present some quantitative time-uniform convergence theorems, i.e. time-uniform mean error bounds, yielding what seems to be the first results of this type for Wong-Zakai diffusion *** Codes 60H35(Primary), 65C30(Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： Brownian movement

Structured time-delay models for dynamical systems with connections to Frenet-Serret frame

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arXiv 2021年

作者： Hirsh, Seth M. Ichinaga, Sara M. Brunton, Steven L. Kutz, J. Nathan Brunton, Bingni W. Department of Physics University of Washington SeattleWA United States Applied and Computational Mathematical Sciences Program University of Washington SeattleWA United States Department of Mechanical Engineering University of Washington SeattleWA United States Department of Applied Mathematics University of Washington SeattleWA United States Department of Biology University of Washington SeattleWA United States

Time-delay embeddings and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition (DMD) on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal with a zero diagonal;for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in Frenet-Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modeling procedure on data from several nonlinear synthetic and real-world examples. © 2021, CC BY.

关键词： Time delay

Hole Statistics of Equilibrium 2D and 3D Hard-Sphere Crystals

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arXiv 2024年

作者： Wang, Haina Huse, David A. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics 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

The probability of finding a spherical "hole" of a given radius r contains crucial structural information about many-body systems. Such hole statistics, including the void conditional nearest-neighbor probability functions GV (r), have been well studied for hard-sphere fluids in d-dimensional Euclidean space Rd. However, little is known about these functions for hard-sphere crystals for values of r beyond the hard-sphere diameter, as large holes are extremely rare in crystal phases. To overcome these computational challenges, we introduce a biased-sampling scheme that accurately determines hole statistics for equilibrium hard spheres on ranges of r that far extend those that could be previously explored. We discover that GV (r) in crystal and hexatic states exhibits oscillations whose amplitudes increase rapidly with the packing fraction, which stands in contrast to GV (r) for fluid states that is monotonic in r. The oscillations in GV (r) for 2D crystals are strongly correlated with the local orientational order metric in the vicinity of the holes, and variations in GV (r) for 3D states indicate a transition between tetrahedral and octahedral holes, demonstrating the power of GV (r) as a probe of local coordination geometry. To further study the statistics of interparticle spacing in hard-sphere systems, we compute the local packing fraction distribution f(l) of Delaunay cells, and find that for d ≤ 3, the excess kurtosis of f(l) switches sign at a certain transitional global packing fraction. Our investigation facilitates the study of structural and bulk properties of materials that involve the creation of rare large holes, such as the solubility of alloys. © 2024, CC BY.

关键词： Spheres

Negative Poisson’s Ratio Materials via Isotropic Interactions

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Physical Review Letters 2008年第8期101卷 085501-085501页

作者： Mikael C. Rechtsman Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics and PRISM Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08544 USA

We show that under tension a classical many-body system with only isotropic pair interactions in a crystalline state can, counterintuitively, have a negative Poisson’s ratio, or auxetic behavior. We derive the conditions under which the triangular lattice in two dimensions and lattices with cubic symmetry in three dimensions exhibit a negative Poisson’s ratio. In the former case, the simple Lennard-Jones potential can give rise to auxetic behavior. In the latter case, a negative Poisson’s ratio can be exhibited even when the material is constrained to be elastically isotropic.

关键词： Lennard-Jones potential

Novel Low-Temperature Behavior in Classical Many-Particle Systems

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Physical Review Letters 2009年第5期103卷 050602-050602页

作者： Robert D. Batten Frank H. Stillinger Salvatore Torquato Department of Chemical Engineering Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Materials Institute Program in Applied and Computational Mathematics Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08544 USA

We show that classical many-particle systems interacting with certain soft pair interactions in two dimensions exhibit novel low-temperature behaviors. Ground states span from disordered to crystalline. At some densities, a large fraction of normal-mode frequencies vanish. Lattice ground-state configurations have more vanishing frequencies than disordered ground states at the same density and exhibit vanishing shear moduli. For the melting transition from a crystal, the thermal expansion coefficient is negative. These unusual results are attributed to the topography of the energy landscape.

关键词： Ground state

Packing hyperspheres in high-dimensional Euclidean spaces

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Physical Review E 2006年第4期74卷 041127-041127页

作者： Monica Skoge Aleksandar Donev Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA PRISM Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton University Princeton New Jersey 08544 USA

We present a study of disordered jammed hard-sphere packings in four-, five-, and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d=4, 5, and 6 to be ϕMRJ≈0.46, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form ϕMRJ=c1∕2d+(c2d)∕2d, where c1=−2.72 and c2=2.56, which appears to be consistent with the high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g2(r) and structure factor S(k) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed “decorrelation principle,” which, among other things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d→∞. As in three dimensions (where ϕMRJ≈0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g2(r) at contact with power-law exponent ≈0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2(r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, ϕF≈0.32 and ϕM≈0.39, respectively, for d=4, and ϕF≈0.20 and ϕM≈0.25, respectively, for d=5. Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension.

关键词： RANDOM CLOSE PACKING 4TH VIRIAL-COEFFICIENTS HARD-PARTICLE PACKINGS ARBITRARY DIMENSIONALITY BOUNDARY-CONDITIONS MONTE-CARLO SPHERES EQUATION STATE SIMULATION

Erratum: Toward the jamming threshold of sphere packings: Tunneled crystals (Journal of applied Physics (2007) 102 (093511))

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Journal of applied Physics 2008年第12期103卷 129902-129902-2页

作者： Torquato, S. Stillinger, F.H. Department of Chemistry Princeton University Princeton NJ 08544 United States Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Princeton Institute for the Science and Technology of Materials Princeton University Princeton NJ 08544 United States Princeton Center for Theoretical Physics Princeton University Princeton NJ 08544 United States

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