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On singularity formation in a Hele-Shaw model

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arXiv 2017年

作者： Constantin, Peter Elgindi, Tarek Nguyen, Huy Vicol, Vlad Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Uc San Diego San DiegoCA92093 United States Program In Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in [CDG+93] and widely studied since. The model consists of a single one dimensional evolution equation for the thickness 2h = 2h(x, t) of a thin neck of fluid, ∂th + ∂x(h ∂x3h) = 0, for x ∈ (−1, 1) and t ≥ 0. The boundary conditions fix the neck height and the pressure jump: h(±1, t) = 1, ∂x2h(±1, t) = P > 0. We prove that starting from smooth and positive h, as long as h(x, t) > 0, for x ∈ [−1, 1], t ∈ [0, T ], no singularity can arise in the solution up to time T. As a consequence, we prove for any P > 2 and any smooth and positive initial datum that the solution pinches off in either finite or infinite time, i.e., inf[−1,1]×[0,T∗) h = 0, for some T∗ ∈ (0, ∞]. These facts have been long anticipated on the basis of numerical and theoretical studies. August 30, 2017.35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

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Group synchronization on grids

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arXiv 2017年

作者： Abbe, Emmanuel Massoulie, Laurent Montanari, Andrea Sly, Allan Srivastava, Nikhil Program in Applied and Computational Mathematics EE Department Princeton University Inria MSR-Inria Joint Centre Department of Electrical Engineering Department of Sttistics Stanford University Department of Mathematics Princeton University Department of Mathematics University of California Berkeley United States

Group synchronization requires to estimate unknown elements (θv)v∈V of a compact group G associated to the vertices of a graph G = (V, E), using noisy observations of the group differences associated to the edges. This model is relevant to a variety of applications ranging from structure from motion in computer vision to graph localization and positioning, to certain families of community detection problems. We focus on the case in which the graph G is the d-dimensional grid. Since the unknowns θv are only determined up to a global action of the group, we consider the following weak recovery question. Can we determine the group difference θ−u1θv between far apart vertices u, v better than by random guessing? We prove that weak recovery is possible (provided the noise is small enough) for d ≥ 3 and, for certain finite groups, for d ≥ 2. Viceversa, for some continuous groups, we prove that weak recovery is impossible for d = 2. Finally, for strong enough noise, weak recovery is always impossible. Copyright © 2017, The Authors. All rights reserved.

关键词： Recovery

Sea-ice distribution and mixed-layer depths in fram strait

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arXiv 2017年

作者： Agarwal, Sahil Grae Worster, M. Program in Applied Mathematics Yale University New Haven United States Institute of Theoretical Geophysics Department of Applied Mathematics and Theoretical Physics University of Cambridge Wilberforce Road CambridgeCB3 0WA United Kingdom

In an effort to understand the dynamics of the Arctic sea-ice edge, we present a simple model of heat and mass transfer in the Fram Strait that reveals some fundamental mechanisms controlling sea-ice extent in the marginal seas and the depth and properties of the Arctic mixed layer. We identify and study key mechanisms relating to the sea-ice wedge described by Untersteiner, a boundary-layer structure near the ice edge, demonstrating how ice thickness and extent depend on ice-export rates, atmospheric forcing and the properties of incoming warm and salty Atlantic water in the West Spitsbergen Current. Our time-dependent results demonstrate a seasonal asymmetry between the rates of ice advance and retreat and explain the significant variations in the Southerly extent of sea ice across the Fram Strait, with a long ice tongue corresponding with the East Greenland Current. Our simple model indicates that thinning of the Arctic sea-ice cover will lead to warming and freshening of the North Atlantic, which would give a de-stabilizing feedback to the Arctic ice cover, leading to a slowdown of the Atlantic Meridional Overturning Circulation. Copyright © 2017, The Authors. All rights reserved.

关键词： Sea ice

Data-driven model reduction and transfer operator approximation

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arXiv 2017年

作者： Klus, Stefan Nüske, Feliks Koltai, Péter Wu, Hao Kevrekidis, Ioannis Schütte, Christof Noé, Frank Department of Mathematics and Computer Science Freie Universität Berlin Germany Department of Chemical and Biological Engineering & Program in Applied and Computational Mathematics Princeton University United States Zuse Institute Berlin Germany

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis (TICA), dynamic mode decomposition (DMD), and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods. Copyright © 2017, The Authors. All rights reserved.

关键词： Dynamical systems

Hyperuniformity of quasicrystals

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Physical Review B 2017年第5期95卷 054119-054119页

作者： Erdal C. Oğuz Joshua E. S. Socolar Paul J. Steinhardt 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 08540 USA

Hyperuniform systems, which include crystals, quasicrystals, and special disordered systems, have attracted considerable recent attention, but rigorous analyses of the hyperuniformity of quasicrystals have been lacking because the support of the spectral intensity is dense and discontinuous. We employ the integrated spectral intensity Z(k) to quantitatively characterize the hyperuniformity of quasicrystalline point sets generated by projection methods. The scaling of Z(k) as k tends to zero is computed for one-dimensional quasicrystals and shown to be consistent with independent calculations of the variance, σ2(R), in the number of points contained in an interval of length 2R. We find that one-dimensional quasicrystals produced by projection from a two-dimensional lattice onto a line of slope 1/τ fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, Z(k)∼k4; for all others, Z(k)∼k2. This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.

关键词： Microstructure Structural properties

Phenomenological friction equation for turbulent flow of Bingham fluids

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Physical Review E 2017年第2期96卷 023107-023107页

作者： H. R. Anbarlooei D. O. A. Cruz F. Ramos Cecilia M. M. Santos A. P. Silva Freire Mechanical Engineering Program COPPE Federal University of Rio de Janeiro Rio de Janeiro Rio de Janeiro 21941-450 Brazil Department of Applied Mathematics Institute of Mathematics Federal University of Rio de Janeiro Rio de Janeiro Rio de Janeiro 21941-450 Brazil

Most discussions in the literature on the friction coefficient of turbulent flows of fluids with complex rheology are empirical. As a rule, theoretical frameworks are not available even for some relatively simple constitutive models. In the present work, a formula is proposed for the evaluation of the friction coefficient of turbulent flows of Bingham fluids. The developments combine a fresh analysis for the description of the microscales of Kolmogorov and the phenomenological turbulence model of Gioia and Chakraborty [G. Gioia and P. Chakraborty, Phys. Rev. Lett. 96, 044502 (2006)]. The resulting Blasius-type friction equation is tested against some experimental data and shows good agreement over a significant range of Hedstrom and Reynolds numbers. Comments on pressure measurements in yielding fluids are made. The limits of the proposed model are also discussed.

关键词： Non-Newtonian fluids Turbulence Viscoplasticity

A search for pulsars around SgrA∗ in the first Event Horizon Telescope dataset

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arXiv 2023年

作者： Torne, Pablo Liu, Kuo Eatough, Ralph P. Wongphechauxsorn, Jompoj Cordes, James M. Desvignes, Gregory De Laurentis, Mariafelicia Kramer, Michael Ransom, Scott M. Chatterjee, Shami Wharton, Robert Karuppusamy, Ramesh Blackburn, Lindy Janssen, Michael Chan, Chi-Kwan Crew, Geoffrey B. Matthews, Lynn D. Goddi, Ciriaco Rottmann, Helge Wagner, Jan Sánchez, Salvador Ruiz, Ignacio Institut de Radioastronomie Millimétrique Avenida Divina Pastora 7 Local 20 GranadaE-18012 Spain Max-Planck-Institut für Radioastronomie Auf dem Hügel 69 BonnD-53121 Germany National Astronomical Observatories Chinese Academy of Sciences 20A Datun Road Chaoyang District Beijing100101 China Cornell Center for Astrophysics and Planetary Science Cornell University IthacaNY14853 United States Lesia Observatoire de Paris Université Psl Cnrs Sorbonne Université Université de Paris 5 place Jules Janssen Meudon92195 France Dipartimento di Fisica 'E. Pancini' Universitá di Napoli 'Federico II' Compl. Univ. di Monte S. Angelo Edificio G Via Cinthia NapoliI-80126 Italy Institut für Theoretische Physik Goethe-Universität Frankfurt Max-von-Laue-Strase 1 Frankfurt am MainD-60438 Germany Infn Sez. di Napoli Compl. Univ. di Monte S. Angelo Edificio G Via Cinthia NapoliI-80126 Italy Jodrell Bank Centre for Astrophysics School of Physics and Astronomy The University of Manchester ManchesterM13 9PL United Kingdom National Radio Astronomy Observatory 520 Edgemont Road CharlottesvilleVA22903 United States Jet Propulsion Laboratory California Institute of Technology PasadenaCA91109 United States Black Hole Initiative Harvard University 20 Garden Street CambridgeMA02138 United States Center for Astrophysics Harvard & Smithsonian 60 Garden Street CambridgeMA02138 United States Steward Observatory Department of Astronomy University of Arizona 933 N. Cherry Ave. TucsonAZ85721 United States Data Science Institute University of Arizona 1230 N. Cherry Ave. TucsonAZ85721 United States Program in Applied Mathematics University of Arizona 617 N. Santa Rita TucsonAZ85721 United States Massachusetts Institute of Technology Haystack Observatory 99 Millstone Road WestfordMA01886 United States Instituto de Astronomia Geofísica e Ciências Atmosféricas Universidade de São Paulo R. do Matão 1226 SP São Paulo05508-090 Brazil Dipartimento di Fisica

The Event Horizon Telescope (EHT) observed in 2017 the supermassive black hole at the center of the Milky Way, Sagittarius A∗ (Sgr A∗), at a frequency of 228.1 GHz (λ = 1.3 mm). The fundamental physics tests that even a single pulsar orbiting Sgr A∗ would enable motivate searching for pulsars in EHT datasets. The high observing frequency means that pulsars - which typically exhibit steep emission spectra - are expected to be very faint. However, it also negates pulse scattering, an effect that could hinder pulsar detections in the Galactic Center. Additionally, magnetars or a secondary inverse Compton emission could be stronger at millimeter wavelengths than at lower frequencies. We present a search for pulsars close to Sgr A∗ using the data from the three most-sensitive stations in the EHT 2017 campaign: the Atacama Large Millimeter/submillimeter Array, the Large Millimeter Telescope and the IRAM 30m Telescope. We apply three detection methods based on Fourier-domain analysis, the Fast-Folding-Algorithm and single pulse search targeting both pulsars and burst-like transient emission;using the simultaneity of the observations to confirm potential candidates. No new pulsars or significant bursts were found. Being the first pulsar search ever carried out at such high radio frequencies, we detail our analysis methods and give a detailed estimation of the sensitivity of the search. We conclude that the EHT 2017 observations are only sensitive to a small fraction ( 2.2%) of the pulsars that may exist close to Sgr A∗, motivating further searches for fainter pulsars in the region. © 2023, CC BY.

关键词： Inverse problems

Pre-processing and classification of hyperspectral imagery via selective inpainting

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Pre-processing and classification of hyperspectral imagery v...

IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Victoria Chayes Kevin Miller Rasika Bhalerao Jiajie Luo Wei Zhu Andrea L. Bertozzi Wenzhi Liao Stanley Osher California Research Training Program in Computational and Applied Mathematics United States of America University of California Los Angeles Department of Mathematics United States of America Ghent University Department of Telecommunications and Information Processing Belgium

ISBN: (纸本)9781509041183

We propose a semi-supervised algorithm for processing and classification of hyperspectral imagery. For initialization, we keep 20% of the data intact, and use Principal Component Analysis to discard voxels from noisier bands and pixels. Then, we use either an Accelerated Proximal Gradient algorithm (APGL), or a modified APGL algorithm with a penalty term for distance between inpainted pixels and endmembers (APGL Hyp), on the initialized datacube to inpaint the missing data. APGL and APGL Hyp are distinguished by performance on datasets with full pixels removed or extreme noise. This inpainting technique results in band-by-band datacube sharpening and removal of noise from individual spectral signatures. We can also classify the inpainted cube by assigning each pixel to its nearest endmember via Euclidean distance. We demonstrate improved accuracy in classification over data-mining techniques like k-means, unmixing techniques like Hierarchical Non-Negative Matrix Factorization, and graph-based methods like Non-Local Total Variation.

关键词： Hyperspectral imagery (HSI) data preprocessing image inpainting image classification image enhancement Data preprocessing hyperspectral imagery Image classification image enhancement pixel Pretreatment matrix decomposition initialization Euclidean distance

Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics

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arXiv 2017年

作者： Zhang, Linfeng Han, Jiequn Wang, Han Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China CAEP Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Data Science Beijing International Center for Mathematical Research Peking University Beijing Institute of Big Data Research Beijing100871 China

We introduce a scheme for molecular simulations, the Deep Potential Molecular Dynamics (DeePMD) method, based on a many-body potential and interatomic forces generated by a carefully crafted deep neural network trained with ab initio data. The neural network model preserves all the natural symmetries in the problem. It is "first principle-based" in the sense that there are no ad hoc components aside from the network model. We show that the proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and molecules. In all these cases, DeePMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size. Copyright © 2017, The Authors. All rights reserved.

关键词： Molecular dynamics