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Scaling positive random matrices: Concentration and asymptotic convergence

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arXiv 2020年

作者： Landa, Boris Program in Applied Mathematics Department of Mathematics Yale University United States

It is well known that any positive matrix can be scaled to have prescribed row and column sums by multiplying its rows and columns by certain positive scaling factors (which are unique up to a positive scalar). This procedure is known as matrix scaling, and has found numerous applications in operations research, economics, image processing, and machine learning. In this work, we investigate the behavior of the scaling factors and the resulting scaled matrix when the matrix to be scaled is random. Specifically, letting Ã∈ RM×N be a positive and bounded random matrix whose entries assume a certain type of independence, we provide a concentration inequality for the scaling factors of Ã around those of A = E[Ã]. This result is employed to bound the convergence rate of the scaling factors of Ã to those of A, as well as the concentration of the scaled version of Ã around the scaled version of A in operator norm, as M, N → ∞. When the entries of Ã are independent, M = N, and all prescribed row and column sums are 1 (i.e., doubly-stochastic matrix scaling), both of the previously-mentioned bounds are O(√log N/N) with high probability. We demonstrate our results in several *** Codes 60B20, 60F10, 65F35 Copyright © 2020, The Authors. All rights reserved.

关键词： Operations research

Optimal reduced-order modeling for nonlinear distributed parameter systems

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Optimal reduced-order modeling for nonlinear distributed par...

American Control Conference (ACC)

作者： J. Borggaard Department of Mathematics and the Interdisciplinary Center for Applied Mathematics Virginia Technology Blacksburg VA USA

A method to develop reduced-order models for nonlinear distributed parameter systems is studied. The method is based on Galerkin projection, but the reduced-basis vectors are optimal for the dynamic model, found by minimizing the error between given full-order simulation data and the reduced-order model. This is achieved by formulating the basis selection problem as an optimal control problem with the reduced-order model as a constraint. This methodology allows a natural extension of reduced-order modeling ideas to nonlinear systems. A numerical experiment comparing the optimal reduced-order model to the popular proper orthogonal decomposition method is provided

关键词： Distributed parameter systems Reduced order systems Hydrogen Nonlinear systems Weather forecasting Large-scale systems Contracts mathematics Differential equations Vectors

On collisions times of 'self-sorting' interacting particles in one-dimension with random initial positions and velocities

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arXiv 2017年

作者： Lega, Joceline Sethuraman, Sunder Young, Alexander L. Department of Mathematics University of Arizona Program in Applied Mathematics University of Arizona

We investigate a one-dimensional system of N particles, initially distributed with random positions and velocities, interacting through binary collisions. The collision rule is such that there is a time after which the N particles do not interact and become sorted according to their velocities. When the collisions are elastic, we derive asymptotic distributions for the final collision time of a single particle and the final collision time of the system as the number of particles approaches infinity, under different assumptions for the initial distributions of the particles' positions and velocities. For comparison, a numerical investigation is carried out to determine how a non-elastic collision rule, which conserves neither momentum nor energy, affects the median collision time of a particle and the median final collision time of the *** Codes 82C22, 62G30 (Primary) 82C23, 62-04, 62E20 (Secondary) Copyright © 2017, The Authors. All rights reserved.

关键词： Velocity

On the relationship between r and R₀ and its role in the bifurcation of stable equilibria of Darwinian matrix models

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Journal of Biological Dynamics 2011年第3期5卷 277-297页

作者： Cushing, J.M. Department of Mathematics Interdisciplinary Program in Applied Mathematics University of Arizona Tucson AZ 85721 United States

If the demographic parameters in a matrix model for the dynamics of a structured population are dependent on a parameter u, then the population growth rate r = r(u) and the net reproductive number R₀ = R₀(u) are functions of u. For a general matrix model, we showthat r and R₀ share critical values and extrema at values u = u^* for which r(u^*) = R₀(u^*) = 1. This allows us to re-interpret, in terms of the more analytically tractable quantity R₀, a fundamental bifurcation theorem for non-linear Darwinian matrix models from the evolutionary game theory that concerns the destabilization of the extinction equilibrium and creation of positive equilibria. Two illustrations are given: a theoretical study of trade-offs between fertility and survivorship in the evolution of an evolutionarily stable strategies and an application to an experimental study of the evolution to a genetic polymorphism. © 2011 Taylor & Francis.

关键词： Bifurcation Darwinian matrix models Equilibria Net reproductive number The evolutionary game theory

An expectation-maximization algorithm for continuous-time hidden Markov models

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arXiv 2021年

作者： Wang, Qingcan Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University United States

We propose a unified framework that extends the inference methods for classical hidden Markov models to continuous settings, where both the hidden states and observations occur in continuous time. Two different settings are analyzed: hidden jump process with a finite state space, and hidden diffusion process with a continuous state space. For each setting, we first estimate the hidden states given the observations and model parameters, showing that the posterior distribution of the hidden states can be described by differential equations in continuous time. We then consider the estimation of unknown model parameters, deriving the continuous-time formulas for the expectation-maximization algorithm. We also propose a Monte Carlo method based on the continuous formulation, sampling the posterior distribution of the hidden states and updating the parameter *** Codes 60J25, 65C40 Copyright © 2021, The Authors. All rights reserved.

关键词： Continuous time systems

Dynamical properties of particulate composites derived from ultradense stealthy hyperuniform sphere packings

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arXiv 2025年

作者： Vanoni, Carlo Kim, Jaeuk Steinhardt, Paul J. Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Stealthy hyperuniform (SHU) many-particle systems are distinguished by a structure factor that vanishes not only at zero wavenumber (as in "standard" hyperuniform systems) but also across an extended range of wavenumbers near the origin. We generate disordered SHU packings of identical and ‘nonoverlapping’ spheres in d-dimensional Euclidean space using a modified collective-coordinate optimization algorithm that incorporates a soft-core repulsive potential between particles in addition to the standard stealthy pair potential. Compared to SHU packings without soft-core repulsions, these SHU packings are ultradense with packing fractions ranging from 0.67-0.86 for d = 2 and 0.47-0.63 for d = 3, spanning a broad spectrum of structures depending on the stealthiness parameter χ. We consider two-phase media composed of hard particles derived from ultradense SHU packings (phase 2) embedded in a matrix phase (phase 1), with varying stealthiness parameter χ and packing fractions . Our main objective is the estimation of the dynamical physical properties of such two-phase media, namely, the effective dynamic dielectric constant and the time-dependent diffusion spreadability, which is directly related to nuclear magnetic relaxation in fluid-saturated porous media. We show through spreadability that two-phase media derived from ultradense SHU packings exhibit faster interphase diffusion due to the higher packing fractions achievable compared to media obtained without soft-core repulsion. The imaginary part of the effective dynamic dielectric constant of SHU packings vanishes at a small wavenumber, implying perfect transparency for the corresponding wavevectors. While a larger packing fraction yields a smaller transparency interval, we show that it also displays a reduced height of the attenuation peak. We also obtain cross-property relations between transparency characteristics and long-time behavior of the spreadability for such two-phase media, showing that one leads to infor

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Level sets of quantum control landscapes

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Level sets of quantum control landscapes

4th International Symposium on Communications, Control, and Signal Processing, ISCCSP-2010

作者： Beltrani, Vincent Dominy, Jason Ho, Tak-San Rabitz, Herschel Department of Chemistry Princeton University NJ 08544 United States Program in Applied and Computational Mathematics Princeton University NJ 08544 United States

ISBN: (纸本)9781424462858

A controlled quantum system possesses a search landscape defined by the observable value as a functional of the control field. Within the search landscape, there exist level sets of controls giving the same observable value. This paper focuses on level sets of the transition probability P i→f. For transition probabilities 0 i→fi→f = 1, a second order D-MORPH algorithm is presented that can explore the perfect control level set. D-MORPH is utilized to identify level set members that exhibit certain desirable secondary characteristics, e.g., minimal pulse ftuence. Numerical simulations for finite level systems are presented to illustrate the variety of control behavior found across level set members. ©2010 IEEE.

关键词： Quantum optics

Inertially driven buckling and overturning of jets in a Hele-Shaw cell

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Physical Review E 2003年第5期68卷 056305-056305页

作者： Adriana I. Pesci Martin A. Porter Raymond E. Goldstein Department of Physics and Program in Applied Mathematics University of Arizona Tucson Arizona 85721 USA

We study a fluid jet descending through stratified surroundings at low Reynolds number in Hele-Shaw flow. The jet buckles and overturns inside a conduit of entrained fluid which supports smooth or unstable traveling waves. A model of the recirculating flow within the conduit shows that buckling and waves arise from Kelvin-Helmholtz instabilities and quantitatively accounts for the main experimental observations. Beyond the onset of the instability, a damped, forced Burgers’ equation obtained from corrections to Darcy’s law for small Reynolds number governs the interface dynamics and supports singularities corresponding to the observed jet overturning and unstable waves.

关键词： Jets

Low Back Muscle Electrical Activity in Pregnant Women of the Second and Third Trimesters 1

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Low Back Muscle Electrical Activity in Pregnant Women of the...

1st International Conference on Computer, Science, Engineering and Technology, ICComSET 2018

作者： Purnamasari, K.D. Widyawati, M.N. Suryono, S. Postgraduate Applied Science Midwifery Program Poltekkes Kemenkes Semarang Indonesia Department of Physics Faculty of Science and Mathematics Diponegoro University Semarang Indonesia

Low back pain during pregnancy causes axial or parasagittal discomfort. These problems are experienced by more than two-thirds of pregnancies. Clinically the measurement of pain was carried out using the Visual Analogue Scale (VAS) method. Unfortunately, the method has low accuracy and is subjective. This paper proposes a method of measuring pain by using electromyography tools so that pain level measurement has a high level of accuracy and is non-invasive to patients. The study was carried out by installing electromyography electrodes on L4 and L5 muscles. The signal is recorded on a digital storage oscilloscope to find out the electrical activity. The results of the recording were evaluated for the action potential and frequency *** study was conducted on 15 respondents with lower back muscle pain in mothers monitored for 1 minute with electromyography. The monitoring results are calculated on average frequency and action potential and then presented in the form of trends. The results of the analysis of pain level measurement were compared with the results of VAS measurements. From the results of this study indicate that the trend of the action potential and signal frequency using electromyography shows different results when respondents experience the same pain scale on VAS. The value generated in the measurement using VAS is only a pain number scale that is felt by the respondent where the value is subjective. © 2019 IOP Publishing Ltd. All rights reserved.

关键词： Cathode ray oscilloscopes