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Approximating algebraic space curves by circular arcs 1

7th International Conference on Curves and Surfaces, Curves and Surfaces 2010

作者： Béla, Szilvia Jüttler, Bert Doctoral Program in Computational Mathematics Johannes Kepler University 4040 Linz Austria Institute of Applied Geometry Johannes Kepler University Altenberger Str. 69 4040 Linz Austria

ISBN: (数字)9783642274138

ISBN: (纸本)9783642274121

We introduce a new method to approximate algebraic space curves. The algorithm combines a subdivision technique with local approximation of piecewise regular algebraic curve segments. The local technique computes pairs of polynomials with modified Taylor expansions and generates approximating circular arcs. We analyze the connection between the generated approximating arcs and the osculating circles of the algebraic curve. © 2012 Springer-Verlag.

关键词： Approximation algorithms

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ON THE CONVERGENCE OF GRADIENT DESCENT TRAINING FOR TWO-LAYER RELU-NETWORKS IN THE MEAN FIELD REGIME

arXiv

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arXiv 2020年

作者： Wojtowytsch, Stephan Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544

We describe a necessary and sufficient condition for the convergence to minimum Bayes risk when training two-layer ReLU-networks by gradient descent in the mean field regime with omni-directional initial parameter distribution. This article extends recent results of Chizat and Bach to ReLU-activated networks and to the situation in which there are no parameters which exactly achieve MBR. The condition does not depend on the initalization of parameters and concerns only the weak convergence of the realization of the neural network, not its parameter *** Codes 35Q68, 68T07, 49Q22, 35F20 Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

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Mesoscopic model for microscale hydrodynamics and interfacial phenomena: Slip, films, and contact-angle hysteresis

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Physical Review E 2013年第1期87卷 013302-013302页

作者： Carlos E. Colosqui Michail E. Kavousanakis Athanasios G. Papathanasiou Ioannis G. Kevrekidis Department of Chemical and Biological Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We present a model based on the lattice Boltzmann equation that is suitable for the simulation of dynamic wetting. The model is capable of exhibiting fundamental interfacial phenomena such as weak adsorption of fluid on the solid substrate and the presence of a thin surface film within which a disjoining pressure acts. Dynamics in this surface film, tightly coupled with hydrodynamics in the fluid bulk, determine macroscopic properties of primary interest: the hydrodynamic slip; the equilibrium contact angle; and the static and dynamic hysteresis of the contact angles. The pseudo-potentials employed for fluid-solid interactions are composed of a repulsive core and an attractive tail that can be independently adjusted. This enables effective modification of the functional form of the disjoining pressure so that one can vary the static and dynamic hysteresis on surfaces that exhibit the same equilibrium contact angle. The modeled fluid-solid interface is diffuse, represented by a wall probability function that ultimately controls the momentum exchange between solid and fluid phases. This approach allows us to effectively vary the slip length for a given wettability (i.e., a given static contact angle) of the solid substrate.

关键词： HYDRODYNAMICS INTERFACES (Physical sciences) THIN films HYSTERESIS MATHEMATICAL models LATTICE Boltzmann methods DYNAMICAL systems WETTING

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Spectral bounds for independent cascade model with sensitive edges

Spectral bounds for independent cascade model with sensitive...

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Annual Conference on Information Sciences and Systems (CISS)

作者： Eun Jee Lee Sudeep Kamath Emmanuel Abbe Sanjeev R. Kulkarni Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University

This paper studies independent cascade models where influence propagates from seed-nodes along edges with independent probabilities. Upper-bounds for the expected number of influenced nodes were previously proposed using the spectral norm of a Hazard matrix. However, these bounds turn out loose in many cases, in particular with respect to sensitive edges such as bottlenecks, seed adjacent, and high probability edges. This paper proposes a similar bound that improves in such cases by handling sensitives edges more carefully.

关键词： Hazards Upper bound Stochastic processes Information science Merging computational modeling Mathematical model

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Pseudo prolate spheroidal functions 11

Pseudo prolate spheroidal functions

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11th International Conference on Sampling Theory and Applications, SampTA 2015

作者： Daniel Abreu, Luis Pereira, Joao M. Austrian Academy of Sciences Acoustics Research Institute Wohllebengasse 12-14 ViennaA-1040 Austria Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

ISBN: (纸本)9781467373531

Let Dt and BΩ denote the operators which cut the time content outside T and the frequency content outside Ω, respectively. The prolate spheroidal functions are the eigen-functions of the operator Ρτ,Ω = DtBΩDt. With the aim of formulating in precise mathematical terms the notion of Nyquist rate, Landau and Pollack have shown that, asymptotically, the number of such functions with eigenvalue close to one is ≈ TΩ/2π. We have recently revisited this problem with a new approach: instead of counting the number of eigenfunctions with eigenvalue close to one, we count the maximum number of orthogonal Ε-pseudoeigenfunctions with Ε-pseudoeigenvalue one. Precisely, we count how many orthogonal functions have a maximum of energy e outside the domain T x Ω, in the sense that PT, Ωf - f2 ≤ Ε. We have recently discovered that the sharp asymptotic number is ≈ (1 - Ε)-1TΩ/2π. The proof involves an explicit construction of the pseudoeigenfunctions of PT, Ω. When T and Ω are intervals we call them pseudo prolate spheroidal functions. In this paper we explain how they are constructed. © 2015 IEEE.

关键词： Orthogonal functions

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Correlations in interacting electron liquids: Many-body statistics and hyperuniformity

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Physical Review B 2024年第10期110卷 104201页

作者： Haina Wang Rhine Samajdar Salvatore Torquato Department of Chemistry Department of Physics Princeton Center for Theoretical Science Princeton Materials Institute Program in Applied and Computational Mathematics

Disordered hyperuniform many-body systems are exotic states of matter with novel optical, transport, and mechanical properties. These systems are characterized by an anomalous suppression of large-scale density fluctuations compared to ordinary liquids. The structure factor of disordered hyperuniform systems often obeys the scaling relation S(k)∼Bkα with B,α>0 in the limit k→0. Ground states of d-dimensional free fermionic gases, which are fundamental models for many metals and semiconductors, are key examples of quantum disordered hyperuniform states with important connections to random matrix theory. However, the effects of electron-electron interactions as well as the polarization of the electron liquid on hyperuniformity have not been explored thus far. In this paper, we systematically address these questions by deriving the analytical small-k behaviors (and, associated, α and B) of the total and spin-resolved structure factors of quasi-one-dimensional, two-dimensional, and three-dimensional electron liquids for varying polarizations and interaction parameters. We validate that these equilibrium disordered ground states are hyperuniform, as dictated by the fluctuation-compressibility relation. Interestingly, free fermions, partially polarized interacting fermions, and fully polarized interacting fermions are characterized by different values of the small-k scaling exponent α and coefficient B. In particular, partially polarized fermionic liquids exhibit a unique form of multihyperuniformity, in which the net configuration exhibits a stronger form of hyperuniformity (i.e., larger α) than each individual spin component. The detailed theoretical analysis of such small-k behaviors enables the construction of corresponding equilibrium classical systems under effective one- and two-body interactions that mimic the pair statistics of quantum electron liquids. Our paper thus reveals that highly unusual hyperuniform and multihyperuniform states can be achieved in simple

关键词： Electronic structure Quantum many-body systems Two-dimensional electron gas

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Positive- and negative-effective-viscosity phenomena in isotropic and anisotropic Beltrami flows

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Physical Review A 1986年第1期34卷 381-381页

作者： B. J. Bayly V. Yakhot Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton New Jersey 08544

A field-theoretic approach, analogous to Kraichnan’s direct-interaction approximation, to the stability theory of complex three-dimensional flows is developed. The long-wavelength stability of a class of Beltrami flows in an unbounded, viscous fluid is considered. We examine two flows in detail, to illustrate the effects of strong isotropy versus strong anisotropy in the basic flow. The effect of the small-scale flow on the long-wavelength perturbations may be interpreted as an effective viscosity. Using diagrammatic techniques, we construct the first-order smoothing and direct-interaction approximations for the perturbation dynamics. It is argued that the effective viscosity for the isotropic flow is always positive, and approaches a value independent of the molecular viscosity in the high-Reynolds-number limit; this flow is thus stable to long-wavelength disturbances. The anisotropic flow has negative effective viscosity for some orientations of the disturbance, and is therefore unstable, when its Reynolds number exceeds √2 .

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

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

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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

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Landauʼs necessary density conditions for the Hankel transform

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Journal of Functional Analysis 2012年第4期262卷 1845-1866页

作者： Luís Daniel Abreu Afonso S. Bandeira Department of Mathematics of University of Coimbra 3001-454 Coimbra Portugal Program in Applied and Computational Mathematics Princeton University NJ 08544 USA

We will prove an analogue of Landauʼs necessary conditions [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier–Bessel frames are obtained.

关键词： Sampling and interpolation Beurling–Landau density Hankel transform Bessel functions Fourier–Bessel frames

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Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

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Communications in computational Physics 2014年第3期15卷 692-711页

作者： Xu Qian Yaming Chen Songhe Song Department of Mathematics and Systems Science and State Key Laboratory of High Performance ComputingNational University of Defense TechnologyChangsha 410073P.R.China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA School of Mathematical Sciences Queen Mary University of LondonLondon E14NSUK

In this paper,we propose a wavelet collocation splitting(WCS)method,and a Fourier pseudospectral splitting(FPSS)method as comparison,for solving onedimensional and two-dimensional Schrödinger equations with variable coefficients in quantum *** two methods can preserve the intrinsic properties of original problems as much as *** splitting technique increases the computational ***,the error estimation and some conservative properties are *** is proved to preserve the charge conservation *** global energy and momentum conservation laws can be preserved under several *** experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

关键词： Schrödinger equation wavelet collocation method splitting technique conservative property

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