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  • Equivalence of the Ellipticity Conditions for Geometric Variational Problems
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2020-03-17
    Antonio De Rosa; Sławomir Kolasiński

    We exploit the so‐called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin and proved to be necessary and sufficient for the validity of the anisotropic counterpart of the Allard rectifiability theorem. In particular, we address an open question of this seminal work, showing that the atomic condition implies the strict Almgren geometric ellipticity condition. © 2020 Wiley Periodicals

    更新日期:2020-03-20
  • Inchworm Monte Carlo Method for Open Quantum Systems
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2020-03-06
    Zhenning Cai; Jianfeng Lu; Siyao Yang

    We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro‐differential equation formulation for open quantum systems, which illuminates the mathematical structure of the inchworm algorithm. This new formulation

    更新日期:2020-03-20
  • Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2020-02-26
    Matteo Bonforte; Alessio Figalli

    We investigate the homogeneous Dirichlet problem for the fast diffusion equation ut = Δum, posed in a smooth bounded domain Ω ⊂ ℝN, in the exponent range ms = (N − 2)+/(N + 2) < m < 1. It is known that bounded positive solutions extinguish in a finite time T > 0, and also that they approach a separate variable solution u(t, x) ∼ (T − t)1/(1 − m)S(x) as t → T−, where S belongs to the set of solutions

    更新日期:2020-03-20
  • Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2020-01-10
    Federico Camia; Jianping Jiang; Charles M. Newman

    We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two‐point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near‐critical scaling limit Euclidean magnetization field. For the lattice model with a = 1

    更新日期:2020-03-20
  • Determination of Space‐Time Structures from Gravitational Perturbations
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2020-01-01
    Gunther Uhlmann; Yiran Wang

    We study inverse problems for the Einstein equations with source fields in a general form. Under a microlocal linearization stability condition, we show that by generating small gravitational perturbations and measuring the responses near a freely falling observer, one can uniquely determine the background Lorentzian metric up to isometries in a region where the gravitational perturbations can travel

    更新日期:2020-03-20
  • Singular Abreu Equations and Minimizers of Convex Functionals with a Convexity Constraint
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-12-24
    Nam Q. Le

    We study the solvability of second boundary value problems of fourth‐order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the monopolist's problem in economics. The right‐hand

    更新日期:2020-03-20
  • Feeble Fish in Time‐Dependent Waters and Homogenization of the G‐equation
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-12-03
    Dmitri Burago; Sergei Ivanov; Alexei Novikov

    We study the following control problem. A fish with bounded aquatic locomotion speed swims in fast waters. Can this fish, under reasonable assumptions, get to a desired destination? It can, even if the flow is time dependent. Moreover, given a prescribed sufficiently large time t, it can be there at exactly the time t. The major difference from our previous work is the time dependence of the flow.

    更新日期:2020-03-20
  • A Proof of Sznitman's Conjecture about Ballistic RWRE
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-11-27
    Enrique Guerra; Alejandro F. Ramı́rez

    We consider random walk in a uniformly elliptic i.i.d. random environment in ℤd for d ≥ 2. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, Sznitman defined the so‐called conditions (T) and (T′)

    更新日期:2020-03-20
  • Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-11-27
    Peter Bella; Mathias Schäffner

    We study local regularity properties for solutions of linear, nonuniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger. We then apply the deterministic regularity results to the corrector equation in

    更新日期:2020-03-20
  • Geometry and Temperature Chaos in Mixed Spherical Spin Glasses at Low Temperature: The Perturbative Regime
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-11-19
    Gérard Ben Arous; Eliran Subag; Ofer Zeitouni

    We study the Gibbs measure of mixed spherical p‐spin glass models at low temperature, in (part of) the 1‐RSB regime, including, in particular, models close to pure in an appropriate sense. We show that the Gibbs measure concentrates on spherical bands around deep critical points of the (extended) Hamiltonian restricted to the sphere of radius , where is the rightmost point in the support of the overlap

    更新日期:2020-03-20
  • Theoretical Justification and Error Analysis for Slender Body Theory
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-11-18
    Yoichiro Mori; Laurel Ohm; Daniel Spirn

    Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three‐dimensional fiber, in part due to the fact that simply prescribing

    更新日期:2020-03-20
  • Explicit Construction of RIP Matrices Is Ramsey‐Hard
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-11-11
    David Gamarnik

    Matrices Φ ∈ ℝn × p satisfying the restricted isometry property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for n = logO(1)p, the explicit deteministic construction of such matrices defied the repeated efforts, and most of the known approaches hit the so‐called sparsity bottleneck. The notable

    更新日期:2020-03-20
  • Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-10-30
    Marta Lewicka; Danka Lučić

    We study the noneuclidean (incompatible) elastic energy functionals in the description of prestressed thin films at their singular limits (Γ‐limits) as h → 0 in the film's thickness h. First, we extend the prior results to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the “nonoscillatory” case). Second, we analyze a more general class of incompatibilities

    更新日期:2020-03-20
  • Global Bifurcation of Rotating Vortex Patches
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-08-05
    Zineb Hassainia; Nader Masmoudi; Miles H. Wheeler

    We rigorously construct continuous curves of rotating vortex patch solutions to the two‐dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [30, 38] of singular limiting patches with 90 corners at

    更新日期:2020-03-20
  • Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-06-30
    Elia Brué; Daniele Semola

    We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N) metric measure spaces; regularity is understood with respect to a newly defined quasi‐metric built from the Green function of the Laplacian. Its main application is that RCD(K, N) spaces have constant dimension. In this way we generalize to such an abstract framework a result proved by Colding‐Naber for Ricci

    更新日期:2020-03-20
  • On the Relation between Enhanced Dissipation Timescales and Mixing Rates
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-05-16
    Michele Coti Zelati; Matias G. Delgadino; Tarek M. Elgindi

    We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect that causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a timescale faster than the purely

    更新日期:2020-03-20
  • Lipschitz Bounds and Nonuniform Ellipticity
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-12-24
    Lisa Beck; Giuseppe Mingione

    We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve

    更新日期:2020-03-20
  • On the Global Behavior of Weak Null Quasilinear Wave Equations
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-12-24
    Yu Deng; Fabio Pusateri

    We consider a subclass of those quasilinear wave equations in 3 + 1 space‐time dimensions that satisfy the “weak null condition” as defined by Lindblad and Rodnianski , and study the large‐time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is . Global solutions for such equations have been constructed by Lindblad and Alinhac. Our main results are the

    更新日期:2020-03-20
  • Quantization in Geometric Pluripotential Theory
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2019-07-23
    Tamás Darvas; Chinh H. Lu; Yanir A. Rubinstein

    The space of Kähler metrics, on the one hand, can be approximated by subspaces of algebraic metrics, while, on the other hand, it can also be enlarged to finite‐energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite‐dimensional spaces of Fubini‐Study metrics of Kähler

    更新日期:2020-03-20
  • Nonlinear Dynamics of Neuronal Excitability, Oscillations, and Coincidence Detection.
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2013-09-01
    John Rinzel,Gemma Huguet

    We review some widely studied models and firing dynamics for neuronal systems, both at the single cell and network level, and dynamical systems techniques to study them. In particular, we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades: single-cell excitability and bursting. We review the mathematical framework for three types of excitability

    更新日期:2019-11-01
  • Vector Diffusion Maps and the Connection Laplacian.
    Comm. Pure Appl. Math. (IF 3.138) Pub Date : 2012-08-01
    A Singer,H-T Wu

    We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat

    更新日期:2019-11-01
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