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Continuity Equation and Characteristic Flow for Scalar Hencky Plasticity Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220723
J.F. Babadjian, G. A. FrancfortWe investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form curlμσ=0 where 𝜇 is a nonnegative measure and 𝜎 a twodimensional divergencefree unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic

Trainability and Accuracy of Artificial Neural Networks: An Interacting Particle System Approach Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220721
Grant Rotskoff, Eric VandenEijndenNeural networks, a central tool in machine learning, have demonstrated remarkable, high fidelity performance on image recognition and classification tasks. These successes evince an ability to accurately represent highdimensional functions, but rigorous results about the approximation error of neural networks after training are few. Here we establish conditions for global convergence of the standard

Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220721
Peter Humphries, Maksym RadziwiłłWe asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a “microsquare”

Proof of Modulational Instability of Stokes Waves in Deep Water Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220715
Huy Q. Nguyen, Walter A. StraussIt is proven that smallamplitude steady periodic water waves with infinite depth are unstable with respect to longwave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and selfcontained approach to prove the spectral modulational instability

Large ∣k∣ Behavior of Complex Geometric Optics Solutions to dbar Problems Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220706
Christian Klein, Johannes Sjöstrand, Nikola StoilovComplex geometric optics solutions to a system of dbar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable DaveyStewartson II equations are studied for large values of the spectral parameter k. For potentials q∈⟨⋅⟩−2Hs(C)q∈⋅−2Hsℂ for some s∈]1,2]s∈1,2 , it is shown that the solution converges as the geometric series in 1/ks−11/ks−1

Global WellPosedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220615
Wilfrid Gangbo, Alpár R. MészárosThis manuscript constructs global in time solutions to master equations for potential mean field games. The study concerns a class of Lagrangians and initial data functions that are displacement convex, and so this property may be in dichotomy with the socalled Lasry–Lions monotonicity, widely considered in the literature. We construct solutions to both the scalar and vectorial master equations in

Viscosity Limits for ZerothOrder Pseudodifferential Operators Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220615
Jeffrey Galkowski, Maciej ZworskiMotivated by the work of Colin de Verdière and SaintRaymond on spectral theory for zerothorder pseudodifferential operators on tori, we consider viscosity limits in which zerothorder operators, P, are replaced by P + iν Δ, ν > 0. By adapting the Helffer–Sjöstrand theory of scattering resonances, we show that, in a complex neighbourhood of the continuous spectrum, eigenvalues of P + iν Δ have limits

Sobolev Inequalities in Manifolds with Nonnegative Curvature Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220606
Simon BrendleWe prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a MichaelSimon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold. © 2022 Wiley Periodicals LLC.

A PDE Approach to the Prediction of a Binary Sequence with Advice from Two HistoryDependent Experts Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220606
Nadejda Drenska, Robert V. KohnThe prediction of a binary sequence is a classic example of online machine learning. We like to call it the “stock prediction problem,” viewing the sequence as the price history of a stock that goes up or down one unit at each time step. In this problem, an investor has access to the predictions of two or more “experts,” and strives to minimize her finaltime regret with respect to the bestperforming

Formation of Points Shocks for 3D Euler Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220527
Tristan Buckmaster, Steve Shkoller, Vlad VicolWe consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolevclass initial data that are a small L∞ perturbation

Shock Formation and Vorticity Creation for 3d Euler Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220525
Tristan Buckmaster, Steve Shkoller, Vlad VicolWe analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which

Linear Stability of Pipe Poiseuille Flow at High Reynolds Number Regime Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220515
Qi Chen, Dongyi Wei, Zhifei ZhangIn this paper, we prove the linear stability of the pipe Poiseuille flow for general perturbations at high Reynolds number regime. This has been a longstanding problem since the experiments of Reynolds in 1883. Our work lays a foundation for the theoretical analysis of hydrodynamic stability of pipe flow, which is one of the oldest yet unsolved problems in fundamental fluid dynamics. © 2022 Wiley

On the YauTianDonaldson Conjecture for Generalized KählerRicci Soliton Equations Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220514
Jiyuan Han, Chi LiLet (X,D) be a polarized log variety with an effective holomorphic torus action, and Θ be a closed positive torus invariant (1,1) current. For any smooth positive function g defined on the moment polytope of the torus action, we study the MongeAmpère equations that correspond to generalized and twisted KählerRicci gsolitons. We prove a version of the YauTianDonaldson (YTD) conjecture for these

Corrigendum and Addendum: Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220422
Chiranjib Mukherjee, S. R. S. VaradhanThe proof of [7, theorem 4.8] for large coupling constant α has a gap (while the proof remains correct for small α). We correct this error by giving a direct proof of [7, theorem 4.8], which holds for any coupling parameter α > 0. Consequently, the main results of [7] on existence and identification of the Polaron measure and its CLT now hold for any α > 0. © 2022 Wiley Periodicals LLC.

Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220417
Anya Katsevich, Afonso S. BandeiraWe derive an asymptotic expansion for the loglikelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signaltonoise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as ExpectationMaximization (EM). We show that likelihood optimization in the

IsotropicNematic Phase Transition and Liquid Crystal Droplets Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220416
Fanghua Lin, Changyou WangLiquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (or OseenFrank energy minimizers in general), free interfaces, and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations.

Online Prediction with HistoryDependent Experts: The General Case Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220405
Nadejda Drenska, Jeff CalderWe study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an

Plateau's Problem as a Singular Limit of Capillarity Problems Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220322
Darren King, Salvatore Stuvard, Francesco MaggiSoap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection

SISTA: Learning Optimal Transport Costs under Sparsity Constraints Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220320
Guillaume Carlier, Arnaud Dupuy, Alfred Galichon, Yifei SunIn this paper, we describe a novel iterative procedure called SISTA to learn the underlying cost in optimal transport problems. SISTA is a hybrid between two classical methods, coordinate descent (“S”inkhorn) and proximal gradient descent (“ISTA”). It alternates between a phase of exact minimization over the transport potentials and a phase of proximal gradient descent over the parameters of the transport

Uniqueness of TwoBubble Wave Maps in High Equivariance Classes Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220320
Jacek Jendrej, Andrew LawrieThis is the second part of a twopaper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions.

Large Deviations for Intersections of Random Walks Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220310
Asselah Asselah, Bruno SchapiraWe prove a large deviations principle for the number of intersections of two independent infinitetime ranges in dimension 5 and greater, improving upon the moment bounds of Khanin, Mazel, Shlosman, and Sinaï [9]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen, and den Hollander [15], who analyzed this question for the Wiener sausage in the finitetime horizon. The

Existence and Uniqueness of Green's Functions to Nonlinear Yamabe Problems Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220310
Yanyan Li, Luc NguyenFor a given finite subset S of a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M\S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As

Local Minimizers with Unbounded Vorticity for the 2D GinzburgLandau Functional Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220301
Andres Contreras, Robert L. JerrardA central focus of GinzburgLandau theory is the understanding and characterization of vortex configurations. On a bounded domain Ω⊆ℝ2, global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit ϵ→0, where ϵ>0 is the inverse of the GinzburgLandau parameter. A notable open problem is whether there are solutions of the GinzburgLandau

Fast Reaction Limit with Nonmonotone Reaction Function Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220223
Benoît Perthame, Jakub SkrzeczkowskiWe analyse the fast reaction limit in the reactiondiffusion system with nonmonotone reaction function and one nondiffusing component. As speed of reaction tends to infinity, the concentration of the nondiffusing component exhibits fast oscillations. We identify precisely its Young measure which, as a byproduct, proves strong convergence of the diffusing component, a result that is not obvious from

Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220212
Massimiliano Berti, Roberto Feola, Fabio PusateriWe consider the gravity water waves system with a periodic onedimensional interface in infinite depth and give a rigorous proof of a conjecture of DyachenkoZakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a longtime

Generalized TAP Free Energy Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220126
WeiKuo Chen, Dmitry Panchenko, Eliran SubagWe consider the mixed pspin meanfield spin glass model with Ising spins and investigate its free energy in the spirit of the TAP approach, named after Thouless, Anderson, and Palmer [67]. More precisely, we define and compute the generalized TAP correction, and establish the corresponding generalized TAP representation for the free energy. In connection with physicists’ replica theory, we introduce

Dissipative Euler Flows for Vortex Sheet Initial Data without Distinguished Sign Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220113
Francisco Mengual, László SzékelyhidiWe construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a “turbulence” zone which grows linearly

Prescribing Morse Scalar Curvatures: Pinching and Morse Theory Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211227
Andrea Malchiodi, Martin MayerWe consider the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimension . We prove new existence results using Morse theory and some analysis on blowingup solutions under suitable pinching conditions on the curvature function. We also provide new nonexistence results showing the sharpness of some of our assumptions, both in terms of the dimension

Emergence of Concentration Effects in the Variational Analysis of the NClock Model Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211222
Marco Cicalese, Gianluca Orlando, Matthias RufWe investigate the relationship between the Nclock model (also known as planar Potts model or model) and the XY model (at zero temperature) through a Γconvergence analysis of a suitable rescaling of the energy as both the number of particles and N diverge. We prove the existence of rates of divergence of N for which the continuum limits of the two models differ. With the aid of Cartesian currents

Upper Tail Large Deviations of Regular Subgraph Counts in ErdősRényi Graphs in the Full Localized Regime Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211222
Anirban Basak, Riddhipratim BasuFor a regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in , an ErdősRényi graph on n vertices with edge probability p, has generated significant interest. For and , where is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation

Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211222
Shixiao Willing Jiang, John HarlimIn this paper, we extend the class of kernel methods, the socalled diffusion maps (DM) and its local kernel variants to approximate secondorder differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the ghost point diffusion maps (GPDM) estimator on an extended manifold, identified by the set of point clouds

A Sharp Inequality on the Exponentiation of Functions on the Sphere Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211220
SunYung Alice Chang, Changfeng GuiIn this paper we show a new inequality that generalizes to the unit sphere the LebedevMilin inequality of the exponentiation of functions on the unit circle. It may also be regarded as the counterpart on the sphere of the second inequality in the Szegö limit theorem on the Toeplitz determinants on the circle. On the other hand, this inequality is also a variant of several classical inequalities of

Likelihood landscape and maximum likelihood estimation for the discrete orbit recovery model Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211215
Zhou Fan, Yi Sun, Tianhao Wang, Yihong WuWe study the nonconvex optimization landscape for maximum likelihood estimation in the discrete orbit recovery model with Gaussian noise. This is a statistical model motivated by applications in molecular microscopy and image processing, where each measurement of an unknown object is subject to an independent random rotation from a known rotational group. Equivalently, it is a Gaussian mixture model

Regularity of Free Boundary Minimal Surfaces in Locally Polyhedral Domains Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20220124
Nick Edelen, Chao LiWe prove an Allardtype regularity theorem for freeboundary minimal surfaces in Lipschitz domains locally modeled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate freeboundary plane, then the surface is 𝐶1,𝛼C1,α graphical over this plane. We apply our theorem to prove partial regularity results for freeboundary minimizing hypersurfaces, and relative

NavierStokes Equations in Gas Dynamics: Green's Function, Singularity, and WellPosedness Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211216
TaiPing Liu, ShihHsien YuThe purpose of the present article is to study weak solutions of viscous conservation laws in physics. We are interested in the wellposedness theory and the propagation of singularity in the weak solutions for the initial value problem. Our approach is to convert the differential equations into integral equations on the level of weak solutions. This depends on exact analysis of the associated linear

C2 Regularity of the Surface Tension for the ∇ϕ Interface Model Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211216
Scott Armstrong, Wei WuWe consider the ∇ϕ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least C2,β for some β > 0. We also prove a fluctuationdissipation relation by identifying its Hessian with

Central Limit Theorem for Linear Eigenvalue Statistics of NonHermitian Random Matrices Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211201
Giorgio Cipolloni, László Erdős, Dominik SchröderWe consider large nonHermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements

Thin Loewner Carpets and Their Quasisymmetric Embeddings in S2 Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211123
Jeff Cheeger, Sylvester ErikssonBiqueA carpet is a metric space that is homeomorphic to the standard Sierpiński carpet in , or equivalently, in S2. A carpet is called thin if its Hausdorff dimension is . A metric space is called QLoewner if its Qdimensional Hausdorff measure is QAhlfors regular and if it satisfies a Poincaré inequality. As we will show, QLoewner planar metric spaces are always carpets, and admit quasisymmetric embeddings

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211108
Xavier TolsaLet be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in and continuous in , which vanishes in a relatively open subset ; moreover, the normal derivative vanishes in a subset of with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics

Fully Nonlinear Equations with Applications to Grad Equations in Plasma Physics Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211108
Luis A. Caffarelli, Ignacio TomasettiIn this paper we generalize an equation studied by Mossino and Temam in [7], to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad in [4], to model the behavior of plasma confined in a toroidal vessel called TOKAMAK. We prove existence of a viscosity solution and regularity up to for any (we improve this regularity

Separable Hamiltonian PDEs and Turning Point Principle for Stability of Gaseous Stars Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211108
Zhiwu Lin, Chongchun ZengWe consider stability of nonrotating gaseous stars modeled by the EulerPoisson system. Under general assumptions on the equation of states, we proved a turning point principle (TPP) that the stability of the stars is entirely determined by the mass–radius curve parametrized by the center density. In particular, the stability can only change at extrema (i.e., local maximum or minimum points) of the

SubGaussian Matrices on Sets: Optimal Tail Dependence and Applications Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20211029
Halyun Jeong, Xiaowei Li, Yaniv Plan, Ozgur YilmazRandom linear mappings are widely used in modern signal processing, compressed sensing, and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to . Thus, the performance of these mappings is usually

On Ill and WellPosedness of Dissipative Martingale Solutions to Stochastic 3D Euler Equations Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210928
Martina Hofmanová, Rongchan Zhu, Xiangchan ZhuWe are concerned with the question of wellposedness of stochastic, threedimensional, incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak–strong uniqueness; (iii) nonuniqueness in law; (iv) existence of a strong Markov solution; (v) nonuniqueness of strong Markov solutions: all hold true within this class. Moreover

The ContinuousTime Lace Expansion Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210904
David Brydges, Tyler Helmuth, Mark HolmesWe derive a continuoustime lace expansion for a broad class of selfinteracting continuoustime random walks. Our expansion applies when the selfinteraction is a sufficiently nice function of the local time of a continuoustime random walk. As a special case we obtain a continuoustime lace expansion for a class of spin systems that admit continuoustime random walk representations.

The Batchelor Spectrum of Passive Scalar Turbulence in Stochastic Fluid Mechanics at Fixed Reynolds Number Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210907
Jacob Bedrossian, Alex Blumenthal, Samuel PunshonSmithIn 1959 Batchelor predicted that the stationary statistics of passive scalars advected in fluids with small diffusivity k should display a power spectrum along an inertial range contained in the viscousconvective range of the fluid model. This prediction has been extensively tested, both experimentally and numerically, and is a core prediction of passive scalar turbulence.

Plateau's Problem as a Singular Limit of Capillarity Problems Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210830
Darren King, Francesco Maggi, Salvatore StuvardSoap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection

Universality Near the Gradient Catastrophe Point in the Semiclassical SineGordon Equation Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210825
BingYing Lu, Peter MillerWe study the semiclassical limit of the sineGordon (sG) equation with below threshold pure impulse initial data of KlausShaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava, and Klein, we found that in a neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are

Mathematical Modeling and Analysis of Spatial Neuron Dynamics: Dendritic Integration and Beyond Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210824
Songting Li, David W. McLaughlin, Douglas ZhouNeurons compute by integrating spatiotemporal excitatory (E) and inhibitory (I) synaptic inputs received from the dendrites. The investigation of dendritic integration is crucial for understanding neuronal information processing. Yet quantitative rules of dendritic integration and their mathematical modeling remain to be fully elucidated. Here neuronal dendritic integration is investigated by using

Uniformly Positive Correlations in the Dimer Model and Macroscopic Interacting SelfAvoiding Walk in ℤd, d ≥ 3 Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210818
Lorenzo TaggiOur first main result is that correlations between monomers in the dimer model in do not decay to 0 when . This is the first rigorous result about correlations in the dimer model in dimensions greater than 2 and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied

Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210816
Jess Banks, Archit Kulkarni, Satyaki Mukherjee, Nikhil SrivastavaA matrix is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E. B. Davies: for each , every matrix is at least close to one whose eigenvectors have condition number at worst , for some depending only on n. We

EntropyBounded Solutions to the OneDimensional Heat Conductive Compressible NavierStokes Equations with Far Field Vacuum Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210802
Jinkai Li, Zhouping XinIn the presence of vacuum, the physical entropy for polytropic gases behave singularly, and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the onedimensional

Arnold Diffusion, Quantitative Estimates, and Stochastic Behavior in the ThreeBody Problem Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210729
Maciej J. Capiński, Marian GideaWe consider a class of autonomous Hamiltonian systems subject to small, timeperiodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is nonzero.

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210722
Andrew Lawrie, Jonas Lührmann, SungJin Oh, Sohrab ShahshahaniWe consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this

Moderately Discontinuous Homology Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210722
Javier Fernández de Bobadilla, Sonja Heinze, María Pe Pereira, José Edson SampaioWe introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow “moderately discontinuous” chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group

Minimizers for the Thin OnePhase Free Boundary Problem Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210716
Max Engelstein, Aapo Kauranen, Martí Prats, Georgios Sakellaris, Yannick SireWe consider the “thin onephase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in plus the area of the positivity set of that function in . We establish full regularity of the free boundary for dimensions , prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the

Upper Tail Large Deviations in First Passage Percolation Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210614
Riddhipratim Basu, Allan Sly, Shirshendu GangulyFor first passage percolation on with i.i.d. bounded edge weights, we consider the upper tail large deviation event, i.e., the rare situation where the first passage time between two points at distance n is macroscopically larger than typical. It was shown by Kesten [24] that the probability of this event decays as . However, the question of existence of the rate function, i.e., whether the logprobability

The Generalization Error of Random Features Regression: Precise Asymptotics and the Double Descent Curve Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210606
Song Mei, Andrea MontanariDeep learning methods operate in regimes that defy the traditional statistical mindset. Neural network architectures often contain more parameters than training samples, and are so rich that they can interpolate the observed labels, even if the latter are replaced by pure noise. Despite their huge complexity, the same architectures achieve small generalization error on real data.

Mean Convex Mean Curvature Flow with Free Boundary Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210602
Nick Edelen, Robert Haslhofer, Mohammad N. Ivaki, Jonathan J. ZhuIn this paper, we generalize White's regularity and structure theory for meanconvex mean curvature flow [45, 46, 48] to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish such a bound via the maximum principle for a tripleapproximation scheme

Spatially Inhomogeneous Evolutionary Games Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210513
Luigi Ambrosio, Massimo Fornasier, Marco Morandotti, Giuseppe SavaréWe introduce and study a mean‐field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion.

Quantitative Linearization Results for the MongeAmpère Equation Comm. Pure Appl. Math. (IF 2.774) Pub Date : 20210502
Michael Goldman, Martin Huesmann, Felix OttoThis paper is about quantitative linearization results for the MongeAmpère equation with rough data. We develop a largescale regularity theory and prove that if a measure μ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson