We 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 two-dimensional divergence-free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic

Neural 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 high-dimensional 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

We 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”

It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave 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 self-contained approach to prove the spectral modulational instability

Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson 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/|k|s−11/ks−1

This 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 so-called Lasry–Lions monotonicity, widely considered in the literature. We construct solutions to both the scalar and vectorial master equations in

Motivated by the work of Colin de Verdière and Saint-Raymond on spectral theory for zeroth-order pseudodifferential operators on tori, we consider viscosity limits in which zeroth-order 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

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon 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.

The 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 final-time regret with respect to the best-performing

We 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 Sobolev-class initial data that are a small L∞ perturbation

We 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

In this paper, we prove the linear stability of the pipe Poiseuille flow for general perturbations at high Reynolds number regime. This has been a long-standing 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

Let (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 Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci g-solitons. We prove a version of the Yau-Tian-Donaldson (YTD) conjecture for these

The 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.

We derive an asymptotic expansion for the log-likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal-to-noise 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 Expectation-Maximization (EM). We show that likelihood optimization in the

Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (or Oseen-Frank 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.

We 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

Soap 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

In 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

This is the second part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions.

We prove a large deviations principle for the number of intersections of two independent infinite-time 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 finite-time horizon. The

For 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

A central focus of Ginzburg-Landau 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 Ginzburg-Landau parameter. A notable open problem is whether there are solutions of the Ginzburg-Landau

We analyse the fast reaction limit in the reaction-diffusion 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 by-product, proves strong convergence of the diffusing component, a result that is not obvious from

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko-Zakharov [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 long-time

We consider the mixed p-spin mean-field 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

We 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

We 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 blowing-up 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

We investigate the relationship between the N-clock model (also known as planar Potts model or -model) and the XY model (at zero temperature) through a Γ-convergence 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

For a -regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in , an Erdős-Rényi 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

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and its local kernel variants to approximate second-order 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

In this paper we show a new inequality that generalizes to the unit sphere the Lebedev-Milin 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

We 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

We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modeled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free-boundary plane, then the surface is 𝐶1,𝛼C1,α graphical over this plane. We apply our theorem to prove partial regularity results for free-boundary minimizing hypersurfaces, and relative

The purpose of the present article is to study weak solutions of viscous conservation laws in physics. We are interested in the well-posedness 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

We 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 fluctuation-dissipation relation by identifying its Hessian with

We consider large non-Hermitian 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

A 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 Q-Loewner if its Q-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a -Poincaré inequality. As we will show, Q-Loewner planar metric spaces are always carpets, and admit quasisymmetric embeddings

Let 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

In 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

We consider stability of nonrotating gaseous stars modeled by the Euler-Poisson 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

Random 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

We are concerned with the question of well-posedness of stochastic, three-dimensional, 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

We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations.

In 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 viscous-convective range of the fluid model. This prediction has been extensively tested, both experimentally and numerically, and is a core prediction of passive scalar turbulence.

Soap 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

We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw 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

Neurons 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

Our 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

A 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

In 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 one-dimensional

We consider a class of autonomous Hamiltonian systems subject to small, time-periodic 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 non-zero.

We 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

We 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

We consider the “thin one-phase" 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

For 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 log-probability

Deep 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.

In this paper, we generalize White's regularity and structure theory for mean-convex 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 triple-approximation scheme

We 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.

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale 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