Sharp Estimate of Global Coulomb Gauge

Let A be a W^{1, 2} -connection on a principal SU(2) -bundle P over a compact 4 -manifold M whose curvature F_A satisfies . Our main result is the existence of a global section σ : M → P with finite singularities on M such that the connection form σ^*A satisfies the Coulomb equation d^*(σ^*A) = 0 and admits a sharp estimate . Here ℒ^{4, ∞} is a new function space we introduce in this paper that satisfies L⁴(M) ⊊ ℒ^{4, ∞}(M) ⊊ L^{4 − ϵ}(M) for all ϵ > 0 . More precisely, ℒ^{4, ∞}(M) is the collection of measurable function u such that , where L^{4, ∞} is the classical Lorentz space and s_u is the L⁴ -integrability radius function associated to u defined by Briefly speaking, we achieve the estimate of by showing that σ^*A is effectively L⁴ -integrable away from controllably many points on M . © 2020 Wiley Periodicals LLC     

Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P^′(t,ω)=A(t,ω)(1?P(t,ω))P(t,ω), t∈[t₀,T], P(t₀,ω)=P₀(ω), where ω is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121–138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P₀ to approximate the density function f₁(p,t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P₀, and the density of P₀, , is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L²([t₀,T]×Ω), so that we include other expansions such as random power series. We only require absolute continuity for P₀, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For , only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence of density functions in terms of expectations that tends to f₁(p,t) pointwise. Numerical examples illustrate our theoretical results.     

Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

This paper presents some progress toward an open question proposed by P.-L. Lions [26] concerning the propagation of regularities of density patches for viscous inhomogeneous incompressible flow. We first establish the global-in-time well-posedness of the two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density . Here is any pair of positive constants and Ω₀ is a bounded, simply connected domain. We then prove that for any positive time t, the density , with the domain Ω(t) preserving the -boundary regularity. © 2018 Wiley Periodicals, Inc.     

Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line

We prove the leading order of a conjecture by Fyodorov, Hiary, and Keating about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as T → ∞ for a set of t ∊ [T, 2T] of measure (1–o(1)) T, we have © 2018 Wiley Periodicals, Inc.     

Upper Bounds on Liouville First-Passage Percolation and Watabiki's Prediction

Given a planar continuum Gaussian free field h^𝒰 in a domain 𝒰 with Dirichlet boundary condition and any δ > 0, we let be a real-valued smooth Gaussian process where is the average of h^𝒰 along a circle of radius δ with center v. For γ > 0, we study the Liouville first-passage percolation (in scale δ), i.e., the shortest path distance in 𝒰 where the weight of each path P is given by . We show that the distance between two typical points is for all sufficiently small but fixed γ > 0 and some constant c^* > 0. In addition, we obtain similar upper bounds on the Liouville first-passage percolation for discrete Gaussian free fields, as well as the Liouville graph distance, which roughly speaking is the minimal number of euclidean balls with comparable Liouville quantum gravity measure whose union contains a continuous path between two endpoints. Our results contradict some reasonable interpretations of Watabiki's prediction (1993) on the random distance of Liouville quantum gravity at high temperatures.© 2019 Wiley Periodicals, Inc.     

Sigma Delta Quantization for Images

In signal quantization, it is well-known that introducing adaptivity to quantization schemes can improve their stability and accuracy in quantizing bandlimited signals. However, adaptive quantization has only been designed for one-dimensional signals. The contribution of this paper is two-fold: (i) we propose the first family of two-dimensional adaptive quantization schemes that maintain the same mathematical and practical merits as their one-dimensional counterparts, and (ii) we show that both the traditional 1-dimensional and the new 2-dimensional quantization schemes can effectively quantize signals with jump discontinuities, which immediately enable the usage of adaptive quantization on images. Under mild conditions, we show that by using adaptivity, the proposed method is able to reduce the quantization error of images from the presently best $O\left(\sqrt{P}\right)$ to the much smaller $O\left(\sqrt{s}\right)$, where s is the number of jump discontinuities in the image and P ($P\gg s$) is the total number of samples. This $\sqrt{P/s}$-fold error reduction is achieved via applying a total variation norm regularized decoder, whose formulation is inspired by the mathematical super-resolution theory in the field of compressed sensing. Compared to the super-resolution setting, our error reduction is achieved without requiring adjacent spikes/discontinuities to be well-separated, which ensures its broad scope of application. We numerically demonstrate the efficacy of the new scheme on medical and natural images. We observe that for images with small pixel intensity values, the new method can significantly increase image quality over the state-of-the-art method. © 2022 Wiley Periodicals, Inc.     

Online Prediction with History-Dependent Experts: The General Case

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 adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of $n\ge 2$ experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as $N\to \mathrm{\infty }$ at a rate of $O\left(N^{-1/6}\right)$ to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V. Kohn [14] for $n=2$ experts and $d\le 4$ days of history. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Normalized Harmonic Map Heat Flow

For any closed Riemannian manifold N we propose the normalized harmonic map heat flow as a means to obtain nonconstant harmonic maps , m ≥ 3 . © 2019 Wiley Periodicals, Inc.     

A Proof of Sznitman's Conjecture about Ballistic RWRE

We consider random walk in a uniformly elliptic i.i.d. random environment in ℤ^d 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′) . The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width L . The second one is the requirement that for all γ ∈ (0, 1) condition (T)_γ is satisfied, which in turn is defined as the requirement that the decay is like for some C > 0 . In this article we prove a conjecture of Sznitman from 2002, stating that (T) and (T′) are equivalent. Hence, this closes the circle proving the equivalence of conditions (T) , (T′) , and (T)_γ for some γ ∈ (0, 1) as conjectured by Sznitman, and also of each of these ballisticity conditions with the polynomial condition (P)_M for M ≥ 15d + 5 introduced by Berger, Drewitz, and Ramı́rez in 2014. © 2019 Wiley Periodicals, Inc.     

Regularity for Shape Optimizers: The Degenerate Case

We consider minimizers of (1) where F is a function nondecreasing in each parameter, and λ_k(Ω) is the k^th Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often-studied F=λ_N. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler-Lagrange equation for the minimizers that is not otherwise available. © 2019 Wiley Periodicals, Inc.     

Min-Oo Conjecture for Fully Nonlinear Conformally Invariant Equations

In this paper we show rigidity results for supersolutions to fully nonlinear, elliptic, conformally invariant equations on subdomains of the standard n -sphere under suitable conditions along the boundary. We emphasize that our results do not assume concavity on the fully nonlinear equations we will work with. This proves rigidity for compact, connected, locally conformally flat manifolds (M, g) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere ∂D(r) , where D(r) denotes a geodesic ball of radius r ∈ (0, π/2] in , and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, (M, g) must be isometric to the closed geodesic ball . As a side product, in dimension 2 our methods provide a new proof to Toponogov's theorem about rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's theorem is equivalent to a rigidity theorem for spherical caps in the hyperbolic three-space ℍ³ . In fact, we extend it to obtain rigidity for supersolutions to certain Monge-Ampère equations. © 2019 Wiley Periodicals, Inc.     

A Liouville Theorem for the Complex Monge-Ampère Equation on Product Manifolds

Let Y be a closed Calabi-Yau manifold. Let ω be the Kähler form of a Ricci-flat Kähler metric on . We prove that if ω is uniformly bounded above and below by constant multiples of , where is the standard flat Kähler form on and ω_Y is any Kähler form on Y, then ω is a product Kähler form up to a certain automorphism of . © 2018 Wiley Periodicals, Inc.     

Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map

We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star‐shaped domain in ?^d, d ≥ 2. Conventional boundary‐based methods require a root search in eigenfrequency k, hence take O(N³) effort per eigenpair found, where N = O(k^d?1) is the number of unknowns required to discretize the boundary. Our method is O(N) faster, achieved by linearizing with respect to k the spectrum of a weighted interior Neumann‐to‐Dirichlet (NtD) operator for the Helmholtz equation. Approximations to the square roots k_j of all O(N) eigenvalues lying in [k ? ?, k], where ? = O(1), are found with O(N³) effort. We prove an error estimate with C independent of k. We present a higher‐order variant with eigenvalue error scaling empirically as O(?⁵) and eigenfunction error as O(?³), the former improving upon the “scaling method” of Vergini and Saraceno. For planar domains (d = 2), with an assumption of absence of spectral concentration, we also prove rigorous error bounds that are close to those numerically observed. For d = 2 we compute robustly the spectrum of the NtD operator via potential theory, Nyström discretization, and the Cayley transform. At high frequencies (400 wavelengths across), with eigenfrequency relative error 10^?10, we show that the method is 10³ times faster than standard ones based upon a root search. © 2014 Wiley Periodicals, Inc.     

A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation which are periodic in the d − 1 last variables (living on the torus 𝕋^d−1) and globally minimize the corresponding energy in Ω = ℝ × 𝕋^d−1, i.e., Namely, we find a class of nonlinear potentials W ≥ 0 such that any global minimizer u of E connecting two zeros of W as x₁ → ± ∞ is one-dimensional; i.e., u depends only on the x₁ -variable. In particular, this class includes in dimension d = 2 the nonlinearities with w being a harmonic function or a solution to the wave equation, while in dimension d ≥ 3 , this class contains a perturbation of the Ginzburg-Landau potential as well as potentials W having d + 1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of E for general potentials W providing in particular compactness results for uniformly finite energy maps u in Ω connecting two wells of W as x₁ → ± ∞ . © 2019 Wiley Periodicals, Inc.     

Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times

We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure with respect to ℙ that governs the law of the increments of the three-dimensional Brownian motion on a finite interval [−T, T] , and Z_{α, T} is the partition function or the normalizing constant and α > 0 is a constant, or the coupling parameter. The polaron measure reflects a self-attractive interaction. According to a conjecture of Pekar that was proved in [9], exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for under and obtain an expression for the limiting variance. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, LLC.     

List colorings of multipartite hypergraphs

Let χ_l(G) denote the list chromatic number of the r‐uniform hypergraph G. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if G is simple and d‐regular, then . To see how close this inequality is to best possible, we examine χ_l(G) when G is a random r‐partite hypergraph with n vertices in each class. The value when r = 2 was determined by Alon and Krivelevich; here we show that almost surely, where d is the expected average degree of G and . The function g(r,α) is defined in terms of “preference orders” and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on χ_l(G) for r = 2 and r = 3, but, perhaps surprisingly, apparently not for r ≥ 4.     

Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t\approx {\epsilon }^{-2}$. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.     

On the kernel space for an L0‐linear function

A random normed module is a random generalization of an ordinary normed space, and it is the randomization that makes a random normed module possess rich stratification structures. On the basis of these stratification structures, this paper shows that either the kernel space N(f) for an L⁰‐linear function f from a random normed module S to the algebra is a closed submodule or N(f) on some specifical stratification is a dense proper submodule of S, which generalizes the classical case. In the meantime, a characterization for the kernel space N(f) to be closed is also given.     

Robustness of Liouville Measure under a Family of Stable Diffusions

Consider a C^∞ closed connected Riemannian manifold (M, g) with negative sectional curvature. The unit tangent bundle SM is foliated by the (weak) stable foliation of the geodesic flow. Let Δ^s be the leafwise Laplacian for and let X be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each ρ , the operator generates a diffusion for . We show that, as ρ → − ∞ , the unique stationary probability measure for the leafwise diffusion of converge to the normalized Liouville measure on SM . © 2020 Wiley Periodicals LLC     

The Isotropic Semicircle Law and Deformation of Wigner Matrices

We analyze the spectrum of additive finite‐rank deformations of N × N Wigner matrices H. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue d_i of the deformation crosses a critical value ± 1. This transition happens on the scale . We allow the eigenvalues d_i of the deformation to depend on N under the condition . We make no assumptions on the eigenvectors of the deformation. In the limit N → ∞, we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the isotropic local semicircle law, which establishes optimal high‐probability bounds on where m(z) is the Stieltjes transform of Wigner's semicircle law and v , w are arbitrary deterministic vectors.© 2013 Wiley Periodicals, Inc.