Communications on Pure and Applied Mathematics最新文献

We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super-Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right-hand side exhibiting a recursive structure in terms of lower-order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean-field limits, quasi-neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected $�N^{�\frac{s}{d}�-�1�}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows.

Extended dynamic mode decomposition (EDMD) is a data-driven algorithm for approximating spectral data of the Koopman operator associated to a dynamical system, combining a Galerkin method with $�N$ functions and a quadrature method with $�M$ quadrature nodes. Spectral convergence of this method subtly depends on an appropriate choice of the space of observables. For chaotic analytic full branch maps of the interval, we derive a constraint between $�M$ and $�N$ guaranteeing spectral convergence of EDMD. In particular, the computed eigenvalues converge exponentially fast (in $�N$) to the eigenvalues of the Koopman operator, taken to act on the dual space of a certain Banach space of analytic functions.

We provide a complete solution to the problem of infinite quantum signal processing (QSP) for the class of Szegő functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a QSP representation. We do so by introducing a new algorithm called the Riemann–Hilbert–Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szegő function. The proof of stability involves solving a Riemann–Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.

Recently, a class of algorithms combining classical fixed-point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $��{10}^{108}�\times �{10}^{108}�$. So far, a complete mathematical explanation for this success has proven elusive.

The family of methods has not yet been extended to the important case of linear system solves. In this paper, we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution is too large to store as a dense vector.

We prove $�\epsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya–Tonegawa. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $��\frac{\pi }{2}�$, then it coincides with a $�C^{�1�,�\alpha �}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than 1.

We study gradient flow on the multi-index regression problem for high-dimensional Gaussian data. Multi-index functions consist of a composition of an unknown low-rank linear projection and an arbitrary unknown, low-dimensional link function. As such, they constitute a natural template for feature learning in neural networks. We consider a two-timescale algorithm, whereby the low-dimensional link function is learnt with a non-parametric model infinitely faster than the subspace parametrizing the low-rank projection. By appropriately exploiting the matrix semigroup structure arising over the subspace correlation matrices, we establish global convergence of the resulting Grassmannian gradient flow dynamics, and provide a quantitative description of its associated “saddle-to-saddle” dynamics. Notably, the timescales associated with each saddle can be explicitly characterized in terms of an appropriate Hermite decomposition of the target link function.

We consider the $��3�d�$ energy critical nonlinear Schrödinger equation with data distributed according to the Gaussian measure with covariance operator $�{�(�1�-�\Delta �)�}^{�-�s�}$, where $�\Delta$ is the Laplace operator and $�s$ is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from $��1�d�$ to higher dimensions.

In this article we prove the orbifold version of the Bogomolov–Gieseker inequality for stable $�Q$-sheaves on log terminal Kähler threefolds.