相关论文: On the long neck principle and width estimates for…

We discuss the implementation, to the case of compact manifolds, of the perturbative method of Friedrich-Butscher for the construction of solutions to the vaccum Einstein constraint equations. This method is of a perturbative nature and…

When numerically solving Einstein's equations for binary black holes (BBH), we must find initial data on a three-dimensional spatial slice by solving constraint equations. The construction of initial data is a multi-step process, in which…

Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups,…

In this manuscript, we put forth a general scheme for defining initial value problems from Einstein's equations of General Relativity constrained by homogeneous and isotropic expansion. The cosmological models arising as solutions are…

Consider a compact Riemannian manifold with boundary. Assume all maximally extended geodesics intersect the boundary at both ends. Then to each maximal geodesic segment one can form a triple consisting of the initial and final vectors of…

Inspired by the recent work [MRT21], we prove a non-universal non-central Moderate Deviation principle for the nodal length of arithmetic random waves (Gaussian Laplace eigenfunctions on the standard flat torus) both on the whole manifold…

We develop and analyze a lattice difference equation (LDE) framework to model the spatial dynamics of invasion in populations. This framework extends beyond classical integro-difference and reaction-diffusion models by incorporating spatial…

We obtain a uniform boundary Harnack principle (BHP) on any open sets for a large class of non-local operators on metric measure spaces under a jump measure comparability and tail estimate condition, and an upper bound condition on the…

A review of the treatment of boundaries in general relativity is presented with the emphasis on application to the formulations of Einstein's equations used in numerical relativity. At present, it is known how to treat boundaries in the…

We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schr\"odinger operator on a bounded or unbounded domain, second, a perturbation and lifting estimate…

We develop a code length principle which is invariant to the choice of parameterization on the model distributions. An invariant approximation formula for easy computation of the marginal distribution is provided for gaussian likelihood…

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the…

Despite the practical success of deep neural networks, a comprehensive theoretical framework that can predict practically relevant scores, such as the test accuracy, from knowledge of the training data is currently lacking. Huge…

We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to…

A minimal geodesic on a Riemannian manifold is a geodesic defined on $\mathbb{R}$ that lifts to a globally distance minimizing curve on the universal covering. Bangert proved that there is a lower bound for the number of geometrically…

This paper studies higher-order inference properties of nonparametric local polynomial regression methods under random sampling. We prove Edgeworth expansions for $t$ statistics and coverage error expansions for interval estimators that (i)…

A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincar\'e inequality for scalar functions, also satisfies an analogous inequality for functions taking values…

We numerically verify and analytically prove a winding number invariant that correctly predicts the number of edge states in one-dimensional, nearest-neighbor (between unit cells), two-band models with any complex couplings and open…

We obtain lower and upper bounds on the heat kernel and Green functions of the Schroedinger operator in a random Gaussian magnetic field and a fixed scalar potential. We apply stochastic Feynman-Kac representation, diamagnetic upper bounds…

We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to…