Constructs a support-preserving homotopy for differential forms with boundary decay estimates.
problem Non-uniqueness of chain homotopies in de Rham complexes with boundary decay properties.
method Constructs a specific chain homotopy with desirable support propagation and boundary decay estimates.
result Obtains a support-preserving right inverse of the divergence operator with optimal decay estimates.
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
A random walk wn on a separable, geodesic hyperbolic metric space X converges to the boundary ∂X with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when …
Develops calculus for QFB metrics, proving Fredholm properties and decay of harmonic forms.
problem Analyzing differential operators on quasi-fibered boundary metrics.
method Introduces pseudodifferential calculus, principal symbols, and Fredholm theory.
result Hodge-deRham operator is Fredholm on QFB Sobolev spaces and L2 harmonic forms decay. Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
In this paper, we examine the dependence of standard gluing process for pseudoholomorphic curves under the change of the length T of the neck-region with respect to the cylindrical metrics associated to the given analytic coordinates near the punctures in the setting of bordered open Riemann surface with boundary pun…
We study the asymptotic Dirichlet problem for f-minimal graphs in Cartan-Hadamard manifolds M. f-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of f-minimal graphs with pre…
In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragmén-Lindelöf principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of t…
The fundamental properties of J-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a J-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We …
Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.
problem Establishing inequalities for radial functions on hyperbolic spaces without zero boundary conditions.
method Novel approach considering both bounded and unbounded domains, focusing on weighted Sobolev and Adams-Trudinger-Moser embeddings.
result Theorems 1.2, 1.3, and 1.4 for weighted Sobolev embedding theorems, and Theorems 1.5 and 1.6 for Adams-Trudinger-Moser type embedding theorems.
New method uses entropy dissipation to prove isoperimetric inequalities.
problem Proving isoperimetric inequalities in geometric settings.
method Information-theoretic approach based on entropy dissipation under heat flow.
result New proof of Euclidean isoperimetric inequality with sharp constant.
Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.
problem Area minimizing currents in Riemannian manifolds with moduli.
method Structural results and uniqueness theorem, inspired by Simon's techniques.
result Uniqueness and decay towards tangent cones for area minimizing currents.
We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its C2-distance fro…
Study on biharmonic heat equation on manifolds with curvature constraints.
problem Analyzing entire solutions of biharmonic heat equation on manifolds.
method Exponential decay estimates for biharmonic heat kernel under Ricci curvature and noncollapsing conditions. Proving uniqueness criteria for Cauchy problem.
result Conservation law for biharmonic heat kernel and uniform L-infinity estimate for entire solutions.
Upper bound found for Steklov eigenvalues counting function.
problem Counting Steklov eigenvalues on compact manifolds with boundary.
method Used Weyl's law and Pólya's Conjecture in the Steklov case.
result Obtained an upper bound for the counting function.
Tian and Yau constructed a complete Ricci-flat Kähler metric on the complement of an ample and smooth anticanonical divisor. We inquire into the behaviour of this metric towards the boundary divisor and prove a slow decay rate of the difference to an appropriate explicitely given referential metric.
The study proves uniqueness of large isoperimetric sets in specific noncompact manifolds.
problem Proving uniqueness of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature.
method Analyzing properties of complete Riemannian manifolds with specific curvature conditions.
result There exists a set of volumes with density 1 at infinity where isoperimetric sets are unique and strictly volume preserving stable.
We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary fa…
The existence and nonexistence of λ-harmonic functions in unbounded domains of Hn are investigated. We prove that if the (n−1)/2 Hausdorff measure of the asymptotic boundary of a domain Ω is zero, then there is no bounded λ-harmonic function of Ω for λ∈[0,λ1(Hn)], where $λ_1(\mathb…
Paper develops an online learning algorithm for functional data models.
problem Recovering slope functions or predictors in functional data models.
method Online regularized learning algorithm in reproducing kernel Hilbert spaces with polynomially decaying step-size.
result Established fast convergence rates for estimation error without capacity assumption.
The paper proves density and positive mass theorems for incomplete manifolds.
problem Proving density and positive mass theorems for manifolds with incomplete ends.
method Using harmonic asymptotics and quantitative positive mass theorem improvements.
result Improved quantitative positive mass theorem in dimensions 3 to 7.
Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
Motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes, we initiate the study of massive scalar waves satisfying □gψ−μψ=0 on the interior of Anti-de Sitter (AdS) black holes. We prescribe initial data on a spacelike hypersurface of a Reissner--Nordström--AdS black hole and impose…
New method improves smoothness of minimizing currents near singular points.
problem Improving smoothness of minimizing currents near singular points.
method New method to estimate the full singular set of the foliation by minimizers and proof of superlinear decay of closeness.
result Generic smoothness of minimizers improved to n−9−εn for n≥11. New indices defined for manifolds with boundary, generalizing previous results.
problem Defining indices for manifolds with boundary.
method Defining higher Atiyah-Patodi-Singer C^*-indices and using them to introduce higher genera.
result Established a higher index formula for new C^*-indices.
Given an asymptotically cylindrical special Lagrangian submanifold L in an asymptotically cylindrical Calabi-Yau 3-fold X, we determine conditions on a decay rate gamma which make the moduli space of (local) special Lagrangian deformations of L in X a smooth manifold and show that it has dimension equal to the dimensio…
New techniques compute L2-cohomology of quasi-fibered metrics.
problem Compute L2-cohomology of quasi-fibered boundary metrics. method Developed new techniques to compute weighted L2-cohomology. result Computed reduced L2-cohomology for various classes of QFB-metrics. Global stability proved for Navier-Stokes equations on hyperbolic space.
problem Stability of the Navier-Stokes equations on hyperbolic space.
method Proved global stability with exponential decay rate for small initial data.
result Exponential decay rate of $μλ_\Def^{(3)}$ for Navier-Stokes equations on hyperbolic space.
This article proves a uniform exponential decay estimate for Seiberg-Witten equations on non-compact 4-manifolds with exact symplectic ends of bounded geometry. This is an extension of the analysis for asymptotically flat almost Kähler (AFAK) structures by Kronheimer and Mrowka. As an application, we construct an invar…
We derive explicit formulas for time decay, for the European call and put options at expiry, and use them to calculate analytical approximations to the price of the American put and early exercise boundary near expiry. We show that for many families of non-Gaussian processes used in empirical studies of financial marke…
Extends Minkowski stability proof to minimal decay assumptions.
problem Global stability of Minkowski spacetime with minimal decay.
method Extends Christodoulou-Klainerman's proof to minimal decay assumptions.
result Exterior stability of Minkowski holds with borderline decay.
New method creates vacuum data at minimal and borderline decay thresholds.
problem Creating vacuum initial data at specific decay thresholds.
method Conical solution-operator method applied to vacuum asymptotically flat initial data.
result Demonstrates global and exterior stability of Minkowski spacetime.
The energy in a square membrane Ω subject to constant viscous damping on a subset ω⊂Ω decays exponentially in time as soon as ω satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate τ(ω) of this decay satisfies τ(ω)=2min(−μ(ω),g(ω)) (see Lebeau [Math. Phys. Stud. …
We study coordinate-invariance of some asymptotic invariants such as the ADM mass or the Chruściel-Herzlich momentum, given by an integral over a "boundary at infinity". When changing the coordinates at infinity, some terms in the change of integrand do not decay fast enough to have a vanishing integral at infinity; bu…
We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if M^4 is a manifold with a CR-boundary at infinity, an adapted spinc-…
Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
Study proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.
problem Analyzing spacelike CMC surfaces near future null-infinity in Schwarzschild spacetime.
method Proves asymptotic hyperbolicity, derives boundary data expressions, and shows compatibility conditions.
result Compatibility conditions and asymptotic behaviors of spacelike CMC surfaces near future null-infinity.
New findings on flatness of certain metrics with fast decay.
problem Rigidity of positive mass theorem under fast metric decay.
method Considered metrics with nonnegative scalar curvature and rapid decay at infinity.
result Any such metric is necessarily flat in dimensions 4 and higher if decay rate exceeds Schwarzschild metric.
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
In this global study of solutions to the linear wave equation on Schwarzschild de Sitter spacetimes we attend to the cosmological region of spacetime which is bounded in the past by cosmological horizons and to the future by a spacelike hypersurface at infinity. We prove an energy estimate capturing the expansion of th…
Cautious Weight Decay modifies weight decay for better optimization.
problem Improving optimization in deep learning models.
method Applies weight decay selectively based on parameter sign alignment.
result Consistently improves model performance across various tasks and scales.
New invariants from quantum group theory for hyperbolic 3-manifolds.
problem Computing invariants for hyperbolic 3-manifolds with boundary.
method Using modular doubles of quantum sl(2;R) and 6j-symbols. result Invariants decay exponentially with hyperbolic volume and 1-loop terms.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.
problem Analyzing decay and non-decay rates of solutions to the massless Vlasov equation on Reissner-Nordström spacetimes.
method Quantitative analysis of geodesic flow and comparison to wave equation instability results.
result Exponential decay rates in subextremal cases and polynomial rates in extremal cases, with non-decay of transversal derivatives in extremal cases.
Introduces gradient decay in Softmax for better generalization.
problem Improving generalization performance in neural networks.
method Gradient decay hyperparameter in Softmax for varying gradient rates based on probability.
result Gradient decay rate affects generalization performance and can be tuned for better optimization.
Study examines wave equation decay and Strichartz estimates on conic manifolds.
problem Analyzing wave equation behavior on conic spaces with critical electromagnetic potentials.
method Established decay and Strichartz estimates through localized spectral measure construction.
result Extended and improved previous results on wave equation behavior with critical potentials.
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.