The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.
problem Nonlocal isoperimetric problem in hyperbolic space.
method Investigates minimization of a functional with perimeter and a nonlocal term derived from the negative power of distance.
result Geodesic balls are unique minimizers for small volumes in hyperbolic space.
Approximates nonlocal curvature of curves using splines.
problem Approximating nonlocal curvature of planar curves.
method Extending nonlocal curvature definition, using incomplete beta function, and linear interpolating spline approximation.
result Nonlocal curvature of a planar curve can be approximated by a spline.
Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.
problem Minimizing an energy functional with capillarity, nonlocal repulsion, and gravity.
method Quantitative isoperimetric inequalities applied to capillarity problem in a half-space.
result Existence and nonexistence of minimizers for various nonlocal kernels and masses.
A new image interpolation model using sparse representation and nonlocal linear regression.
problem Image interpolation without blurring and noise.
method Sparse representation, nonlocal self-similarity, nonlocal linear regression, adaptive sub-dictionary learning, weighted encoding.
result Our method outperforms state-of-the-art methods in quantitative measures and visual quality.
In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…
The study proves that sets with constant nonlocal curvature are composed of equal balls under certain conditions.
problem Characterizing sets with constant nonlocal curvature.
method Analyzing measurable sets in R^d with constant nonlocal h-mean curvature under a suitable integrability assumption.
result Finite unions of equal balls are the only sets with constant nonlocal curvature under the given conditions.
Nonlocal neural networks have been proposed and shown to be effective in several computer vision tasks, where the nonlocal operations can directly capture long-range dependencies in the feature space. In this paper, we study the nature of diffusion and damping effect of nonlocal networks by doing spectrum analysis on t…
Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.
problem Existence of global minimizers for a free energy functional on negatively curved manifolds.
method Investigation of Carlson-Levin type inequalities for Cartan-Hadamard manifolds.
result Establishes necessary and sufficient conditions for the existence of global energy minimizers.
Study shows global invertibility in nonlinear elasticity with vanishing self-repulsion term.
problem Global invertibility in nonlinear elasticity with a vanishing nonlocal self-repulsion term.
method Proves global invertibility in the Γ-limit of elastic energy with a vanishing nonlocal self-repulsion term. result Global invertibility can be obtained in the Γ-limit of the elastic energy with a vanishing nonlocal self-repulsion term. INEUS solves high-dimensional PIDEs efficiently with neural networks.
problem Solving high-dimensional partial integro-differential equations (PIDEs) efficiently.
method INEUS uses iterative neural networks to replace nonlocal integrals with sampling and reformulates PIDE solving as recursive regression.
result INEUS delivers accurate and scalable solutions for high-dimensional linear and nonlinear PIDEs.
Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.
problem Convexity of minimizers under mass constraint.
method Nonlocal free energy with nonlocal perimeter and convex potential.
result Quantitative stability theorem for nonlocal free energy assuming symmetry on the potential.
Paper connects Bäcklund transformations to nonlocal pseudosymmetries.
problem Finding Bäcklund transformations for differential equations.
method Factorization with nonlocal pseudosymmetries to determine Bäcklund transformations as C-morphisms. result Bäcklund transformations are determined by nonlocal pseudosymmetries' invariants.
The paper proves removable singularity for nonlocal minimal graphs.
problem Proving removable singularities for nonlocal minimal graphs.
method Analyzing (s,1)-capacity zero compact sets to ensure graphs are minimal in the entire domain. result Nonlocal minimal graphs are removable in the entire domain if they are minimal in a set of (s,1)-capacity zero. The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.
problem Proving the nonlocal version of the Alexandrov Theorem for sets with smooth boundaries.
method Formulated a necessary and sufficient condition for the theorem to hold, used a specific formula for the tangential derivative of the nonlocal mean curvature, and applied the method of moving planes.
result The only set with smooth boundary and constant nonlocal mean curvature is an Euclidean ball.
Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
Study on symmetric hyperbolic systems with nonlocal potentials, proving well-posedness and existence of solutions.
problem Initial value problem for symmetric hyperbolic systems with nonlocal potentials.
method Analysis on globally hyperbolic Lorentzian manifolds, proving existence, uniqueness, and regularity of solutions.
result Established well-posedness of the Cauchy problem for symmetric hyperbolic systems with nonlocal potentials.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
We study the geometric flow of a planar curve driven by its curvature and the normal derivative of its capacity potential. Under a convexity condition that is natural to our problem, we establish long term existence and large time asymptotics of this flow.
Nonlocal Bayesian modeling for continuous spatio-temporal dynamics
problem Handling irregular time points, sparse observations, and nonlocal interactions in spatio-temporal forecasting
method Hierarchical Bayesian framework with coordinate-based spatial basis expansion and continuous-time ODE
result Strong forecasting and uncertainty calibration
Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a flock, and use a fractional PDE to model the evo…
Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.
problem Determine Riemannian manifolds up to isometry using local source-to-solution maps.
method Comprehensive spectrum analysis and semigroup theory for nonlocal parabolic operators.
result Can determine Riemannian manifold up to isometry using local source-to-solution maps in a small open cylinder.
Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.
problem Approximating the Willmore functional using nonlocal methods.
method Gamma-convergence and fractional Laplacian analysis in Fermi coordinates.
result Proves Γ-limsup estimate for the proposed nonlocal approximation. Study of nonlocal mean curvature and symmetry of surfaces.
problem Symmetry of surfaces with ordered nonlocal mean curvature.
method Generalization of Alexandrov's moving plane method.
result Similar result to classical setting proved in nonlocal setting.
Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.
problem Time inconsistency in stochastic differential games.
method Proves existence and uniqueness of solutions to nonlocal fully-nonlinear parabolic systems.
result Generalizes stochastic game theory to include time-inconsistent preferences.
New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.
problem Proving Yau's conjecture for nonlocal minimal surfaces.
method Introducing nonlocal minimal surfaces and applying min-max variational methods.
result Construction of infinitely many nonlocal s-minimal surfaces on closed manifolds. The paper characterizes stochastic completeness on Riemannian manifolds using nonlocal conditions.
problem Stochastic completeness on complete Riemannian manifolds.
method Proves nonlocal characterizations and provides several new conditions equivalent to stochastic completeness.
result Stochastic completeness is equivalent to genuinely nonlocal conditions, including the zero-mean identity and uniqueness of solutions to fractional equations.
For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.
MetaNOR learns common nonlocal kernels for efficient metamaterial modeling.
problem Efficiently modeling wave propagation in new metamaterials.
method Meta-learns a common nonlocal kernel from existing tasks and transfers this knowledge to new tasks with minimal data.
result Substantial improvements in sampling efficiency for new metamaterials.
We are concerned with hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing sph…
We study hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in RN, N≥2, all of th…
We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the ca…
In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solv…
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
For any α>0, we study kα-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in C∞-norm. Other relevant kα-type nonlocal flow is also discussed when α≥1.
Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.
problem Proving uniqueness of solutions for a specific nonlocal Liouville equation.
method Connection to Calogero--Moser derivative NLS and ground state solitons.
result Uniqueness of solutions in the Gaussian case and general positive, symmetric-decreasing K. In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on Rn. We show that the problem has infinite positive solutions in Cτ(Rn)⋂Hlocα(Rn). Moreover each of these solutions tends to some positive constant limit at infinity. We extend Lin's result to the nonlocal problem on …
The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.
problem Characterizing equilibrium strategies and value functions for time-inconsistent stochastic control problems.
method Method of continuity and Banach's fixed point arguments, with Schauder prior estimates.
result Global well-posedness of nonlocal fully nonlinear PDEs with sharp a-priori estimates.
Develops a nonlocal PINN framework using PDDO for better solution of PDEs with sharp gradients.
problem Dealing with sharp gradients in solutions of PDEs using traditional PINN approaches.
method Integrates long-range interactions (nonlocality) into PINN using Peridynamic Differential Operator (PDDO).
result Nonlocal PINN approach improves solution accuracy and parameter inference for problems with sharp gradients.
Smoothness of graphs evolving by fractional mean curvature is proven.
problem Evolution of graphs by fractional mean curvature.
method Analytic semigroup approach to nonlocal quasilinear evolution equation.
result Short time existence, uniqueness, and optimal Hölder regularity of classical solutions.
First variation of fractional k-dimensional measure for submanifolds
problem Computing the first variation of a fractional k-dimensional measure for submanifolds method First variation computation
result Definition of a nonlocal mean-curvature vector for embedded submanifolds
NKN deep neural network learns governing equations and classifies images.
problem Learning governing equations and classifying images with deep neural networks.
method Nonlocal kernel network (NKN) that is resolution independent, deep, and handles various tasks.
result NKN outperforms baseline methods in learning governing equations and image classification tasks.
We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of compl…
We show that Chern-Simons gauge theory with appropriate cutoffs is equivalent, term by term in perturbation theory, to a Fermionic theory with a nonlocal interaction term. When an additional cutoff is placed on the Fermi fields, this Fermionic theory gives rise to a convergent perturbation expansion. This leads us to c…
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 confirms optimal minimax rate for nonlocal interaction kernel estimation.
problem Estimating nonlocal interaction kernels in interacting particle systems.
method Introduced tamed least squares estimator (tLSE) achieving optimal convergence rate.
result Optimal minimax rate of convergence confirmed for β≥1/4. Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.
problem Properties of solutions to fractional Allen-Cahn equation and stationary nonlocal minimal surfaces.
method Quantitative stratification principle applied to fractional Allen-Cahn equation, leading to optimal estimates.
result Sharp potential energy and perimeter estimates for fractional Allen-Cahn equation and nonlocal minimal surfaces.
We consider the class of measurable functions defined in all of Rn that give rise to a nonlocal minimal graph over a ball of Rn. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known…
New method learns kernels in nonlocal operators robustly.
problem Learning kernels in nonlocal operators is ill-posed.
method Nonparametric regression with Tikhonov regularization.
result Robust estimator of kernel yields homogenized model.