arXiv research

Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.

Study nonlocal minimal surfaces for minimal surfaces in 3D manifolds.

This paper develops a new nonlocal approximation method for minimal surfaces, proving robust estimates and separation properties.

The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

Solves optimal liquidation problem for stock price following geometric Brownian motion.

In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on $R^n$. We show that the problem has infinite positive solutions in $C^τ(R^n)\bigcap H^α_{loc}(R^n)$. Moreover each of these solutions tends to some positive constant limit at infinity. We extend Lin's result to the nonlocal problem on …

Study of nonlocal mean curvature and symmetry of surfaces.

Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.

Sharp stability in Almgren problem solved in any dimension.

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…

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 paper proves well-posedness of nonlocal PDEs related to stochastic control problems.

In this paper we discuss various minimality properties for the orthogonal product of two 1-dimensional $\Y$ sets, and some related problems. This is motivated by an attempt to give the classification of singularities for 2-dimensional Almgren-minimal sets in $\R^4$.

Approximates nonlocal curvature of curves using splines.

Smoothness of graphs evolving by fractional mean curvature is proven.

Paper connects Bäcklund transformations to nonlocal pseudosymmetries.

A new image interpolation model using sparse representation and nonlocal linear regression.

Study confirms optimal minimax rate for nonlocal interaction kernel estimation.

The paper proves removable singularity for nonlocal minimal graphs.

The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.

We prove existence, uniqueness, and regularity of viscosity solutions to the stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. We give sufficient conditions on the coefficients of the operator to obtain Hölder and Lipschitz con…

Proposes a new nonlocal curvature tensor concept.

This research proves that two min-max theories for hypersurfaces are equivalent.

In this study, we introduce an explicit trading-volume process into the Almgren-Chriss model, which is a standard model for optimal execution. We propose a penalization method for deriving a verification theorem for an adaptive optimization problem. We also discuss the optimality of the volume-weighted average-price st…

Study on symmetric hyperbolic systems with nonlocal potentials, proving well-posedness and existence of solutions.

Constructs area-minimizing submanifolds with fractal singularities.

New theorem for nonlocal minimal surfaces in any dimension.

Proves spectra equivalence for Riemannian manifolds.

We consider an optimal liquidation problem with infinite horizon in the Almgren-Chriss framework, where the unaffected asset price follows a Levy process. The temporary price impact is described by a general function which satisfies some reasonable conditions. We consider an investor with constant absolute risk aversio…

We propose two methods to obtain exact solutions for the Almgren-Chriss model about optimal execution of portfolio transactions. In the first method we rewrite the Almgren-Chriss equation and find two exact solutions. In the second method, employing a general reparametrized time, we show that the Almgren-Chriss equatio…

We show that wealth processes in the block-shaped order book model of Obizhaeva/Wang converge to their counterparts in the reduced-form model proposed by Almgren/Chriss, as the resilience of the order book tends to infinity. As an application of this limit theorem, we explain how to reduce portfolio choice in highly-re…

Nonlocal Bayesian modeling for continuous spatio-temporal dynamics

We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.

Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.

New method learns kernels in nonlocal operators robustly.

We solve the problem of reducing to the simplest and convenient for our purposes, canonical form for an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature in order to get an effective construction of the integrable hierarchies related to all…

Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.

New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.

The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear dif…

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in \cite[Section 5.11]{Federer74} to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a…

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.

The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demo…

MetaNOR learns common nonlocal kernels for efficient metamaterial modeling.

This paper optimizes brokerage contracts for multiple clients trading a single asset.

We are concerned with hypersurfaces of $\mathbb{R}^N$ 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 $\mathbb{R}^N$ 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 $\mathbb{R}^N$, $N\geq 2$, all of th…

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…