arXiv research

Trapezoids uniquely identified by their Dirichlet Laplace spectrum.

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

New upper bound found for nodal sets of Laplace eigenfunctions.

Corrected and improved simulation methods for Dirichlet-Laplace prior.

In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.

Anisotropic metric on manifolds uniquely determined by boundary data.

Fundamental gap vanishes for convex domains in hyperbolic space.

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in $\mathbb{R}^3$ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinde…

The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a flat disc than on any other surface of revoltuion immersed in Euclidean space with the same boundary.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Efficiently approximates uncertainty in classification models using Dirichlet distributions.

Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.

Study heat profiles and eigenfunctions using Brownian motion.

The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided.

Formula for Laplacian determinants on polygonal domains with slits.

We prove an adiabatic decomposition formula of the zeta-determinant of the Laplace type operator with respect to Dirichlet boundary condition. We allow the non-invertible tangential operator. As a result, our adiabatic decomposition formula involves the scattering matrix over the manifold with cylindrical end. We also …

We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.

Dirichlet-Neumann duality for Riemannian submersions

The paper extends graph-based semi-supervised learning to infinite-dimensional Wasserstein space.

The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.

Generalizes crystallographic properties to all dimensions.

Ribbon: Scalable Approximation and Robust Uncertainty Quantification

The paper proves a bound on eigenvalues for surfaces embedded in 3D space.

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $Ω$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion un…

Study Hodge Laplacians for manifold data, improving error bounds.

It is proved the existence of entire solutions of the Laplace's and minimal hypersurface's PDEs on a Hadamard manifold $M$ under certain curvature conditions by investigating the asymptotic Dirichlet's problems for these PDEs. In the harmonic case it is obtained an existence result which assumes the same growth conditi…

The Dirichlet mechanism protects privacy while minimizing KL divergence.

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

Proposes a new method for robust uncertainty quantification in regression tasks.

Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.

New neural network approach solves Poisson equations efficiently.

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (…

We consider vector fields $X$ on a closed manifold $M$ with rest points of Morse type. For such vector fields we define the property of exponential growth. A cohomology class $ξ\in H^1(M;\mathbb R)$ which is Lyapunov for $X$ defines counting functions for isolated instantons and closed trajectories. If $X$ has exponent…

The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.

We consider an optimization problem for the first Dirichlet eigenvalue of the $p$-Laplacian on a hypersurface in $\mathbb{R}^{2n}$, with $n \ge 2$. If $p \ge 2n-1$, then among hypersurfaces in $\mathbb{R}^{2n}$ which are $O(n) \times O(n)$-invariant and have one fixed boundary component, there is a surface which maximi…

Latent Dirichlet Allocation models discrete data as a mixture of discrete distributions, using Dirichlet beliefs over the mixture weights. We study a variation of this concept, in which the documents' mixture weight beliefs are replaced with squashed Gaussian distributions. This allows documents to be associated with e…

Hot spots conjecture proven for small eigenvalue domains.

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

A Dirichlet $k$-partition of a domain $U \subseteq \mathbb{R}^d$ is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs with applications in data analysis. Both versions admit va…

Fast approximate inference for non-Gaussian data.

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg …

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use t…

The study proves constant-curvature analogues of hot spots conjecture for triangles.

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, w…

We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimens…

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb H^2$, showing that for some of them $λ_2 - λ_1 < \frac{3π^2}{D^2}$, where $D$ is the diameter of the domain and $λ_1$, $λ_2$ are the first and second Dirichlet eigenvalues of the Laplace operat…