arXiv research

Neural Laplace models diverse DEs in the Laplace domain for better dynamics.

Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.

Fundamental gap vanishes for convex domains in hyperbolic space.

New upper bound found for nodal sets of Laplace eigenfunctions.

The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.

Estimates the index of the Laplace operator on planar domains with Robin boundary condition.

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n \geq 2$, with volume close to the volume of the manifold. If the first (positive) eigenfunction $φ_0$ of the Laplace-Beltrami operator over the manifold is a nonconst…

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of t…

The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.

Formula for Laplacian determinants on polygonal domains with slits.

John Conway created pairs of domains that sound the same for a special kind of music.

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.

In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a…

Transforms game optimization dynamics into frequency domain for precise hyperparameter analysis.

Study critical points of Laplace eigenfunctions in polygons.

Study small perturbations on low energy Laplace eigenfunctions.

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…

Study on $p$-Laplace equation in convex cones, proving rigidity under specific conditions.

This purpose of this write-up is to share an idea for accurate computation of Laplace eigenvalues on a broad class of smooth domains. We represent the eigenfunction $u$ as a linear combination of eigenfunctions corresponding to the common eigenvalue $ρ^{2}$:\EQN{6}{1}{}{0}{\RD{\CELL{u(r,θ) =\sum_{n=0}^{N}P_{n}J_{n}(ρ) …

LaLoRA prevents forgetting in LoRA fine-tuning.

Hot spots conjecture proven for small eigenvalue domains.

Previous research has shown that computation of convolution in the frequency domain provides a significant speedup versus traditional convolution network implementations. However, this performance increase comes at the expense of repeatedly computing the transform and its inverse in order to apply other network operati…

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the …

Generalizes crystallographic properties to all dimensions.

We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely…

Estimates spectral gap for Brownian motion on sticky-reflecting domains.

LLapDiff models irregular multivariate time series without step-by-step integration.

Study on Lawson surfaces' first Laplace eigenvalue using symmetry and algebraic methods.

Improved HMoE models using Laplace gating function enhance expert specialization and performance.

In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus with holes.

Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.

Laplace approximation improves deep learning efficiency without sacrificing performance.

By Cheeger-Colding's almost splitting theorem, if a domain in a Ricci flat manifold is pointed-Gromov-Hausdorff close to a lower dimensional Euclidean domain, then there is a harmonic almost splitting map. We show that any eigenfunction of the Laplace operator is almost constant along the fibers of the almost splitting…

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklo…

Sharp estimate for nodal domains intersecting a ball on a Riemannian manifold.

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

By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the …

We prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero …

Negative curvature restricts the gap between the first and second eigenvalues of convex domains.

New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.

Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically bounded by the 4-th root of the eigenvalue. It turns out that the number of nodal domains where the eigenfunction has an extre…

The paper develops quantitative estimates for holomorphic sections over bounded domains.

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

Study heat profiles and eigenfunctions using Brownian motion.

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.

The paper bounds Cheeger ratios of eigenfunctions and their level sets.

The paper finds large Steklov eigenvalues on manifolds using homogenization.