arXiv research

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stre…

The paper examines special Q-nets that terminate after a finite number of Laplace steps.

We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations to the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of…

Formula derived for Laplace-Beltrami on Stiefel manifold.

Enhances predictive performance in Bayesian deep learning via generalized Laplace approximation.

Paper studies Laplace operator estimates in harmonic map heat flows.

Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.

Let $x : M \to E^m$ be an isometric immersion of a Riemannian manifold $M$ into a Euclidean $m$-space. Denote by $Δ$ the Laplace operator of $M$. Then $Δ$ gives rise to a differentiable map $L :M \to E^m$, called the Laplace map, defined by $L(p)=(Δx)(p)$, $p\in M$. We call $L(M)$ the Laplace image, and the transformat…

Extends geometric structures to manifolds with new operators.

Modeling stock returns and volatility using a bivariate gamma generalized Laplace law.

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

Introduces a new elliptic operator with positive eigenvalue.

Study of line congruences for Appell's rank-4 hypergeometric functions.

In this paper, classical isometric helicoidal and rotational surfaces are studied, and generalized by Bour's theorem in three dimensional Euclidean space. Moreover, the third Laplace-Beltrami operators of two classical surfaces are obtained.

Revisits online Laplace methods for neural networks, showing they are sound under certain conditions.

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rr…

Trapezoids uniquely identified by their Dirichlet Laplace spectrum.

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

The study bounds quantum eigenfunctions on complex manifolds.

New methods improve Laplace approximations for deep neural networks by selecting key parameters.

A new method combines Laplace and Variational Bayes for scalable inference.

Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.

In this note, we concentrate on the sub-Laplace on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by $n$ Brownian motions and their $\frac{n(n-1)}2$ Lévy area processes, which is the simple extension of the sub-Laplace on the Heisenberg group $\mathbb{H}$. In order …

The paper studies graph Laplace operator behavior near isolated singularities.

QLA improves Bayesian uncertainty estimation for DNNs without increasing computational cost.

We consider generalized Hodge-Laplace operators $αd δ+ βδd$ for $α, β> 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

Geometrically reformulates the Laplace method for optimal transport.

We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the cur…

Study finds metrics maximizing one Laplace eigenvalue on 3D and higher manifolds.

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

Improves Laplace approximation for Bayesian inference on Riemannian manifolds.

Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.

Graph Laplace operators uniquely identify metrics and densities on manifolds.

The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator for forms defined on the prolongation of the algebroid. All of the Laplace opera…

We propose a generalized double Pareto prior for Bayesian shrinkage estimation and inferences in linear models. The prior can be obtained via a scale mixture of Laplace or normal distributions, forming a bridge between the Laplace and Normal-Jeffreys' priors. While it has a spike at zero like the Laplace density, it al…

The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.

We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.

In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal th…

We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space $\mathbb{R}^{3}$. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve…

Improved Kuznecov remainder estimates for generic metrics.

We prove inequalities for Laplace eigenvalues of Kaehler manifolds generalising to higher eigenvalues the classical inequality for the first Laplace eigenvalue due to Bourguignon, Li, and Yau in 1994. We also obtain similar inequalities for analytic varieties in Kaehler manifolds.

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

Study on hyperbolic manifolds finds measures of Laplace eigenfunctions restricted to cosphere bundles.

Variational Laplace improves Bayesian neural networks performance.

The 1993 Laplace transform approach of Geman and Yor is a celebrated advance in valuing Asian options. Its insights are fundamental from both a mathematical and a financial perspective. In this paper, we discuss two observations regarding the financial relevance of its results. First, we show that the Geman and Yor Lap…

We discuss questions of isospectrality for hyperbolic orbisurfaces, examining the relationship between the geometry of an orbisurface and its Laplace spectrum. We show that certain hyperbolic orbisurfaces cannot be isospectral, where the obstructions involve the number of singular points and genera of our orbisurfaces.…

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the gen…

Variational Laplace improves Bayesian neural network performance without sampling.