Neural Laplace models diverse DEs in the Laplace domain for better dynamics.
problem Inadequate ODEs for long-range dependencies and discontinuities.
method Unified framework in Laplace domain, using stereographic map for smoothness.
result Superior performance in diverse DEs, including complex history dependency and abrupt changes.
Paper introduces p-Laplace equations for curvature in conformal geometry.
problem Study of geometry and topology of manifolds.
method Introducing p-Laplace equations for intermediate Schouten curvature.
result Nonnegative intermediate Schouten curvature leads to estimates on Hausdorff dimension of singular sets and vanishing of homotopy groups.
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…
Classifies positive solutions to critical p-Laplace equation.
problem Classifying positive solutions to a specific type of partial differential equation.
method Analyzes solutions on \(\mathbb{R}^n\) with various energy growth conditions and infinity behavior.
result Provides classification under different conditions, including rigidity in some cases.
Study on p-Laplace equation in convex cones, proving rigidity under specific conditions.
problem Overdetermined problem for p-Laplace equation in convex cones. method Established properties of capacitary potential, used P-function, isoperimetric inequality, and Heintze-Karcher inequality. result Rigidity result under orthogonal intersection assumption.
In this paper, we obtain two rigidity results for p-Laplace type equation and n-Laplace equation with exponential nonlinearity on n-dimensional compact Riemannian manifolds by using of nonlinear flow and the carré du champ methods, respectively, where rigidity means that the PDE has only constant solution when a …
In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating vari…
The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.
problem Calibrating and pricing options in polynomial Ornstein-Uhlenbeck volatility models.
method Analyzes Fourier-Laplace transforms, connects to Riccati equations, and develops numerical schemes.
result Establishes existence and solution for Riccati equations and provides efficient numerical methods.
New study proves no strictly positive solutions to a specific Laplace equation on certain manifolds.
problem Existence of strictly positive solutions to a critical Laplace equation on manifolds with nonnegative Ricci curvature.
method Analyzed a suitable function defined along the level sets of the solution.
result No strictly positive solutions exist unless the manifold is isometric to R^n and the solution is a Talenti function.
For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geo…
Due to the isotropy d-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the R-radius hyperboloid model of d-dimensional hyperbolic geometry with R>0 and d≥2, we compute azimuthal Fourier expansions for a fundamental so…
In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on (1,1)-forms. The method is effective in proving an optimal result when M has nonnegative bisectional curvature. It also provides …
The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.
problem Computing conformal mappings between Riemannian surfaces.
method Adapting the conjugate function method to Riemannian surfaces using hp-adaptive finite element methods. result Highly accurate numerical computations of conformal mappings on surfaces, including complex geometries.
This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
New solutions found for Ginzburg-Landau equations on complex manifolds.
problem Finding solutions to Ginzburg-Landau equations on closed manifolds.
method Bifurcation theory applied to Laplace-type operator eigenvalues.
result First nonminimal and irreducible solutions on nontrivial line bundles.
Study of line congruences for Appell's rank-4 hypergeometric functions.
problem Understanding line congruences for Appell's rank-4 hypergeometric functions.
method Derived original formulae for Laplace transform of rank-4 system, applied to geometry of surfaces defined by these functions.
result Natural line congruences for Laplace transforms of Appell's rank-4 functions form a W-congruence.
Extend classical theory of affine processes to path-dependent setting
problem Path-dependent affine processes
method Introduce path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform
result Define path-dependent affine processes through their exponential-affine Fourier--Laplace transform and establish a characterization theorem
What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic repr…
Paper studies viscosity solutions in unique Martinet spaces.
problem Properties of viscosity solutions in Martinet spaces.
method Established properties and proved uniqueness of solutions.
result Uniqueness of viscosity solutions in Martinet spaces.
The Laplace equation in the two-dimensional Euclidean plane is considered in the context of the inverse stereographic projection. The Lie algebra of the conformal group as the symmetry group of the Laplace equation can be represented solely in terms of the solutions and derivatives of the solutions of the Laplace equat…
Study critical quasilinear equations on Riemannian manifolds with curvature constraints.
problem Investigate critical quasilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature.
method Utilize a new nonlinear Kato inequality and Cheng-Yau type gradient estimates for positive solutions.
result Classify positive solutions to the critical p-Laplace equation and show rigidity concerning the ambient manifold. Unified analytical tool for non-Markovian jump processes.
problem Analyzing history-dependent jump processes with non-Markovian behavior.
method Developed a standard form of master equations using Laplace-space embedding and asymptotic solution.
result Unified analytical toolset for general non-Markovian processes, leading to the GLE approximation.
This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that i…
The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point cloud…
New method calculates geometric Brownian motion with affine drift and its integral.
problem Calculating the distribution of geometric Brownian motion with affine drift and its integral.
method Laplace transform approach and Heun differential equation.
result Joint distribution of geometric Brownian motion with affine drift and its integral can be determined.
We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and…
Bayesian nonparametric models get better posterior estimates via SPDE methods.
problem Estimating posterior distributions in nonparametric Bayesian models.
method Extending diffusion methods to SPDEs on Hilbert spaces for posterior contraction and Laplace approximation.
result Derivation of posterior contraction rates and finite-sample Bernstein von Mises results.
Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.
problem Understanding harmonic morphisms and their relationship to minimal submanifolds.
method Characterization of harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations, derivation of reduction properties for other co-dimensions, application to find novel area-minimising hypercones.
result Novel family of degree 4 area-minimising hypercones in R^m, m≥32.
The article characterizes a hemisphere using a Laplace operator and a differential equation.
problem Characterizing a hemisphere in a Riemannian manifold with boundary.
method Using the de-Rham Laplace operator and a nontrivial solution of the Fischer-Marsden equation.
result Proves the cosmic no-hair conjecture under a given integral condition.
Study asymptotically almost periodic solutions on real hyperbolic manifolds.
problem Existence and asymptotic behavior of solutions to parabolic equations.
method Dispersion and smoothing estimates, fixed point argument.
result Existence and uniqueness of asymptotically almost periodic solutions.
Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary condition…
This paper, focusing on the growth rate of the measure, gives pointwise bounds of solutions of eigenvalue equations of the Laplace-Beltrami operator on noncompact Riemannian manifolds.
Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.
problem Estimating eigenvalues on quaternion-Kähler manifolds.
method Lower bounds derived from modulus of continuity estimates for heat equation solutions and Laplace comparison theorem.
result Established bounds for first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.
Transforms game optimization dynamics into frequency domain for precise hyperparameter analysis.
problem Analyzing convergence of hyperparameters in game optimization.
method Frequency-domain framework using High-Resolution Differential Equations (HRDEs) and Laplace transforms.
result Derives precise convergence criteria for the Lookahead algorithm.
The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.
problem Analyzing heat-type equations on manifolds with specific boundary conditions.
method Proving Schauder estimates for the Laplace-Beltrami operator on manifolds with fibered boundaries and a Φ-metric.
result The proof of parabolic Schauder estimates for the Laplace-Beltrami operator.
We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.
problem Analyzing mass concentration and nodal domains of Laplace eigenfunctions.
method Heat diffusion technique to study eigenfunctions and their nodal sets.
result Discovers new insights into the decay and behavior of Laplace eigenfunctions.
The paper studies geometric structures in perfect fluid spacetimes with specific metrics.
problem Analyzing the geometric properties of perfect fluid spacetimes with specific metrics.
method Investigates conditions for conformal Ricci-Yamabe soliton and derives Laplace equations.
result Conditions for expanding, steady, or shrinking conformal Ricci-Yamabe solitons are identified.
Generalizes crystallographic properties to all dimensions.
problem Analytic eigenfunctions in crystallographic groups.
method Algebraic, geometric, and analytic proofs.
result Equivalent conditions for real analytic eigenfunctions in crystallographic polytopes.
Bayesian PINNs optimize loss weights for PDEs and data.
problem Optimizing loss weights in physics-informed neural networks.
method Laplace approximation for efficient model evidence computation.
result Unified Bayesian setting for PDEs and noisy measurements.
We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This…
We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger equation on n-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of R-separability in PDE (Laplace equation on E3…
The paper extends Huber's theorem to higher dimensions using n-Laplace equations.
problem Proving finite point conformal compactification for general dimensions.
method Using n-Laplace equations and strengthened Arsove-Huber's theorem.
result Established finite point conformal compactification theorem for manifolds.
We study the problem of utility maximization from terminal wealth in which an agent optimally builds her portfolio by investing in a bond and a risky asset. The asset price dynamics follow a diffusion process with regime-switching coefficients modeled by a continuous-time finite-state Markov chain. We consider an inves…
Bayesian neural networks approximate Gaussian, this method adapts to non-Gaussian posteriors.
problem Bayesian neural networks struggle with non-Gaussian posteriors, leading to poor performance.
method Proposes a Riemannian Laplace approximation to adapt to the shape of the true posterior.
result Consistently improves over conventional Laplace approximation across tasks.
Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.
problem Connection between Yang-Mills fields and modified Lévy Laplacians on 4-manifolds.
method Analysis of modified Lévy Laplacians and their relation to Yang-Mills equations under nontrivial holonomy groups.
result Existence of a modified Lévy Laplacian related to Yang-Mills self-duality equations.
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
Solves sigma model on U(3)/U(1)^3, describing geodesics and spectrum.
problem Classical and quantum problems for 1D sigma model with specific target space.
method Mapping to Gaudin model, solving polynomial equations.
result Explicit description of geodesics and spectrum found.