Harmonic functions on one side of a quasicircle on a compact Riemann surface can be uniquely extended to the other side.
problem Transmission of harmonic functions across a quasicircle boundary on compact Riemann surfaces.
method Analyzing the properties of quasicircles and applying them to harmonic function boundary values.
result A unique harmonic function can be defined on the other side of a quasicircle boundary, preserving boundary values.
New method improves AI system's uncertainty estimation.
problem Precise uncertainty estimation in AI predictions.
method Information Aware Max-Norm Dirichlet Networks.
result Outperforms state-of-the-art neural networks for uncertainty estimation.
Extends Hess-Schrader-Uhlenbrock inequality for 1-forms in tamed Dirichlet spaces.
problem Establishing a new inequality for 1-forms in tamed Dirichlet spaces.
method Developed a vector calculus for tamed Dirichlet spaces and applied it to establish the inequality.
result Established the Hess-Schrader-Uhlenbrock inequality for 1-forms in L2-cotangent module. A Bayesian approach termed BAyesian Least Squares Optimization with Nonnegative L1-norm constraint (BALSON) is proposed. The error distribution of data fitting is described by Gaussian likelihood. The parameter distribution is assumed to be a Dirichlet distribution. With the Bayes rule, searching for the optimal parame…
Griffiths extremal metrics solve complex Finsler equations and quantify Kähler geometry.
problem Interpolation of norms and complex Finsler geometry.
method Introduced Griffiths extremal Finsler metrics and solved their Dirichlet problem.
result Griffiths extremal Finsler metrics quantize solutions to a PDE in Kähler geometry.
Study heat profiles and eigenfunctions using Brownian motion.
problem Investigate heat profiles and eigenfunctions of Laplace equations.
method Probabilistic tools based on Brownian motion and Feynman-Kac formulae.
result Supremum norm bounds for ground state Dirichlet eigenfunctions and comparison of maximum temperatures.
Paper proves stability and Dirichlet problem for translating hypersurfaces.
problem Stability and Dirichlet problem for translating hypersurfaces.
method Analyzes translating solitons in en+k, proves stability conditions, and studies Dirichlet problem. result Proves the infimum of mean curvature is zero for translating solitons and conditions for stability.
Study shows continuity of non-orientable surface determination from Dirichlet-to-Neumann map.
problem Determining non-orientable surfaces from Dirichlet-to-Neumann map.
method Proving closeness of Dirichlet-to-Neumann maps implies near-conformal diffeomorphism.
result Established continuity of determination Λ↦[(M,g)] and quantitative estimates of dT([(M,g)],[(M′,g′)]). Solves a complex equation on manifolds with boundary.
problem Solving a Dirichlet problem for k-Hessian equations on complex manifolds. method Second order a priori estimate and blow-up argument.
result Control over all necessary norms of the solution.
New approach to solving minimal surface system Dirichlet problem on smooth domains.
problem Solving Dirichlet problem for minimal surface system on smooth domains.
method Using mean curvature flow (MCF) with boundary conditions and non-negative Ricci curvature assumption.
result Existence of long-time mean curvature flow and existence result for exterior Dirichlet problem.
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
New PINNs method improves accuracy in computing Mean Escape Time from bounded domains.
problem Computing Mean Escape Time from bounded domains with high accuracy.
method Boundary-adapted Physics-Informed Neural Networks (PINNs) with exact Dirichlet boundary enforcement.
result Derivation of H2(Ω) a priori error bounds for PINNs with normalized distance approximations. We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds M×ϱR. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean cur…
Method learns Dirichlet-to-Neumann maps on graphs using Gaussian processes.
problem Coupling multiphysics simulations on graphs with conservation constraints.
method Gaussian processes combined with discrete exterior calculus and maximum likelihood estimation.
result Data-driven predictions with uncertainty quantification on entire graph.
The article characterizes gradient ρ-Einstein solitons under specific conditions.
problem Characterizing gradient ρ-Einstein solitons with certain properties.
method Analyzing solitons with vector fields of bounded norm, finite weighted Dirichlet integral, and specific Ricci curvature restrictions.
result Non-trivial complete gradient ρ-Einstein solitons with finite weighted Dirichlet integral and certain Ricci curvature restrictions are of constant scalar curvature and steady.
New tensorization theorem for Sobolev spaces on product spaces.
problem Characterize Sobolev spaces on product metric measure spaces.
method Showed two descriptions of Sobolev space on product spaces coincide.
result Norm equivalence and density results for Sobolev spaces.
The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.
problem Analyzing Laplace learning for infinite-dimensional Gaussian measure data.
method Minimizes Dirichlet energy on a graph constructed from the full dataset.
result Proves pointwise convergence of the graph Dirichlet energy for Gaussian measure data.
In reinforcement learning, temporal difference (TD) is the most direct algorithm to learn the value function of a policy. For large or infinite state spaces, exact representations of the value function are usually not available, and it must be approximated by a function in some parametric family. However, with \emph{no…
Sharp bounds derived for eigenvalues on specific geometric spaces.
problem Eigenvalue problems on asymptotically hyperbolic manifolds and submanifolds.
method Sharp bounds derived for three types of eigenvalue problems: p-Dirichlet, polyharmonic, and weakly Poincaré-Einstein. result Sharp bounds and their implications for asymptotic sectional curvatures and mean curvature.
Deep neural solvers can approximate harmonic functions with low error.
problem Harmonic functions in Barron space are not regular.
method Elliptic regularity theory applied to Barron functions.
result Approximation of harmonic functions by Barron functions with low error.
Abstract: Generalizes SGMs to infinite-dimensional Hilbertian setting.
problem Difficulties in extending SGMs to infinite-dimensional settings.
method Uses Gamma and Malliavin Calculus, Dirichlet forms, Wiener chaoses, and time-reversal formula.
result Generalized SGMs to Hilbertian setting with finite-dimensional entropic convergence bounds.
The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the Laplacian converges in a norm-resolvent sense to a Schroedinger operator on the limit…
In this paper, we extend the work in \cite{D}\cite{ChrusLiWe}\cite{ChrusCo}\cite{Co}. We weaken the asymptotic conditions on the second fundamental form, and we also give an L6−norm bound for the difference between general data and Extreme Kerr data or Extreme Kerr-Newman data by proving convexity of the renormali…
The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
The paper improves interpolation in generative models by using specific base distributions.
problem Unexpected side effects in linear interpolations of normalizing flows.
method Enforces a specific manifold using Dirichlet and von Mises-Fisher base distributions.
result Superior performance in terms of bits per dimension, FID, and KID scores for interpolation.
New measure shows various training techniques control model complexity.
problem Understanding how to control model complexity in deep learning.
method Developed geometric complexity measure and demonstrated its effectiveness.
result Many training techniques control geometric complexity, providing a unified framework.
Improved VB algorithm for NIG mixtures outperforms Gaussian mixtures for non-Gaussian data.
problem Clustering non-Gaussian data, especially heavy-tailed and asymmetric.
method Proposed an improved VB algorithm for NIG mixture models and extended Dirichlet process mixture models.
result Outperforms Gaussian mixtures and existing NIG mixture models, especially for highly non-normative data.
We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm ∣∇R∣ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determi…
AIR-Net adapts low-rank regularization dynamically for better image completion.
problem Fixed low-rank regularization limits adaptability to different images.
method AIR-Net uses adaptive and implicit regularization parameterized by a dynamic Laplacian matrix.
result AIR-Net enhances implicit regularization and outperforms fixed methods in non-uniform missing data scenarios.
We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by K>0 and dimension bounded above by N∈(1,∞) in a synthetic sense, the so called CD(K,N) spaces. We first establish a Polya-Szego type inequality stating that the $W^{…
Investigates a conjugate prior for Dirichlet distribution.
problem No specific problem stated; focuses on mathematical investigation.
method Investigates a conjugate class for the Dirichlet distribution within the exponential family.
result Identifies a conjugate prior for the Dirichlet distribution.
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
Solves a specific Dirichlet problem on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds.
method Solves the Dirichlet problem for Monge-Ampère type equations for (n−1)-plurisubharmonic functions on Hermitian manifolds. result Solves a specific Dirichlet problem on Hermitian manifolds.
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
New algorithm for computing Veech groups from translation surfaces.
problem Computing Veech groups for translation surfaces.
method Infinite translation surface containing copies of all surfaces in a stratum; associated affine automorphisms of the infinite surface map marked segments to other pairs of segments.
result Explicit hyperbolic ball condition for Fuchsian groups to agree with their Dirichlet domain.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
Study classifies ruled surfaces critical to Dirichlet energy.
problem Identifying ruled surfaces critical to Dirichlet energy.
method Explicit parametrization of ruled surfaces.
result Classification of ruled surfaces as critical points of Dirichlet energy.
LDTA expands LDA's topic modeling capacity with tree-structured priors.
problem Limited expressiveness of Dirichlet priors in LDA for complex topic relationships.
method Introduces Latent Dirichlet-Tree Allocation (LDTA) with Dirichlet-Tree (DT) priors, and develops universal mean-field variational inference and Expectation Propagation.
result LDTA enables expressive, tree-structured priors over topic proportions, expanding modeling capacity of LDA.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
The paper studies higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues.
problem Analyzing eigenvalues of higher order Dirichlet-to-Neumann maps on graphs.
method Introducing and studying higher order Dirichlet-to-Neumann maps on graphs, deriving estimates on eigenvalues.
result Raulot-Savo-type estimates on the eigenvalues of the DtN maps.
Solves a specific Dirichlet problem on Riemannian manifolds.
problem Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds.
method Derives existence of C1,1-solutions under appropriate assumptions. result Existence of C1,1-solutions. Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.
Enhances topic models to better handle polysemous words.
problem Lack of polysemy handling in Gaussian latent Dirichlet allocation.
method Introduces a hierarchical structure to capture polysemy in Gaussian latent Dirichlet allocation.
result Significantly improves polysemy detection and provides more parsimonious topic representations.
TPBS models improve robustness to overfitting with localized Dirichlet energy regularization.
problem Global Dirichlet energy-based regularization fails for TPBS models due to perfect interpolation.
method Propose local Dirichlet energy regularization and two inference estimators.
result TPBS models outperform neural networks in overfitting regimes and maintain competitive performance otherwise.
Study on geometry of Dirichlet distributions using Fisher-Rao metric.
problem Understanding the geometry of Dirichlet distributions.
method Analysis of Fisher-Rao metric on Dirichlet distribution parameter space.
result Geodesic completeness and negative sectional curvature of the space.
We prove an analogue of Sogge's local Lp estimates for Lp norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov, Hu, and Chen-Sogge. The improvements are logarithmic on negatively curved m…
This paper is devoted to a proof of a generalized Ray-Singer conjecture for a manifold with boundary (the Dirichlet and the Neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary). The Ray-Singer conjectur…
Graph Laplacians adapt to different manifold dimensions, while Dirichlet energies converge to a tensorized Dirichlet energy.
problem Understanding machine learning methods for data with varying intrinsic dimensions.
method Γ-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on intersecting manifolds of varying dimensions.
result Normalized Dirichlet energy converges to a tensorized Dirichlet energy that adapts to all dimensions simultaneously.