We use persistent homology along with the eigenfunctions of the Laplacian to study similarity amongst triangulated 2-manifolds. Our method relies on studying the lower-star filtration induced by the eigenfunctions of the Laplacian. This gives us a shape descriptor that inherits the rich information encoded in the eigen…
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48 results for “eigenfunction analysis”
Estimates for eigenfunctions and quasimodes on compact manifolds.
problem Characterizing eigenfunctions and quasimodes on compact manifolds.
method Sharp -estimates for log-quasimodes, focusing on small Lebesgue exponents.
result No characterization possible for .
This study provides a new mathematical structure for Koopman eigenfunctions.
problem Understanding and representing nonlinear dynamics as linear.
method Theoretical, analytical, and numerical approaches to Koopman eigenfunction space.
result Equivalence of minimal generating set and maximal independent set, defining conditions for independence.
From Varadhan's Limit to Eigenmaps: A Guide to the Geometric Analysis behind Manifold Learningmath.DG
Overview of geometric analysis for manifold learning.
problem Analyzing high-dimensional data via spectral embeddings.
method Heat kernel and eigenfunctions on Riemannian manifolds.
result Uniform control of spectral embeddings on key classes of manifolds.
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.
Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
problem Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
method Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
result Identifying the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization
We present Spectral Inference Networks, a framework for learning eigenfunctions of linear operators by stochastic optimization. Spectral Inference Networks generalize Slow Feature Analysis to generic symmetric operators, and are closely related to Variational Monte Carlo methods from computational physics. As such, the…
Researchers found the first and second eigenvalues are Courant-sharp on a Möbius strip.
problem Determining Courant-sharp eigenvalues on a Möbius strip.
method Analyzing the eigenvalues and nodal patterns of the Möbius strip.
result Only the first and second eigenvalues are Courant-sharp on the Möbius strip.
New method for minimal submanifolds in spheres using eigenfunctions.
problem Constructing minimal submanifolds in spheres.
method Eigenfunctions of the Laplace-Beltrami operator to describe minimal submanifolds.
result New proof of -eigenfunctions on spheres.
Study eigenvalues and eigenfunctions of fourth-order operators in annuli, proving optimal estimates and non-radiality.
problem Eigenvalue and eigenfunction analysis of fourth-order operators in degenerating annuli.
method Optimal estimates and non-radiality results for eigenfunctions in annuli.
result Nigh optimal estimate for the first eigenvalue and non-radiality of eigenfunctions in degenerating annuli.
Kernel method approximates dynamical operators from data.
problem Estimating eigenfunctions of dynamical operators from data.
method Kernel-based approach in reproducing kernel Hilbert spaces.
result Eigenfunctions estimated via matrix eigenvalue problems.
Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.
problem Finding a unified scheme for complex-valued eigenfunctions on Riemannian symmetric spaces.
method Employing the Cartan embedding for classical compact Riemannian symmetric spaces and quaternionic Grassmannians.
result Construction of new eigenfunctions on quaternionic Grassmannians.
Study critical points of Laplace eigenfunctions in polygons.
problem Characterize critical points of Laplace eigenfunctions in polygonal domains.
method Analyze components of the critical set with codimension 1.
result For simply connected polygons, if a second Neumann eigenfunction has infinitely many critical points, the polygon must be a rectangle.
Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.
problem Quantifying the Hessian of log-concave eigenfunctions on convex domains.
method Analyzes log-concavity properties of the first Dirichlet eigenfunction on convex domains.
result Obtains quantitative estimates for the Hessian of .
The study counts critical points of Steklov eigenfunctions on manifolds.
problem Counting critical points of Steklov eigenfunctions on manifolds.
method Established an identity relating indexes of eigenfunctions and their restrictions to the boundary, and used it to count critical points.
result A precise count of interior critical points of Steklov eigenfunctions in terms of manifold's Euler characteristic and boundary sign changes.
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.
Eigenfunction gradients on curved spaces imply rigid structure.
problem Eigenfunction gradient estimates on curved manifolds.
method Sharp Li-Yau type gradient estimates for Neumann or Dirichlet eigenfunctions.
result Compact manifolds with specific curvature properties are rigidly structured.
New method creates minimal submanifolds using complex-valued eigenfunctions.
problem Creating minimal submanifolds in compact Riemannian manifolds.
method Employing complex-valued eigenfunctions.
result Manufactured minimal submanifolds in compact Riemannian manifolds.
We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequa…
Study on variance of Laplace eigenfunctions on manifolds.
problem Investigating the variance of Laplace eigenfunctions on compact manifolds.
method Combining Kac-Rice formula, Wiener-Itô chaos decompositions, and pointwise Weyl law analysis.
result Established a quantitative bound for the fluctuations of nodal volumes, improving existing results.
The paper bounds Cheeger ratios of eigenfunctions and their level sets.
problem Understanding geometric features of Riemannian manifolds through eigenfunctions.
method Constructive upper bounds on Cheeger constants using eigenvalues and eigenfunctions.
result Upper bounds on Cheeger ratios of eigenfunction level sets and their superlevel sets.
Survey of kernels, RKHS, and their applications in machine learning.
problem Understanding kernels and their applications in machine learning.
method Review of historical context, mathematical definitions, and practical applications of kernels.
result Comprehensive overview of kernels, RKHS, and their applications.
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.
We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace--Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of …
Let us fix two different radial eigenfunctions of a hyperbolic Laplacian and assume that both of them have the same value at the origin. Both eigenvalues can be complex numbers. The main goal of this paper is to estimate the lower bound for the interval (0,T], where these two eigenfunctions must assume different values…
The paper explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.
problem Existence and properties of eigenfunctions for spherical conical metrics.
method Application of multivalued harmonic maps to spheres and algebraic constructions.
result New criteria and examples of metrics with many 2-eigenfunctions.
Study small perturbations on low energy Laplace eigenfunctions.
problem Understanding small changes in low energy Laplace eigenfunctions.
method Investigates nodal geometry and topology, focusing on low frequency regimes and small perturbations.
result Highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions.
Paper connects probability density cuts to graph theory eigenfunctions.
problem Developing sparse cuts for probability densities.
method Defines sparse cuts and principal eigenfunctions for probability densities, proving Cheeger and Buser inequalities.
result No such inequalities hold for prior definitions, proving new inequalities for probability densities.
Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost and it is difficult to design nonstationary GP priors in practice. In this paper, we propose a sparse Gaussian process model, EigenGP, based on the Karhunen-Loeve (KL) expan…
Eigenfunction maxima inside high-d nodal domains.
problem Understanding eigenfunction maxima in high-dimensional nodal domains.
method Proving eigenfunction maxima inside nodal domains of high-dimensional manifolds.
result Eigenfunction maxima are within a specific radius of the eigenvalue and dimension.
We give an upper bound for the -dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic Riemannian manifolds. This is the analog of H. Donnely and C. Fefferman result on nodal set of eigenfunctions.
In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures correspondin…
In this paper, we study eigenvalues and eigenfunctions of -Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of -Laplacian, as we ident…
Polynomial growth bounds for eigenfunctions on non-compact spaces.
problem Growth of eigenfunctions on non-compact spaces with Gaussian weight.
method Analyzing non-compact manifolds with Gaussian weight and drift Laplacian.
result Showed same polynomial growth bounds as Euclidean space for general tensors.
We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for -norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.
In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick applicatio…
Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.
problem Bounding the measure of zero sets of Neumann Laplace eigenfunctions.
method Analyzes nodal sets in domains with boundaries.
result Hausdorff measure bound of nodal sets is .
Zonal states and improved bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfacesmath.AP
Improved bounds for eigenfunctions on hyperbolic surfaces found.
problem Establishing improved bounds for eigenfunctions of magnetic Laplacians.
method Using explicit eigenstates called magnetic zonal states.
result Explicit eigenstates called magnetic zonal states found.
The study bounds quantum eigenfunctions on complex manifolds.
problem Restricting quantum eigenfunctions on complex manifolds.
method Analytic continuation and FBI transform for Laplace eigenfunctions.
result Upper and lower bounds for eigenfunctions.
Machine learning finds Z/2 eigenfunctions on a sphere.
problem Finding Z/2 eigenfunctions on the sphere.
method Created a multivalued neural network and used JAX to implement it. Fixed branch points at tetrahedron and cube vertices, and allowed AI to move them in the third case.
result Found Z/2 eigenfunctions for three cases.
Study on -eigenfunctions on compact manifolds, showing manifold properties and eigenfamily dimensions.
problem Characterizing compact manifolds with -eigenfunctions and understanding their eigenfamilies.
method Analyzing -eigenfamilies on compact Riemannian manifolds, showing that any such manifold is a mapping torus and any -eigenfamily is one-dimensional.
result Any compact manifold admitting a -eigenfunction is a mapping torus and any -eigenfamily is one-dimensional.
In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present…
Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.
problem Proving log-concavity of eigenfunctions on curved surfaces.
method Analyzing the Laplacian eigenfunctions on positively curved surfaces.
result Strong log-concavity of the first eigenfunction on positively curved surfaces.
We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.
Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.
problem Understanding covariant derivatives of eigenfunctions on curved spaces.
method Analyzing the Laplace-Beltrami operator on Riemannian manifolds with constant curvature.
result Covariant derivatives of eigenfunctions are scalar multiples of the functions, and these scalars are polynomials in the eigenvalue.
Study eigenfunctions of Laplacian on sphere with even point removals.
problem Behavior of eigenfunctions and eigenvalues on modified spheres.
method Analysis of eigenfunctions and eigenvalues of Laplacian on sphere with even point removals.
result Eigenfunctions and eigenvalues change with point removals.
Unified framework for constructing kernels for transport equations and Koopman eigenfunctions.
problem Constructing kernels for transport equations and Koopman eigenfunctions.
method Three methods: variational principle, Green's function, and resolvent operator.
result Kernels constructed via these methods are identical under mild assumptions.
Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.
problem Establishing bounds for eigenfunctions on product spaces.
method Combining asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.
result Sharp bounds for eigenfunctions on products of rank-one symmetric spaces.