New examples challenge extended Courant property for linear combinations of eigenfunctions.
problem Extended Courant property for linear combinations of eigenfunctions.
method Numerical computations and examples of equilateral rhombus and regular hexagon.
result Counterexamples to the Extended Courant property for linear combinations of eigenfunctions.
The Extended Courant Property is disproven for certain linear combinations of eigenfunctions.
problem Disproving the Extended Courant Property for specific cases.
method Simple and explicit examples of domains (convex, with cracks, sphere, torus) are provided.
result The Extended Courant Property is not universally true for linear combinations of eigenfunctions.
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.
A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and othe…
Study of loops in sums of Laplace eigenfunctions on surfaces.
problem Uniform bound for the number of nested loops in sums of Laplace eigenfunctions.
method Real-analytic category analysis and biharmonic function construction.
result Uniform bound for the number of rooted double nests in terms of surface, root, and spectral cutoff.
Kernel method approximates Koopman operator eigenfunctions.
problem Complexity of computing Koopman operator spectra.
method Kernel-based approach to construct principal eigenfunctions.
result Principal eigenfunctions match linearization eigenvalues.
The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.
We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimens…
Sharp uncertainty principle for nodal sets in singular spaces.
problem Estimating the size of nodal sets in non-smooth spaces.
method Uncertainty principle applied to eigenfunctions in metric measure spaces with synthetic Ricci curvature bounds.
result New lower bounds on nodal set sizes in non-smooth spaces.
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.
Eigenfunction level sets can have infinitely many connected components.
problem Estimating the number of connected components of eigenfunction level sets.
method Analyzing Neumann eigenfunctions on triangles, polygons, and surfaces.
result There exist eigenfunctions with infinitely many connected components in their level sets.
Spectral Inference Networks learn eigenfunctions from data using optimization.
problem Learning eigenfunctions of linear operators from data.
method Spectral Inference Networks generalize Slow Feature Analysis to generic symmetric operators and use stochastic optimization.
result Spectral Inference Networks accurately recover eigenfunctions and discover interpretable representations from video data.
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 ρ2:\EQN{6}{1}{}{0}{\RD{\CELL{u(r,θ) =\sum_{n=0}^{N}P_{n}J_{n}(ρ) …
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 Lp bounds for eigenfunctions on products of rank-one symmetric spaces. The paper derives inequalities for eigenvalues and eigenfunction norms on manifolds.
problem Eigenvalue inequalities and eigenfunction norms on manifolds.
method Combining Milman's and Cheng-Li's work.
result Universal inequalities and upper bounds for eigenvalues and eigenfunction norms.
This article concerns upper bounds for L∞-norms of random approximate eigenfunctions of the Laplace operator on a compact aperiodic Riemannian manifold (M,g). We study fλ chosen uniformly at random from the space of L2-normalized linear combinations of Laplace eigenfunctions with eigenvalues in the inte…
We show that on a compact Riemmanian manifold (M,g), nodal sets of linear combinations of any p+1 smooth functions form an admissible p−sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a resul…
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in R2. method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
problem Existence of contractible domains with specific boundary conditions.
method Local bifurcation argument around geodesic disks, anisotropic Hölder spaces, computer-assisted techniques.
result Existence of nontrivial contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
Proposes a framework to extract ordered eigenfunctions from contextual kernels.
problem Lack of exact spectral decomposition in existing methods.
method Modular building blocks for compatibility with contextual kernels and scalability.
result Extracted eigenfunctions provide effective importance scores for feature selection.
Deep learning discovers intrinsic coordinates for nonlinear dynamics.
problem Identifying coordinate transformations for strongly nonlinear dynamics.
method Modified auto-encoder network to discover Koopman eigenfunctions.
result Compact and efficient embedding of intrinsic rank for nonlinear systems.
In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragmén-Lindelöf principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of t…
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.
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on L2--Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates. result Almost sharp local Lp--Bernstein inequalities for p∈[1,∞]. Kernel operators help detect patterns in complex data.
problem Detecting long-lived coherent patterns in high-dimensional time-series data.
method Dominant eigenfunctions of kernel transfer operators combined with gradient-based optimization.
result Effective detection of long-lived coherent patterns in high-dimensional time-series data.
Sharp estimate for nodal domains intersecting a ball on a Riemannian manifold.
problem Local bounds for nodal domains on Riemannian manifolds.
method Combining Remez inequality for eigenfunctions and Landis growth lemma in narrow domains.
result Proved a sharp estimate of nodal domains intersecting a ball.
Improves bounds on eigenfunctions using microlocal averages in phase space.
problem Improving Lp bounds on eigenfunctions in high frequency limit. method Develops sufficient conditions for microlocal averages in nonpositive curvature and partially hyperbolic flows.
result Improves microlocal averages for eigenfunctions in more general settings.
Paper proposes a new optimization framework for learning eigenfunctions of operators.
problem Computing eigenvalue decomposition of high-dimensional operators.
method Operator SVD with Neural Networks via Nested Low-Rank Approximation.
result Proposed method efficiently learns top-L singular values and functions in the correct order.
Paper proposes a new method to optimize feature coordinates for better image classification.
problem Improving feature extraction for better machine learning classification.
method Mutual-energy inner product optimization method.
result The method enhances low-frequency features and suppresses high-frequency noise, leading to better classification results.
Study on linear independence of Poincaré series for anti-de Sitter 3-manifolds.
problem Linear independence of generalized Poincaré series for anti-de Sitter 3-manifolds.
method Analysis of eigenfunctions and Laplacian on anti-de Sitter 3-manifolds.
result Unbounded multiplicities of eigenvalues for L2-eigenfunctions and stable L2-eigenvalues. Eigenfunctions on manifolds concentrate near nodal sets.
problem Understanding concentration of eigenfunctions on manifolds.
method Volume measure analysis and Colding-Minicozzi method.
result Exponential concentration of volume measure around nodal sets.
The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.
problem Understanding the relationship between Fisher information matrices and neural tangent kernels for 2-layer ReLU networks.
method Analyzes Fisher information matrices and neural tangent kernels for 2-layer ReLU networks with random hidden weights, focusing on spectral decomposition and eigenfunctions.
result Obtained an approximation formula for functions represented by 2-layer neural networks.
Identifies optimal base features for zero-shot adaptation in reinforcement learning.
problem Unclear what constitutes a good set of base features for a wide range of downstream tasks.
method Identifies optimal base features based on downstream performance, without assuming downstream tasks are linear.
result Optimal base features are the same across three task families, differing from Laplacian eigenfunctions.
Optimizes metrics on surfaces for eigenvalues.
problem Finding optimal metrics for eigenvalues on surfaces.
method Combining constructions of Palais-Smale-like sequences and techniques from Karpukhin et al.
result Existence of optimal metrics for various eigenvalues on surfaces.
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 (λ,μ)-eigenfunctions on spheres. 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
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 logu. Characterizes specific harmonic manifolds using Laplacian eigenfunctions.
problem Characterizing harmonic manifolds.
method Using radial eigenfunctions of the Laplacian.
result Characterizes space forms, complex hyperbolic spaces, and quaternionic hyperbolic spaces as harmonic manifolds.
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.
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.
Method uses Laplacian eigenfunctions and persistent homology for shape similarity.
problem Studying similarity among triangulated 2-manifolds.
method Lower-star filtration induced by Laplacian eigenfunctions.
result Shape descriptors inherit rich information from eigenfunctions.
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.
Develops methods for spectral estimation and rare-event prediction in complex systems.
problem Challenges in understanding dynamics in complex systems with many degrees of freedom.
method Inexact iterative numerical linear algebra methods for spectral estimation and rare-event prediction.
result Demonstrates methods on low-dimensional and high-dimensional models, showing their effectiveness.
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.
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.