Machine learning models solve inverse eigenvalue problems for symmetric potentials and refractive indices.
problem Solving inverse eigenvalue problems for symmetric potentials and refractive indices.
method Supervised regression models (k-Nearest Neighbours, Random Forests, Multi-Layer Perceptron) trained on eigenvalue datasets.
result Machine learning methods can numerically solve inverse eigenvalue problems under appropriate parameter tuning.
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for…
We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degener…
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…
The paper compares Steklov and Laplacian eigenvalues on graphs.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on graphs.
method Analyzing eigenvalues and discussing rigidity.
result Obtained Lichnerowicz-type estimates and combinatorial estimates for Steklov eigenvalues.
Study finds only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.
problem Determining Courant-sharp eigenvalues for compact flat surfaces.
method Analyzing flat Klein bottle and cylinders, proving only first and second eigenvalues are Courant-sharp.
result Only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.
The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlat…
Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.
problem Isometry and eigenvalue gap in Finslerian models.
method Presented models and isometry proofs.
result First eigenvalue gapless for all three models.
In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…
The paper explores inequalities between eigenvalues on Riemannian manifolds.
problem Investigating relationships between eigenvalues on Riemannian manifolds.
method Constructing gradient estimates for a first eigenfunction to derive inequalities.
result Obtained some relationships between weighted p-Laplacian first eigenvalues. Sharp bounds and rigidity theorems for eigenvalues on manifolds.
problem Estimating eigenvalues and characterizing rigidity on manifolds.
method Volume comparison, Escobar-type eigenvalue comparisons, and Reilly formula.
result Sharp bounds and rigidity conditions for eigenvalues on manifolds.
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
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.
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.
The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
problem Eigenvalues of the Laplace operator and clamped plate problem.
method Sharp lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
result Sharp lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
Paper finds how Steklov eigenvalues change on graphs and trees.
problem Understanding how Steklov eigenvalues vary on graphs and trees.
method Analyzes monotonicity of Steklov eigenvalues on graphs and trees.
result Extends Steklov eigenvalue results to higher eigenvalues and trees.
The paper extends a spectral evolution model for link prediction in evolving networks.
problem Link prediction in evolving networks.
method Approximated eigenvalue trajectories using Rayleigh quotient and extrapolation.
result Learning algorithms based on approximated trajectories outperform traditional methods.
The paper bounds eigenvalues of specific operators on certain manifolds.
problem Bounding eigenvalues of Paneitz and third-order boundary operators on locally conformally flat manifolds.
method Proof based on conformal equivalence to canonical models, showing injectivity of developing maps, and explicit computations on canonical models.
result Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds.
Eigenvalue estimate for shrinkers in mean curvature flow.
problem Eigenvalue estimates on shrinkers for mean curvature flow.
method Generalized earlier work of Ding and Xin to noncompact cases.
result Eigenvalue estimate holds on every properly embedded shrinker.
Item-item collaborative filtering (CF) models are a well known and studied family of recommender systems, however current literature does not provide any theoretical explanation of the conditions under which item-based recommendations will succeed or fail. We investigate the existence of an ideal item-based CF method a…
The exact meaning of the noise spectrum of eigenvalues of the covariance matrix is discussed. In order to better understand the possible phenomena behind the observed noise, the spectrum of eigenvalues of the covariance matrix is studied under a model where most of the true eigenvalues are zero and the parameters are n…
In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an n-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the kth eigenvalue and for isoparametric minimal hypersurfaces in the unit sphe…
For a bounded domain Ω with a piecewise smooth boundary in an n-dimensional Euclidean space Rn, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Lap…
Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.
problem Eigenvalue rigidity of random hyperbolic surface covers.
method Selberg trace formula and polynomial method.
result Distribution of eigenvalues converges to hyperbolic plane's spectral measure.
The paper provides estimates for eigenvalues of elliptic differential problems.
problem Computing eigenvalue estimates for elliptic differential problems.
method Analytical computation of eigenvalues for specific types of elliptic differential equations.
result Universal estimates of eigenvalues and gaps between consecutive eigenvalues are derived.
The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.
problem Eigenvalue comparisons on graphs.
method Analytical comparisons and discussions of eigenvalues and their applications.
result Extensions of eigenvalue estimates for Dirichlet and Neumann eigenvalues.
Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.
problem Finding bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
method Sharp lower and upper bounds derived using the support function and distance function to the origin of the boundary.
result Sharp bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
Let $\om $ be a bounded domain in an n-dimensional Euclidean space Rn. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &Δ{\mathbf u}+ α{\rm grad}(\text{div}{\mathbf u})=-σ{\mathbf u}, \ \text{in $Ω$}, &{\mathbf u}|_{\partial Ω}={\mathbf 0}. \aligned . $$ Estimate…
We consider classifiers for high-dimensional data under the strongly spiked eigenvalue (SSE) model. We first show that high-dimensional data often have the SSE model. We consider a distance-based classifier using eigenstructures for the SSE model. We apply the noise reduction methodology to estimation of the eigenvalue…
We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold P of CPn which depends only on the radius of the tube, the degrees of the polynomials defining P and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the mo…
Paper uses autoencoders for efficient reduced-order modeling of eigenvalue problems.
problem Efficiently modeling eigenvalue problems in high dimensions.
method Autoencoder-based reduced-order modeling for eigenvalue problems.
result Autoencoder-based models outperform standard POD-Galerkin methods in neutron diffusion applications.
Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.
problem Lower bounds for higher eigenvalues of the poly-Laplacian operator.
method Sharp inequalities and eigenvalue bounds in low and arbitrary dimensions.
result Improved lower bounds for eigenvalues of the poly-Laplacian in arbitrary dimensions.
Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
problem Finding lower bounds for eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
method Analytical proofs for both Neumann and Dirichlet boundary conditions.
result Established lower bounds for eigenvalues on compact quaternionic Kähler manifolds.
We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we p…
The paper studies eigenvalues of Xin-Laplacian on Riemannian manifolds.
problem Eigenvalue problems related to Xin-Laplacian on Riemannian manifolds.
method Establishing general formulas and applying Chen-Cheng type results.
result Sharp estimates for the upper bound of the second nonzero eigenvalue of the Laplace-Beltrami operator.
The paper finds new inequalities for Laplacian and biharmonic eigenvalues on manifolds.
problem Eigenvalue inequalities for Laplacian and biharmonic operators on submanifolds.
method Using Sobolev inequalities to establish new eigenvalue inequalities.
result Established new inequalities for Laplacian and biharmonic eigenvalues.
In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minim…
Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.
problem Eigenvalue problem for complex Hessian operator on pseudoconvex manifolds.
method Established C1,1-regularity and uniqueness of the first eigenfunction, derived variational formula for the first eigenvalue. result Derivation of a bifurcation-type theorem and geometric bounds for the eigenvalue.
Study on second Robin eigenvalue for Laplacian on manifolds.
problem Maximizing the second Robin eigenvalue for geodesic balls in nonpositively curved space forms.
method Comparison theorem and maximization analysis for the second Robin eigenvalue.
result Geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue.
Paper finds principal eigenvalue for infinity Laplacian in metric spaces.
problem Finding the principal eigenvalue of the infinity Laplacian in metric spaces.
method Direct PDE approach and Perron's method to establish existence of solutions.
result Existence of solutions to the infinity eigenvalue problem in metric spaces.
Estimates for eigenvalues on Riemannian manifolds using classical inequalities.
problem Estimating eigenvalues of the Dirichlet Laplacian on Riemannian manifolds.
method Building on Li-Yau's and Yang's inequalities, deriving upper and lower bounds.
result Explicit estimates on lower bounds for eigenvalues of the Dirichlet Laplacian on projective spaces and their minimal submanifolds.
Lower bounds for eigenvalues on Bakry-Emery manifolds proven.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth-order elliptic operators on Riemannian manifolds.
method Proves inequalities using Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds.
result Generalizes eigenvalue inequalities for the clamped plate problem to complete Riemannian manifolds.
In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.
The paper proves stability of eigenvalue inequalities on surfaces.
problem Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces.
method Employing eigenvalues of measures and Sobolev space W−1,2, the paper proves stability estimates for the first and second nonzero Laplace eigenvalues on surfaces. result Metrics almost maximizing the normalized eigenvalue are W−1,2-close to a maximal metric. Study on Kähler manifolds shows rigidity of eigenvalues with positive Ricci bound.
problem Optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
method Established optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
result Complex projective space is the only Kähler manifold with the largest multiplicity of the first eigenvalue.
Estimates small eigenvalues for geometrically finite manifolds.
problem Estimating small eigenvalues of Schrödinger operators.
method Geometrically finite manifolds, Riemannian vector bundles.
result Estimates the number of small eigenvalues.
The paper finds universal inequalities for eigenvalues on hyperbolic spaces.
problem Eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds.
method Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.
result Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.