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.
Study proves inequalities for eigenvalues of symmetric domains in space forms.
problem Eigenvalue problem for Laplacian on symmetric domains in space forms.
method Symmetry assumptions on domain, proof of inequalities for eigenvalues.
result Proves Szegö-Weinberger type inequalities for first n positive Neumann eigenvalues.
Study finds lower bounds for energy on fibred manifolds using fiberwise symmetrization.
problem Finding lower bounds for energy functionals on fibred manifolds.
method Established a framework for fiberwise symmetrization to find lower bounds.
result Proved a comparison theorem for the first eigenvalue of the Laplacian on warped product manifolds.
Inverse problem solved for rotationally symmetric manifolds using eigenvalues and resonances.
problem Determining the rotation radius of a manifold from its eigenvalues and resonances.
method Unitary equivalence to one-dimensional Schrödinger operators, non-linear real analytic isomorphism between Hilbert spaces.
result The rotation radius is uniquely determined by its eigenvalues and resonances.
We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
Study of Dirac operator eigenvalues on symmetric spaces.
problem Finding the first eigenvalue of the Dirac operator on compact spin symmetric spaces.
method Analyzing algebraic data of groups involved to derive formulas for eigenvalues.
result Explicit expression for the first eigenvalue of outer compact spin symmetric spaces.
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
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.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Study proves inequality for eigenvalues in symmetric spaces.
problem Proving an inequality for Steklov eigenvalues in symmetric spaces.
method Analyzes eigenvalues on bounded domains in noncompact rank-1 symmetric spaces.
result Extends previous results to non-Euclidean spaces.
Paper proves eigenvalue inequality for Hopf-symmetric domains.
problem Eigenvalue inequality for Hopf-symmetric domains in non-compact symmetric spaces.
method Used geometric and spectral analysis on non-compact rank one symmetric spaces.
result Eigenvalue inequality for bounded Hopf-symmetric domains in non-compact symmetric spaces.
Study on stability of quaternion-Kähler manifolds using eigenvalue estimates.
problem Stability problem for positive quaternion-Kähler manifolds.
method Description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and symmetric 2-tensors. Improved eigenvalue estimates for the Hodge-Laplacian on 2-forms.
result Sharp lower bound for the first non-zero eigenvalue on the parallel subbundle Sym^2 E of the 2-form bundle.
In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
New method to bound Laplacian eigenvalues of geodesic balls.
problem Computing upper bounds for the first eigenvalue of Laplacian on geodesic balls.
method Transforming metric tensor into rotationally symmetric form preserving geodesic sphere areas.
result Upper bound for Laplacian eigenvalues is sharp and computable using geodesic sphere areas.
Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose fi…
Study eigenvalues of CROSS spaces with various metrics.
problem Eigenvalue of Laplace-Beltrami operator on CROSSes.
method Explicit formulae for the first eigenvalue of CROSSes with homogeneous metrics.
result Homogeneous metrics on CROSSes are isospectral if and only if they are isometric.
In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use t…
Solves numerical computation of Killing and conformal Killing vector fields on compact Riemannian manifolds.
problem Overdetermined systems of PDE make numerical computation difficult.
method Reduces to symmetric eigenvalue problem solved by finite element techniques.
result Valid in any dimension and for arbitrary compact Riemannian manifolds.
The study proves no L2-eigenvalues for higher rank locally symmetric spaces.
problem Absence of principal eigenvalues for higher rank locally symmetric spaces.
method Derives dynamical assumptions on the Γ-action on geodesics and Satake compactifications.
result Generalization of Patterson's result to higher rank locally symmetric spaces.
Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.
problem Investigating the first eigenvalue of the Laplace operator on 1-forms in compact inner symmetric spaces.
method Analyzing the Casimir eigenvalue of the highest root for the isotropy representation.
result The first eigenvalue of the Laplace operator on 1-forms is the Casimir eigenvalue of the highest root.
Study properties of semi-symmetric Lorentzian spaces, foliated manifolds.
problem Properties of semi-symmetric pseudo-Riemannian manifolds.
method Investigate foliated manifolds with Lorentzian metrics and analyze Ricci operator eigenvalues.
result Ricci operator has only real eigenvalues for Lorentzian metrics.
The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.
problem Maximizing the first normalized Laplace-Beltrami eigenvalue on tori.
method Constructing equivariant harmonic maps to spheres and analyzing their properties.
result Rotationally symmetric critical metrics for the first eigenvalue are found and characterized.
Estimates eigenvalues of Jacobi operator for harmonic spaces.
problem Estimating eigenvalues of Jacobi operator.
method Using density function of a harmonic space.
result Sharp estimates imply symmetric Osserman space.
We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We …
We give lower and upper bounds for the first eigenvalue of geodesic balls in spherically symmetric manifolds. These lower and upper bounds are C0-dependent on the metric coefficients. It gives better lower bounds for the first eigenvalue of spherical caps than those from Betz-Camera-Gzyl.
We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space G/K.
Develops a game-theoretic approach to solve SGEP efficiently.
problem Efficiently solving the symmetric generalized eigenvalue problem for large datasets.
method Formulates SGEP as a Nash equilibrium in a game-theoretic context and develops a parallelizable algorithm.
result Achieves O(dk) runtime complexity, making it feasible for large-scale problems. Minimal submanifolds in matrix spaces proven for specific ranks.
problem Minimal submanifolds in matrix spaces.
method Proving semialgebraic sets of matrices are minimal.
result Rectangular, skew-symmetric, and symmetric matrices with prescribed eigenvalues are minimal.
We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only C0 dependent on the metric coefficients.
Let G/K be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of G sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of G. As an example of application, we give the list of the …
We review some recent results concerning lower eigenvalues estimates for the Dirac operator [6, 7]. We show that Friedrich's inequality can be improved via certain well-chosen symmetric tensors and provide an application to Sasakian spin manifolds.
We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigen…
When a Riemannian manifold (M,g) is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to −r1, where r stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rota…
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
Study eigenvalues of drift Laplacian on symmetric self-shrinkers in R^3.
problem Estimating the first eigenvalue of the drift Laplacian on symmetric self-shrinkers.
method Analyzing the dihedral and prismatic groups to prove the first eigenvalue is 1/2.
result Proved that the first eigenvalue of the drift Laplacian is 1/2 for symmetric self-shrinkers.
We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let B1 be an open ball in Rn and B0 be a ball contained in B1. Let ν be the outward unit normal on ∂B1. Then the first eigenvalue o…
The study classifies solutions to a specific eigenvalue problem and identifies the critical catenoid.
problem Eigenvalue problem on the sphere with boundary conditions.
method Classifying positive solutions as rotationally symmetric and analyzing boundary conditions.
result Characterization of the critical catenoid as the only embedded free boundary minimal annulus.
We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.
We consider the problem of conformally deforming a metric to one with a prescribed symmetric function of the eigenvalues of the Ricci tensor, in the case of negative curvature.
Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.
problem Bounding Laplacian eigenvalues on manifolds with non-negative scalar curvature.
method Investigation of invariant spectrum on compact Riemannian manifolds with large isometry groups.
result Upper bounds for eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
New divergence identity for scalar curvature helps prove rigidity of tensors.
problem Proving rigidity of Codazzi tensors under curvature and invariant conditions.
method Derived a divergence identity for a vector field and applied it to tensor rigidity.
result New proof of Tang-Yan theorem on constant eigenvalues for tensors.
In this paper, we study eigenvalues and eigenfunctions of p-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 p-Laplacian, as p→1, we ident…
RIG extends IG to Riemannian manifolds for explainable AI.
problem Lack of explainability in AI models.
method Extension of Integrated Gradients to Riemannian manifolds.
result RIG restricts to IG in Euclidean space.
Unified treatment of eigenvalue processes using Riemannian geometry.
problem Eigenvalue processes in various settings.
method Riemannian submersion and gradient flow of isospectral orbits.
result Eigenvalue processes are projections of Brownian motion through Riemannian submersions.
We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere Sn. The spaces of symmetric eigentensors are explicitly given.
The paper compares eigenvalues of Laplacians on fibred manifolds using symmetrization techniques.
problem Comparing eigenvalues of Laplacians on fibred Riemannian manifolds.
method Using fiberwise spherical and Euclidean symmetrization, the paper proves various comparison theorems.
result Eigenvalues of fibred manifolds are compared to their base manifolds under certain curvature conditions.
Improved convergence speed of principal component analysis through modified learning rules.
problem Slow convergence for covariance matrices with close eigenvalues.
method Introduced an additional term to the objective function to mitigate convergence issues.
result Significantly improved convergence speed confirmed through simulations.