The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.
problem Finding extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
method Characterizations obtained under suitable geometric constraints.
result Characterizations of extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
We prove an Obata-type rigidity result for the spherical cap and apply it for an eigenvalue problem with mixed boundary condition.
Optimizing shapes for a specific eigenvalue problem involving two balls.
problem Optimizing shapes for the first mixed Steklov-Dirichlet eigenvalue.
method Geometric proof based on Newton's shell theorem.
result Geometric insight into eigenvalue optimization.
The study characterizes mixed super quasi-Einstein manifolds with Ricci-Bourguignon solitons.
problem Characterizing mixed super quasi-Einstein manifolds with Ricci-Bourguignon solitons.
method Exploring properties of mixed super quasi-Einstein manifolds, including conformal Ricci pseudosymmetry and Einstein's field equation. Characterizing manifolds that admit Ricci-Bourguignon solitons and providing a detailed eigenvalue problem characterization.
result Characterization of mixed super quasi-Einstein manifolds with Ricci-Bourguignon solitons, including a detailed eigenvalue problem and an example construction.
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…
New estimator for overlapping community detection in graphs.
problem Reconstructing community relations in graphs with overlapping memberships.
method Proposes a new estimator for Mixed-Membership Stochastic Block Model (MMSB) under diverging spiked eigenvalues condition.
result New estimator matches theoretical minimax lower bound on estimation error.
Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.
problem Understanding the speed of mixing in Anosov diffeomorphisms.
method Investigates Ruelle-Pollicott resonances on manifolds of any dimension, connecting them to cohomology eigenvalues of a quasi-compact transfer operator.
result Established a cohomological bound for the speed of mixing of Anosov diffeomorphisms.
The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.
problem Finding the domain with the largest first eigenvalue for a given volume and boundary conditions.
method Shape optimization for the first eigenvalue of the p-Laplace operator in hyperbolic space.
result The concentric annular region maximizes the first eigenvalue among multiply-connected domains.
Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.
problem Prove nullity of biharmonic maps from flat 2-torus to round 2-sphere.
method Use spectral geometry, construct polynomial isomorphism with elliptic curve, determine rational points on spectral curve.
result Nullity of every map in the family is 5, confirming conjecture.
Hot spots conjecture proven for small eigenvalue domains.
problem Hot spots conjecture for hyperbolic planar domains with small eigenvalues.
method Proved a variant of Rauch's hot spots conjecture.
result Second Neumann Laplace eigenfunctions have no interior critical points on large convex domains.
We address the problem of estimating the mixing time tmix of an arbitrary ergodic finite-state Markov chain from a single trajectory of length m. The reversible case was addressed by Hsu et al. [2019], who left the general case as an open problem. In the reversible case, the analysis is greatly facilita…
The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.
problem Constructing free boundary minimal surfaces in product spaces of balls.
method Extremal eigenvalue approach involving mixed Steklov-Neumann eigenvalues.
result No absolute maximum exists for the problem in product spaces.
New framework for studying eigenvalue functionals of metrics.
problem Understanding critical points of eigenvalue functionals.
method Clarke subdifferential theory to unify previous research.
result Unified understanding of critical metrics and new examples.
Generalizes Cheeger inequality to Carnot-Carathéodory spaces.
problem Lower bounds on eigenvalues of Laplacians in complex spaces.
method Geometric approach, including Neumann and mixed boundary conditions.
result Concrete method to lower bound Cheeger constant.
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
A new method using negative-shifted gradient descent improves overparameterized linear regression by avoiding structural limitations of negative ridge endpoints.
problem Structural limitations of negative ridge endpoints in overparameterized linear regression.
method Negative-shifted gradient descent, which avoids the pole constraint of negative ridge endpoints.
result The method improves over all admissible endpoints by a polynomial factor in risk under explicit conditions.
Study proves radial symmetry in convex cones using subharmonic functions.
problem Proving radial symmetry in convex cones with boundary conditions.
method Using maximum principle and integral identities for subharmonic functions.
result Proves radial symmetry and Serrin-type results for partially overdetermined problems.
Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this …
In this paper, we give a new sharp generalization bound of lp-MKL which is a generalized framework of multiple kernel learning (MKL) and imposes lp-mixed-norm regularization instead of l1-mixed-norm regularization. We utilize localization techniques to obtain the sharp learning rate. The bound is characterized by the d…
Our aim in this paper is to study local rigidity for metrics defined on a compact manifold M with boundary satisfying constant scalar curvature on M and constant mean curvature on ∂M. We present some geometrical hypotheses ensuring local rigidity for both, the general Riemannian and the warped metric case…
We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\cal R}_i = U_i{\cal R}_1U_i^{-1}…
Study finds the minimum number of finite Gaussian mixtures for best approximation.
problem Finding the minimum number of finite Gaussian mixtures for best approximation.
method Local moment matching for upper bound and spectral analysis for lower bound.
result Corrects a previous lower bound in the case of Gaussian mixing distributions.
Spectral clustering approaches have led to well-accepted algorithms for finding accurate clusters in a given dataset. However, their application to large-scale datasets has been hindered by computational complexity of eigenvalue decompositions. Several algorithms have been proposed in the recent past to accelerate spec…
This paper proposes exact and approximation algorithms for Sparse PCA, improving interpretability and scalability.
problem Selecting a prespecified-size principal submatrix from a covariance matrix to maximize its largest eigenvalue.
method Proposes two exact mixed-integer SDPs and a mixed-integer linear program (MILP) for SPCA, analyzes theoretical optimality gaps, and develops approximation algorithms.
result The proposed algorithms achieve strong theoretical optimality and effective scalability, with continuous relaxations close to optimality and MILP solving small to medium-size instances.
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 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.
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…
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.
New biharmonic Steklov problem on forms yields eigenvalue estimates.
problem Eigenvalue estimates for differential forms with curvature quantities.
method Introduced a new biharmonic Steklov problem and proved existence of a discrete spectrum.
result Established Kuttler-Sigillito inequalities connecting eigenvalues of differential forms.
We consider the SL(2,R) action on moduli spaces of quadratic differentials. If μ is an SL(2,R)-invariant probability measure, crucial information about the associated representation on L2(μ) (and in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded …
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.
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.
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.
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…
In study of eigenvalue problems, a classical problem is the Stekloff eigenvalue problem. There are many estimates of the first non- zero Stekloff eigenvalue, including a sharp estimate on surfaces, obtained by Escobar in "The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997)". In this paper…
Sharp bounds found for Steklov-type eigenvalues on surfaces.
problem Finding bounds for the first eigenvalue of Steklov-type problems on compact surfaces.
method Proved bounds using Gaussian curvature constraints and properties of geodesic curvature.
result Sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces.
Study on biharmonic Steklov problem on differential forms.
problem Characterize and estimate eigenvalues of biharmonic Steklov problem.
method Introduce boundary conditions, prove properties, derive inequalities.
result Characterize smallest eigenvalue and prove spectrum properties.
Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.
problem Unified Minkowski problem for (p,q)-mixed quermassintegrals.
method Introducing (p,q)-mixed quermassintegrals and (p,q)-dual mixed curvature measure to study the Minkowski problem.
result Derivation of important properties and geometric inequalities for (p,q)-mixed quermassintegrals.
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.
Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.
problem Biharmonic Steklov problems with Neumann boundary conditions.
method Introduced a biharmonic Steklov problem and proved its well-posedness. Established eigenvalue estimates using Kuttler-Sigillito inequalities.
result Eigenvalue estimates for the biharmonic Steklov problem with Neumann boundary conditions.
New algorithm samples from Ising models efficiently, even with outliers.
problem Sampling from Ising models with general interaction matrices.
method Combines MCMC and variational inference techniques.
result First polynomial time sampling algorithms for low-rank Ising models.
Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.
problem Eigenvalue continuity and shape optimization for Laplace and Steklov problems.
method Variational eigenvalue analysis, Sobolev space convergence, shape optimization techniques.
result Sharp isoperimetric inequalities for Steklov eigenvalues, upper bound 8πk for k-th perimeter-normalized eigenvalue. Paper finds inequalities for eigenvalues of buckling problems on special metric spaces.
problem Eigenvalue inequalities for buckling problems of drifting Laplacian.
method Investigated on bounded domains in complete smooth metric measure spaces (SMMSs) with special functions.
result General inequalities for eigenvalues derived under curvature constraints.
We investigate the eigenvalues of the buckling problem of arbitrary order on compact domains in Euclidean spaces and spheres. We obtain universal bounds for the kth eigenvalue in terms of the lower eigenvalues independently of the particular geometry of the domain.
Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.
problem Generating free boundary minimal surfaces using Steklov eigenvalue problems.
method Maximizing Steklov eigenvalues over a class of metrics, using conformal uniformization and gradient-based optimization.
result Numerical solutions for free boundary minimal surfaces with various boundary components.
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.
The paper studies eigenvalue problems on manifolds and recovers known inequalities.
problem Eigenvalue problems on complete compact Riemannian manifolds with Dirichlet boundary conditions.
method Cheng comparison estimates, Faber-Krahn inequality, Cheeger estimates.
result Eigenvalue bounds and convergence to Cheeger's constant as p,qo1,1. Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
problem Finding upper bounds for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
method Proving upper bounds using the weighted p-Laplace operator and (p,q)-Laplacian on submanifolds. result Reilly-type upper bounds for the first eigenvalues of Steklov and (p,q)-Laplacian problems.