Universal inequalities for Laplacian eigenvalues on discrete groups.
problem Proving inequalities for Laplacian eigenvalues on discrete groups.
method Analyzing Laplacian eigenvalues with Dirichlet boundary conditions on subsets of discrete groups.
result Yang-type universal inequalities for Cayley graphs of amenable groups and the d-regular tree.
Study eigenvalues of Toeplitz operators on symplectic manifolds with discrete wells.
problem Understanding eigenvalues of Toeplitz operators on symplectic manifolds.
method Semiclassical analysis of Toeplitz operators on high tensor powers of a positive line bundle.
result Upper bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical limit.
Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.
problem Approximating eigenvalues of Laplace-Beltrami on manifolds with bounded Ricci curvature.
method Graph discretization of Riemannian manifolds with (ε,ρ)-approximation, proving eigenvalue convergence. result Graph Laplacian eigenvalues converge uniformly to manifold Laplacian eigenvalues as parameters approach zero.
Study Steklov eigenvalues on hyperbolic triangle-tiling graphs.
problem Analyzing Steklov eigenvalues on specific hyperbolic graph structures.
method Introduced a graph roughly isometric to hyperbolic plane, used discretization to transfer bounds.
result Steklov eigenvalues tend to zero proportionally to the inverse of the domain size.
The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.
problem Spectral analysis of connection Laplacian on tori.
method Employing parallel orthonormal basis in pullback bundle, examining eigenvalues of connection Laplacian on real and discrete tori.
result Eigenvalues of connection Laplacian on discrete tori converge to those on real torus, with unique twist in torsion matrix.
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.
The paper proves inequalities for lattice eigenvalues, answering a question.
problem Finding bounds for Dirichlet Laplace eigenvalues on integer lattices.
method Proving analogues of existing inequalities for eigenvalues.
result Answers a question posed by Chung and Oden about eigenvalues on integer lattices.
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
The paper bounds higher Steklov eigenvalues of graphs on surfaces.
problem Bounding higher Steklov eigenvalues of graphs on surfaces.
method Using metrical deformation via probability flows, the upper bound is derived.
result The upper bound of higher Steklov eigenvalues is established.
Study on Lawson surfaces' first Laplace eigenvalue using symmetry and algebraic methods.
problem Yau's conjecture on first eigenvalue of minimal hypersurfaces in the sphere.
method Symmetry-based approach exploiting discrete reflection symmetries and algebraic structure of reflection groups.
result Equality λ1(ξ_{m,k})=2 for Lawson surfaces with m and k even.
Estimates the first eigenvalue of a Laplacian on self-shrinkers in Ricci shrinkers.
problem Estimating the first eigenvalue of a Laplacian on self-shrinkers in Ricci shrinkers.
method Analyzes the drifted Laplacian on hypersurfaces in Ricci shrinkers, proving a lower bound for the first nonzero eigenvalue.
result Provides a lower bound for the first nonzero eigenvalue of the drifted Laplacian on embedded f-minimal hypersurfaces.
Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.
problem Estimating eigenvalues of poly-Laplace operator on subgraphs of lattice graphs.
method Introduced discrete poly-Laplace operator, derived upper and lower bounds for eigenvalues.
result Poly-Laplace eigenvalues are at least squares of lower-order poly-Laplace eigenvalues.
The study shows how discrete graphs can resemble hypercube structures under certain curvature conditions.
problem Understanding the structure of graphs with specific curvature conditions.
method Analyzing weighted graphs with lower Ricci curvature bounds and eigenvalue closeness to establish structural similarity.
result Discrete graphs with specific curvature conditions are close to hypercube structures in terms of Frobenius distance and eigenfunctions.
Paper describes eigenvalues of genus 3 surfaces graphs.
problem Understanding eigenvalues of genus 3 surfaces.
method Analyzes graphs derived from pair of pants decompositions.
result Complete description of eigenvalue sets for genus 3.
Cycloids, hipocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete doub…
Study eigenvalues of Bochner Laplacian on symplectic manifolds.
problem Understanding low-lying eigenvalues of Bochner Laplacian on symplectic manifolds.
method Analyzes high tensor powers of positive line bundles on symplectic manifolds.
result Asymptotic expansions for low-lying eigenvalues.
Study eigenvalues and functions on specific Heisenberg manifolds.
problem Eigenvalues and eigenfunctions of a specific operator on Heisenberg Bieberbach manifolds.
method Analysis of eigenvalues and eigenfunctions of the Folland-Stein operator on Heisenberg Bieberbach manifolds.
result Characterized eigenvalues and eigenfunctions of the Folland-Stein operator on specific manifolds.
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.
Optimizes eigenvalues on surfaces with symmetries.
problem Maximizing Laplace and Steklov eigenvalues on Riemann surfaces with symmetries.
method Simplifies existing techniques for conformal class optimization.
result Proves existence and regularity of maximizers for Laplace and Steklov eigenvalues.
The paper examines the rigidity of eigenvalues in shrinking Ricci solitons.
problem Rigidity of eigenvalues in shrinking Ricci solitons.
method Analysis of the drifted Laplacian on shrinking Ricci solitons, showing eigenvalue bounds and rigidity results.
result If the nextth eigenvalue is close to a lower bound, the n-soliton must be the trivial Gaussian soliton. We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…
Study shows how a strip's twisting increases at infinity, affecting its spectrum.
problem Understanding the spectrum of a strip with diverging twisting.
method Analyzing the Dirichlet Laplacian in a two-dimensional strip with segments rotating at increasing velocity.
result Essential spectrum forms a three-dimensional tube at infinity, with discrete eigenvalues possible if the tube's cross-section is a disk.
The paper studies higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues.
problem Analyzing eigenvalues of higher order Dirichlet-to-Neumann maps on graphs.
method Introducing and studying higher order Dirichlet-to-Neumann maps on graphs, deriving estimates on eigenvalues.
result Raulot-Savo-type estimates on the eigenvalues of the DtN maps.
In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator Λ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there.We inves…
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In…
In this note we partially answer a question posed by Colbois, Dryden, and El Soufi. Consider the space of constant-volume Riemannian metrics on a connected manifold M which are invariant under the action of a discrete Lie group G. We show that the first eigenvalue of the Laplacian is not bounded above on this space, pr…
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. We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their asymptotic s…
As is well-known for compact Riemann surfaces, eigenvalues of the Laplacianbare distributed discretely and most of eigenvalues vary viewed as functions on the Teichmuller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguished features is th…
Discrete analogues of classical spectral geometric inequalities and extremal eigenvalue problems on graphs.
problem Extremal eigenvalue problems on graphs
method Developing nodal domain methods for adjacency matrices
result Establishing sharp extremal characterizations across diverse graph classes
Sharp bounds on diameter and eigenvalues for amply regular graphs.
problem Finding bounds for amply regular graphs' diameter and eigenvalues.
method New ideas relating discrete Ricci curvature to local matching properties, including a novel construction of a regular bipartite graph.
result Sharp diameter and eigenvalue bounds for amply regular graphs.
Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
problem Finding upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
method Discretizing a bounded domain and using comparison theorems.
result The $k^{\mbox{th}}$ eigenvalue tends to 0 proportionally to 1/∣B∣d−11. We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an upper bound on the error that depends on upper bounds on the diameter and the sect…
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
The cross correlation matrix between equities comprises multiple interactions between traders with varying strategies and time horizons. In this paper, we use the Maximum Overlap Discrete Wavelet Transform to calculate correlation matrices over different timescales and then explore the eigenvalue spectrum over sliding …
We give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as well as new direct methods which translate curvature to combinatorial properties.…
Study on CR Paneitz operator on non-embeddable CR manifolds.
problem Understanding the CR Paneitz operator on non-embeddable CR manifolds.
method Analysis of the CR Paneitz operator's properties on non-embeddable CR manifolds.
result CR Paneitz operator on the Rossi sphere has infinitely many negative eigenvalues.
Study discrete analog of zeta-determinant maximization on triangulated surfaces.
problem Maximizing zeta-determinant for discrete Laplacian on triangulated surfaces.
method Analogous to Osgood, Phillips, and Sarnak's theorem, study stationary points of determinants for discrete cotan-Laplacian.
result Discrete metrics of constant discrete Gaussian curvature are stationary points of the determinant, suggesting minima.
This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary'' at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an (n+1)-dimensional manifold is the ray [n2/4,∞), with no …
The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.
problem Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator.
method Trace asymptotics formula and Weyl type asymptotic formula for eigenvalue counting function.
result The spectrum of Hp in the gap is discrete. Existence and uniqueness of discrete Einstein metrics on trees proven.
problem Existence and uniqueness of discrete Einstein metrics on trees.
method Using Perron-Frobenius theory and Lin-Lu-Yau Ricci curvature.
result Existence and uniqueness of discrete Einstein metrics on trees established.
Develops a new quadrature method for Lebesgue integrals.
problem Finding optimal values and weights for non-Gaussian processes.
method Solves a generalized eigenvalue problem to find value-nodes and weights.
result Advantages in analyzing irregular and stochastic processes.
Estimates eigenvalues using Bessel functions on manifolds.
problem Estimating eigenvalues of differential operators on manifolds.
method Using mean value lemma and curvature assumptions, derive differential inequalities involving Bessel functions.
result Establishes new estimates for eigenvalues involving positive roots of Bessel functions.
This paper studies how adding leaves to a tree affects its spectral properties.
problem Investigating the asymptotic behavior of tree spectra under leaf attachment.
method Analyzing the Ricci matrix and its largest eigenvalue for trees with pendant edges added.
result The sequence of largest eigenvalues converges to a limit that depends on local branch data.
An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of bal…
The paper discretizes Riemannian manifolds with lower Ricci bounds and compares their spectra.
problem Spectral comparison of discretized Riemannian manifolds with lower Ricci bounds.
method Introduces a weighted combinatorial Laplacian on ε-discretizations and proves spectral comparison theorems.
result Eigenvalues of the weighted graph Laplacian are uniformly comparable to those of the Laplace-Beltrami operator.
Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.
problem Understanding how edge subdivision impacts the Perron eigenvalue of tree Ricci matrices.
method Compressing branches into scalar feedback functions via Schur complement, reducing the spectral problem to a one-dimensional Chebyshev equation.
result Edge subdivision can decrease, preserve, or increase the Perron eigenvalue of tree Ricci matrices.