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 classifies biconservative Lorentz hypersurfaces with complex eigenvalues.
problem Classifying biconservative Lorentz hypersurfaces with complex eigenvalues.
method Analyzing biharmonic and biconservative submanifolds in E1n+1. result Every biconservative Lorentz hypersurface M1n in E1n+1 with complex eigenvalues has constant mean curvature. The paper optimizes Dirac eigenvalues on surfaces and connects them to harmonic maps into complex projective spaces.
problem Optimizing the k-th positive Dirac eigenvalue on surfaces with fixed area and conformal class. method Connecting the problem to the maximization of Laplacian eigenvalues and using critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces.
result The first nonzero Dirac eigenvalue on a torus is minimized by the flat metric.
Study of H-eigenvalues for complex tensors and their applications in differential geometry.
problem Characterizing H-eigenvalues of Hermitian tensors. method Introduced H-eigenvalues, derived inclusion sets, and established criteria for definiteness. result Determined inclusion sets and criteria for Hermitian and CPS tensors.
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.
It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci…
Eigenvalues of 1-form Laplacian on hyperbolic manifolds relate to geodesic cycles.
problem Understanding the geometry of small eigenvalues on hyperbolic manifolds.
method Relating eigenvalues to cycle complexity and geodesics.
result Small eigenvalues correspond to closed geodesics with low genus surfaces.
Complex frequency generalizes eigenvalues in LTI systems.
problem Characterizing dynamics of signals with complex values.
method Geometric frequency interpretation and transformation analysis.
result Complex frequencies in LTI systems match eigenvalues.
Study small eigenvalues of Toeplitz operators and their relation to Mabuchi geodesics.
problem Analyzing small eigenvalues of Toeplitz operators on complex projective manifolds.
method Proving the existence of exponentially decaying eigenvalues for Toeplitz operators with specific symbols, and establishing a connection to Mabuchi geodesics.
result Logarithmic distribution of small eigenvalues correlates with Mabuchi geodesics between polarizations.
In Kähler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.
Proposes a faster Isomap algorithm by reducing eigenvalue decomposition complexity.
problem High computational complexity of Isomap, especially in eigenvalue decomposition stage.
method Introduces a projection operator to reduce the complexity of the eigenvalue decomposition stage to linear order.
result Reduces Isomap's computational complexity to linear order while preserving structural information.
Eigenvalue problem for Kähler metrics on compact manifolds.
problem Eigenvalue problem for the Laplacian on Kähler manifolds.
method Introducing λk-extremal Kähler metrics and deducing conditions for extremality. result Conditions for a Kähler metric to be λk-extremal. Introduces differential forms to study inequalities between eigenvalues.
problem Eigenvalue inequalities for Laplacian and other operators.
method Uses differential forms and the de Rham complex.
result Shows differential forms are central to Rohleder's work.
Proves existence of eigenvalue and eigenfunction for complex Monge-Ampère operator.
problem Eigenvalue problem for complex Monge-Ampère operator on bounded domains.
method Follows P.L. Lions' strategy for real case, proves new existence theorem for complex degenerate equations, uses a priori estimates and variational approach.
result Existence of first eigenvalue and eigenfunction with specified properties.
We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second…
Paper defines multiplicities for quaternion eigenvalues without complex matrix concepts.
problem Defining multiplicities for quaternion eigenvalues without traditional matrix concepts.
method Introduces two definitions for algebraic and geometric multiplicities equivalent to classical definitions.
result Definitions are equivalent to classical ones and prove all properties easily.
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…
Defined Ricci curvature on simplicial complexes and proved bounds.
problem No specific problem stated; generalization of graph Ricci curvature to simplicial complexes.
method Modified Ricci curvature definition for simplicial complexes and proved bounds.
result Upper and lower bounds of Ricci curvature on simplicial complexes.
Positive entropy automorphisms have virtual eigenvalues outside the unit circle.
problem Understanding the dynamics of surface automorphisms with positive entropy.
method Investigating virtual homological eigenvalues of surface automorphisms.
result Existence of virtual eigenvalues outside the unit circle for automorphisms with positive entropy.
We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacob…
Paper studies eigenvalue bounds for complex curves on Kähler surfaces.
problem Investigates eigenvalue bounds for complex curves on Kähler surfaces.
method Analyzes second variation of a conformally invariant Willmore-type functional to derive bounds.
result Derives lower bound Λ1≥2Ric for Kähler surfaces, with equality for low genus curves. Researchers found M-eigenvalues for higher dimensional conformal flat manifolds.
problem Finding M-eigenvalues for Riemann curvature tensor in higher dimensions.
method Generalized Xiang, Qi and Wei's results to higher dimensions and provided expressions for M-eigenvalues and eigenvectors.
result M-eigenvalues uniquely determine the Riemann curvature tensor and can be complex.
The extragradient method accelerates convergence in complex game dynamics.
problem Complex interactions in game dynamics cause simple methods to diverge, necessitating more sophisticated approaches.
method A polynomial-based analysis to identify three scenarios for accelerated convergence of the momentum extragradient method.
result The momentum extragradient method achieves faster convergence under specific eigenvalue conditions.
The paper studies stability and instability of minimal submanifolds in complex Einstein spaces.
problem Stability and instability of minimal submanifolds in complex Einstein spaces.
method Computation of index and nullity, investigation of stability, and algorithm for higher eigenvalues.
result Criterion for instability of minimal submanifolds in some cases.
The Yamabe invariant is linked to static potentials and eigenvalues.
problem The relationship between Yamabe invariant and static potentials/eigenvalues.
method Analyzes the Yamabe invariant in the context of static potentials and eigenvalues of the Laplacian.
result The Yamabe invariant is closely tied to static potentials and the first eigenvalue of the Laplacian.
Study estimates the first eigenvalue on Kähler manifolds with specific curvature conditions.
problem Estimating the first eigenvalue of Laplacian on Kähler manifolds with holomorphic sectional curvature constraints.
method Developed a Bochner-Kodaira type identity for holomorphic sectional curvature to prove eigenvalue estimates.
result First eigenvalue of Laplacian on Kähler manifolds is bounded from below under certain curvature conditions.
In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which affirmatively answers to a conjecture proposed by Chen-Zheng-Yang. In addition,…
Estimates the first non-zero eigenvalue using Ricci curvature on graph edges.
problem Estimating the first non-zero eigenvalue of the Laplacian on graph edges.
method Defining edge distance, studying coarse Ricci curvature, and using Jost-Horak's Laplacian definition.
result Obtained an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.
Improved eigenvalue distribution method for financial data.
problem Noise and complexity in financial markets.
method Matrix H theory, hierarchical structure, informational cascade.
result Captures a larger fraction of data variance in financial markets.
We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic Kähler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex hyperbolic space. As a corollary we obtain explicit lower bounds for the first eigenvalu…
Motivated by a sharp eigenvalue estimate for the Kohn Laplacian, we prove a theorem that characterizes the CR sphere in terms of the existence of a non-trivial complex-valued function satisfying a certain overdetermined system.
The paper finds geodesics in Kähler potentials with no degeneration.
problem Finding geodesics in Kähler potentials without degeneration.
method Establishing a lower bound estimate for eigenvalues of complex Hessian.
result Geodesics can connect close points in Kähler potentials without degeneration.
The paper estimates eigenvalues of submanifolds in symmetric spaces.
problem Estimating eigenvalues of submanifolds in symmetric spaces.
method Combining spherical embeddings and conformal test-function arguments.
result Sharp estimates for the second eigenvalue of submanifolds in symmetric spaces.
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
The paper studies the spectrum of Laplace-Beltrami operators on complex spaces.
problem Analyzing the spectrum of Laplace-Beltrami operators on compact complex spaces.
method Examined the Friedrichs extension of Laplace-Beltrami and Hodge-Kodaira Laplacians, providing estimates for eigenvalues and trace-class properties.
result Discrete spectrum and trace-class properties of Laplace-Beltrami operators on compact complex spaces.
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…
Two new algorithms reduce online kernel regression's computational cost while maintaining optimal regret bounds.
problem Trade-off between regret and computational cost in online kernel regression.
method AOGD-ALD and NONS-ALD algorithms dynamically maintain nearly orthogonal basis to approximate kernel mapping and control approximate error.
result Achieves nearly optimal regret bounds at sublinear computational complexity.
Gradient and eigenvalue estimates for Kähler manifolds' canonical bundle.
problem Estimating Hodge Laplacian on (m,0) forms for Kähler manifolds. method New Bochner type formula involving Ricci curvature and scalar curvature gradient.
result Gradient and eigenvalue estimates depend only on Ricci curvature bound.
Study reveals signatures of market crashes through eigenvalue analysis of stock return matrices.
problem Understanding the complexity and dynamics of market crashes.
method Cross-correlation structures and eigenspectra of stock return matrices were analyzed over different epochs.
result The smallest eigenvalue can distinguish between internal and external market instabilities.
The study determines if eigenvalues of a Laplacian can reveal the curvature of Kähler manifolds.
problem Can eigenvalues of the Laplacian determine the holomorphic sectional curvature of Kähler manifolds?
method Analyzes cohomologically Einstein and Fano Einstein conditions, showing constancy of curvature for most pairs (p,n).
result Characterizes standard complex projective spaces using a single spectral set under cohomological Einstein conditions.
Abstract: Characterizes special Kähler manifolds with specific properties.
problem Characterizing Kähler manifolds with special properties.
method Analyzes properties of functions and gradients on manifolds.
result Characterizes manifolds supporting certain functions and gradients.
Study reveals how spectral bias affects learnability on real-world data.
problem Understanding how well complex datasets can be learned using kernel methods.
method Use eigenvalues and eigenfunctions from idealized data to reveal spectral bias on real-world data.
result Bound learnability on real-world data using symmetries of realistic kernels.
In this paper we deal with the classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on coefficients of the polynomial integral we consider the region (so called elliptic regions) where there are complex-conjugate eigenvalues. We show that for …
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…
The study proves a tube theorem for complex hyperbolic manifolds.
problem Understanding the geometry of complex hyperbolic manifolds.
method Tubular neighborhood theorem and geometric combination theorem.
result Explicit estimates and bounds for tube widths in complex hyperbolic manifolds.
In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherica…
In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain …
Investigates kernel regression rates without assuming polynomial eigenvalue decay.
problem Achieving minimax rates without strict assumptions on kernel eigenvalue decay.
method Examines kernel regularization methods under weak assumptions on eigenvalue decay.
result Achieves minimax convergence rates under less restrictive conditions.