Researchers prove spectral uniqueness of complex/quaternionic structures on manifolds.
problem Spectral uniqueness of complex/quaternionic structures on manifolds.
method Explicit expression for smallest positive eigenvalue of Laplace-Beltrami operator.
result Irreducible symmetric spaces are spectrally unique within families of homogeneous metrics.
Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.
problem Sarnak's spectral gap question for hyperbolic packings.
method Analysis of Patterson-Sullivan base eigenfunctions and spectral gaps.
result Unique square-integrable eigenfunction has maximal spectral gap.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
Paper proves uniqueness of solutions to a geometric inequality problem.
problem Uniqueness of solutions to the isotropic Lp Minkowski problem. method Analysis of the Hilbert-Brunn-Minkowski operator LK to derive stability estimates. result Uniqueness of S2-isotropic solutions to the isotropic Lp Minkowski problem in Rn for specific ranges of p. Paper presents a unique method to recover signals from their bispectrum.
problem Retrieving signals accurately from their bispectrum.
method Two-step trust region algorithm that minimizes a non-convex objective function.
result Signals with finite spectral or temporal support can be recovered from at least 3B measurements of their bispectrum.
Symmetric spaces have unique spectra under certain group actions.
problem Identifying unique spectral properties of symmetric spaces.
method Analyzing the spectrum of metrics under group actions of G2 and Spin(7). result Non-flat compact irreducible symmetric spaces are spectrally unique.
In this short note, we prove that a bi-invariant Riemannian metric on Sp(n) is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on Sp(n). In other words, on any of these compact simple Lie groups, every left-invariant metric which is n…
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
Proves uniqueness of certain spacetime solutions with extremal horizons.
problem Proving uniqueness of extremal Schwarzschild de Sitter spacetime solutions.
method Analytic proof in four and higher dimensions, spectral problem for hyperbolic surfaces.
result Proves extremal Schwarzschild de Sitter solutions are unique up to identifications.
Study proves spectral determination of triangles and quadrilaterals, with restrictions on higher-order polygons.
problem Determining the geometry of convex polygons from their Steklov spectra.
method Analysis of characteristic polynomial and spectral properties of Steklov spectrum.
result Almost all triangles and certain quadrilaterals are uniquely determined by their Steklov spectra.
This paper is devoted to an inverse Steklov problem for a particular class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.
Spectral regularization simplifies sequence models by focusing on grammatical simplicity.
problem Sequence modeling challenges in learning tasks.
method Introduces spectral regularization based on Hankel matrices and trace norm, addressing bi-infinite matrices with an unbiased estimator.
result Demonstrates spectral regularization's potential benefits on Tomita grammars.
The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.
problem Determining metrics on negatively curved manifolds using spectral data.
method Analyzing conjugacy classes and marked length spectra.
result Sparse sets exist that uniquely determine metrics on negatively curved manifolds.
We show that any equation from the Davey--Stewartson hierarchy induces an infinite family of geometrically different deformations of tori in R4 preserving the Willmore functional. We expose a derivation of the Weierstrass representation for surfaces in the four-space which is not unique in difference from the case …
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
This work develops a model to distinguish network and covariate information.
problem Identifying unique network and covariate information.
method Low-rank model with two-step estimation: spectral method followed by refinement.
result The method accurately recovers joint and individual components.
Researchers prove unique connection and curvature for Podleś quantum sphere.
problem Calculating curvature and Weitzenbock formula for Podleś quantum sphere.
method Using spectral triple and Dabrowski-Sitarz framework, they computed curvature tensors and proved a generalized Weitzenbock formula.
result The scalar curvature of Podleś sphere converges to 2 as q approaches 1.
Spectral flow connects manifold geometry to rigidity criteria.
problem Tackling rigidity of simply-connected closed manifolds.
method Spectral deformation flow and invariant-based approach.
result Spherical profile is the unique manifold-compatible asymptotic realization.
Corners can be identified by a drum's sound spectrum.
problem Determining the presence of corners in a drum's shape from its sound.
method Proving spectral invariance of corners in domains with Lipschitz, piecewise smooth boundaries.
result Corners are uniquely determined by a drum's spectrum among domains with fixed genus.
We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous fa…
Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.
problem Quantify spectral gaps for Teichmüller geodesics.
method Bounding spectral gaps in terms of geometric quantities on flat surfaces.
result Quantitative non-uniform hyperbolicity of Teichmüller geodesic flow.
The paper studies magnetic field effects on surface eigenvalues and spectral properties.
problem Understanding magnetic effects on surface eigenvalues and spectral properties.
method Provided precise spectral asymptotics expansion for the magnetic Dirichlet-to-Neumann map on surfaces.
result The spectrum of the magnetic Dirichlet-to-Neumann map uniquely determines the number and length of boundary components, parallel transport, and magnetic flux.
Study Higgs bundles on curves with punctures, extending spectral correspondence.
problem Classify Higgs bundles on punctured curves with logarithmic structures.
method Logarithmic Hecke compactification, spectral conditions, and sheaf classification.
result Logarithmic spectral correspondence extended to punctured curves.
We study the Laplace spectra of the intrinsic instantaneous metrics on the event and cosmological horizons of a Kerr-Newman de Sitter space-time and prove that the spectral data from these horizons uniquely determine the space-time. This is accomplished by exhibiting formulae relating the parameters of the space-time m…
The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.
problem Understanding spectral encodings under Weyl growth conditions.
method Analyzing geometric Weyl bulk-density exponent and proving spectral rigidity.
result The geometric Weyl bulk-density exponent (d−2)/2 rigidifies spectral encodings in the O-regularly varying class, leading to unique admissible exponents and scaling laws. The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In t…
Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analys…
New framework detects directional influence in multivariate time series.
problem Detecting directional influence in multivariate time series.
method Order-constrained spectral non-invariance.
result Unique diagnostic functional for directional influence.
On curved spaces, viscous fluids reach equilibrium quickly.
problem Thermalization of viscous fluids on negatively curved manifolds.
method Stochastic Navier-Stokes equations with kinematically selected deformation Laplacian.
result Exponential thermalization rate of $2νλ_\Def$.
In this paper, the elastic Dirichlet-to-Neumann map Ξg is studied for the stationary elasticity system in a compact Riemannian manifold (Ω,g) with smooth boundary ∂Ω. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map Ξg. We …
We show that a bi-invariant metric on a compact connected Lie group G is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric g0 on G there is a positive integer N such that, within a neighborhood of g0 in the class of left-invariant metrics of a…
The paper classifies ancient ovals in higher dimensions and proves their symmetry and uniqueness.
problem Classifying compact ancient noncollapsed mean curvature flows in arbitrary dimensions.
method Analyzing k-ovals and using spectral ratio parameters to prove symmetry and uniqueness. result Ancient k-ovals are uniquely determined by (k−1)-dimensional spectral ratio parameters and are Z2kimesO(n+1−k)-symmetric. Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function …
The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of Kähler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the…
We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain Dm and fixed {\it intermediate} domain DW. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying well-posed…
Recursive training of generative models can lead to model collapse, and the recursion converges to a unique limiting distribution.
problem Model collapse in recursive training of generative models
method Recursive training on their own outputs
result Recursive training converges to a unique limiting distribution
The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
problem Proving strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
method Establishing a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption of geometry near the base point being close to a generalized cylinder. result Proves strong uniqueness of generalized cylindrical tangent flows and shows that the subset of points with rectifiable Sqck(N) is horizontally parabolic. Inverse spectral theory reveals shapes from sound.
problem Can the shape of a drum be determined by its sound?
method Inverse isospectral techniques applied to specific shapes.
result The regular n-gon can be uniquely determined by its eigenvalues.
The paper investigates how activation functions impact the training of Neural ODEs, leading to global convergence.
problem Challenges in training Neural ODEs, particularly gradient computation accuracy and convergence analysis.
method Investigates the impact of activation functions on the training dynamics of Neural ODEs.
result Establishes global convergence of Neural ODEs under gradient descent in overparameterized regimes.
We uniquely and explicitly reconstruct the instantaneous intrinsic metric of the Kerr-Newman Event Horizon from the spectrum of its Laplacian. In the process we find that the angular momentum parameter, radius, area; and in the uncharged case, mass, can be written in terms of these eigenvalues. In the uncharged case th…
Study identifies cancer genes through graph anomaly analysis of protein interactions.
problem Insufficient modeling of biological information in protein interaction networks for cancer gene identification.
method Proposes HIerarchical-Perspective Graph Neural Network (HIPGNN) to detect weight heterogeneity and spectral flattening in cancer gene nodes.
result HIPGNN detects weight heterogeneity and spectral flattening, leading to improved cancer gene identification.
We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Lapl…
Study magnetic potentials on Anosov manifolds using spectral data.
problem Recover magnetic potentials from spectral data on Anosov manifolds.
method Utilize principal wave trace invariants and magnetic Schrödinger operator.
result Spectral data uniquely determines magnetic and electric potentials on Anosov manifolds.
Analyzes how diffusion models learn, revealing a spectral bias in structure mastery.
problem Understanding the learning dynamics and bias in diffusion models.
method Developed an analytical framework using a Gaussian-equivalence principle to solve gradient-flow dynamics and integrate probability-flow ODEs.
result Exposes a universal inverse-variance spectral law: high-variance structure is mastered faster than low-variance detail.
In this paper, we study an insurer's reinsurance-investment problem under a mean-variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative Lévy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthe…
Study magnetic Laplacian eigenvalues on contact manifolds.
problem Characterize spectral properties of magnetic fields on contact manifolds.
method Analyze first eigenvalue of magnetic horizontal Laplacian, provide upper bounds, and use topological conditions.
result Equality in upper bounds implies Heisenberg left-invariant nilmanifold structure and unique determination of manifold Chern class.
A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, w…
Community detection has been one of the central problems in network studies and directed network is particularly challenging due to asymmetry among its links. In this paper, we found that incorporating the direction of links reveals new perspectives on communities regarding to two different roles, source and terminal, …