Upper bound found for Steklov eigenvalues counting function.
problem Counting Steklov eigenvalues on compact manifolds with boundary.
method Used Weyl's law and Pólya's Conjecture in the Steklov case.
result Obtained an upper bound for the counting function.
Eigenvalues on spheres are compared to the unit round sphere, proving a sharp bound and equality condition.
problem Comparing eigenvalues of spheres under different metrics.
method Analyzing Laplace eigenvalues and using Alexandrov spaces.
result Equality of eigenvalues forces metrics to be isometric to the unit round sphere.
Solves nonlinear problems on metric structures through eigenvalue counting.
problem Nonlinear equations on metric structures
method Counting large eigenvalues of linearized operators
result Solves fully nonlinear Loewner-Nirenberg and Yamabe problems
In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an n-dimensional Riemannian manifold.
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
The paper improves count data regression models for overdispersed data.
problem Improving regression models for overdispersed count data.
method Double ℓ1-regularized negative binomial regressions. result Oracle inequalities and consistency for Lasso estimators of partial regression coefficients.
We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimen…
Global propagator for massless Dirac operator defined and analyzed.
problem Analyzing the massless Dirac operator on 3-manifolds.
method Constructing propagator as sum of oscillatory integrals, providing global definitions and small time expansions.
result Explicit calculation of propagators' symbols and coefficients in eigenvalue counting functions.
Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.
problem Counting negative eigenvalues of magnetic Pauli operator.
method Reduction to boundary Dirac operator, Atiyah-Patodi-Singer index theory, Benjamin-Ono equation conservation law.
result New formula on the number of eigenvalues of magnetic Neumann Laplacian in semi-classical limit.
We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Gi…
The paper studies eigenvalues of Xin-Laplacian on Riemannian manifolds.
problem Eigenvalue problems related to Xin-Laplacian on Riemannian manifolds.
method Establishing general formulas and applying Chen-Cheng type results.
result Sharp estimates for the upper bound of the second nonzero eigenvalue of the Laplace-Beltrami operator.
Paper finds a counter-example invalidating a spectral asymptotic algorithm.
problem Invalidation of spectral asymptotic algorithm for elastic eigenvalues.
method Discussion of a counter-example for elastic eigenvalues.
result Most conclusions in Yu. Safarov and D. Vassiliev's book are fundamentally wrong.
Efficiently infers sparse networks from count data with reduced memory usage.
problem Sparse network inference for count data with high dimensions and dependencies.
method Improved Bigraphical Lasso using eigenvalue decomposition of Cartesian product graph.
result Reduced computational complexity from O(n2p2) to O(n2+p2). Temporal coarse-graining of multi-sector default count data generates effective correlation matrices and rank copulas.
problem Explaining the difference in default dependence between monthly and annual aggregation.
method Dynamic low-rank state-space model with AR(1) latent credit-state factors.
result Effective correlation matrices and rank copulas are generated from monthly default count data.
Let Ω be a bounded domain with C∞ boundary in an n-dimensional C∞ Riemannian manifold, and let ϱ be a non-negative bounded function defined on ∂Ω. It is well-known that for the biharmonic equation Δ2u=0 in Ω with the 0-Dirichlet boundary condition, there exists an infinite se…
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. The study counts units and eigenvalue patterns in SL_n(Z) and Sp_{2n}(Z) in thin tubes.
problem Counting totally real units and eigenvalue patterns in SL_n(Z) and Sp_{2n}(Z) in thin tubes.
method Analyzes directional entropy of logarithmic embeddings and eigenvalue data in thin tubes around rays.
result The number of objects grows exponentially with the directional entropy, providing bounds for conjugacy classes.
We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if M is compact, is defined for w≥0 and T>0 by $$N_{M,w}(T) = \sum\…
This paper models default data to capture dynamic dependence across sectors.
problem Static models fail to explain monthly default dependence.
method Dynamic low-rank state-space model for monthly multi-sector default-count data.
result Effective correlation matrices and copulas are induced from monthly data.
This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenval…
We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M=(0,∞)×Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…
Let M⊂Sn+1⊂Rn+2 be a compact cmc rotational hypersurface of the (n+1)-dimensional Euclidean unit sphere. Denote by ∣A∣2 the square of the norm of the second fundamental form and J(f)=−Δf−nf−∣A∣2f the stability or Jacobi operator. In this paper we compute the spectra of the…
We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in Pn. We deduce as special cases algebro-geome…
We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this…
The paper studies eigenvalues and stability of hypersurfaces in spheres.
problem Finding eigenvalues and stability of hypersurfaces in spheres.
method Derives equations for mean curvature and uses numerical methods to compute eigenvalues.
result Numerical computation of eigenvalues and stability indices for specific hypersurfaces.
In this note, we consider a fixed vector field V on S2 and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where V is also tangent. We show that the expected value of the corresponding counting function is asymptotic to the eigenvalue with a leading coefficient that i…
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 …
Graph Neural Networks outperform the Weisfeiler-Lehman algorithm in representation power.
problem Limited representation power of Graph Neural Networks compared to the Weisfeiler-Lehman algorithm.
method Algebraic analysis using eigenvalue decomposition of graph operators.
result Graph Neural Networks produce more discriminative representations than the Weisfeiler-Lehman algorithm.
The paper describes correlations of spectra for higher rank Anosov representations.
problem Understanding correlations of spectra for Anosov representations of higher rank groups.
method Relates correlation problem to counting projections in truncated hypertubes.
result Extends previous work on rank one representations to higher rank.
The paper extends Weyl's law to CROSSes, showing sharpness and polynomial improvement.
problem Understanding the error term in Weyl's law for different types of manifolds.
method Analyzing the Laplacian eigenvalues on Compact Rank One Symmetric Spaces (CROSSes).
result For CROSSes, the error term in Weyl's law is sharp, and for products of CROSSes, it can be polynomially improved.
The study examines numerical aspects of Karhunen-Loève expansions for stochastic processes.
problem Constructing Karhunen-Loève expansions for second-order stochastic processes.
method Spectral decomposition of covariance operator via Fredholm integral equation, discretization, singular value decomposition of weight-scaled sample matrix.
result Consistent solutions for model-based and data-driven KLE construction, characterized by convergence of SVD-based eigenvalue estimates and KL coefficients distributions.
The study bounds harmonic functions on manifolds with nonnegative Ricci curvature.
problem Bounding harmonic functions on manifolds with specific curvature properties.
method Analyzing the asymptotic volume ratio and eigenvalue counting function.
result Sharp upper bounds for harmonic functions with polynomial growth.
Counting tripods on a flat torus using lattice point counting.
problem Counting finite BPS webs in flat torus geometry.
method Lattice point counting techniques in C2. result Asymptotic counting result for tripods on the torus.
Flow Matching for count data improves sample quality and efficiency.
problem Mapping between count distributions across batches or time points in high-dimensional count data.
method count-FM, a flow-matching framework based on a continuous-time birth-death process with local unit jumps.
result count-FM achieves better sample quality than representative baselines while using fewer parameters.
Corrects errors in previous work on spectral asymptotics in elasticity.
problem Two-term asymptotics of elastic eigenvalues on Riemannian manifolds.
method Strongly continuous semigroups and pseudodifferential operators.
result Theorem 1.1 in \cite{Liu-21} is rigorously proven.
New theorem counts curves on orbifolds.
problem Counting curves on surfaces.
method Applied Mirzakhani's theorem to orbifolds.
result Curve counting theorem extends to orbifolds.
A new method, Count-MORL, improves offline reinforcement learning by using state-action frequency.
problem Improving offline reinforcement learning performance.
method Integrates count-based conservatism into model-based offline reinforcement learning.
result The learned policy is near-optimal and outperforms existing methods.
Proposes a method to reconcile count time series forecasts.
problem No formal framework for probabilistic reconciliation of count time series.
method Generalizes Bayes' rule for reconciling real-valued and count variables.
result Improves forecast accuracy for count variables compared to Gaussian reconciliation.
Study geodesic paths on flat surfaces, comparing length and singularity counts.
problem Comparing geometric length and singularity counts on geodesic paths.
method Apply counting limit laws to infinite graphs and then to flat surfaces.
result Statistical comparison of geometric length and singularity counts on geodesic paths.
Counts arcs in surfaces, proving convergence of geodesic currents.
problem Counting arcs of the same type in compact surfaces and related geometries.
method Derives convergence of geodesic currents to prove arc counts.
result Proves convergence of geodesic currents, leading to arc counting results.
Deviance-style normalization for sparse, jointly overdispersed count matrices
problem Jointly overdispersed count matrices
method Dirichlet-multinomial deviance residualization
result Preserves exact sparsity, evaluates in constant time, recovers multinomial residual
The paper proposes count echo state networks for forecasting graduate student enrollments.
problem Forecasting graduate student enrollments from historical data.
method Developed hierarchical count echo state networks and compared them to Poisson autoregressions and negative binomial models.
result Hierarchical negative binomial based echo state network is the superior model.
Counted essential surfaces in a knot's exterior, finding a unique pattern.
problem Counting essential surfaces in a knot's exterior.
method Counted essential surfaces by genus, using Euler totient function. Showed normal surfaces are connected by counting their components. Used Agol, Hass, and Thurston's tools to convert component counting into orbit counting.
result Found a unique pattern in the number of essential surfaces by genus.
Counting objects in digital images is a process that should be replaced by machines. This tedious task is time consuming and prone to errors due to fatigue of human annotators. The goal is to have a system that takes as input an image and returns a count of the objects inside and justification for the prediction in the…
We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an i…
Proposes a robust EM algorithm for analyzing incomplete panel count data.
problem Missing reports in panel count data.
method Functional EM algorithm for non-parametric counting process mean function estimation.
result Robust to misspecification of Poisson process assumption and missing completely at random.
Calegari, Marques, and Neves count minimal surfaces in hyperbolic manifolds.
problem Counting minimal surfaces in hyperbolic manifolds.
method Using a laminar measure concept.
result An idea of a proof for counting minimal surfaces.
Better neural arithmetic logic units improve cell counting model generalization.
problem Neural networks struggle with high cell counts outside training data range.
method Introduced Neural Arithmetic Logic Units (NALU) for arithmetic operations in existing architectures.
result Improved cell counting accuracy for higher numeric ranges with better generalization.