Short proof of heat kernel asymptotics and convolution approximation.
problem Short time asymptotics and heat kernel approximation for Laplace type operators.
method Short time asymptotic expansion and convolution approximation of heat kernels.
result Approximation of heat kernel using repeated convolutions.
The paper studies heat kernel asymptotics and proves Morse inequalities.
problem Analyzing the asymptotic behavior of heat kernels near critical points.
method Localization and scaling techniques in semi-classical analysis.
result The heat kernel near critical points is approximated by harmonic oscillator kernels, leading to Morse inequalities.
Kernel smoothing on unknown manifolds with bounds and asymptotic normality.
problem Data on unknown manifolds without boundaries.
method Finite sample bounds and asymptotic normality for kernel smoothing and its derivatives.
result Established finite sample bounds and asymptotic normality for kernel smoothing.
The paper improves asymptotic polybalanced kernels for extremal Kaehler metrics.
problem Stability and metrics on algebraic manifolds.
method Asymptotic polybalanced kernels associated to extremal Kaehler metrics.
result Stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds.
Study Szegő kernel on non-compact CR manifolds with specific conditions.
problem Analyzing Szegő kernel on non-compact CR manifolds.
method Establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex CR manifolds with transversal CR R \mathbb{R} R -action under natural geometric conditions. result Szegő kernel asymptotic expansions established on non-compact CR manifolds.
Study on kernel tests for high-dimensional data, focusing on MMD and CLT.
problem Asymptotic behavior of kernel two-sample tests in high dimensions and large samples.
method Maximum mean discrepancy (MMD) with isotropic kernels, deriving asymptotic expansions and CLT.
result Interplay between moment discrepancy and dimension-and-sample orders in kernel tests.
Study on Bergman kernels near analytic hypersurfaces with exponential decay.
problem Analyzing the behavior of Bergman kernels near analytic hypersurfaces.
method Uniform estimates and asymptotic analysis of partial Bergman kernels for sections vanishing along hypersurfaces.
result Uniform estimate and asymptotic behavior of Bergman kernels near vanishing locus.
Study asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
problem Asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
method Sharp expansions derived for the Poisson kernel and Green's functions near singularities.
result Sharp expansions of the Green's functions solve the first part of Kim-Musso-Wei's conjecture.
Study reproducing kernels on complex Kepler manifolds.
problem Complex geometry of generalized Kepler manifolds.
method Introduce Hilbert spaces of holomorphic functions and find asymptotic expansions of reproducing kernels.
result Complete asymptotic expansions of reproducing kernels found for Kähler potentials.
Study on symplectic manifolds, focusing on Bergman kernels and their asymptotic behavior.
problem Asymptotic behavior of Bergman kernels on symplectic manifolds of bounded geometry.
method Establish off-diagonal exponential estimates, full asymptotic expansions, and Berezin-Toeplitz quantization.
result Improved remainder estimates for the asymptotic expansion of Bergman kernels.
Master thesis proves Bergman kernel asymptotics for positive line bundles.
problem Proving asymptotic expansion of Bergman kernel for positive line bundles.
method Introduced a semi-classical symbol space and symbolic calculus.
result Established pointwise asymptotic expansion on positive parts of certain semi-positive line bundles.
Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.
problem Asymptotic behavior of Bergman kernels for high tensor powers of positive line bundles.
method Analyzing the Schwartz kernel of the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector, proving exponential estimates and asymptotic expansions.
result Explicit asymptotic expansions for the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector.
We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/040…
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
New insights into simple kernel smoothing reveal surprising asymptotics.
problem Understanding precise asymptotics of Nadaraya-Watson kernel smoothing.
method Using ideas from the random energy model in statistical physics.
result Sharp asymptotics for the NW predictor on the sphere.
Study on symplectic manifolds finds asymptotic expansions for Bergman kernels.
problem Analyzing asymptotic expansions of Bergman kernels on symplectic manifolds.
method Established asymptotic expansions for renormalized Bochner Laplacians on high tensor powers of positive line bundles.
result Full off-diagonal asymptotic expansion for generalized Bergman kernels.
We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the k k k -th tensor powers of a positive line bundle L L L in a 1 k \frac{1}{\sqrt{k}} k 1 -neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential k φ k\varphi k φ …
The paper studies heat kernel asymptotics for Kohn Laplacians on CR manifolds.
problem Analyzing heat kernel asymptotics for Kohn Laplacians on CR manifolds.
method Establishing asymptotics of heat kernels and equivariant heat kernels on CR manifolds.
result Heat kernel asymptotics for Kohn Laplacians on CR manifolds are derived.
Heat kernel analysis on a specific sub-Riemannian manifold yields precise asymptotics.
problem Analyzing heat kernel asymptotics on a sub-Riemannian manifold with symmetries.
method Adapting Molchanov's technique to obtain heat kernel asymptotics at the cut locus.
result Exact structure of the cut locus and complete small-time asymptotics for the heat kernel on the bi-Heisenberg group.
Study on heat kernel asymptotics and path integrals on Riemannian manifolds.
problem Investigating the short-time expansion of heat kernel on compact Riemannian manifolds.
method Formally expressing the heat kernel as a path integral and using Laplace's method.
result The lowest order term of the heat kernel's short-time expansion is given by the Fredholm determinant of the Hessian of the energy functional.
We study the asymptotic of the Bergman kernel of the spin c ^c c Dirac operator on high tensor powers of a line bundle.
Optimal kernel improves estimation accuracy in modal statistical methods.
problem Estimation accuracy of kernel-based modal statistical methods depends on the kernel used.
method The study theoretically shows an optimal kernel that minimizes asymptotic error criterion.
result An optimal kernel minimizes the error criterion when using an optimal bandwidth.
Optimal Biweight kernel and computationally efficient Epanechnikov kernel for modal linear regression.
problem Finding the best kernel for modal linear regression.
method Refined analysis of asymptotic statistical behavior and IRLS algorithm convergence.
result Biweight kernel minimizes asymptotic mean squared error, Epanechnikov kernel guarantees IRLS convergence.
Study on CR manifolds with S 1 S^1 S 1 action finds Szegő kernel asymptotics.
problem Analyzing Szegő kernel on CR manifolds with S 1 S^1 S 1 action. method Established asymptotic expansion for the m m m -th Fourier component of the Szegő kernel function. result Explicit formulas for the first three coefficients of the expansion.
New tools for analyzing Kähler manifolds, proving operator algebra and asymptotic kernel.
problem Analyzing Berezin-Toeplitz operators on Kähler manifolds.
method Introducing new tools for analytic microlocal analysis.
result Space of analytic Berezin-Toeplitz operators is an algebra.
Study shows quantum behavior near infinity in metric asymptotics.
problem Quantum behavior of metrics near infinity on quasi-projective manifolds.
method Analysis of Bergman kernel function near smooth divisor at infinity of Cheng-Yau metric.
result Quantum phenomenon observed for points very close to the divisor at infinity.
New method for spectral and Bergman kernels under local spectral gap condition.
problem Analyzing spectral and Bergman kernels for complex manifolds.
method Developed a new scaling method to study spectral and Bergman kernels.
result Established pointwise asymptotics of spectral and Bergman kernels.
Study Bergman kernel metrics on degenerating hyperelliptic surfaces.
problem Asymptotic behavior of Bergman kernels near singularities.
method Taylor expansion for Abelian differentials and period matrices.
result Explicit coefficients in asymptotic formulas for Bergman kernels.
The paper analyzes heat kernel asymptotics for real powers of Laplacians on manifolds.
problem Analyzing the small-time behavior of heat kernels for real powers of Laplacians.
method Analyzes asymptotics on the diagonal and away from it, proving non-triviality and non-locality of coefficients.
result Logarithmic terms appear only if the manifold dimension is odd and the power is rational with even denominator.
The Bergman-Szegő kernel is analyzed for weakly pseudoconvex CR manifolds of finite type.
problem Analyzing the Bergman-Szegő kernel for specific CR manifolds.
method Constructing a parametrix for the Szegő kernel, extending earlier results.
result Extending Fefferman's boundary asymptotics to weakly pseudoconvex domains in \(\mathbb{C}^{2}\).
In this paper we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We then explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in …
Study on manifolds with kinks and Gaussian kernel behavior.
problem Understanding the asymptotic behavior of graph Laplacian on manifolds with singularities.
method Introduced manifolds with kinks, derived asymptotic behavior of Graph Laplacian with Gaussian kernel, and validated results numerically.
result Asymptotic behavior of the Graph Laplacian is determined by the inward sector of the tangent space.
The study computes Bergman kernels and point process asymptotics on Kähler manifolds.
problem Computing asymptotics of Bergman kernels and point process distributions on Kähler manifolds.
method Equivariant and partial Bergman kernels, determinantal point processes, asymptotic analysis.
result The distribution of linear statistics converges to a centered normal variable with specific variances.
We analyzed optimism in linear and kernel regression models.
problem Understanding predictive complexity in regression models.
method Derived closed-form asymptotic optimism for linear and kernel regression models.
result Scaled optimism is a useful measure for model complexity.
We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part − N μ N μ -\N^μ\N_μ − N μ N μ . Our objective is to obtain information on the asymptotic expansions of the corresponding r…
Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.
problem Analyzing sub-Riemannian heat kernels and their derivatives on incomplete manifolds.
method Localized asymptotic analysis, focusing on minimizing geodesics and the non-abnormal cut locus.
result Uniform bounds and expansions for heat kernels and their derivatives on compacts, including the diffusion bridge measure.
We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line b…
The paper solves heat kernel asymptotics on non-degenerate CR manifolds.
problem Existence of small-time asymptotics for the heat kernel of the Kohn Laplacian on CR manifolds.
method Analytic methods and spectral theory for CR manifolds.
result Established small-time asymptotics for the heat kernel and analytic torsion on non-degenerate CR manifolds.
In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate poi…
We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the short-time asymptotic expansion of the heat kernel is calculated explicitly.
We compute the leading and sub-leading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.
Study the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
problem Understanding the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
method Detailed study of the geometry, derivation of the horizontal Laplacian and subelliptic heat kernel formulas, derivation of small time asymptotics.
result Explicit formulas for the horizontal Laplacian and subelliptic heat kernel of the quaternionic anti-de Sitter fibration.
New test for conditional independence using kernel embeddings.
problem Testing conditional independence in high-dimensional settings.
method Analytic kernel embeddings, asymptotic distribution.
result New test outperforms existing methods in high-dimensional settings.
The paper studies local heat kernel properties on smooth manifolds.
problem Understanding heat kernel properties in open convex sets of smooth Riemannian manifolds.
method Utilizes path integral formulation to investigate properties like uniqueness, symmetry, and asymptotics.
result Uniqueness and symmetry of Seeley-DeWitt coefficients are established.
The study examines Bergman kernels on complex manifolds with boundary and their asymptotic expansions.
problem Analyzing Bergman kernels on complex manifolds with boundary and their asymptotic behavior.
method Establishing asymptotic expansions of partial Bergman kernels for high-frequency Fourier modes on R \mathbb{R} R -symmetric complex manifolds with boundary. result Established R \mathbb{R} R -equivariant extension results for biholomorphic maps between weakly pseudoconvex domains. Study integral kernels on complex symmetric spaces and their Dyson Brownian Motion applications.
problem Analysis of integral kernels on complex symmetric spaces.
method Simple new method of alternating sum formulas to construct W W W -invariant kernels and their asymptotic behavior. result Obtained asymptotic behavior of integral kernels and applied to Dyson Brownian Motion.
The paper analyzes hypoelliptic heat kernels near a manifold's cut locus.
problem Analyzing hypoelliptic heat kernels near a manifold's cut locus.
method Probabilistic approach using S. Watanabe's distributional Malliavin calculus and T. Lyons' rough path theory.
result Obtained a short time asymptotic expansion of hypoelliptic heat kernels up to any order.
Formula for Toeplitz operator kernel on CR manifolds.
problem Analyzing Toeplitz operators on CR manifolds.
method Formula for the symbol of the kernel, asymptotic expansions.
result Formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel.