The paper studies graph Laplace operator behavior near isolated singularities.
problem Investigating asymptotics of graph Laplace operator near isolated singularities.
method Analyzing curvature growth and conformal modifications to understand operator behavior.
result The graph Laplace operator converges to a weighted Laplace-Beltrami operator as bandwidth decreases, or behaves like \(O(\frac{1}{\sqrt{t}})\) if curvature grows too fast.
We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansion…
Asymptotic Laplace transform for geometric Brownian motion applied to bond pricing.
problem Laplace transform of geometric Brownian motion time integral.
method Asymptotic analysis for σ2To0 limit. result Approximation for zero coupon bond prices in Dothan model.
We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.
problem Foliation of an asymptotically flat end by critical hypersurfaces.
method Constructing hypersurfaces as critical points of a functional, solving an over-determined boundary value problem.
result Solutions to the Laplace-Beltrami operator over a foliation of critical spheres.
New algorithms improve likelihood of finding global optima in Bayesian inference.
problem Finding global optima in Bayesian inference is difficult due to nonconvexity.
method Developed two algorithms: consistent Laplace approximation (CLA) and consistent stochastic variational inference (CSVI).
result Both CSVI and CLA improve likelihood of obtaining global optima compared to standard methods.
Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.
problem Understanding the geometry and regularity of submanifolds in hyperbolic space.
method Analyzing asymptotic geometry and regularity properties near the ideal boundary, computing essential spectra.
result Computed essential spectra of the Laplace operator on certain submanifolds.
Study finds eigenvalue patterns on rough manifolds with measurable metrics.
problem Eigenvalue patterns on rough Riemannian manifolds with measurable metrics.
method Demonstrated a Weyl law for eigenvalues of Laplacian and weighted Laplace equations.
result Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds.
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 −NμNμ. Our objective is to obtain information on the asymptotic expansions of the corresponding r…
The paper calculates heat asymptotics for nonminimal Laplace type operators and applies it to noncommutative tori.
problem Analyzing heat asymptotics for nonminimal Laplace type operators.
method Computing the asymptotics of the trace of the heat kernel for a specific class of operators.
result The modular scalar curvature for noncommutative tori is calculated.
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space R3. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve…
Researchers describe the mass of conformal differential operators in terms of their asymptotic expansions.
problem Understanding the mass of conformal differential operators and its invariance under conformal transformations.
method Explicit description of the full asymptotic expansion of the Schwartz kernel of complex powers of m-Laplace type operators. result The mass of conformal differential operators is a conformal invariant in odd dimensions when the kernel is trivial.
We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results obtained earlier for the scalar Laplacian and present closed formulas for all heat in…
Study Laplace operators in adiabatic limit of fibre bundles.
problem Understanding Laplace-type operators in the adiabatic limit of complex vector bundles.
method Analyse the adiabatic limit of fibre bundles with compact fibres, proving existence of effective operators providing asymptotics.
result Existence of effective operators providing asymptotics to any order in ε for Laplace-type operators H on an almost-invariant subspace of L^2(E).
Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on Rn endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to [0,+∞).
Study new spectral invariants for Laplace and Dirac operators on manifolds.
problem Understanding spectral invariants of elliptic operators on manifolds.
method Introduced new spectral invariants depending on eigenvalues and eigensections, computed asymptotic expansion.
result Computed first two coefficients of the asymptotic expansion of the new spectral invariant.
We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.
We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of correspondin…
Unified analytical tool for non-Markovian jump processes.
problem Analyzing history-dependent jump processes with non-Markovian behavior.
method Developed a standard form of master equations using Laplace-space embedding and asymptotic solution.
result Unified analytical toolset for general non-Markovian processes, leading to the GLE approximation.
Efficient Bayesian inference via Gaussian approximations.
problem Parameter estimation and model selection in Bayesian statistics.
method Variational-Laplace approach using Gaussian approximations.
result Novel theoretical results on asymptotic convergence of VL schemes.
Study investigates ruin probability with random premiums and risky investments.
problem Ruin probability with random premiums and risky investments.
method Laplace transform applied to a model with geometric Brownian motion.
result Asymptotic behavior of ruin probability for large initial capital values.
Study small perturbations on low energy Laplace eigenfunctions.
problem Understanding small changes in low energy Laplace eigenfunctions.
method Investigates nodal geometry and topology, focusing on low frequency regimes and small perturbations.
result Highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions.
New positive mass theorem for hyperbolic 3-manifolds using Green functions.
problem Proving a positive mass theorem for asymptotically hyperbolic 3-manifolds.
method Using a monotonicity formula for the Green function of the Laplace operator.
result Established a new positive mass theorem for three-dimensional manifolds.
Study on p-Laplace equation in convex cones, proving rigidity under specific conditions.
problem Overdetermined problem for p-Laplace equation in convex cones. method Established properties of capacitary potential, used P-function, isoperimetric inequality, and Heintze-Karcher inequality. result Rigidity result under orthogonal intersection assumption.
We investigate the short-time expansion of the heat kernel of a Laplace type operator on a compact Riemannian manifold and show that the lowest order term of this expansion is given by the Fredholm determinant of the Hessian of the energy functional on a space of finite energy paths. This is the asymptotic behavior to …
Study asymptotically almost periodic solutions on real hyperbolic manifolds.
problem Existence and asymptotic behavior of solutions to parabolic equations.
method Dispersion and smoothing estimates, fixed point argument.
result Existence and uniqueness of asymptotically almost periodic solutions.
Paper introduces p-Laplace equations for curvature in conformal geometry.
problem Study of geometry and topology of manifolds.
method Introducing p-Laplace equations for intermediate Schouten curvature.
result Nonnegative intermediate Schouten curvature leads to estimates on Hausdorff dimension of singular sets and vanishing of homotopy groups.
The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.
problem Analyzing heat-type equations on manifolds with specific boundary conditions.
method Proving Schauder estimates for the Laplace-Beltrami operator on manifolds with fibered boundaries and a Φ-metric.
result The proof of parabolic Schauder estimates for the Laplace-Beltrami operator.
A trace on the C^*-algebra A of quasi-local operators on an open manifold is described, based on the results in \cite{RoeOpen}. It allows a description `a la Novikov-Shubin \cite{NS2} of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is pr…
New methods for computing volumes and constructing Fano fibrations.
problem Understanding Fano fibrations and their weighted volumes.
method New methods for computing weighted volumes, including Laplace transforms and incomplete Gamma-functions. Conjectural construction of Fano fibrations.
result Conjectural construction of Fano fibrations with asymptotically conical bases from degenerating Fano fibrations.
In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on (1,1)-forms. The method is effective in proving an optimal result when M has nonnegative bisectional curvature. It also provides …
Study on variance of Laplace eigenfunctions on manifolds.
problem Investigating the variance of Laplace eigenfunctions on compact manifolds.
method Combining Kac-Rice formula, Wiener-Itô chaos decompositions, and pointwise Weyl law analysis.
result Established a quantitative bound for the fluctuations of nodal volumes, improving existing results.
Study heat trace expansion on manifolds with conic points.
problem Analyzing heat diffusion on manifolds with sharp corners.
method Using the Singular Asymptotics Lemma to derive an expansion.
result Detailed asymptotic expansion reveals geometric insights.
Improved Kuznecov remainder estimates for generic metrics.
problem Estimating period integrals of Laplace eigenfunctions on manifolds.
method Two-term asymptotic expansion and elimination of oscillatory second term.
result Improved remainder estimates for Baire-generic metrics.
The paper solves an optimal investment problem using M-CEV model and provides explicit strategies.
problem An optimal investment problem under M-CEV model with power utility function.
method Laplace transform and confluent hypergeometric functions to derive explicit expressions.
result Explicit strategies and asymptotic formulas derived for analysis of model parameters.
New method calculates geometric Brownian motion with affine drift and its integral.
problem Calculating the distribution of geometric Brownian motion with affine drift and its integral.
method Laplace transform approach and Heun differential equation.
result Joint distribution of geometric Brownian motion with affine drift and its integral can be determined.
We propose a generalized double Pareto prior for Bayesian shrinkage estimation and inferences in linear models. The prior can be obtained via a scale mixture of Laplace or normal distributions, forming a bridge between the Laplace and Normal-Jeffreys' priors. While it has a spike at zero like the Laplace density, it al…
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.
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds…
Improves Laplace approximation for Bayesian inference on Riemannian manifolds.
problem Inaccurate Gaussian approximations for complex targets and finite-data posteriors.
method Develops alternative variants of the Laplace approximation using a Riemannian metric.
result Exact approximations at the limit of infinite data, improving practical performance.
In this thesis we study the geometry of the fixed point set Σ of a smooth mapping Φ:M→M on a smooth compact Riemannian manifold M without boundary by computing the asymptotic expansion of the deformed heat trace $\Trace Φ\exp(tΔ)$ of the Laplace operator Δ on M. We assume that the fixed point set Σ is a…
Analytic methods for option pricing under a novel Lévy model.
problem Developing pricing models for exotic options under a hyperexponential Lévy process.
method Series expansions and Laplace transform techniques.
result Analytic expressions for option prices and Greeks, including an asymptotic expansion of implied volatility.
This paper computes a heat coefficient for nonminimal Laplace type operators.
problem Computing the heat coefficient for nonminimal Laplace type operators.
method Analyzing the asymptotics of the trace of operator products and their integral representations.
result Computation of the heat coefficient a4 for nonminimal Laplace type operators. The article calculates the most-likely path for Asian option pricing in local volatility models.
problem Approximating the price of Asian options in local volatility models.
method Path-integral approach using Brownian bridge and Laplace asymptotic formula.
result The most-likely path (MLP) is found to approximate the option price in the limit of small sampling time.
Study of spectral geometry on noncommutative tori using functional metrics.
problem Understanding the spectral properties of noncommutative tori.
method Introduction of functional metrics and analysis of their Laplace type operators and spectral invariants.
result Explicit computation of scalar curvature and total scalar curvature for certain functional metrics.
Bayesian nonparametric models get better posterior estimates via SPDE methods.
problem Estimating posterior distributions in nonparametric Bayesian models.
method Extending diffusion methods to SPDEs on Hilbert spaces for posterior contraction and Laplace approximation.
result Derivation of posterior contraction rates and finite-sample Bernstein von Mises results.
Proposes sampling from reverse diffusion posteriors for contextual bandits.
problem Complex distributions in contextual bandits.
method Approximate posterior sampling with a diffusion model prior using Laplace approximation.
result Empirically consistent and efficient approximations for contextual bandits.
On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eig…