Study on second Robin eigenvalue for Laplacian on manifolds.
problem Maximizing the second Robin eigenvalue for geodesic balls in nonpositively curved space forms.
method Comparison theorem and maximization analysis for the second Robin eigenvalue.
result Geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue.
The paper compares heat kernels on manifolds with Robin boundary conditions.
problem Comparing heat kernels on manifolds with different boundary conditions.
method Proving comparison theorems for heat kernels on geodesic balls and minimal submanifolds.
result Eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds.
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
Study on pseudo-Einstein 3-manifolds for a specific inequality, introducing Robin mass.
problem Existence of contact structures on pseudo-Einstein CR manifolds.
method Introduced Robin mass and used it to study the variation of total mass under conformal change.
result Existence of a minimizer for total mass yielding the classical LHLS inequality.
Paper discusses gluing formula for zeta-determinants with Robin boundary condition.
problem Computing zeta-determinants with Robin boundary condition.
method Uses BFK type gluing formula and computes differences with Dirichlet boundary condition.
result Computes zeta-determinant on a cylinder with Robin boundary condition.
Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.
problem Comparing solutions of Poisson equations on Riemannian manifolds with Robin boundary.
method Using Schwarz rearrangement and isoperimetric inequalities.
result Extends results on Poisson equations with Ric≥(n−1)κ. Study eigenvalues of p-Laplacian on manifolds with Robin boundary conditions.
problem Investigate the first Robin eigenvalue of the p-Laplacian on Riemannian manifolds.
method Prove eigenvalue comparison theorems and establish sharp bounds.
result Sharp lower bounds for the first Robin eigenvalue of the p-Laplacian.
New manifolds found with equal Robin and Steklov spectra.
problem Finding manifolds with equal Robin and Steklov spectra.
method Used Sunada method and torus action method to construct manifolds.
result Constructs manifolds with equal Robin and Steklov spectra under various boundary conditions.
Paper proves Faber-Krahn inequalities for weighted Laplacian eigenvalues.
problem Proving inequalities for eigenvalues of weighted Laplacian.
method Analyzing Robin boundary conditions on Rn and Hn. result Optimal domain for eigenvalues is a ball centered at the origin.
Upper bounds for second Robin eigenvalue on Riemannian surfaces.
problem Bounding the second Robin eigenvalue of Schrödinger operators on Riemannian surfaces.
method Geometric upper bound via Hersch balancing argument on capped surfaces.
result Sharp geometric restrictions for minimal surfaces in negatively curved manifolds.
New findings on Obata equation with Robin boundary conditions on manifolds.
problem Analyzing the Obata equation with Robin boundary conditions on manifolds.
method Investigation of the equation with Robin boundary condition ∂ν∂f+af=0 on manifolds with boundary. result New manifolds for both positive and negative a values were discovered. Paper finds eigenvalue bounds for hyperbolic space domains.
problem Finding eigenvalue bounds for Robin Laplacian in hyperbolic space.
method Lower and upper bounds derived for eigenvalues.
result Geodesic ball maximizes eigenvalue in negative boundary parameter case.
Proposes a flexible tournament design combining knockout and round-robin.
problem Designing a tournament that eliminates participants linearly.
method Combines knockout and round-robin structures for flexible elimination.
result Flexible tournament design can eliminate participants linearly.
For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-…
New estimator stabilizes higher-order influence functions for bilinear forms.
problem Stability issues in estimating bilinear forms using higher-order influence functions.
method Proposes a new stabilized higher-order estimator for a class of bilinear forms without sample splitting.
result New estimator exhibits more stable finite-sample performance compared to the empirical higher-order estimator.
New estimator stabilizes higher-order influence functions for stable statistical inference.
problem Numerical instability in estimating inverse population Gram matrix.
method Proposes a new stabilized higher-order estimator without sample splitting.
result Stabilized estimator exhibits more stable performance and similar statistical guarantees.
The study examines metrics that extremize eigenvalues of a specific map on manifolds with boundary.
problem Variational properties of the spectrum of the Dirichlet-to-Robin map on manifolds with boundary.
method Analysis of the extremal metrics for the first and second normalized eigenvalues of the Dirichlet-to-Robin map.
result Existence and characterization of extremal metrics for the first and second eigenvalues of the Dirichlet-to-Robin map.
Estimates the index of the Laplace operator on planar domains with Robin boundary condition.
problem Index estimates for planar domains with Robin boundary condition
method Combines conformal and spectral techniques with topology of the domain.
result Lower bounds for the index in terms of the number of boundary components.
Critical spherical catenoids have Robin nullity and asymptotic radius determined.
problem Analyzing the critical spherical catenoids in hyperbolic space.
method Analytic results using Sturm-Liouville theory, Beta-function evaluation, and Laplace asymptotic analysis.
result Robin nullity and asymptotic radius of the critical spherical catenoid are determined.
Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.
problem Characterizing umbilical hypersurfaces in space forms.
method Using a Serrin-type partially overdetermined problem with inhomogeneous Robin boundary condition.
result Any contact angle θ ∈ (0, π) can be achieved, generalizing previous results.
Let M be a complete connected Riemannian manifold with boundary $\pp M$, Q a bounded continuous function on $\pp M$, and $L= \DD+Z$ for a C1-vector field Z on M. By using the reflecting diffusion process generated by L and its local time on the boundary, a probabilistic formula is presented for the semigro…
Estimates eigenvalues using Bessel functions on manifolds.
problem Estimating eigenvalues of differential operators on manifolds.
method Using mean value lemma and curvature assumptions, derive differential inequalities involving Bessel functions.
result Establishes new estimates for eigenvalues involving positive roots of Bessel functions.
Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.
problem Rigidity of weak solutions for anisotropic N-Laplacian equations with boundary conditions.
method Established a key integral inequality involving anisotropic gradient and second fundamental form, proving rigidity under natural monotonicity assumptions.
result All weak solutions to Neumann boundary problems are constant without a priori boundedness assumption.
Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.
problem Analyzing self-adjoint extensions of Dolbeault Laplacians on Riemann surfaces.
method Defined ζ-regularized determinants, introduced Robin mass, derived comparison formulas. result Explicit expressions for Robin mass in spinor bundles and scalar cases.
Study proves a sharp upper bound for the zero set area of a static manifold's potential.
problem Proving a sharp upper bound for the zero set area of a static manifold's potential.
method Proved a rigidity theorem for the Euclidean closed unit ball in R^3.
result Sharp upper bound for the area of the zero set of the potential.
Analytic formulae for point vortex dynamics on surfaces with symmetry.
problem Understanding point vortex behavior on surfaces with continuous symmetry.
method Derived analytic formulae for Green and Robin functions on surfaces with hydrodynamic Killing vector fields.
result Unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry.
Paper solves capillary Lp-Minkowski problem for p>1.
problem Finding capillary convex bodies with prescribed Lp-surface area measures.
method Reduction to Monge-Ampère equation with Robin boundary condition.
result Solved capillary Lp-Minkowski problem in smooth category for p>1.
The paper studies a 1D diffusion equation with nonlinear Robin boundary conditions and finds conditions for global and finite time blow-up or blow-down.
problem Investigating the behavior of solutions to a specific diffusion equation with nonlinear Robin boundary conditions.
method Analyzing the Ricci flow on a cylinder and applying it to the diffusion equation.
result Conditions for global and finite time blow-up or blow-down of solutions.
Paper proves inequality for p-Laplacian eigenvalues on curved spaces.
problem Eigenvalue inequalities for p-Laplacian on curved manifolds.
method Robin boundary conditions, lower Ricci bounds, positive asymptotic volume ratio.
result Bossel-Daners inequality extends to compact submanifolds.
The paper identifies a new geometric and spectral phenomenon in the critical hyperbolic catenoid family.
problem The study investigates the critical hyperbolic catenoid family and its geometric and spectral properties.
method The approach involves analyzing the critical hyperbolic catenoid family, identifying parameter-criticality, and studying the Robin spectrum.
result The paper proves that at a parameter-critical value a♯, the Robin nullity of Σa♯ is at least 3, with an additional kernel element in mode k=0. On a compact Riemannian manifold M with boundary, we give an estimate for the eigenvalues (λ_k(τ,α))_k of the magnetic Laplacian with the Robin boundary conditions. Here, τ is a positive number that defines the Robin condition and α is a real differential 1-form on M that represents the magnetic field. We e…
Extends robust methods for causal inference, improving estimator performance.
problem Estimating causal effects in the presence of latent confounders.
method Minimax kernel machine learning for doubly robust functionals.
result Proposed method leads to robust and high-performance estimators.
Causal inference from observational data is hard due to discontinuous causal effects.
problem Causal inference from observational data is hard due to discontinuous causal effects.
method The problem is tackled by showing that many standard point estimates can be read as point summaries of multimodal distributions over the space of structural causal models.
result Many standard point estimates can be discontinuous summaries, while explicit posterior means and medians are continuous.
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
Improved fourth-order compact scheme for option valuation with Robin boundary condition.
problem Lower convergence rates in numerical methods for American options.
method High-order compact scheme, Robin boundary condition, coupled nonlinear PDEs.
result Fourth-order convergence rate achieved without mesh refinement.
The paper solves a specific Minkowski problem for capillary hypersurfaces.
problem Finding capillary convex bodies with prescribed dual curvature measures.
method Reduction to a Monge-Ampère type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution for θ∈(0,2π). Study on convex capillary hypersurfaces with Lp curvature in half-space.
problem Prescribed Lp curvature for convex capillary hypersurfaces.
method Reduction to Hessian quotient equation with Robin boundary condition.
result Existence and uniqueness of smooth admissible solutions.
Proves well-posedness of mixed Robin boundary value problem on manifolds with bounded geometry.
problem Well-posedness of mixed Robin boundary value problem on manifolds with boundary and bounded geometry.
method Analyzes operators on sections of a vector bundle with bounded geometry, proving regularity and well-posedness in Sobolev spaces.
result Main result is well-posedness in Sobolev spaces Hs(M;E) for s≥0 on non-compact manifolds. Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k-th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
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.
Classical learning assumes the learner is given a labeled data sample, from which it learns a model. The field of Active Learning deals with the situation where the learner begins not with a training sample, but instead with resources that it can use to obtain information to help identify the optimal model. To better u…
The functional determinants of the GJMS scalar operators, P_{2k}, on even-dimensional spheres are computed via Barnes multiple gamma functions relying on the numerical availability of the digamma function. For the critical k=d/2 case, it is necessary to calculate the Stirling moduli. The multiplicative anomalies are gi…
New method reveals corners of drum shapes.
problem Determining the shape of drum corners from its sound.
method Locality principle and calculations of heat kernels.
result Corners are spectral invariants of the Laplacian.
Study optimal adaptive allocation for multi-armed bandits with Markovian rewards.
problem Optimal adaptive allocation for multi-armed bandits with Markovian rewards.
method Round-robin Kullback-Leibler upper confidence bounds for optimal adaptive allocation.
result Logarithmic dependence of regret on time horizon, asymptotically optimal.
Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.
problem Understanding topological constraints for stable free boundary CMC surfaces in negatively curved settings.
method Established intrinsic area-length-topology inequalities via a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator.
result Explicit topological restrictions for stable free boundary CMC surfaces, showing low genus and few boundary components.
New stable HOIF estimators for statistical functionals.
problem Constructing numerically stable HOIF estimators for statistical functionals.
method Developed new sHOIF estimators with provable guarantees.
result 2nd order sHOIF estimators were validated in synthetic experiments.