Algorithm transforms weakly negative plumbing trees to negative definite ones.
problem Transforming weakly negative plumbing trees to negative definite ones.
method Combining plumbing calculus with diagonalization algorithm to systematically eliminate positive eigenvalues.
result Explicit algorithm reduces weakly negative definite plumbing trees to negative definite ones.
We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed …
Study series invariants of plumbed 3-manifolds using root lattices.
problem Understanding invariants of plumbed 3-manifolds twisted by root lattices.
method Use formal series to study invariants, decompose Z ^ ( q ) \widehat{Z}(q) Z ( q ) , and compute in specific cases. result Show that Z ^ ( q ) \widehat{Z}(q) Z ( q ) is unique and decomposes into related series invariant under five Neumann moves. Study shows continuity of non-orientable surface determination from Dirichlet-to-Neumann map.
problem Determining non-orientable surfaces from Dirichlet-to-Neumann map.
method Proving closeness of Dirichlet-to-Neumann maps implies near-conformal diffeomorphism.
result Established continuity of determination Λ ↦ [ ( M , g ) ] Λ\mapsto [(M,g)] Λ ↦ [( M , g )] and quantitative estimates of d T ( [ ( M , g ) ] , [ ( M ′ , g ′ ) ] ) d_T([(M,g)],[(M',g')]) d T ([( M , g )] , [( M ′ , g ′ )]) . New upper bound for Neumann Laplacian eigenvalues on convex domains.
problem Bounding Neumann eigenvalues on convex domains.
method Deriving a new upper bound for eigenvalues.
result Universal inequalities for Neumann eigenvalues derived from the upper bound.
Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.
problem Finding the minimum value of the first Neumann eigenvalue for convex domains.
method Proved the sharp lower bound using diameter and width.
result Sharp lower bound for the first Neumann eigenvalue established.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.
problem Bounding the measure of zero sets of Neumann Laplace eigenfunctions.
method Analyzes nodal sets in domains with C 1 , 1 C^{1,1} C 1 , 1 boundaries. result Hausdorff measure bound of nodal sets is C λ C \sqrt{\lambda} C λ . Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
We prove the following: Let 2 p + 1 2p + 1 2 p + 1 be no less than 5 and p p p be a natural number. Let K K K and J J J be closed, oriented, ( 2 p + 1 ) (2p+1) ( 2 p + 1 ) -dimensional connected, ( p − 1 ) (p-1) ( p − 1 ) -connected, simple submanifolds of the standard ( 2 p + 3 ) (2p+3) ( 2 p + 3 ) -sphere. Then K K K is equivalent to J J J if and only if a Seifert matrix associated with a simple Seifert …
The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.
problem Eigenvalue comparisons on graphs.
method Analytical comparisons and discussions of eigenvalues and their applications.
result Extensions of eigenvalue estimates for Dirichlet and Neumann eigenvalues.
Geodesic disks maximize the first non-trivial Neumann eigenvalue on spheres.
problem Maximizing the first non-trivial Neumann eigenvalue on spheres.
method Proving maximizers are geodesic disks.
result Geodesic disks maximize the first non-trivial Neumann eigenvalue.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Neumann's work connects 3-manifold invariants to quantum topology.
problem Understanding invariants of 3-manifolds.
method Ideal triangulations, volume, Chern-Simons invariants, Bloch group.
result Neumann's work bridges classical invariants to quantum topology.
Recently, the first named author together with Xinan Ma \cite{ma2015neumann}, have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniforml…
Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.
problem Recover stiffness tensor and density from Dirichlet-to-Neumann map.
method Analyze invariance under coordinate transformations and gauge freedoms.
result Present gauge freedoms in the Dirichlet-to-Neumann map for Riemannian elastic wave equation.
Study solves a mathematical problem related to elliptic Schroedinger-to-Neumann maps.
problem Solving a Cherrier-Escobar problem for elliptic Schroedinger-to-Neumann maps.
method Using algebraic topological argument of Bahri-Coron, assuming positive eigenvalue and Green function.
result Solvability of the extended problem under specified conditions.
Universal inequalities for Laplacian eigenvalues on convex domains.
problem Eigenvalue distribution of the Laplacian on convex domains.
method Established two universal inequalities.
result Two new inequalities for Laplacian eigenvalues.
Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.
problem Biharmonic Steklov problems with Neumann boundary conditions.
method Introduced a biharmonic Steklov problem and proved its well-posedness. Established eigenvalue estimates using Kuttler-Sigillito inequalities.
result Eigenvalue estimates for the biharmonic Steklov problem with Neumann boundary conditions.
Study uniquely determines Riemannian metric derivatives from boundary data.
problem Determining Riemannian metric derivatives from boundary data.
method Computing the full symbol of the elastic Dirichlet-to-Neumann map.
result The elastic Dirichlet-to-Neumann map uniquely determines all partial derivatives of the Riemannian metric on the boundary.
Two geodesic balls maximize the third Neumann eigenvalue in hyperbolic space.
problem Maximizing the third eigenvalue of the Neumann Laplacian in hyperbolic space.
method Using the disjoint union of two geodesic balls to prove maximality.
result The third eigenvalue is maximal for the union of two geodesic balls.
New inequalities for planar convex domains' Laplacian eigenvalues.
problem Neumann eigenvalues of the Laplacian on planar convex domains.
method Established two new universal inequalities.
result New inequalities for Laplacian eigenvalues on convex domains.
Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
problem Calculating spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
method Established an effective procedure to calculate all coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map.
result Explicitly provided the first four coefficients of the spectral asymptotic formula.
Study Neumann problem for special Lagrangian type equations.
problem Neumann problem for special Lagrangian type equations.
method Uniform a priori estimates, continuity method, direct proof of boundary double normal derivative estimates.
result Existence result for Neumann problem of special Lagrangian type equations.
Study relative commutants in von Neumann algebras using contraction notions.
problem Understanding relative commutants in group and tracial crossed product von Neumann algebras.
method Introducing contraction notions to study relative commutants.
result Results applied to negatively curved groups and SL(d, Z).
In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-W…
Neumann eigenmaps improve landmark-based diffusion map embeddings.
problem Landmark-based diffusion map embeddings can be computationally inefficient and unstable.
method NeuMaps use a renormalized Neumann Laplacian for eigendecomposition, incorporating landmarks as a subgraph.
result NeuMaps offer a computationally efficient and stable embedding method.
Sharp inequality outside ball proved using Neumann method.
problem Anisotropic isoperimetric inequality for domains outside an Euclidean ball.
method Applied ABP method to Neumann boundary value problem.
result Proved sharp anisotropic isoperimetric inequality.
Solves Neumann problem on CR manifold boundary.
problem Neumann problem on CR manifold boundary.
method Analyzes CR Yamabe operator and contact forms.
result Solves Neumann problem and finds contact form.
Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.
problem Determining geometric data from boundary symbol of connection Laplacian.
method Analyze symbol of Dirichlet-to-Neumann operator associated with connection Laplacian.
result Geometric data on boundary and normal derivatives are determined by symbol.
We study new heat kernel estimates for the Neumann heat kernel on a compact manifold with positive Ricci curvature and convex boundary. As a consequence, we obtain new lower bounds for the Neumann eigenvalues which are consistent with Weyl's asymptotics.
Paper derives formulas for static Einstein spaces, linking Neumann data to stability.
problem Stability of conformally compact static spaces.
method First and second variation formulas for renormalized area.
result Negativity of Neumann data implies instability.
In this paper, we prove long time existence and convergence results for a class of general curvature flows with Neumann boundary condition. This is the first result for the Neumann boundary problem of non Monge-Ampere type curvature equations. Our method also works for the corresponding elliptic setting.
Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.
problem Asymptotic expansion of heat trace for thermoelastic Dirichlet-to-Neumann map.
method Provided a method to obtain all coefficients of the asymptotic expansion.
result Explicitly gave the first two coefficients involving volume and total mean curvature of the boundary.
Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
problem Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
method Constructs a metric on a compact manifold to demonstrate the nonexistence of Courant-type bounds.
result Provides a negative answer to the existence of Courant-type nodal domain bounds.
Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well a…
It is shown that the Novikov inequalities for critical points of closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a corollary we obtain a vanishing theorem for L 2 L^2 L 2 cohomology, generalizing a theorem of W. Lueck. We also prove that von Neumann Betti numbers coincide with the Nov…
We study integrable geodesic flows on Stiefel varieties V n , r = S O ( n ) / S O ( n − r ) V_{n,r}=SO(n)/SO(n-r) V n , r = S O ( n ) / S O ( n − r ) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on V n , r V_{n,r} V n , r with the above metrics and proves their integrability in the non-commutative sense b…
Twisted Neumann--Zagier matrices for quantum invariants.
problem Constructing quantum invariants from ideal triangulations.
method Define and compute twisted Neumann--Zagier matrices from combinatorics.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial.
Study proves inequalities for eigenvalues of symmetric domains in space forms.
problem Eigenvalue problem for Laplacian on symmetric domains in space forms.
method Symmetry assumptions on domain, proof of inequalities for eigenvalues.
result Proves Szegö-Weinberger type inequalities for first n positive Neumann eigenvalues.
In this paper, we consider the global regularity for Monge-Ampère type equations with the Neumann boundary conditions on Riemannian manifolds. It is known that the classical solvability of the Neumann boundary value problem is obtained under some necessary assumptions. Our main result extends the main theorem from the …
The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.
problem Finding surfaces with commuting boundary Laplacian and Dirichlet-to-Neumann map.
method Investigated 2D submanifolds of R^2, classified surfaces of genus 0 or with k≥3 boundary components.
result One-parameter family of 2D submanifolds with commuting Laplacian and Dirichlet-to-Neumann map.
Abstract: Expresses zeta-determinant of Dirichlet-to-Neumann operator on forms.
problem Express zeta-determinant of Dirichlet-to-Neumann operator on forms.
method Expresses zeta-determinant as difference of Laplacian determinants with boundary conditions.
result Computes terms explicitly for dimensions 2 and 3.
Paper generalizes kernel mean embedding to von Neumann-algebra-valued measures.
problem Analyzing complex multivariate distributions and quantum mechanics.
method Generalizes kernel mean embedding to von Neumann-algebra-valued measures in reproducing kernel Hilbert modules.
result Injectivity and universality of the generalized KME are confirmed.
Graph Neural Networks solve topology problems in simple 3D models.
problem Deciding homeomorphism of 3-manifolds described by plumbing graphs.
method Supervised and reinforcement learning with Graph Neural Networks.
result High accuracy in determining homeomorphic 3-manifolds.
We present a new and direct proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below.
The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.
problem Classifying solutions to a Liouville equation with a nonlinear Neumann boundary condition.
method Analyzing the n n n -Laplacian Liouville equation on the half-space R + n \mathbb{R}^{n}_{+} R + n with positive nonlinear Neumann boundary condition. result The classification of solutions extends previous results for n = 2 n=2 n = 2 and p = n p=n p = n . Study reveals how boundary wave operator determines metric properties on anti-de Sitter spacetimes.
problem Determining metric properties on anti-de Sitter spacetimes.
method Analysis of Klein-Gordon equation and Dirichlet-to-Neumann map.
result Determines the Taylor series of the bulk metric at the boundary.