A new invariant from smooth 4-manifolds using von Neumann algebras.
problem Constructing a von Neumann algebra from smooth 4-manifolds.
method Geometric construction of von Neumann algebra from smooth structure, preserving unitary equivalence under diffeomorphisms.
result A new invariant of smooth 4-manifolds, the cosmological constant, can be estimated topologically.
Sharp pseudospectral bounds prevent transient amplification in coupled gradient descent.
problem Transient amplification in coupled gradient descent systems.
method Developed a sharp pseudospectral theory for block-triangular Jacobians, proving Kreiss constant bounds and matching minimax lower bounds.
result Obtained a finite-horizon iteration-complexity bound of O(K(J)2log(1/δ)) for stochastic coupled descent. The paper defines new constants for p-Laplacian on manifolds.
problem Bounding eigenvalues of the p-Laplacian on compact manifolds. method Introducing Steklov and Neumann isocapacitary constants.
result Two-sided bounds for (p,α)-Sobolev constants and eigenvalues. The paper shows connections can be uniquely determined by their boundary data.
problem Determining unique connections from boundary measurements.
method Defined a Dirichlet-to-Neumann map for twisted Dirac Laplacians and showed its pseudodifferential properties.
result Equal Dirichlet-to-Neumann maps imply locally gauge equivalent connections.
Direct proof of Neumann isoperimetric inequality for convex domains.
problem Neumann isoperimetric inequality on convex domains.
method Direct proof using Riemannian manifold properties.
result New proof of Neumann isoperimetric inequality.
Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.
problem Estimating eigenvalues and spectrum for graph substructures.
method Introducing Cheeger type constants via isocapacitary constants to estimate eigenvalues and spectrum.
result Estimates for first Dirichlet, Neumann, and Steklov eigenvalues, as well as the bottom of the spectrum of the Laplace operator and Dirichlet-to-Neumann operator.
Study Cheeger inequalities for Riemannian manifolds with boundary.
problem Estimating Steklov eigenvalues on Riemannian manifolds with boundary.
method Establish Cheeger-type inequalities using isocapacitary constants.
result Cheeger inequalities for Steklov eigenvalues on compact and non-compact manifolds.
We study integrable geodesic flows on Stiefel varieties Vn,r=SO(n)/SO(n−r) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on Vn,r with the above metrics and proves their integrability in the non-commutative sense b…
The paper finds a lower bound for a Neumann eigenvalue of surfaces with flat ends.
problem Finding a lower bound for the first Neumann eigenvalue of surfaces with asymptotically flat ends.
method Integration of Bouchner's formula with contributions from A. Lichnerowicz, S. Brendle, R. Tsiamis, and Poincare's constant.
result Obtains a lower bound for the first Neumann eigenvalue of properly embedded surfaces.
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.
Paper bounds the A-hat genus using curvature and isoperimetric constants.
problem Bounding the A-hat genus of Riemannian manifolds.
method Spectral analysis of the Dirac operator, scalar curvature lower bounds, and isoperimetric constants.
result Proves an upper bound on the A-hat genus using manifold properties.
The Ham Sandwich Theorem helps find bounds on Laplacian eigenvalues.
problem Finding bounds on Laplacian eigenvalues for convex domains.
method Applied Gromov's ham sandwich method to get monotonicity and inequalities.
result Universal inequalities for Neumann eigenvalues of Laplacian on convex domains.
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a f…
The study proves constant-curvature analogues of hot spots conjecture for triangles.
problem Proving the hot spots conjecture in constant curvature domains.
method Analyzing geodesic triangles of constant negative curvature and using Killing fields.
result First mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points in constant curvature triangles.
Proves existence and uniqueness of CMC solutions in product manifolds.
problem Existence and uniqueness of solutions to CMC equation with Neumann boundary data.
method Analyzes product manifold MnimesR with specific curvature conditions. result Proves existence and uniqueness of solutions.
Study on nonlinear equations on spheres and hemispheres with zero Neumann boundary condition.
problem Finding conditions for constant solutions to Brezis-Nirenberg type problems.
method Developed a study involving nonlinear partial differential equations on spheres and hemispheres with zero Neumann boundary condition.
result Conditions for equations to have only constant solutions.
Method learns Dirichlet-to-Neumann maps on graphs using Gaussian processes.
problem Coupling multiphysics simulations on graphs with conservation constraints.
method Gaussian processes combined with discrete exterior calculus and maximum likelihood estimation.
result Data-driven predictions with uncertainty quantification on entire graph.
Liouville theorem for minimal graphs on manifolds with specific properties.
problem Characterizing positive minimal graphic functions on specific Riemannian manifolds.
method Using volume doubling property and uniform Neumann-Poincaré inequality.
result Positive minimal graphic functions on the manifold are constants.
Let Ω be an open, bounded domain in the plane with connected and smooth boundary, and ω an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue μ>0. If the boundary value of ω is a nonzero constant along the boundary, denoting 0=μ1(Ω)<μ2(Ω)<=... the set of all Neumann eigen…
Upper bound found for first nonzero Neumann eigenvalue.
problem Finding an upper limit for the first nonzero Neumann eigenvalue in Riemannian manifolds.
method Proved an upper bound using sectional curvature, Ricci curvature, and geodesic balls.
result First nonzero Neumann eigenvalue is bounded by a constant times the eigenvalue of a geodesic ball.
The paper proves a Neumann eigenvalue sum inequality in non-Euclidean space forms.
problem Proving an inequality involving Neumann eigenvalues in non-Euclidean spaces.
method Analyzing space forms with constant curvature and using geodesic balls.
result Proves a conjecture about Neumann eigenvalues in non-Euclidean spaces.
The paper improves RBP for stable and efficient training of recurrent neural networks.
problem Stability issues in RBP for training recurrent neural networks.
method Proposed two variants of RBP: CG-RBP and Neumann-RBP, and compared them with BPTT and TBPTT.
result Neumann-RBP is more efficient in terms of memory usage compared to TBPTT.
New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
problem Existence of contractible domains with specific boundary conditions.
method Local bifurcation argument around geodesic disks, anisotropic Hölder spaces, computer-assisted techniques.
result Existence of nontrivial contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
Existence of YMH fields with boundary conditions proven.
problem Existence of Yang--Mills--Higgs fields with boundary conditions.
method Study of convergence and blow-up behavior of Sacks-Uhlenbeck type α-YMH fields as α→1, regularity theorem for coupled systems.
result Existence of smooth YMH fields up to the boundary under certain conditions.
In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…
Bayesian optimization helps find best nuclear interaction parameters.
problem Finding best coupling constants in complex nuclear interaction models.
method Bayesian optimization applied to chiral effective field theory.
result Bayesian optimization performs well in low-dimensional parameter domains.
Rotationally symmetric hypersurfaces converge to cylinders under area-preserving flow.
problem Convergence of rotationally symmetric hypersurfaces to cylinders under area-preserving mean curvature flow.
method Geometric properties and maximal principle used for gradient and curvature estimates, leading to long-time existence and convergence.
result Rotationally symmetric hypersurfaces converge to cylinders under area-preserving mean curvature flow.
We construct a cut-off version of nonpertubative closed Bosonic string field theory in the light-cone gauge with imaginary string coupling constant. We show that the partition function is a continuous function of the string coupling constant, and conjecture a relation between the formal power series expansion of this p…
Eigenfunction level sets can have infinitely many connected components.
problem Estimating the number of connected components of eigenfunction level sets.
method Analyzing Neumann eigenfunctions on triangles, polygons, and surfaces.
result There exist eigenfunctions with infinitely many connected components in their level sets.
Study on Kähler metrics on ruled surfaces, proving existence and non-existence.
problem Existence and non-existence of Kähler metrics on minimal ruled surfaces.
method Analysis of twisted and coupled constant scalar curvature Kähler metrics.
result Bound for Chen-Cheng invariant on ruled surfaces.
Local corner-factor conjecture for Neumann jump determinants supported by models.
problem Determining the determinant of Neumann jump operator on piecewise curves.
method Formulated conjecture, supported by three model calculations, and discussed connections.
result Support for the conjectural determinant formula \(\Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^Nα_j^{-1/2}\).
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 Kähler metrics with constant scalar curvature using coupled equations.
problem Finding Kähler metrics with constant scalar curvature.
method Solving a system of elliptic equations for a Kähler metric and a closed (1,1)-form, proving higher order estimates and smooth convergence.
result Smooth convergence to a cscK metric coupled to a harmonic (1,1)-form under uniform estimates.
Generalizes Cheeger inequality to Carnot-Carathéodory spaces.
problem Lower bounds on eigenvalues of Laplacians in complex spaces.
method Geometric approach, including Neumann and mixed boundary conditions.
result Concrete method to lower bound Cheeger constant.
Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions f∈ℓ2(Γ) which act by left convolution on ℓ2(Γ). Let Tr be a totally geodesic flat torus of dimension r in M and let $Γ_0\cong\bb Z^r$ be t…
Study proves no nontrivial minimal surfaces in half-space with specific boundary conditions.
problem Proving the nonexistence of minimal surfaces in half-space with certain boundary conditions.
method Analyzes minimal surface equations in half-space with specific boundary conditions.
result Establishes Liouville type theorems for minimal surfaces in half-space.
Estimates higher-order Poincaré constants for weighted manifolds.
problem Estimating constants for weighted manifolds and their applications.
method Introducing higher-order Poincaré constants and estimating them from above.
result Upper bounds for eigenvalues and isoperimetric constants.
This paper analyzes microstructure dynamics in coupled markets using CFMMs.
problem Quantifying contributions of CFMMs to market dynamics in coupled markets.
method Examined constant function market makers (CFMMs) in coupled markets, focusing on basket inflation/deflation.
result CFMMs contribute significantly to basket inflation/deflation in coupled markets.
Paper proposes a coupling-based diagnostic for SGD stepsize optimization.
problem Optimizing stepsize for SGD convergence.
method Coupling-based convergence diagnostic for monitoring stationarity.
result Proposed stepsize scheme achieves superior performance across convex and non-convex problems.
The paper studies geometric constants under modified Ricci flows with variable parameters.
problem Understanding geometric constants under variable coupling parameters in Ricci flows.
method Introduced modified Ricci flows with variable coefficients, derived evolution formulas, and proved monotonicity conditions.
result Conditions for maintaining monotonicity of geometric constants under modified Ricci flows.
Paper solves Serrin problem for ring-shaped domains, showing velocity has finitely many maxima.
problem Characterizing rotationally symmetric solutions to a specific PDE on a ring-shaped domain.
method Introduced new arguments in the spirit of comparison geometry to overcome the lack of monotonicity.
result Simplest conditions are not sufficient; rotational symmetry requires finitely many maxima.
Formula for α-Futaki character on toric manifolds.
problem Obstruction to Kähler-Yang-Mills equations existence.
method Provided a formula and computed values for α-Futaki character.
result No solutions with α>0 on certain ample line bundles over toric manifolds.
On any given compact (n+1)-manifold M with non-empty boundary, it is proved that the moduli space of Einstein metrics on M is a smooth, infinite dimensional Banach manifold under a mild condition on the fundamental group. Thus, the Einstein moduli space is unobstructed. The Dirichlet and Neumann boundary maps to data o…
Let (M,g,J) be a compact Hermitian manifold with a smooth boundary. Let Δp and Dp be the realizations of the real and complex Laplacians on p forms with either Dirichlet or Neumann boundary conditions. We generalize previous results in the closed setting to show that (M,g,J) is Kaehler if and only if $Spec(Δ_p)=S…
We review the fundamentals of coupling constant metamorphosis (CCM) and the Stäckel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the…
Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.
problem Understanding Stäckel equivalence in superintegrable systems.
method Using invariant quadrics to determine Stäckel classes of superintegrable systems.
result Stäckel classes of superintegrable systems can be derived from associated invariant quadrics.
The paper calculates Morse indices and nullities for embedded networks on spheres.
problem Computing Morse indices and nullities for embedded networks on spheres.
method Using the Dirichlet-to-Neumann map and properties of eigenvalues and eigenfunctions.
result For all stationary triple junction networks in S2, there is only one eigenvalue -1. Analyzes solutions to non-elliptic equations on bounded domains.
problem Analyzes solutions to non-elliptic equations on bounded domains.
method Analyzes solutions to non-elliptic equations on bounded domains.
result Analyzes solutions to non-elliptic equations on bounded domains.