arXiv research

Euler's elastica with monotone curvature is uniquely minimal.

Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.

Extends plate problems to differential forms on manifolds.

Proves Payne conjecture for buckling and membrane eigenvalues.

New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.

Smooth compactness theorem for elasticae, except straight segments.

Paper proves rigidity estimates for hyperbolic shells and applies them to \(Γ\)-limit theory.

Analyzes Willmore flow for graphs with boundary data, proving existence and convergence.

Affirm Lord Rayleigh's conjecture on curved spaces for clamped plates.

In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.

We examine the effect of clamping variables for approximate inference in undirected graphical models with pairwise relationships and discrete variables. For any number of variable labels, we demonstrate that clamping and summing approximate sub-partition functions can lead only to a decrease in the partition function e…

In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.

We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -κ^2$ for some $κ\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounde…

Recent work (Pennington et al, 2017) suggests that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning. Motivated by this, we study the distribution of singular values of the Jacobian of the generator in Generative Adversarial Networks (GANs). We find th…

For a bounded domain $Ω$ in a complete Riemannian manifold $M^n$, we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.

This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap betwe…

CLAMP uses neural manifold packing to improve self-supervised learning.

The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.

Sharp bounds derived for eigenvalues on specific geometric spaces.

Study on hyperbolic elastic flow, proving convergence and quantifying singularities.

The Poisson problem consists in finding an immersed surface $Σ\subset\mathbb{R}^m$ minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates originating f…

Paper studies eigenvalues of a specific operator on Riemannian manifolds.

Neural networks are commonly trained to make predictions through learning algorithms. Contrastive Hebbian learning, which is a powerful rule inspired by gradient backpropagation, is based on Hebb's rule and the contrastive divergence algorithm. It operates in two phases, the forward (or free) phase, where the data are …

New method improves model reconstruction using counterfactuals and polytope theory.

Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensiona…

Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.

In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherica…

The paper bridges stochastic control and deep hedging for European call options with transaction costs.

Near-optimal private tests for simple and MLR hypotheses developed under Gaussian differential privacy.

The paper explores inequalities between eigenvalues on Riemannian manifolds.

Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.

Classifies local boundary conditions for Dirac-type operators on manifolds.

Proof of local well-posedness for a specific boundary condition in general relativity.

Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.

New boundary conditions solve Cauchy problem for Dirac operators on spacetimes.

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …

The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonst…

We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the curvature of a the Lorentz boundary manifold.

Novel boundary conditions for Ricci flow to deform compact manifolds.

Proves well-posedness for Einstein equations with specific boundary conditions.

PINN-FEM combines PINNs and FEM for accurate Dirichlet boundary condition enforcement.

Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.

Study constant mean curvature surfaces with integrable boundary conditions.

Probabilistic graphical models are traditionally known for their successes in generative modeling. In this work, we advocate layered graphical models (LGMs) for probabilistic discriminative learning. To this end, we design LGMs in close analogy to neural networks (NNs), that is, they have deep hierarchical structures a…