Related papers: Computing the Maslov index for large systems

Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points.…

We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges super-exponentially fast, then it must coincide with the cylinder itself. We also show that the result is sharp with…

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive…

In this note the interrelations between several natural morphisms on the $\pi_1$ of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (non-linear) Maslov index of loops of quantomorphisms of…

The Maslov index is used to compute the spectra of different boundary value problems for Schr\"{o}dinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold $M$ divided into components…

Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing…

We consider the stability of nonlinear traveling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard…

In this paper, we revisit our paper "Matrix Riccati Equations, Kaluza-Klein, Finsler Spaces, and Mapping Among Manifolds"[1]. We will build mapping among generalized quadratic Hamiltonians and we construct Calabi's Riemmannian Line Elements…

We construct a four-parameter family of Markov processes on infinite Gelfand-Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinite-dimensional boundary…

The Steklov spectrum of a smooth compact Riemannian manifold $(M,g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map) This article is devoted to the Steklov spectral inverse…

We give an overview of the recursive characterisations of random matrix ensembles that are currently at the forefront of random matrix theory by way of studying two classes of ensembles using two different types of recursive schemes:…

We obtain geometric insight into the stability of traveling pulses for reaction-diffusion equations with skew-gradient structure. For such systems, a Maslov index of the traveling wave can be defined and related to the eigenvalue equation…

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…

The isometric embedding problem for Riemannian manifolds, which connects intrinsic and extrinsic geometry, is a central question in differential geometry with deep theoretical significance and wide-ranging applications. Despite extensive…

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain $R$. By exploiting a connection with symmetric determinantal representations, we characterize the image of the…

Correlation matrices are fundamental summaries of functional brain networks, yet standard analyses often treat entries independently, ignoring the curved geometry of correlation space. Existing geometric methods frequently lack closed-form…

This paper is the second in a series of studies on developing efficient artificial intelligence-based approaches to pathfinding on extremely large graphs (e.g. $10^{70}$ nodes) with a focus on Cayley graphs and mathematical applications.…

We give a unified method for the general equivalence problem of extrinsic geometry, on the basis of our formulation of a general extrinsic geometry as that of an osculating map $\varphi\colon (M,\mathfrak f) \to L/L^0 \subset…

We study the dynamics near the unstable Lagrangian points in galactic bar models using dynamical system tools in order to determine the global morphology of a barred galaxy. We aim at the case of non-autonomous models, in particular with…

This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks,…