arXiv research

Formulates Index III lemma and Rauch III theorem with applications.

This note explores comparison geometry concepts and theorems.

The paper proves inequalities for submanifolds in Riemannian manifolds.

We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2 metrics

The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.

Develops a novel ML smoothing method for incomplete data in state-space models.

In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite dimensional Riemannian geometry such as Rauch and Berger Theorems and the Topogono…

Hot spots conjecture proven for small eigenvalue domains.

The paper establishes eigenvalue inequalities for a specific operator on curved spaces.

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniquene…

In this paper, we define a new capacity which allows us to control the behaviour of the Dirichlet spectrum of a compact Riemannian manifold with boundary, with "small" subsets (which may intersect the boundary) removed. This result generalizes a classical result of Rauch and Taylor ("the crushed ice theorem"). In the s…

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modifie…

We generalize the notion of planar bicycle tracks -- a.k.a. one-trailer systems -- to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the domains that are swept out by any given tractor/tractrix system. These …

Study hot spots on warped product manifolds and infinite cones.

Factor graphs have recently gained increasing attention as a unified framework for representing and constructing algorithms for signal processing, estimation, and control. One capability that does not seem to be well explored within the factor graph tool kit is the ability to handle deterministic nonlinear transformati…

We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_λ$ over geodesics go to zero at the rate of $O((\logλ)^{-1/2})$ if $λ$ are their frequencies. As discussed in \cite{CSPer}, no …

TASC improves synthetic control for time-series data with trends.

The energy in a square membrane $Ω$ subject to constant viscous damping on a subset $ω\subset Ω$ decays exponentially in time as soon as $ω$ satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate $τ(ω)$ of this decay satisfies $τ(ω)= 2 \min(-μ(ω), g(ω))$ (see Lebeau [Math. Phys. Stud. …

Generative dynamic texture models (GDTMs) are widely used for dynamic texture (DT) segmentation in the video sequences. GDTMs represent DTs as a set of linear dynamical systems (LDSs). A major limitation of these models concerns the automatic selection of a proper number of DTs. Dirichlet process mixture (DPM) models w…

New method for robust fixed-point smoothing without state augmentation.

We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a tar…

Reliable 4D aircraft trajectory prediction, whether in a real-time setting or for analysis of counterfactuals, is important to the efficiency of the aviation system. Toward this end, we first propose a highly generalizable efficient tree-based matching algorithm to construct image-like feature maps from high-fidelity m…

In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov'…

State-space smoothing has found many applications in science and engineering. Under linear and Gaussian assumptions, smoothed estimates can be obtained using efficient recursions, for example Rauch-Tung-Striebel and Mayne-Fraser algorithms. Such schemes are equivalent to linear algebraic techniques that minimize a conv…

A new method for state estimation in state-space models using incomplete data.

The expected low market penetration of connected vehicles (CVs) in the near future could be a constraint in estimating traffic flow parameters, such as average travel speed of a roadway segment and average space headway between vehicles from the CV broadcasted data. This estimated traffic flow parameters from low penet…

Proposes a framework to handle missing data in traffic forecasting with sensor blackouts.