Train Bi-Control Problem on Riemannian Setting

This article refers to the optimization of the energy consumption of guided traction rails, such as those used for electric trains (including subway electric units), railcars, locomotives, and trams, in a Riemannian framework. The proposed optimization strategy takes into account the compliance time drive and aims at improving the transport system for given operation conditions. Our study has five targets: (1) improving the optimal control techniques; (2) establishing a strategy for the operating conditions of the vehicle; (3) formulating and solving additional problems of optimal movement; (4) improving automatic systems for vehicle traction to optimize energy consumption in a Riemannian context; (5) formulating and solving a problem of maximizing the profit of the train. Some significant figures and formulas obtained by Maple procedures clarify the problems.

Pseudo-Riemannian geometry encodes information geometry in optimal transport

Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate invariant properties of statistical inference. Their relations and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential-geometric relation between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, which provides a geometric perspective on the fundamental Ma–Trudinger–Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the framework of c-divergence under which divergences are defined by optimal transport maps. As a by-product, we obtain a new information-geometric interpretation of the MTW tensor on the graph of the transport map. This relation sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the logarithmic $$L^{(\alpha )}$$ L ( α ) -divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. In these cases we give a geometric interpretation of the information-geometric curvature in terms of the divergence between a primal-dual pair of geodesics.

Thermodynamic and Kinetic Characterization of Protein Conformational Dynamics within a Riemannian Framework

We have formulated a Riemannian framework for describing the geometry of collective variable spaces of biomolecules within the context of molecular dynamics (MD) simulations. The formalism provides a theoretical framework to develop enhanced sampling techniques, path-finding algorithms, and transition rate estimators consistent with a Riemannian treatment of the collective variable space, where the quantities of interest such as the potential of mean force (PMF) and minimum free energy path (MFEP) remain invariant under coordinate transformation. Specific algorithms within this framework are discussed such as the Riemannian umbrella sampling, the Riemannian string method, and a Riemannian-Bayesian estimator of free energy and diffusion constant, which can be used to estimate the transition rate along an MFEP.

Structural connectivity analysis using Finsler geometry

In this work we demonstrate how Finsler geometry—and specifically the related geodesic tractography— can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated con-nectivity measures, based on closely related works on the Riemannian framework for diffusion MRI. Using data from the Human Connectome Project, as well as population data from an autism spectrum disorder study, we demonstrate that this new Finsler metric, together with the new connectivity measures, results in connectivity maps that are much closer to known tract anatomy compared to previous geodesic connectivity methods. Our implementation can be used to compute geodesic distance and connectivity maps for segmented areas, and is publicly available.