C. Udriste | Searcharxiv

arXiv2015

Geometric properties of reliability polynomials

Geometric modeling of multivariate reliability polynomials is based on algebraic hypersurfaces, constant level sets, rulings etc. The solved basic problems are: (i) find the reliability polynomial using the Maple and Matlab software environment; (ii) find restrictions of reliability polynomial via equi-reliable components; (iii) how should the reliability components linearly depend on time, so that the reliability of the system be linear in time? The main goal of the paper is to find geometric methods for analysing the reliability of electric systems used inside aircrafts.

math.OC↗

arXiv2000

From flows and metrics to dynamics

Recall that a vector field on an n-dimensional differentiable manifold M is a mapping X defined on M with values in the tangent bundle TM that assigns to each point $x\in M$ a vector X(x) in the tangent space $T_x M$. A vector field may be interpreted alternatively as the right-hand side of an autonomous system of first-order ordinary differential equations, i.e., a flow. Now we show that any flow can be enveloped by a conservative dynamics using a semi-Riemann metric g on M. This kind of dynamics was called {\it geometric dynamics} [7]-[9]. The given vector field, the initial semi-Riemann metric, the Levi-Civita connection, and an associated (1,1)-tensor field are used to build a new geometric structure (e.g., semi-Riemann-Jacobi, semi-Riemann-Jacobi-Lagrange, semi-Finsler-Jacobi, etc) on the manifold M ensuring that all the trajectories of a geometric dynamics are pregeodesics (Lorentz-Udrişte world-force law). Implicitly, we solved a problem rised first by Poincaré: find a suitable geometric structure that converts the trajectories of a given vector field into geodesics.

math.DS↗