We examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M . In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using techniques from analytic number theory, we address the following problems: Is the commensurability class of...

In Euclidean space, the geodesics on a surface of revolution can be characterized by means of Clairaut’s theorem, which essentially says that the geodesics are curves of fixed angular momentum. A similar result is known for three dimensional Minkowski space for timelike geodesics on surfaces of revolution about the time axis. Here, we extend this result to consider generalizations of surfaces o...

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid Q that are cut out by algebraic surfaces in R. Such geodesics are either connected components of spatial elliptic curves or of rational curves. Our approach is based on elements of the Weierstrass–Poncaré reduction theory for hyperelliptic tangential covers of elliptic curves, the addition...

A geodesic is a parameterized curve on a Riemannian manifold governed by a second order partial differential equation. Geodesics are notoriously unstable: small perturbations of the underlying manifold may lead to dramatic changes of the course of a geodesic. Such instability makes it difficult to use geodesics in many applications, in particular in the world of discrete geometry. In this paper...

Let X be a compact Riemannian manifold with conic singularities, i.e. a Riemannian manifold whose metric has a conic degeneracy at the boundary. Let ∆ be the Friedrichs extension of the Laplace-Beltrami operator on X. There are two natural ways to define geodesics passing through the boundary: as “diffractive” geodesics which may emanate from ∂X in any direction, or as “geometric” geodesics whi...

We study languages of geodesics in lamplighter groups and Thompson’s group F . We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group ...

The condition metric in spaces of polynomial systems has been introduced and studied in a series of papers by Beltrán, Dedieu, Malajovich and Shub. The interest of this metric comes from the fact that the associated geodesics avoid ill-conditioned problems and are a useful tool to improve classical complexity bounds for Bézout’s theorem. The linear case is examined here: Using nonsmooth nonconv...

3 Example: Differentiating the Lotka-Volterra ODE system 2 3.1 The Lotka-Volterra ODE system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 The derivatives of the Lotka-Volterra ODE system . . . . . . . . . . . . . . . . . . 3 3.3 Integration of the original and augmented Lotka-Volterra ODE system with the Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Integrating various suppliers to satisfy market demand is of great importance for e ective supply chain management. In this paper, we consider the ODE-PDE model of supply chain and apply a classical explicit fourth-order Runge-Kutta scheme for the related ODE model of suppliers. Also, the convergence of the proposed method is proved. Finally a numerical example is studied to demonstrate the acc...

Modeling dynamical systems with ordinary differential equations implies a mechanistic view of the process underlying the dynamics. However in many cases, this knowledge is not available. To overcome this issue, we introduce a general framework for nonparametric ODE models using penalized regression in Reproducing Kernel Hilbert Spaces (RKHS) based on operator-valued kernels. Moreover, we extend...