We will investigate the stability behavior of quadratic maps in higher dimensions. To check stability, we will use infinitesimal V -stability of critical points of the map; since the infinitesimal V -stability of a map at all of its critical points is equivalent to the stability of the map. We will establish the connection between infinitesimal V-stability of solutions to the Algebraic Riccati ...

The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia’s reasoning is resolutely geometric in nature but contains “infinitesimal” elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present difficulties that prevent Newton’s proofs from being mechani...

On a Riemannian manifold, a solution of the Killing equation is an infinitesimal isometry. Since the Killing equation is overdetermined, infinitesimal isometries do not exist in general. The complete prolongation of the Killing equation is a PDE on the bundle of 1-jets of vector fields. Restricted to a curve, it becomes an ODE that generalizes the Jacobi equation. A solution of this ODE is call...

In this paper we analyze symmetries, hidden and commensurability classes of $(\epsilon, d_L)$-twisted knot complements, which are the complements knots that have a sufficiently large number twists in each their twist regions. These can be constructed via long Dehn fillings on fully augmented links complements. We show such no implies there at most two other respective classes. Under mild additi...