Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve.

LOCAL NEGATIVITY OF SURFACES WITH NON-NEGATIVE KODAIRA DIMENSION AND TRANSVERSAL CONFIGURATIONS OF CURVES

We give a bound on the H-constants of configurations of smooth curves having transversal intersection points only on an algebraic surface of non-negative Kodaira dimension. We also study in detail configurations of lines on smooth complete intersections $X \subset \mathbb{P}_{\mathbb{C}}^{n + 2}$ of multi-degree d = (d1, …, dn), and we provide a sharp and uniform bound on their H-constants, which only depends on d.

Revisiting the Teixeira Singularity Bifurcation Analysis: Application to the Control of Power Converters

For a discontinuous piecewise smooth (DPWS) dynamical system in [Formula: see text], whose state space is divided into two open regions by a plane acting as the switching manifold, there generically appear two lines of quadratic tangency, one for each involved vector field. When these two tangency lines have a transversal intersection, such a point is called a two-fold singularity. If furthermore, both tangencies are of invisible type, then the two-fold point is known as a Teixeira singularity (TS). The Teixeira singularity can undergo an interesting bifurcation, namely when a pseudo-equilibrium point crosses the two-fold singularity, passing from the attractive sliding region to the repulsive sliding region (or vice versa) and, simultaneously, a crossing limit cycle (CLC) arises. After deriving carefully a local canonical form, we revisit the previous works regarding this bifurcation thus correcting some detected misconceptions. Furthermore, we provide by means of a more direct approach the critical coefficients characterizing the bifurcation, also giving computational procedures for them. The achieved results are applied to some illustrative examples, within the realm of discontinuous piecewise linear (DPWL) systems. This family acts like a normal form for the bifurcation, since DPWL systems are able to reproduce all the unfolded dynamics. The study of TS-points in electronic DC-DC Boost power converters under a sliding mode control (SMC) strategy is addressed. Apart from being a relevant application, it allows to show the real usefulness of the analysis done.

On the Openness of Unique Pure-Strategy Nash Equilibrium

This paper investigates whether small perturbations to a game with continuous strategy spaces and unique Nash equilibrium also yields a game with unique equilibrium. The answer is affirmative for games with smooth payoffs, differentiable strict concavity in own actions, and transversal intersection of best response curves. Though intuitive for games with unique interior equilibrium, this result holds even for equilibria at the boundaries of strategy sets.