Curvatures on the surface are important geometric invariants and are widely used in different area of research. Examples include feature recognition, segmentation, or shape analysis. Therefore, it is of interest to develop an effective algorithm to approximate the curvatures accurately. The classical methods to compute these quantities involve the estimation of the normal and some involve the computation of the second derivatives of the 3 coordinate functions under the param- eterization. Error is inevitably introduced because of the inaccurate approximation of the second derivatives and the normal. In this paper, we propose several novel methods to compute curva- tures on the surface using the conformal parameterization. With the conformal parameterization, the conformal factor function $\lambda$ can be defined on the surface. Mean curvature (H) and Gaussian curvatures (K) can then be computed with the conformal Factor ($\lambda$). It involves computing only the derivatives of the function $\lambda$, instead of the 3 coordinate functions and the normal. We also introduce a technique to compute H from K and vice versa, using the parallel surface.

Call admission and routing controls for loss (circuit-switched) networks with semi- Markovian, multi-class call arrivals and general connection durations, were formulated as optimal stochastic control problems in [12, 13]. Each of the resulting so-called (network) hybrid HJB equations corresponds to a collection of coupled first-order partial differential equations for which, when it exists, the continuously differentiable value function is a solution to the associated hybrid HJB equations. In general, the smoothness of the value functions and uniqueness of the solutions to the hybrid HJB equations may not hold. In this paper, viscosity solutions to a general class of hybrid HJB equations are developed and under mild conditions it is shown that the value function is continuous and, further, any continuous value function is the unique viscosity solution to the hybrid HJB equations.

The current literature on the global state feedback stabilization of nonlinear systems modeled by a perturbed chain of nonlinear integrators, particularly those whose linearization about the origin may contain uncontrollable modes, essentially contains two methods: a smooth controller scheme (only under strict assumptions) and a non-smooth one. The most general of these systems could previously only be globally asymptotically stabilized by continuous time-invariant state feedback controller, where this paper shows that now at least C¹ stabilization can be achieved, upon existence, in this more general setting. This new method can be seen as not only a natural unification of the smooth and nonsmooth methods, but also a generalization to construct smoother stabilizers.