Abstract:
The paper adopts a geometric approach to stabilization and tracking of a spherical robot actuated by three internal rotors mounted on three mutually orthogonal axes inside the robot. The system is underactuated and subject to nonholonomic constraints. Initially, the equations of motion are derived through Euler-Poincar\'{e} reduction. Then two feedback control laws are synthesized: the first control law addresses orientation of the robot alone, keeping the contact point arbitrary; the second control law is a tracking law that addresses the contact position and angular velocity tracking.

Abstract:
We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the $L^1-$estimate between the entropy solution and the geometric optics expansion function is bounded by $O(\varepsilon^2)$, {\it independent of} the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.

Abstract:
Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly non-uniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph, and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specifed degree, and number of components. We also obtain weak laws for functionals of marked point processes, including statistics of Boolean models.

Abstract:
This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of $\mathbb{R}^3$. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.

Abstract:
In classical control theory, tracking refers to the ability to perform measurements and feedback on a classical system in order to enforce some desired dynamics. In this paper we investigate a simple version of quantum tracking, namely, we look at how to optimally transform the state of a single qubit into a given target state, when the system can be prepared in two different ways, and the target state depends on the choice of preparation. We propose a tracking strategy that is proved to be optimal for any input and target states. Applications in the context of state discrimination, state purification, state stabilization and state-dependent quantum cloning are presented, where existing optimality results are recovered and extended.

Abstract:
We address the Riemann and Cauchy problems for systems of $n$ conservation laws in $m$ unknowns which are subject to $m-n$ constraints ($m \geq n$). These constrained systems generalize the system of conservation laws in standard form to a level sufficient to include various examples of conservation laws in Physics and Engineering beyond gas dynamics, e.g., some multi-phase flows in porous media. We prove local well-posedness of the Riemann problem and global existence of the Cauchy problem for initial data with sufficiently small total variation, in one spatial dimension. The key to our existence theory is to generalize the $m\times n$ systems of constrained conservation laws to $n\times n$ systems of conservation laws with states taking values in an $n$-dimensional manifold and to extend Lax's theory for local existence as well as Glimm's random choice method to our geometric framework. Our main objective lies in the applicability of this geometric framework.

Abstract:
In the semiclassical domain the exponent of vortex quantum tunneling is dominated by a volume which is associated with the path the vortex line traces out during its escape from the metastable well. We explicitly show the influence of geometrical quantities on this volume by describing point vortex motion in the presence of an ellipse. It is argued that for the semiclassical description to hold the introduction of an additional geometric constraint, the distance of closest approach, is required. This constraint implies that the semiclassical description of vortex nucleation by tunneling at a boundary is in general not possible. Geometry dependence of the tunneling volume provides a means to verify experimental observation of vortex quantum tunneling in the superfluid Helium II.

Abstract:
In [11] and [5], an error estimate of optimal convergence rates and optimal error propagation (optimal^2) was given for the Runge-Kutta discontinuous Galerkin (RKDG) method solving the scalar nonlinear conservation laws in the case of smooth solutions. This manuscript generalizes the problem to the case of a piecewise smooth solution containing one fully developed shock. A front tracking technique is incorporated in the RKDG scheme to produce a numerical solution with a truly high order error. The numerical smoothness approach of [11] is generalized to this particular case of a discontinuous solution.

Abstract:
Region covariance descriptor recently proposed has been approved robust and elegant to describe a region of interest, which has been applied to visual tracking. We develop a geometric method for visual tracking, in which region covariance is used to model objects appearance; then tracking is led by implementing the particle filter with the constraint that the system state lies in a low dimensional manifold: affine Lie group. The sequential Bayesian updating consists of drawing state samples while moving on the manifold geodesics; the region covariance is updated using a novel approach in a Riemannian space. Our main contribution is developing a general particle filtering-based racking algorithm that explicitly take the geometry of affine Lie groups into consideration in deriving the state equation on Lie groups. Theoretic analysis and experimental evaluations demonstrate the promise and effectiveness of the proposed tracking method.

Abstract:
Region covariance descriptor recently proposed has been approved robust and elegant to describe a region of interest, which has been applied to visual tracking. We develop a geometric method for visual tracking, in which region covariance is used to model objects appearance; then tracking is led by implementing the particle filter with the constraint that the system state lies in a low dimensional manifold: affine Lie group. The sequential Bayesian updating consists of drawing state samples while moving on the manifold geodesics; the region covariance is updated using a novel approach in a Riemannian space. Our main contribution is developing a general particle filtering-based racking algorithm that explicitly take the geometry of affine Lie groups into consideration in deriving the state equation on Lie groups. Theoretic analysis and experimental evaluations demonstrate the promise and effectiveness of the proposed tracking method.