This research builds up on recent results in planar kinematic synthesis with contact direction and curvature constraints on the workpiece. The synthesis of spherical serial chains is considered to guide a rigid body, such that it does not violate normal direction and curvature constraints imposed by contact with objects in the environment. These constraints are transformed into conditions on the velocity and acceleration of points in the moving body to obtain synthesis equations which can be solved by algebraic elimination. Trajectory interpolation formulas yield the movement of the chain with the desired contact properties in each of the task positions. Example shows the application of the developed theory to the failure recovery of a robot manipulator.

Real-valued functions have wide applications in various areas within computer graphics. In this work, we examine three representation of shapes using functions. In particular, we study the classical B-spline representation of piece-wise polynomials in the univariate domain. We provide a generalization of B-spline to the bivariate domain using intuition gained from the univariate construction. We also study the popular scheme of representing 3D density distribution using a uniform, rectilinear grid, where we provide a novel contouring scheme that culls occluded inner geometries. Lastly, we examine a ray-based representation for 3D indicator functions called ray-rep, for which we present a novel meshing scheme with multi-material extensions.

<p>In this dissertation we study scalar curvature rigidity phenomena for the upper hemisphere, and subsets thereof. In particular, we are interested in Min-Oo's conjecture that there exist no metrics on the upper hemisphere having scalar curvature greater or equal to that of the standard spherical metric, while satisfying certain natural geometric boundary conditions.</p><p>While the conjecture as originally stated has recently been disproved, there are still many interesting modications to consider. For instance, it has been shown that Min-Oo's rigidity conjecture holds on sufficiently small geodesic balls contained in the upper hemisphere, for metrics sufficiently close to the spherical metric. We show that this local rigidity phenomena can be extended to a larger class of domains in the hemisphere, in particular finding that it holds on larger geodesic balls, and on certain domains other than geodesic balls (which necessarily have more complicated boundary geometry). We discuss a possible method for finding the largest possible domain on which the local rigidity theorem is true, and give a Morse-theoretic interpretation of the problem.</p><p>Another interesting open question is whether or not such a rigidity statement holds for metrics that are not close to the spherical metric. We find that a scalar curvature rigidity theorem can be proved for metrics on sufficiently small geodesic balls in the hemisphere, provided certain additional geometric constraints are satisfied.</p> / Dissertation

Controllability and stabilisability are two fundamental properties of control systems and it is intuitively appealing to conjecture that the former should imply the latter; especially so when the state of a control system is assumed to be known at every time instant. Such an implication can, indeed, be proven for certain types of controllability and stabilisability, and certain classes of control systems. In the present thesis, we consider real analytic control systems of the form $\Sgr:\dot{x}=f(x,u)$, with $x$ in a real analytic manifold and $u$ in a separable metric space, and we show that, under mild technical assumptions, small-time local controllability from an equilibrium $p$ of \Sgr\ implies the existence of a piecewise analytic feedback \Fscr\ that asymptotically stabilises \Sgr\ at $p$. As a corollary to this result, we show that nonlinear control systems with controllable unstable dynamics and stable uncontrollable dynamics are feedback stabilisable, extending, thus, a classical result of linear control theory. Next, we modify the proof of the existence of \Fscr\ to show stabilisability of small-time locally controllable systems in finite time, at the expense of obtaining a closed-loop system that may not be Lyapunov stable. Having established stabilisability in finite time, we proceed to prove a converse-Lyapunov theorem. If \Fscr\ is a piecewise analytic feedback that stabilises a small-time locally controllable system \mbox{$\Sgr:\dot{x}=f(x,u)$} in finite time, then the Lyapunov function we construct has the interesting property of being differentiable along every trajectory of the closed-loop system obtained by ``applying" \Fscr\ to \Sgr. We conclude this thesis with a number of open problems related to the stabilisability of nonlinear control systems, along with a number of examples from the literature that hint at potentially fruitful lines of future research in the area. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-09-24 10:24:22.51

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In contrast with training algorithms such as Baum-Welch, which produce solutions that are a local optimum of the objective function, this thesis describes the attempt to develop a training algorithm which delivers the global optimum Discrete ICdden Markov Model for a given training sequence. A total of four different methods of attack upon the problem are presented. First, after building the necessary analytical tools, the thesis presents a direct, calculus-based assault featuring Matrix Derivatives. Next, the dual analytic approach known as Geometric Programming is examined and then adapted to the task. After that, a hill-climbing formula is developed and applied. These first three methods reveal a number of interesting and useful insights into the problem. However, it is the fourth method which produces an algorithm that is then used for direct comparison vAth the Baum-Welch algorithm: examples of global optima are collected, examined for common features and patterns, and then a rule is induced. The resulting rule is implemented in *C' and tested against a battery of Baum-Welch based programs. In the limited range of tests carried out to date, the models produced by the new algorithm yield optima which have not been surpassed by (and are typically much better than) the Baum-Welch models. However, far more analysis and testing is required and in its current form the algorithm is not fast enough for realistic application.

Thesis (M.S.)--West Virginia University, 2004. / Title from document title page. Document formatted into pages; contains viii, 138 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 69-72).

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2002. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2003. / Includes bibliographical references. Also available on the Internet.