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Publication Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups(2013-11-09) Gallier, Jean HOne of the main goals of these notes is to explain how rotations in Rn are induced by the action of a certain group, Spin(n), on Rn, in a way that generalizes the action of the unit complex numbers, U(1), on R2, and the action of the unit quaternions, SU(2), on R3 (i.e., the action is denied in terms of multiplication in a larger algebra containing both the group Spin(n) and R(n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra, Cln, the Clifford algebra associated with Rn. Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Cliord algebra Clp;q associated with a nondegenerate symmetric bilinear form of signature (p; q) and culminating in the beautiful \8-periodicity theorem" of Elie Cartan and Raoul Bott (with proofs).Publication Modeling and Analysis of Multi-hop Control Networks(2009-04-13) Rajeev, Alur; Pappas, George James; D'Innocenzo, Alessandro; Weiss, Gera; Johansson, Karl HWe propose a mathematical framework, inspired by the Wireless HART specification, for modeling and analyzing multi-hop communication networks. The framework is designed for systems consisting of multiple control loops closed over a multi-hop communication network. We separate control, topology, routing, and scheduling and propose formal syntax and semantics for the dynamics of the composed system. The main technical contribution of the paper is an explicit translation of multi-hop control networks to switched systems. We describe a Mathematica notebook that automates the translation of multihop control networks to switched systems, and use this tool to show how techniques for analysis of switched systems can be used to address control and networking co-design challenges.Publication Surface Representations Using Spherical Harmonics and Gabor Wavelets on the Sphere(2001-01-01) Bülow, Thomas; Daniilidis, KostasIn this paper we present a new scheme for the representation of object surfaces. The purpose is to model a surface efficiently in a coarse to fine hierarchy. Our scheme is based on the combination of spherical harmonic functions and wavelet networks on the sphere. The coefficients can be estimated from scattered data sampled from a star-shaped object’s surface. Spherical harmonic functions are used to model the coarse structure of the surface, while spherical Gabor wavelets are used for the representation of fine scale detail. Theoretical background on wavelets on the sphere is provided as well as a discussion of implementation issues concerning convolutions on the sphere. Results are presented which show the efficiency of the proposed representation.Publication On the Optimal Assignment of Conference Papers to Reviewers(2008-01-01) Taylor, Camillo JPublication Construction of C∞ Surfaces From Triangular Meshes Using Parametric Pseudo-Manifolds(2008-04-22) Siqueira, Marcelo; Xu, Dianna; Gallier, Jean HWe present a new constructive solution for the problem of fitting a smooth surface to a given triangle mesh. Our construction is based on the manifold-based approach pioneered by Grimm and Hughes. The key idea behind this approach is to define a surface by overlapping surface patches via a gluing process, as opposed to stitching them together along their common boundary curves. The manifold based approach has proved to be well-suited to fit with relative ease, Ck-continuous parametric surfaces to triangle and quadrilateral meshes, for any arbitrary finite k or even k = ∞. Smooth surfaces generated by the manifold-based approach share some of the most important properties of splines surfaces, such as local shape control and fixed-sized local support for basis functions. In addition, the differential structure of a manifold provides us with a natural setting for solving equations on the surface boundary of 3D shapes. Our new manifold-based solution possesses most of the best features of previous constructions. In particular, our construction is simple, compact, powerful, and flexible in ways of defining the geometry of the resulting surface. Unlike some of the most recent manifold-based solutions, ours has been devised to work with triangle meshes. These meshes are far more popular than any other kind of mesh encountered in computer graphics and geometry processing applications. We also provide a mathematically sound theoretical framework to undergird our solution. This theoretical framework slightly improves upon the one given by Grimm and Hughes, which was used by most manifold-based constructions introduced before.Publication Logarithms and Square Roots of Real Matrices(2008-05-02) Gallier, Jean HThe need for computing logarithms or square roots of real matrices arises in a number of applied problems. A significant class of problems comes from medical imaging. One of these problems is to interpolate and to perform statistics on data represented by certain kinds of matrices (such as symmetric positive definite matrices in DTI). Another important and difficult problem is the registration of medical images. For both of these problems, the ability to compute logarithms of real matrices turns out to be crucial. However, not all real matrices have a real logarithm and thus, it is important to have sufficient conditions for the existence (and possibly the uniqueness) of a real logarithm for a real matrix. Such conditions (involving the eigenvalues of a matrix) are known, both for the logarithm and the square root. As far as I know, with the exception of Higham's recent book [18], proofs of the results involving these conditions are scattered in the literature and it is not easy to locate them. Moreover, Higham's excellent book assumes a certain level of background in linear algebra that readers interested in applications to medical imaging may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham [18]. In this paper, I present a unified exposition of these results, including a proof of the existence of the Real Jordan Form, and give more direct proofs of some of these results using the Real Jordan Form.Publication Fast Parallel Deterministic and Randomized Algorithms for Model Checking(1993) Lee, Insup; Rajasekaran, SanguthevarModel checking is a powerful technique for verification of concurrent systems. One of the potential problems with this technique is state space explosion. There are two ways in which one could cope with state explosion: reducing the search space and searching less space. Most of the existing algorithms are based on the first approach. One of the successful approach for reducing search space uses Binary Decision Diagrams (BDDs) to represent the system. Systems with a large number of states (of the order of 5 x 10") have been thus verified. But there are limitations to this heuristic approach. Even systems of reasonable complexity have many more states. Also, the BDD approach might fail even on some simple systems. In this paper we propose the use of parallelism to extend the applicability of BDDs in model checking. In particular we present very fast algorithms for model checking that employ BDDs. The algorithms presented are much faster than the best known previous algorithms. We also describe searching less space as an attractive approach to model checking. In this paper we demonstrate the power of this approach. We also suggest the use of randomization in the design of model checking algorithms.Publication Multi-Arm Manipulation of Large Objects With Rolling Contacts(1992) Paljug, Eric D.Publication Proving Properties of Typed lambda-Terms Using Realizability, Covers, and Sheaves(1994-12-07) Gallier, JeanThe main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. there has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this framework. We are then able to prove a meta-theorem which shows that if a property of realizers satisfies some simple conditions, then it holds for the semantic interpretations of all terms. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. the above approach is applied to the simply-typed λ-calculus (with types →, ×, +, and ⊥) , and to the second-order (polymorphic λ-calculus (with types → and ∀2), for which it yields a new theorem.Publication Exploration of Unknown Mechanical Assemblies Through Manipulation(1989-11-01) Kumar, R. Vijay; Yun, Xiaoping; Bajcsy, RuzenaIf robots must function in unstructured environments, they must also possess the ability to acquire information and construct appropriate models of the unknown environment. This paper addresses the automatic generation of kinematic models of unknown objects with moveable parts in the environment. If the relative motion between moving parts must be observed and characterized, vision alone cannot suffice. An approach in which manipulation is used with vision for sensing is better suited to the task of determining kinematic properties. In this paper, algorithms for constructing models of unknown mechanical assemblies and characterizing the relative motion are developed. Results of a simulation are described to demonstrate the role of manipulation in such an endeavor.