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Book of Abstracts International Advisory Committee R. Ablamowicz A. Jadczyk I. Porteous USA FranceUK P. Anglés B. Jancewicz J. Ryan France Poland USA W. Baylis J. Keller I. Shestakov Canada Mexico Brazil E. Bayro J. Ławrynowicz F. Sommen Mexico Poland Belgium L. Dabrowski A. Micali G. Sommer Italy France Germany T. Dray Z. Oziewicz W. Sprößig USA Mexico Germany B. Fauser J.M. Parra V. Souˇcek Germany Spain Czech Rep. J. Helmstetter M. Pavšiˇc France Slovenia Local Organizing Committee J. C. Gutiérrez R. da Rocha USP UFABC P. Koshlukov W. A. Rodrigues Jr.∗ UNICAMP UNICAMP R. Mosna F. Toppan UNICAMP CBPF E. C. de Oliveira J. Vaz Jr.∗ UNICAMP UNICAMP ∗Chairman ABSTRACTS Abłamowicz . .8 Batard . .9 Baylis . 10 Bayro ......................................... 11 Brachey . 12 Brackx . 13 Conradt . 15 Czachor . 16 De Melo . 17 De Schepper . 18 Degimerci . 19 Demir . 20 Eriksson . 21 Fioresi . 22 Franssens . 23 Gürlebeck . 24 Helmstetter . 25 Hestenes . 26 Hitzer . 27 Hitzer . 28 Hitzer . 29 Hoefel . 32 Jancewicz . 33 Jardim . 34 Krump . 35 Kuznetsova . 36 Lasenby . 37 Lavor ......................................... 39 Ławrynowicz . 40 Leão.......................................... 41 Limoncu . 42 Loya.......................................... 43 Lundholm . 44 Macías . 45 Marmolejo . 46 Martin . 47 Micali . 49 Mosna . 51 Notte . 52 Pavšiˇc......................................... 53 Perotti . 55 Pinotsis . 56 Reséndiz . 57 Rocha . 58 Rochon . 59 Rodrigues . 60 Santhanam . 61 Schmeikal . 62 Selig.......................................... 63 Smid ......................................... 65 Snygg......................................... 79 Sobczyk . 66 Souza......................................... 67 Sprößig . 68 Staples . 69 Stolfi . 70 Sweetser . 72 Tolksdorf . 73 Toppan . 74 Tremblay . 75 Trovon ........................................ 76 Vergara . 77 Wills.......................................... 78 Computation of Non Commutative Gröbner Bases in Grassmann and Clifford Algebras Rafał Abłamowicz It is well known that tensor algebras, Clifford algebras, and Grassmman and su- per Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincaré-Birkhoff-Witt (or, PBW for short) “monomial" basis. The necessary and sufficient condition for an algebra to have such basis have been established by V. Levandovskyy as the so called “nondegeneracy condition". This has led him to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Gröbner-ready algebras) and their classification. It was Teo Mora who already in the 90’s considered a comprehensive and algorithmic approach to Gröb- ner bases for commutative and non-commutative algebras. It was T. Stokes who 18 years ago introduced Gröbner left bases (GLB) and Gröbner left ideal basis, with the latter solving an ideal membership problem. Thus, a natural question is to first seek Gröbner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levan- dovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Gröbner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise that way. Examples of G-algebras include quasi-commutative polynomial rings, such as, for example, the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided ideal, include Grassmann and Clifford algebras, and other finite dimensional associative algebras. After recalling basic con- cepts behind the theory of commutative Gröbner bases, a review of the Gröbner in non-commutative algebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. 8 nD Images as Clifford Bundles Sections - Application to Segmentation Thomas Batard, Christophe Saint Jean and Michel Berthier We present a new theoretical framework for nD image processing using Clifford algebras. Multidimensionnal images are considered as sections of a trivial Clifford bundle (CT(D), p, D), endowed with a riemannian fiber metric. Due to the triviality, any covariant derivative r on this bundle is the sum of the usual derivative with w, a one-form on D with values in End(CT(D)). We show that varying w and derivating well-chosen sections with respect to r provides all the information needed to perform various kind of segmentation. We present several illustrations of our results, dealing in particular with color (n=3) and color/infrared (n=4) images. As an example, let us mention the problem of detecting homogeneous regions of a given hue with constraints on temperature; the segmentation results from the computation of r(IB), where I is the image section and B is the bivector section coding the given hue. 9 Quantum/Classical Interface: A Classical Geometric Origin of Fermion Spin W. E. Baylis, R. Cabrera and D. Keselica Although intrinsic spin is usually viewed as a purely quantum property with no classical analog, we present evidence here that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical me- chanics that uses spinors. Spinors and projectors arise naturally in the Clifford’s geometric algebra of physical space and not only provide powerful tools for solving problems in classical electrodynamics, but also reproduce a number of quantum re- sults. In particular, many properites of elementary fermions, as spin-1/2 particles, are obtained classically and relate spin, the associated g-factor, its coupling to an external magnetic field, and its magnetic moment to Zitterbewegung and de Broglie waves. Spinors are also amplitudes that can undergo quantum-like interference. The relationship of spin and geometry is further strengthened by the fact that physical space and its geometric algebra can be derived from fermion annihilation and cre- ation operators. The approach resolves Pauli’s argument against treating time as an operator by recognizing phase factors as projected rotation operators. 10 Conformal Geometric Algebra for Robot Physics E. Bayro-Corrochano In this talk we use as a mathematical framework the conformal geometric alge- bra for applications in computer vision, graphics engineering, learning, control and robotics. We will show that this mathematical system keeps our intuitions and in- sight of the geometry of the problem at hand and it helps us to reduce considerably the computational burden of the problems. Surprisingly as opposite to the standard projective geometry, in conformal geometric algebra we can deal simultaneously with incidence algebra operations (meet and join) and conformal transformations represented effectively using spinors (quaternions, dual quaternions, etc). In this regard, surprisingly this framework appears promising for dealing with kinematics, dynamics and projective geometry problems without the need to abandon the math- ematical system (as current approaches). We present some real tasks of perception and action treated in a very elegant and efficient way: sensor-body calibration, 3D reconstruction and robot navigation and visually guided 3D object grasping making use of the directed distance , algebra of incidence and conformal transformations. For a real time probabilistic geometric framework in tracking we use the Motor (dual quaternion) extended Kalman filter and for control problems we reformulate the differential geometry and the Jacobian based control rule for n D. O. F. robot arms using conformal geometric algebra. The authors believe that the framework of geometric algebra can be in general of great advantage for applications in image pro- cessing, stereo vision, range data, laser, omnidirectional and odometry based robotic systems, kinematics and dynamics of robot mechanisms, humanoids and advanced nonlinear control techniques. 11 Algorithms for Computation of Gröbner Bases in Grassmann Algebras Troy Brachey Algorithms for computation of Gröbner bases in Grassmann algebras are pre- sented. The author illustrates his own procedures as part of a Maple package for computation in Grassmann algebras. Examples of computation will be shown to demonstrate effectiveness of algorithms and procedures. Emphasis will be placed on the ideal membership problem. Keywords: Gröbner basis, Grassmann algebra, Clifford algebra, ideal membership, Maple, Gröbner left basis, Gröbner left ideal basis 12 Hermitean Clifford Analysis Fred Brackx Euclidean Clifford analysis is nowadays a well established branch of classical anal- ysis centred around the notion of monogenic functions, in particular null solutions of the rotation invariant Dirac operator. Recently so–called Hermitean Clifford analysis emerged as a refinement of Euclidean Clifford analysis. Hermitean Clifford analysis is based on the introduction of an additional datum, a so–called complex structure, in order to bring the notion of monogenicity closer to complex analysis. A complex structure J on a euclidean space E should be compatible with the Euclidean structure 2 on E, i.e. J 2 SO(E), and J = −1E. It is seen at once that the dimension of E is then forced to be even: m = 2n. The subgroup of SO(E) preserving the complex structure,
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