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), π, D), endowed with a riemannian fiber metric.

Due to the triviality, any covariant derivative ∇ on this bundle is the sum of the usual derivative with ω, a one-form on D with values in End(CT(D)). We show that varying ω and derivating well-chosen sections with respect to ∇ 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 ∇(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 ∈ 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, it means commuting with J, turns out to be isomorphic with the unitary group U(n).

The complex structure J induces an associated, so–called twisted, Dirac operator ∂J. Hermitean Clifford analysis then focusses on Hermitean monogenic functions, i.e. simultaneous null solutions of both operators ∂ and ∂J, in this way breaking down the rotational invariance of the Dirac operator, reducing it to U(n)-symmetry for the considered system.

This talk is focussed on the justification of the Hermitean Dirac system. First it is shown how the Hermitean Dirac operators originate quite naturally as gener- alized gradients in the sense of Stein and Weiss, when projecting the gradient on 1 U(n)-invariant subspaces generated by the projection operators 2 (1 ± iJ). Next it is shown how under the action of U(n), the space of spinor valued polynomials on R2n decomposes into a sum of irreducible subspaces of R2n =∼ Cn, which is however not multiplicity free. By complementing the U(n)–action by a new, hidden, symmetry commuting with it, the resulting decomposition becomes multiplicity free; this idea is the well-known Howe Dual Pair from representation theory. It is shown that the decompositions obtained exactly correspond to the fundamental Fischer decompo- sitions for Hermitean monogenic functions.

13 The conceptual meaning of Hermitean monogenicity is further unravelled by study- ing possible splittings of the Hermitean Dirac first order system into independent parts without changing the properties of the solutions. In this way connections with holomorphic functions of several complex variables are established. These connec- tions also become apparent when studying the Cauchy integral formulae. In fact the Hermitean Clifford analysis function theory is in full development. Its current state of affairs is presented: Martinelli-Bochner formula, Hilbert transform, Taylor expansion, Bergman kernel,etc.

14 From Projective to Geometric Algebra

Oliver Conradt

An (outer) algebra without an one element is developed from projective geometry. The main characteristics of projective geometry such as the incidence relations, the operations of connecting and intersecting and the complete principle of duality are reflected by this Projective Algebra. Following Arthur Cayley (1821-1895) and the Erlanger Programm by Felix Klein (1849-1925) a generic metric is introduced making Projective to a Geometric Clifford Algebra. Remaks on the history of mathematics, on space and counterspace and its application to geometry and physics complete the talk.

15 Geometric-Algebra Teleportation of Geometric Structures

Marek Czachor

“Cartoon-computation” is a formalism based on geometric algebra coding that allows for a geometric analogue of quantum computation. The geometric product replaces here the tensor product, and entangled states are replaced by multivectors. One does not need quantum mechanics and yet all the quantum algorithms can be reformulated in this language. In particular, since teleportation protocols can be formulated in terms of networks of elementary gates and all the quantum gates gave geometric-algebra analogues, it follows that teleportation can be formulated in purely geometric terms. I will show on explicit examples how it works.

[1] D. Aerts, M. Czachor, “Cartoon computation: Quantum-like algorithms without quantum me- chanics”, J. Phys. A: Math. Theor. 40, F259-F266 (2007), Fast Track Communication, quant-ph/0611279.

[2] M. Czachor, “Elementary gates for cartoon computation”, J. Phys. A: Math. Theor. 40, F753-F759 (2007), Fast Track Communication, arXiv:0706.0967 [quant-ph].

[3] D. Aerts, M. Czachor, “Tensor-product vs. geometric-product coding”, Phys. Rev. A 77, 012316 (2008), arXiv:0709.1268 [quant-ph].

16 Variational Formulation for Quaternionic Quantum Mechanics

C. A. M. de Melo and B. M. Pimentel

A quaternionic version of Quantum Mechanics is achieved using the Schwinger’s formulation based on measurements and a Variational Principle. Com- mutation relations and evolution equations are provided, and the results are com- pared with other formulations.

17 The Hilbert–Dirac Operator on R2n in Hermitean Clifford Analysis

Hennie De Schepper

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining the standard Euclidean case. It focusses on the simultaneous null solutions, called Her- mitean monogenic functions, of two complex Dirac operators which are invariant under the action of the unitary group. The specificity of the framework, introduced by means of a complex structure creating a Hermitean space, forces the underlying vector space to be even dimensional.

In engineering sciences, and in particular in signal analysis, the Hilbert transform of a real signal u(t) of a one-dimensional time variable t has become a fundamental tool. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the variables separately. As opposed to these tensorial approaches, Clifford analysis is particularly suited for a treatment of multidimensional phenomena, encompassing all dimensions at the same time as an intrinsic feature.

In this contribution, we devote ourselves to the introduction of a Hilbert transform on R2n in the Hermitean setting. Due to the forced even dimension of all vector spaces involved, any Hilbert convolution kernel in R2n should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in R2n+2. We show that the difficulties posed by this inevitable dimensional jump can be overcome by following a matrix approach. The resulting matrix Hermitean Hilbert transform also gives rise, through composition with the matrix Dirac operator, to a Hermitean Hilbert-Dirac convolution operator “factorizing” the Laplacian and being closely related to Riesz potentials.

18 Spinors on 2n-dimensional Manifolds with Structure Group SO(n,C)

N. Degirmenci and ¸S.Karapazar

It is known that spinors are important geometric objects on manifolds as tensors. In this work we construct a new kind of spinors on a 2n-dimensional manifold M with structure group SO(n,C). The complex spin group Spin(n,C) is the universal covering group of the complex orthogonal group SO(n,C). We use the spinor repre- sentation of Spin(n,C) for the construction of the spinors on M. Then we define their covariant derivative and study some properties of them. Lastly we consider Dirac operator on such spinors.

19 Hyperbolic Quaternion Formulation of Electromagnetism

Suleyman Demir, Murat Tanı¸slıand Nuray Candemir

Many papers in literature have been demonstrated that hypercomplex number systems with nonreal square root +1 have a wide potential to investigate the physi- cal theories in different areas. Hyperbolic quaternions are one of the non-associative hyperbolic number systems that are very suitable for the investigation of space-time theories. Unfortunately, this system is 4-dimensional. By using the same idea on the construction of the complex quaternions, we combined two hyperbolic quaternion to express up to 8-dimensional physical quantities. Hyperbolic quaternion formu- lation of electromagnetism was absent in literature. Therefore, this work fills a gap and contains useful results. Maxwell’s equations and relevant field equations are investigated with hyperbolic quaternions, and these equations have been given in compact, simpler and elegant forms. Derived equations are compared with their vectorial, complex quaternionic, dual quaternionic and octonionic representations, as well.

20 Hyperbolic Function Theory

Sirkka-Liisa Eriksson

The aim of this talk is to consider the hyperbolic version of the standard Clifford analysis. The need for such a modification arises when one wants to make sure that the power function xm is included. H. Leutwiler noticed in 1990 that the power function is the conjugate gradient of a harmonic function, defined with respect to the hyperbolic metric of the upper half space. The theory was extended to the total Clifford algebra valued functions called hypermonogenic in 2000 by H. Leutwiler and S.-L. Eriksson. The integral formula in the upper half space was proved in 2004. We give a new formulation of the integral theorem, where the kernel functions are hypermonogenic. We consider also the power series presentations and related results.

21 Supersymmetry and Supergeometry

Rita Fioresi

Supersymmetry has been the driving force to develop supergeometry: how can we express symmetries which go beyond the ordinary groups? In supergeometry the underlying topological space of a supermanifold or a supervariety are only part of the story, we also need a supersheaf to describe the geometric objects.The functor of points approach to supergeometry brings back the geometric intuition and recov- ers all the ordinary constructions in the much richer setting of supergeometry.As examples of this phylosophy we will describe the construction of tangent spaces to a supermanifold and to a Lie supergroup, the quotient of Lie supergroups and the global Frobenious theorem.

22 Clifford Analysis Solution of the Electromagnetic Boundary Value Problem in a Gravitational Background Vacuum

Ghislain R. Franssens

We formulate and solve the boundary value problem for electromagnetic radia- tion in a vacuum with given arbitrary gravitational background, in terms of Clifford

Analysis based on the algebra Cl1,3 (R) over a pseudo-Riemannian manifold (M, g) with signature (1, 3). It is found that the general solution for the full electromag- netic field can be obtained by analytical means, once a fundamental solution of the Laplace-de Rham scalar wave equation is known. As a by product of our method, we obtain the generalization of the Sommerfeld radiation condition in a gravitational background vacuum.

The considered problem is also instructional for fine tuning Clifford Analysis over manifolds, so that it becomes better suited to solve a larger set of physics prob- lems. In particular, we show the naturalness of imposing a Clifford structure on the cotangent bundle, as opposed to the tangent bundle.

23 On Some Mapping Properties of Monogenic Functions

K. Gürlebeck and J. Morais

The aim of this contribution is to study if monogenic functions can be defined by their geometric mapping properties. At first monogenic functions are considered as general quasi-conformal mappings. Dilatations and distortions of these mappings are estimated in terms of the hypercomplex derivative. This includes the description of the interplay between the Jacobian determinant and the hypercomplex deriva- tive of such monogenic functions. It will be shown that both concepts can be used to characterize quasi-conformal monogenic functions. Pointwise estimates from be- low and from above are given by using a generalized Bohr theorem and a Borel- Caratheodory theorem for monogenic functions. Then it will be studied if a subclass of quasi-conformal mappings exists that can be used for a “geometric” definition of monogenic functions via their mapping properties. Main goal is to find out if all functions from this subclass are monogenic and if monogenic functions must be in this subclass of quasi-conformal mappings. Finally it will be checked if these functions belong to some recently studied weighted spaces of monogenic functions.

Keywords: monogenic functions, quasi-conformal mappings, geometric mapping properties

24 Lipschitz Groups in Meson Algebras

Jacques Helmstetter

Let B(M, f ) be the meson algebra associated with the symmetric bilinear form M × M → K over a field K of characteristic 6= 2.

Jacobson discovered the algebra morphism D from B(M, f ) into the non-twisted tensor product C`(M, f ) ⊗ C`(M, f ) that maps every a ∈ M to (a ⊗ 1 + 1 ⊗ a)/2 ; from the knowledge of the graded structure of B(M, f ), a much easier and shorter proof of the injectiveness of D can be derived. Its injectiveness proves that the even subalgebra B0(M, f ) is provided with a parity subgrading B0,0(M, f ) ⊕ B0,1(M, f ) .

We still assume f to be nondegenerate. Every reflection r in (M, f ) is determined by a non-isotropic vector d ∈ M, and for all a ∈ M,

2 f (a, d) d 2d2 r(a) = a − = −zaz−1 if z = − 1 ∈ B (M, f ) . f (d, d) f (d, d) 0,1

The multiplicative group G generated by all these factors z is isomorphic to the orthogonal group GO(M, f ) by an isomorphism that maps every x ∈ G to the trans- −1 −1 formation a 7−→ xax if x ∈ B0,0(M, f ), or a 7−→ −xax if x ∈ B0,1(M, f ).

Two applications follow. First we can define mesonic Lipschitz groups and mesonic Lipschitz monoids that satisfy the usual properties. Secondly we can prove that Duf- fin’s wave equation for meson particles is invariant by the action of GO(M, f ).

25 The Geometry of the Electron Clock

David Hestenes

In his seminal contribution to quantum mechanics, Louis de Broglie conjectured that the electron has an internal clock oscillating with frequency (B = mc2/h). This idea was soon forgotten with the invention of wave mechanics, wherein the de Broglie frequency (B is interpreted exclusively as the frequency of a wave. Re- cent confluence of a new electron theory with experimental evidence suggests that de Broglie may well have been right in the first place. This lecture explains how that came about. Geometric Algebra played a crucial role in creating a new theory wherein the electron is the seat of an electric dipole that oscillates with an ultrahigh frequency called zitter that is close to twice the de Broglie frequency. Direct detec- tion of the zitter appears to be possible by channeling electrons along crystal axes. Theory predicts a resonant interaction between zitter and crystal periodicity at about 80 MeV/c. An exploratory experiment has reported a positive result. If the existence of zitter is confirmed, it will have major implications for quantum mechanics. In particular, it requires a subtle modification of the Dirac equation with a surprising connection to electroweak theory.

26 Geometric Roots of −1

Eckhard Hitzer

It is known that Clifford geometric algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Re- search has been done on the biquaternion roots of -1 (Sangwine, 2006), abandoning the restriction to blades. Biquaternions are isomorphic to Cl(3,0). All these roots of -1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations.

We now extend this research to general algebras Cl(p,q). We will fully derive the geometric roots of -1 for the Clifford geometric algebras with p+q<=3, and explain the resulting solution manifolds.

27 The Interactive 3D Space Group Visualizer Based on Clifford Geometric Algebra Description of Space Groups

Eckhard Hitzer and C. Perwass

A new interactive software tool is demonstrated, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) is a script for the open source visual CLUCalc, which fully supports geometric algebra computation.

Selected generators (Hestenes & Holt, JMP, 2007) form a multivector generator basis of each space group. The approach corresponds to an algebraic implemen- tation of groups generated by reflections (Coxeter and Moser, 4th ed., 1980). The basic operation is the reflection. Two reflections at non-parallel planes yield a rota- tion, two reflections at parallel planes a translation, etc. Combination of reflections corresponds to the geometric product of vectors describing the individual reflection planes.

In our presentation we will first give some insights into the Clifford geometric algebra description of space groups. The symmetry generation data are stored in an XML file, which is read by a special CLUScript in order to generate the visualization. Then we will use the Space Group Visualizer to demonstrate space group selection and give a short interactive computer graphics presentation on how reflections com- bine to generate all 230 three-dimensional space groups.

28 Geometric Algebra Feature Extraction and Classification

Minh Tuan Pham, Kanta Tachibana, Eckhard Hitzer, Sven Buchholz, Tomohiro Yoshikawa and Takeshi Furuhashi

This research proposes to use geometric algebra to systematically extract geomet- ric features from data given in a vector space. We show the results of classification of hand-written digits, which were classified by feature extraction with the proposed n method. Given a set of spatial vectors ξ = {pl ∈ R , l = 1, . . . , m} we extract k- vectors of different grades k; which encode the variations of the features.

Figure 0.1: Examples of the handwritten digit ‘1’, shown with straight line segments, rescaled to square, different from real pen curves.

Assuming ξ is a series of n-dimensional vectors, n0 + 1 feature extractions are de- rived where n0 = min{n, m}. For k = 1, . . . , n0, we write k0 = k − 1,

0 m−1 f0 (ξ) = {hplpl+1i, l = 1, . . . , n − 1} ∈ R , 0 −1 0 0 (m−k )|Ik| fk (ξ) = {hpl ... pl+k0 eI i, I ∈ Ik, l = 1, . . . , n − k } ∈ R .

29 Figure 0.2: Correct classification rate with f1 and mixture of experts.

Figure 0.3: Flow of multi-class classification. The top diagram shows the training of

the GMM for class C ∈ {‘0’, . . . , ‘9’}. The D1C denotes a subset of training samples whose label is C. The f : ξ 7→ x shows feature extraction. Either

of { f1, f2, f0} is chosen as f . The bottom diagram shows estimation by the learned GMMs. The same f chosen for training is used here. The GMMC ∗ outputs p (ξ | C). The final estimation is C = arg maxC p (ξ | C) P (C),

where P (C) is the prior distribution. The set D3 consists of independent test data.

30 Figure 0.4: Mixture of GMMs. Three GMMs via different feature extractions are mixed to yield output p (ξ | C).

|Ik| is the cardinality of the set Ik of all combinations of k elements from n elements. = −1 = For I i1 ... ik, eI eik ... ei2 ei1 .

We performed the feature extractions fk, k ∈ {0, 1, 2} for hand-written digit data of the UCI Repository, see Fig. 0.1. In this Pendigits dataset 7494 samples were written by 30 people, divided into learning data D1, and validation data D2. The 3498 re- maining samples were written by 14 other people and are used as test data D3. The flow of training and estimation of hand-written digit classification, as an example of multi-class classification, is shown in Fig. 0.3. Hyperparameters like the mixture number and the cutoff coefficient for each Gaussian mixture model (GMM) are de- cided by validation with dataset D2. The classification precision for D3 using only coordinate feature extraction f1 decreased remarkably with increasing ε, the random rotation range parameter, Fig. 0.2. On the other hand, the classification precision us- ing mixture of experts (as in Fig. 0.4) did not decrease that much. The rotations had no influence in the cases of f0 and f2, which use inner and outer products, respectively.

Our results confirm that the strategy to mix different GA feature extractions is superior in both classification precision and robustness when compared with pure coordinate value features, which is the most often used conventional method.

Grant-in-Aid for COE program Frontiers of Computational Science (Nagoya University), and for Young Scientists (B) #19700218.

31 On the Cohomology of g-algebras

Eduardo Hoefel

A g-algebra consists a triple (g, A, ρ) where g is a Lie algebra, A is an associative algebra and ρ : g ⊗ A → A is a Lie algebra action by derivations. The cohomology of g-algebras has been introduced by Flato, Gerstenhaber and Voronov in 1995. In this talk I will show how the cohomology of g-algebras can be defined through Koszul operads. The operad of g-algebras is related to Kajiura-Stasheff’s Open- Closed Homotopy Algebra (OCHA) and to Voronov’s swiss-cheese operad

32 The Energy-Momentum Tensor in Premetric Electrodynamics

Bernard Jancewicz

The electromagnetic theory in large part is metric independent. It is called premet- ric electrodynamics. Metric enters the constitutive relation. The energy-momentum is a one-form, the energy-momentum tensor is an energy-momentum three-dimensional density, therefore is has to be a mapping of volume trivectors into one-forms. When F and G are well known two-forms describing the electromagnetic field, V is a volume trivector, the energy momentum tensor is the following linear mapping 1 V → T(V) = 2 [Gb(VbF) − Fb(VbG)].

33 Twisted Dirac Operators on Asymptotically Locally Flat Gravitational Instantons

Marcos Jardim

Asymptotically locally flat (ALF) gravitational instantons are an important class of non-compact manifolds, both from the physical and mathematical point of view. We will discuss instantons over ALF gravitational instantons and the corresponding twisted Dirac operators. In particular, we give a condition for these twisted Dirac operators to be Fredholm and compute their index.

34 Three Dirac Operators in the Stable Rank

Lukas Krump

For many years, there is a constant interest in understanding a structure of a reso- lution starting with the Dirac operator in several Clifford variables. The case of three variables in the stable rank was studied by several methods.

Recently, the Penrose transform method was successfully applied in low dimen- sions and similarly for two variables in higher dimensions. We shall show that the Penrose transform methods can be applied also for three variables in higher dimen- sions, giving comparable results and yielding a perspective way to a general case.

35 Clifford Algebras and Non (anti)commutative Deformations of Supersymmetry

Zhanna Kuznetsova

Supergroups with Grassmann parameters are replaced by odd Clifford parame- ters. The connection with non-anticommutative supersymmetry is discussed. The following topics will be covered: Berezin-like calculus, non (anti)commutative su- persymmetric quantum mechanics, Drinfeld twist deformations.

36 Applications of Geometric Algebra in Cosmology and Physics

Anthony Lasenby

Geometric Algebra (GA) is a powerful tool in many branches of physics and engi- neering. Here we will summarise some selected results obtained in cosmology and physics using a GA approach.

The cosmology topics to be discussed centre on three linked areas: (1) a confor- mal geometric algebra (CGA) approach to Bianchi cosmology, with applications to a novel nonsingular Bianchi IX universe; (2) the use of CGA to give a boundary condition on the total elapsed conformal time in the Universe. This provides an unexpected linkage between the value of the cosmological constant and the number of e-folds of inflation [1]; (3) the role of spinning fluids and torsion in cosmology. In the former category we look at Weyssenhoff fluids[2, 3], and in the latter we examine Elko spinors (as introduced in [4] and discussed further in [5]) from a GA point of view.

The physics topics link to the above via conformal geometric algebra and a back- ground embedding in a scale-free 5-dimensional spacetime that in 4d constitutes de Sitter space. Some aspects of propagators in the curved background space are dis- cussed, and applications made in electromagnetism, quantum mechanics and fluid flow. (A preliminary discussion of applications to electromagnetism was given in [6].) If time permits, some recent applications of CGA in rigid body motion will also be discussed.

[1] A. Lasenby and C. Doran. Closed universes, de Sitter space, and inflation. Phys.Rev.D, 71(6):063502, 2005. astro-ph/0307311.

[2] J. Weyssenhoff and A. Raabe. Relativistic dynamics of spin-fluids and spin-particles. Acta Phys. Pol., 9:7, 1947.

37 [3] S. D. Brechet, M. P. Hobson, and A. N. Lasenby. Weyssenhoff fluid dynamics in general relativity using a 1 + 3 covariant approach. Classical and Quantum Gravity, 24:6329-6348, 2007. arXiv:0706.2367.

[4] D. V. Ahluwalia-Khalilova and D. Grumiller. Spin-half fermions with mass dimension one: the- ory, phenomenology, and dark matter. Journal of Cosmology and Astro-Particle Physics, 7:12, 2005. arXiv:hep-th/0412080.

[5] R. da Rocha and W.A. Rodrigues Jr. Where are ELKO spinor fields in Lounesto spinor field classification? Mod. Phys. Lett A, 21:65-74, 2006. arXiv:math-ph/0506075.

[6] A.N. Lasenby. Recent applications of conformal geometric algebra. In H. Li, P.J. Olver, and G. Sommer, editors, Computer Algebra and Geometric Algebra with Applications (Lecture Notes in Computer Science), page 298. Springer, Berlin, 2005.

38 Clifford Algebra Applied to Grover’s Algorithm

Rafael Alves and Carlile Lavor

Grover’s algorithm is a quantum algorithm for searching in unstructured databases. Due to the properties of quantum mechanics, it provides a quadratic speedup over their classical counterparts. Using Clifford algebra, we present a new way to under- stand and simplify the ideas of Grover’s algorithm.

39 An Approach to Two- and Three-Dimensional Models of Order-Disorder Transition and Simple Orthorhombic Ising Lattices

Julian Ławrynowicz

The famous works by L. Onsager [8] and Z.-D. Zhang [10] give quaternion based two- and three-dimensional models of order-disorder transition and simple orthorhom- bic Ising [2] lattices. The way of applying the quaternion structure can be made more elegant and simple by the use of Clifford structures and the related P.Jordan structures. In particular, four sequences of Jordan algebras can be constructed [3] in relation to R, C, H and O, and the H-sequence appears to be crucial for approaching in four directions: 1) Onsager-Zhang, 2) fractals determining the binary and ternary alloy structures [5, 6, 7], and 3) optimizing the quantum mechanics framework [1, 4]. Besides, the Onsager-Zhang techniques and Clifford-Jordan structures approach in- volve and stimulate a quick development of the Toeplitz forms theory as it had been successfully initiated by G. Szegö [9].

[1] S. I. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford Univ. Press 1966.

[2] E. Ising, Z. Phys. 31 (1925), 253–162.

[3] P.Jordan, J. von Neumann, E. Wigner, Ann. of Math. 35 (1934), 29–64.

[4] J. Ł., Osamu Suzuki, Internat. J. of Theor. Phys. 40 (2001), 387–397.

[5] J. Ł., O. S., Internat. J. of Pure and Appl. Math. 24, no. 2 (2005), 181–209.

[6] J. Ł., S. Marchiafava, S. Nowak-Ke¸pczyk, Internat. J. of Geom. Meth. in Modern Phys. 3 (2006), 1167–1197.

[7] J. Ł., M. N.-K., O. S., Internat. J. of Pure and Appl. Math., to appear.

[8] L. Onsager, Phys. Rev. 65 (1944), 117–149.

[9] G. Szegö, Communications du séminaire mathématique de l’université de Lund, tome supplémentaire dédié à Marcel Riesz (1952), 228–238.

[10] Z.-D. Zhang, Philosophical Magazine 87 (2007), 5309–5419.

40 On the Spectrum of the Twisted Dolbeault Laplacian over Kähler Manifolds

Marcos Jardim and Rafael Leão

We use Dirac operators techniques to improve the estimates for the first eigenval- ues of the Dolbeault Laplacian twisted by a Hermitian-Einstein connection on Kähler manifolds with positive scalar curvature obtained using the Kähler identities for the connection.

Keywords: Twisted Dolbeault Laplacian; Hermitian-Einstein connections; holomor- phic vector bundles.

41 Degenerate Spin Groups as Semi-Direct Products

T. Dereli, ¸S.Koçak and M. Limoncu

Let Q be a symmetric bilinear form on Rn =Rp+q+r with corank r, rank p + q and signature type (p, q), p resp. q denoting positive resp. negative dimensions. We consider the degenerate spin group Spin(Q) = Spin(p, q, r) in the sense of Crumey- rolle and prove that this group is isomorphic to the semi-direct product of the nondegenerate and indefinite spin group Spin(p, q) with the additive matrix group Mat(p + q), r
.

42 Witten’s Holonomy Theorem on Manifolds with Corners

Paul Loya

In the 1980’s Daniel Quillen introduced determinant line bundles and about the same time Edward Witten derived a remarkable formula for the holonomy of the de- terminant line bundle of a Dirac operator using something called the “eta invariant” of Atiyah, Patodi, and Singer. In the physics literature, the holonomy of the determi- nant line bundle is called the “global anomaly". Witten’s derivation was later made rigorous by Bismut and Freed and also by Cheeger.

In this talk I will give an introduction to eta invariants and Witten’s holonomy theorem, and then I will discuss recent work concerning generalizations of this the- orem to situations quite different from the original results. This talk will be suitable for a general audience.

[1] J.-M. Bismut and D.S. Freed, The analysis of elliptic families. Dirac operators, eta invariants, and the holonomy theorem, Comm. Math. Phys.107 (1986), no.1, 103-163.

[2] J. Cheeger, η-invariants, the adiabatic approximation and conical singularities. I. The adiabatic approximation, J. Differential Geom. 26 (1987), no. 1, 175–221.

[3] E. Witten, Global gravitational anomalies, Comm. Math. Phys. 100 (1985), no.2, 197-229.

43 On the Geometry of Supersymmetric Quantum Mechanical Systems

Douglas Lundholm

We consider some simple examples of supersymmetric quantum mechanical sys- tems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.

44 On the Notions of Hyperderivative and n-dimensional Directional Derivatives

M. E. Luna-Elizarrarás, M. A. Macías-Cedeño and M. Shapiro

The aim of this talk is to introduce the notion of n-dimensional directional deriva- tive in the context of Clifford analysis and to establish its relations with the hyper- derivative of a hyperholomorphic function introduced by Gürlebeck and Malonek in [1]. These relations are similar to those existing between the derivative of a holo- morphic function and its one-dimensional directional derivatives, in one complex variable. Thus, there are extended onto the Clifford Analysis case the correspond- ing notions from [2]. All this applies to the problem of the hyperderivability of the Clifford-Cauchy-type integral.

[1] H. Malonek, K. Gürlebeck, A Hypercomplex Derivative of Monogenic Functions in Rn+1 and its Appli- cations. Complex Variables, Vol. 39, pp. 199–228, 1999.

[2] I. Mitelman, M. Shapiro, Differentiation of the Martinelli-Bochner Integrals and the Notion of Hyper- derivability. Math. Nachr. 172, pp. 211–238, 1995.

45 Hardy Spaces, Singular Integrals and the Geometry of Euclidean Domains

Emilio Marmolejo Olea

We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators(SIO’s), such as Riesz transforms as well as Cauchy-Clifford and harmonic double layer operator, on the one hand and, on the other hand the regularity and geometric properties of domains. Among other things, we give several characterizations of Euclidean balls, their complements, and Half-spaces, in terms of the aforementioned SIO’s.

(This is joint work with Steve Hoffman, Marius Mitrea, Salvador Perez-Esteva and Michael Taylor.)

46 Dirac and Semi-Dirac Pairs of Differential Operators

Mircea Martin

n The standard Dirac operator D = Deuc,n on the Euclidean space R , n ≥ 2, is defined as a first-order homogeneous differential operator on Rn with coefficients in n † † the real Clifford algebra An(R) associated with R by assuming that DD = D D = n † ∆euc,n, where ∆euc,n stands for the Laplace operator on R , and D = −D.

As a generalization of this specific class of differential operators we will inves- tigate pairs (D, D†) of first-order homogeneous differential operators on Rn with coefficients in a real unital Banach algebra A, such that either

† † DD = µL∆euc,n, D D = µR∆euc,n, or † † DD + D D = µ∆euc,n,

† where µL, µR, or µ are some elements of A. Every pair (D, D ) that has the for- mer property is called a Dirac pair of differential operators, and each pair (D, D†) with the latter property is called a semi-Dirac pair. The two typical examples of a Dirac, or semi-Dirac pair of differential operators on Rn are given by D = d + d∗ and D† = −(d + d∗), or D = d and D† = −d∗, where d is the operator of exterior differentiation acting on smooth differential forms on Rn, and d∗ is its formal adjoint.

Our main goal is to prove that for any Dirac pair, or semi-Dirac pair, (D, D†), we have two Cauchy-Pompeiu type, and two Bochner-Martinelli-Koppelman type inte- gral representation formulas in several real variables, one for D and, as expected, another for D†, respectively. In addition, we are going to show that the existence of such integral representation formulas characterizes the two classes of pairs of differ- ential operators.

47 As a final comment, we should point out that the concept of a Dirac, or semi-Dirac, pair of differential operators has natural extensions in several complex variables and in the setting of differential operators on a Clifford bundle over an oriented Riemannian manifold.

48 The Graded Structure of Nondegenerate Meson Algebras

Artibano Micali

In 1938 Duffin proposed a wave equation for meson particles which looked like Dirac’s wave equation for electrons:

 1 ∂ ∂ ∂ ∂ imc β − β − β − β + ψ = 0 ; 4 c ∂t 1 ∂x 2 ∂y 3 ∂z h¯ yet the four matrices βj satisfied other relations than Dirac’s relations; if we set 0 0 β4 = β4 and βj = −βj for j = 1, 2, 3, Duffin’s relations are:

0 0 βjβkβl + βl βkβj = δj,kβl + δl,kβj .

Whereas the universal algebra associated with Dirac’s relations is a Clifford algebra, the universal algebras associated with Duffin’s relations and with all analogous re- lations are called Duffin-Kemmer algebras, or shortly meson algebras.

Let M be a vector space of finite dimension n over a field K, and f a symmetric bilinear form M × M → K. The meson algebra B(M, f ) is the associative algebra generated by M (and the unit element 1 ) with the only relations aba = f (a, b) a for all a, b ∈ M. As a consequence of these relations, we also have

abc + cba = f (a, b) c + f (c, b) a .

Like all Clifford algebras, B(M, f ) is provided with a parity grading B(M, f ) =

B0(M, f ) ⊕ B1(M, f ) and with a reversion. The dimensions of B0(M, f ),B1(M, f ) 2n 2n 2n+1 and B(M, f ) are ( n ) , (n−1) and ( n ) .

To get a precise description of B(M, f ) when f is nondegenerate, we define a graded representation B(M, f ) → End(E) with

^ ^ ^ ^ E = (M) ⊕ (M) , E0 = (M) ⊕ 0 and E1 = 0 ⊕ (M) ;

49 with every a ∈ M is associated this odd operator in End(E) :

^ (u, v) 7−→ ( a ∧ v , a y f u ) for all u, v ∈ (M) .

We prove that the image of B(M, f ) in End(E) is the subalgebra of all endomor- phisms of E satisfying these two properties: they leave invariant all Vk(M) ⊕ Vk−1(M) (for k = 0, 1, 2, ..., n + 1), and they commute with the natural action of the graded algebra V 0(M) ⊕ Vn(M) in E.

Some applications to Duffin’s wave equation follow.

50 Latent Symmetries of the Dirac-Kähler Equation and Electroweak Interactions

Ricardo Mosna and Jayme Vaz Jr.

The solutions of the Dirac-Kähler equation in flat spacetime are known to pos- sess a fourfold degeneracy which can be used to define, in a certain sense, four uncoupled Dirac equations on minimal left ideals of the Dirac algebra (complexified Clifford algebra of spacetime). The arbitrariness in choosing this set of ideals gives rise to a global symmetry of the Dirac-Kähler Lagrangian. We gauge this symmetry by considering sets of (minimal left) ideals varying from point to point in space- time. The resulting gauge fields then couple, in an essential way, the different ideals of the algebra. The structure of the interactions imply that the gauge fields only couple states with the same handedness, so that gauge interactions between left- handed and right-handed states are naturally suppressed. Moreover, the formalism automatically gives rise to a term in the Lagrangian corresponding to the associ- ated antiparticles, with the correct handednesses. By restricting the interactions to those conserving electric charge, the resulting model recovers the left-right symmet- ric model of electroweak interactions, provided that we identify the different ideals with leptons (or quarks) of a given generation of particles. When the symmetry is broken, so that the ideals corresponding to the right-handed (left-handed) neutrino (antineutrino) remain fixed, the Glashow-Weinberg-Salam model is recovered. In this context, the Higgs field can be essentially thought of as defining a parametriza- tion of the set of ideals associated with the different particles. We finally consider possible applications of this formalism to lattice field theory.

51 The Square of the Dirac and Spin-Dirac Operators on a Riemann-Cartan Space

Eduardo Notte-Cuello

In this work we introduce the Dirac and spin-Dirac operators associated to a con- nection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac opera- tors associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role in several topics of modern Mathematics. We obtain a generalized Lichnerowicz formula, de- compositions of the Dirac and spin-Dirac operators and their squares in terms of the standard Dirac and spin-Dirac operators and using the fact that spinor fields (sections of a spin-Clifford bundle) have representatives in the Clifford bundle we present also a noticeable relation involving the spin-Dirac and the Dirac operators.

(This work is in conjuntion with W. A. Rodrigues Jr. and Q. A. G. Souza and was partially supported by the DIULS of the La Serena University)

52 On the Unification of Interactions by Clifford Algebra

Matej Pavšiˇc

In current approaches to quantum gravity the starting point is often in assuming that at short distances there exists an underlying structure, based, e.g., on strings and branes, or spin networks and spin foams. It is then expected that the smooth spacetime manifold M4 of classical general relativity will emerge as a sufficiently good approximation at large distances. However, it is feasible to assume that what we have, even at large distances, is in fact not spacetime, but a more general space. One possibility is in assuming that the long distance approximation to a more fun- damental structure is the space of extended events, corresponding to points, lines, areas, 3-volumes, and 4-volumes in M4. All those objects can be elegantly repre- M µ1...µR sented by Clifford numbers x γM ≡ x γµ1...µR , R = 0, 1, 2, 3, 4. This leads to the concept of the so called Clifford space C, a 16-dimensional manifold whose tangent space is Clifford algebra C`(1, 3). We assume that C has in general non vanishing curvature. The connection and vielbein of Clifford space are determined by solu- tions to the generalized Einstein equations, and contain not only the 4-dimensional gravitational field, but also other gauge fields, thus enabling a unification of interac- tions similar to that in Kaluza-Klein theories.

We consider the generalized Dirac equation for the Clifford algebra valued field Ψ(X) that depends on position in C. At every point X ∈ C the field Ψ can be decom- posed into four independent geometric spinors belonging to the left minimal ideals of C`(1, 3). We explore such a system and argue that it is promising for the unifi- cation of the Standard model particles and gauge fields. Usually it is believed that C`(1, 3) is not sufficient, therefore higher dimensional Clifford algebras are consid- ered. But in those approaches the fields depend on position in M4, not in C. Having a 16-dimensional manifold C, we can exploit the possibility that C admits isome- tries. The corresponding conserved charges turn out to have two contributions: one from the ‘orbital’ angular momentum in the ‘internal’ part of C, and the other one

53 from the ‘internal spin’. This brings into the game additional quantum numbers that enlarge the set of basis states for our system.

54 On Directional Quaternionic Hilbert Operators

Alessandro Perotti

The talk discusses harmonic conjugate functions and Hilbert operators in the space of Fueter regular functions of one quaternionic variable. We consider left-regular functions in the kernel of the Cauchy-Riemann operator

∂ ∂ ∂ ∂  ∂ ∂  D = + i + j − k = 2 + j . ∂x0 ∂x1 ∂x2 ∂x3 ∂z¯1 ∂z¯2

2 Let J1, J2 be the complex structures on the cotangent bundle of H ' C induced by left multiplication by i and j, and set J3 = J1 J2. For every complex structure 2 1
Jp = p1 J1 + p2 J2 + p3 J3 (p ∈ S a imaginary unit), let ∂p = 2 d + pJp ◦ d be the Cauchy-Riemann operator w.r.t. the structure Jp.

Let Cp = h1, pi ' C. If Ω satisfies a geometric condition, for every Cp-valued function f1 in a Sobolev space on the boundary ∂Ω, we obtain a function Hp( f1) : ⊥ ∂Ω → Cp , such that f = f1 + Hp( f1) is the trace of a regular function on Ω. The 2 function Hp( f1) is uniquely characterized by L (∂Ω)-orthogonality to the space of CR-functions w.r.t. the structure Jp. In this way we get, for every direction p ∈ S2, a bounded, linear Hilbert operator 2 Hp, with the property that Hp = id − Sp, where Sp is the Szegö projection w.r.t. the structure Jp.

55 Quaternions, Boundary Value Problems and Evaluation of Integrals

Dimitrios Pinotsis

We present two novel applications of the theory of Quaternions: (a) The solution of certain boundary value problems for linear elliptic Partial Differential Equations (PDEs) in four dimensions. (b) The explicit computation of certain three dimensional integrals without integrating with respect to the real variables. Both applications are based on an important formalism in complex analysis, the so called Dbar formalism, and its quaternionic generalizations. The relevant results have been published in [1,2].

[1] D.A. Pinotsis, The Dbar Formalism, Quaternions and Applications, PhD Thesis, University of Cambridge (2006)

[2] A.S. Fokas and D.A. Pinotsis, Quaternions, Evaluation of Integrals and Boundary Value Problems, Computational Methods and Function Theory (to appear)

56 Oscillatory Movements and Dual Quaternions

Rafael Reséndiz

This paper shows how dual quaternions can be a way to describe oscillatory move- ments. Dual quaternions are the algebraic counterpart of screws. This fact enables to get an alternative description of an harmonic oscillatory motion. In this way, we can arrive to model more complicated oscillatory movements for example, it is possible to get the solution of a classic PDE: the Wave Equation. Thus, following this idea we can describe the kinetic behavior (position, velocity and time) by a dual quaternion.

57 Geometric Aspects of ELKO Spinor Fields: Pure Spinors, Supergravity and Flagpoles

Roldão da Rocha

Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong, together with Majorana spinor fields, to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class (5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three cor- responding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. We also investigate and provide the necessary and sufficient conditions to naturally extend the Standard Model to spinor fields possessing mass dimension one. As ELKO is a prime candidate to describe dark matter. Also, we show that the Einstein-Hilbert, the Einstein-Palatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), when the three classes of Dirac spinor fields, under Lounesto spinor field classification, are considered. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. Any other class of spinor field (Weyl, Majorana, flag- pole, or flag-dipole spinor fields) yields a trivial (zero) QSL, up to a boundary term. Finally it is shown how to express ELKO spinor fields uniquely in terms of pure spinor fields.

58 On a Generalized Fatou-Julia Theorem in Multicomplex Spaces

Dominic Rochon

In this talk, we present an overview of the hypercomplex 3D fractals generated from Multicomplex Dynamics. In particular, we give a multicomplex (i.e. bicomplex, tricomplex, etc.) version of the so-called Fatou-Julia theorem. More precisely, we n present a complete topological characterization in R2 of the multicomplex filled- Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. We also present a simple method to explore and infinitely approach these hypercomplex 3D fractals.

59 Killing Vector Fields, Maxwell Equations and Lorentzian Spacetimes

Waldyr A. Rodrigues Jr.

In this talk we first analyze the structure of Maxwell equations in a Lorentzian space when the potential obeys the Lorenz gauge. We show that imposition of the Lorenz gauge can only be done if the spacetime has Killing vector fields, and in this case the potential must be a (dimensional) constant multiple of a the 1-form field physically equivalent to a Killing vector field. Moreover we determine the form of the current associated with this potential showing that it is proportional to the β potential, i.e., given by 2A Rβ, where the Rβ are the Ricci 1-form fields. Finally we study the structure of the spacetime generated by the coupled system consisting of a electromagnetic field F = dA, (with the electromagnetic potential A satisfying the Lorenz gauge) an ideal charged fluid with dynamics described by an action function S and the gravitational field. We show that Einstein equations is then equivalent to Maxwell equations with a current given by f FAF (the product meaning the Clifford product of the corresponding fields), where f is a scalar function which satisfies a well determined algebraic quadratic equation.

60 Generalized Clifford Algebra, Unbiased Quantum States, and Quantum Information

T. S. Santhanam

Weyl’s algebra of unitary operators in ray space is a special case of a General- ized Clifford Algebra. Two unitary operators of Generalized Clifford Algebra are called “complementary” if their eigenvectors (in n-dimensions) satisfy the relation |(e e )| = √1 j k j, k n . independent of and . In other words, the connecting matrix has √1 all elements of modulus n . Such bases are called “mutually unbiased”. These states play a very important role in quantum information and quantum communi- cation. What this means is that for a given input all outputs are equally probable. We will discuss in this talk a method of constructing these mutually unbiased states of quantum mechanics using the representations of Generalized Clifford Algebras.

61 Pregeometry of Extensions and Eigenfields∗

Bernd Schmeikal

In this lecture it is attempted to derive geometric properties of space from those of the fields. This can be done by realizing the meaning of extensions within non- commutative geometry. The mystery lies in a special togetherness of commutativity

and anticommutativity, of 1-norm and 2-norm. We are familar with the Z2 -grading of the Clifford algebra and the double cover of orthogonal groups. With this we

associate the projector equation and decomposition of unity 1 = P1 + P0 according to spin decomposition or chirality. A binary decomposition like that is characteristic for quantumelectrodynamics (qed). It has first been used by John von Neumann and interpreted by von Weizsäcker as logic alternative. Weyl has for some time pondered over the meaning of the Klein-4 group and rays as compared with vec- tors. Then he could not yet realize the importance of a quaternary decomposition of

unity 1 = P0 + P1 + P2 + P3 and the K4-grading. This equation characterizes quan- tumchromodynamics (qcd) in quite general algebras. What a binary decomposition is for qed, the quaternary is for qcd. In this paper the algebraic foundations are given by what is called here a maximal ternary Cartan decomposition in noncommutative al- gebras. The natural norm of a Cartan extension is not derived from the Minkowski metric, but from the fact that, within the extension subspaces, the Clifford product becomes the inner product while the exterior product vanishes.

Keywords: noncommutative geometry, graded algebra, Neumann ring, Weyl field, stochastic graded field, extension, Cartan extension, ternary extension, eigenfield, pure state, standard model, Majorana spinor, Clifford algebra, 1-norm, stratified

space, K4-grading, quantumchromodynamics

∗written in memory of Carl Friedrich von Weizsäcker

62 Exponential and Cayley maps for Dual Quaternions

Jon Selig

Dual quaternions were introduced by Clifford in [1] to transform what he called rotors. These were vectors bound to points in space, essentially directed lines with an associated magnitude. These rotors were intended to model angular velocities and wrenches. The sum of two rotors is in general a motor, what would now be called a twist. In this work the term dual quaternion will be used rather than Clif- ford’s name ‘biquaternion’ since this seems to refer to several possible cases.

In modern notation dual quaternions can be thought of as elements of the degener- ate Clifford algebra C`(0; 2; 1) or perhaps more conveniently as the even subalgebra of C`(0; 3; 1). The Spin group for this algebra is the double cover of the group of proper Euclidean motions. The group of proper Euclidean motions itself can be realised as a quadric in the projectivisation of the Clifford algebra, this quadric is usually known as the Study quadric.

The Lie algebra of both groups groups also lies in the Clifford algebra. Lie algebra elements, sometimes called twists, are represented by dual pure quaternions. That is 2 dual quaternions of the form s = q0 + eq1, where e is the dual unit satisfying e = 0 and q0, q1 are quaternions with no real part.

These twist satisfy a degree 4 relation, namely

s4 + 2θ2s2 + θ4 = 0,

2 − where θ = q0q1 . Using this relation it is possible to find a system of idempotents and nilpotents P+, P−, N+, N− satisfying,

2 2 2 2 P+ = P+, P− = P−, P+N+ = N+, P−N− = N−, N+ = N− = 0,

63 and all other products are zero, [4]. This system can be used to write infinite power series in s as cubic polynomials in s.

In particular the exponential map, defined by the familiar McLauren series but for dual pure quaternions, is a map from the lie algebra to the group. Using the above method we obtain the following formula, 1 1 es = (2 cos θ + θ sin θ) − (θ cos θ − 3 sin θ)s+ 2 2θ 1 1 (sin θ)s2 − (θ cos θ − sin θ)s3, 2θ 2θ3 which is similar to the well known Rodrigues formula for rotations.

Another map from the Lie algebra to the group of proper Euclidean transforma- tions is given by the Cayley map. This is defined as g = (1 − s)(1 + s)−1. In a similar manner this map can also be expressed as a cubic polynomial in the twist s. This map is useful for numerical methods since it does not involve trigonometric functions.

These two maps will be compared and also compared to the Cayley maps derived from different matrix representations of the group, see [3]. The geometry of these Cayley map in relation to the Study quadric will be explored. Next relations for the derivatives of these maps are found, following Hausdorff [2]. Using the system of idempotents and nilpotents these relations are easily inverted to give relations for the derivative of the twist.

Finally, the problem of finding all possible analytic maps from the Lie algebra to the groups is studied.

[1] W.K. Clifford, Preliminary Sketch of the biquaternions. Proc. London Math. Soc. s1-4(1):381.395, 1871. [2] F. Hausdorff. Die Symbolische exponential formel in den gruppen theorie. Berichte de S´lachicen Akademie de Wissenschaften (Math Phys Klasse) vol. 58, pp. 19.48, 1906. [3] J.M. Selig, Cayley Maps for SE(3), .The International Fedaration of Theory of Machines and Mech- anisms 12th World Congress, Besancÿon 2007.

[4] G. Sobczyk. The generalized spectral decomposition of a linear operator. The College Mathematics Journal pp. 27.38, 1997.

64 Polynomial Invariants for Rarita-Schwinger Representations

Dalibor Smid

Rarita-Schwinger operators are generalizations of Dirac operators to higher spin representations. We describe the space of invariant polynomial endomorphisms of such representations, as a first step for establishing the Fischer decomposition of polynomials with values in Rarita-Schwinger representations.

(Joint work with David Eelbode.)

65 Unification of Space-Time-Matter-Energy

Garret Sobczyk and Tolga Yarman

A complete description of space-time, matter and energy is given in terms of the conservation of energy-momentum in Einstein’s special theory of relativity. We derive explicit equations of motion for two falling bodies, based upon the principle that each body must subtract the mass-equivalent for any change in its kinetic energy that is incurred during the fall. In this theory, we find that there are no singularities and consequently no blackholes.

66 Graphical Calculi and Categories with Additional Structure Fernando Souza

Graphical calculi representing categorical structures have been used extensively in various branches of mathematics and other areas, particularly topology, algebra, category theory, logic, classical and quantum information processing, and physics. They provide tools for faithfully visualizing abstract generalizations of tensor prod- ucts, duality, traces, braidings, twists, formal summations, and irreducible represen- tations, among other structures. Through this, they have comprised a powerful way to calculate, proof, and generalize in a variety of contexts. They have also played a key role in the typical application of categories to the establishment of connections between seemingly diverse areas.

There are various approaches to those graphical calculi, developed at different levels of rigor. In this survey, we will review some of the major approaches from a categorical viewpoint, revising the relationship between them. After a short in- troduction to the underlying notions from knot theory (diagrams for links, braids, tangles, and rigid-vertex graphs), we will cover the following treatments: The well- studied case of the categories with additional structure that are freely generated by a given category; categories that have additional structure themselves; Penrose’s ar- row notation and its variations (including operadic ones and non-aligned diagrams), as well as its correspondence to approaches based on words and incidence relations via Penrose’s tensor notation; and the construction of algebraic objects (including some universal ones) in the presence of categorical structures, including a brief dis- cussion of planar algebras.

Our presentation will be as self-contained as possible, emphasizing the usage of these techniques. Important examples from several sources shall be mentioned. Em- phasis will be given to two overlapping, interdisciplinary subjects: Quantum algebra, particularly Hopf-algebra objects and bialgebra objects in various kinds of categories with additional structure; and combinatorial representation theory.

67 Hydrodynamics with Thermal and Magnetic Effects in Complex Quaternionic Setting

Wolfgang Sprößig

The talk is considered with Poisson–Stokes equations where thermal effects are described by Boussinesq approximations. The methods based on the Bergman– Hodge decomposition of the corresponding complex quaternionic Hilbert space. Rothe’ s method is applied for solutions of the time-dependent Boussinesq problem. The same ideas can be transfered for the consideration of problems of magneto- hydrodynamics. Representation formulae can be presented.

68 Clifford Algebras, Graph Problems, and Computational Complexity

G. Stacey Staples

Extending Clifford-algebraic methods to graph theory opens the door to applica- tions in theoretical computer science, symbolic dynamics, and coding theory. For example, defining the adjacency matrix A of a graph whose vertices and edges are labeled with basis vectors of C`p,q allows one to enumerate the graph’s k-cycles by examining the trace of Ak, provided k is odd. More general results are possi- ble by constructing commutative subalgebras of Clifford algebras. Three general- purpose algebras can be constructed within a Clifford algebra of appropriate sig- nature. The algebras are generated by the unit scalar along with elements {xi} 2 2 satisfying xixj = xjxi and one of the following: (i) xi = 0 (null-square); (ii) xi = 1 2 (unipotent); (iii) xi = xi (idempotent).

Combinatorial properties of these algebras make them useful for studying a wide variety of graph problems. In addition, they illustrate a potential reduction in com- putational complexity. The problem of enumerating a graph’s k-cycles is known to be NP-complete. By considering entries of Λk, where Λ is an appropriate “nilpo- tent adjacency matrix” associated with a finite graph on n vertices, the k-cycles in the graph are recovered by performing O(n3 log k) Clifford operations or “C` ops”. While the number of geometric products required is not a natural measure of com- plexity in classical computing, it is natural if one assumes the existence of a Clifford computer that can perform such operations in either constant or polynomial time. A number of applications will be discussed, including processes on geometric random graphs. These processes are related to addition-deletion networks used to model wireless networks.

(Joint work with René Schotty – yIECN and LORIA Universié Henri Poincaré-Nancy I, BP 239, 54506 Vandoeuvre-l¸tes-Nancy, France, email: [email protected])

69 Oriented Projective Geometry

Jorge Stolfi

Oriented Projective Geometry (OPG) [1] is a variant of Real Projective Geometry (RPG) that supports the concepts of orientation, sideness, handedness, convexity, etc., in any dimension. Many algorithms of computational geometry depend on these concepts, and therefore can be formulated in OPG but not in RPG.

On the other hand, OPG retains most of the advantages that RPG has over Eu- clidean Geometry, such as the smooth handling of points at infinity and parallelism, geometric duality, and projective transformation. Thus, geometric algorithms that were developed for Euclidean Geometry often become simpler, more elegant, and more general when recast in OPG.

Mathematically, the n-dimensional oriented projective space Tn is a double cover of the real projective n-space Pn — which is simply the sphere Sn. The lines of Tn are the great circles of Sn, and are endowed with an internal orientation (a sense of travel along the line) and an external orientation (a sense of turning around the line). Planes and higher-dimensional subspaces are also oriented. The fundamental RPG operations, join (the smallest projective subspace enclosing two given subspaces) and meet (the intersection of two subspaces) are replaced by orientation-sensitive and an- ticommutative versions.

Computationally, points in Tn can be represented by n + 1 homogeneous coordi- nates, as in Pn; except that reversing the signs of all coordinates, which is a no-op in Pn, yields a distinct point of Tn. Lines and higher-dimensional subspaces can be ho- mogeneously represented by Plücker coordinates; or, more compactly, by projective frames.

OPG can be viewed as an algebraic structure embeedded in a larger Clifford alge- bra. The elementary entities of OPG (points, lines, planes, etc.) can be identified with

70 certain “pure” elements of a Clifford algebra (products of points), and the join and meet operations of OPG are essentially components of the Clifford algebra’s prod- uct. However, representing an element of a Clifford algebra requires 2n+1 numbers, whereas a point or hyperplane of OPG requires only n + 1 homogeneous coordi- nates, and a line or hyperline requires only n(n + 1)/2. Thus, while Clifford algebra often provides more succint and general formulas than OPG, the latter usually yields more efficient algorithms — and is easier to interpret in geometric terms.

[1] Stolfi, J., Oriented Projective Geometry: A Framework for Geometric Computations, Academic Press (1991).

71 New Insights in the Physics of Spacetime Using Quaternions

Douglas Sweetser

A real 4 × 4 matrix representation of quaternions is shown to be a commuting division algebra so long as zero and quaternions where one element equals the sum of the other three are excluded, as happens in physics for light-like events. A two limit derivative is defined, where first the 3-vector goes to zero, then the scalar, as happens in physics for time-like events. This directional derivative along the scalar or time axis is the well behaved domain of classical physics where events are ordered in time. If the limit process is reversed as happens for space-like events, then only the normed derivative can be calculated. This is the domain of quantum mechanics where one can know all the possible states. Finally, events in spacetime will be animated directly from quaternion equations. Analytic animations of trig functions such as a sine can be surprising. The first visualization of all the groups that make up the standard model - U(1), SU(2), and SU(3) - will be shown.

72 Twisted Clifford Bundles, Gravity and the Mass of the Higgs Boson

Jürgen Tolksdorf

We discuss the basic ideas of “gauge theories of Dirac type” and summarize some of its basic features. In particular, we discuss general relativity from the point of view of (twisted) Clifford bundles and Dirac type first order differential operators. This geometrical description of general relativity allows to relate gravity to sponta- neous symmetry breaking without the use of a Higgs potential. In fact, from the point of view of Dirac type gauge theories the Higgs potential is regarded as the sum of two terms with a rather different geometrical origin. One of these terms is shown to be related to gravity, whereas the other term is related to Yang-Mills gauge theory. In particular, the Higgs potential is not needed to provide the gauge bosons with a non-trivial mass spectrum. The latter can be expressed in terms of a (certain class of) Dirac type first order operators on a twisted Clifford bundle. The basic reason for the occurrence of the Higgs potential in the Standard Model of Particle Physics is to provide the Higgs boson itself with an appropriate mass. We briefly discuss how the Higgs potential of the Standard Model can be derived from Dirac type operators on a twisted Clifford bundle and how this allows to make a predic- tion for the mass of the yet to be find Higgs boson. The predicted value turns out to be compatible with all the known data derived from the Standard Model of Particle Physics. Hence, this predicted value for the mass of the Higgs boson may be verified in the nearby future, for instance, by the LHC accelerator.

Eventually, we shall conclude our discussion with some remarks on possible rela- tions between Dirac type gauge theories and a natural generalization of Maxwell’s equations in Dirac form: 6∂F = 0

73 Clifford Algebras and the Supersymmetric Quantum Mechanics

Francesco Toppan

The algebra of the one-dimensional N-Extended Supersymmetry (the superalge- bra of the Supersymmetric Quantum Mechanics) induces Clifford algebras when formulating the eigenvalue problem. The classification of its irreducible represen- tations realized by linear derivative operators acting on a finite number of bosonic and fermionic fields is presented. The presentation of the irreducible representa- tions in terms of N-colored oriented graphs is discussed. Off-shell invariant actions for 1D sigma models are constructed. For N=1,2,4,8, the division algebra structure constants enter the invariant actions as coupling constants.

74 Hyperbolic Pseudoanalytic Function Theory and the Klein-Gordon Equation

Sébastien Tremblay

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this talk we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-
Gordon equation with a potential:  − ν(x, t) ϕ(x, t) = 0. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua- type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the Klein-Gordon equation. Using hyperbolic pseudo- analytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential.

(This work was done in collaboration with V.V. Kravchenko and D. Rochon)

75 An Algebraic Remark on Clifford Algebras and Differential Operators

Alexandre Trovon

The purpose of this talk is to study, from an algebraic viewpoint, the rule played by differential operators on abstract Clifford algebras. Within this framework it is possible to find a relationship among linear differential operators and a kind of gen- eralized Dirac operator, which points to an investigation on algebraic spin structures.

76 Noncommutative Field Theory and Twisted Symmetries

J. David Vergara

Within the context of quantum field theory, a considerable amount of work has been recently done dealing with quantum field theories in noncommutative space- times (NCQFT). One of the most relevant issues in this area is related to the sym- metries under which these noncommutative systems are invariant. The most recent contention being that NCQFT are invariant under global “twisted symmetries”. This criterion has been extended to the case of the twisting of local symmetries, such as diffeomorphisms by Wess, et al, and this has been used to propose some noncom- mutative theories of gravity. Another possible extension of this idea is to consider the construction of noncommutative gauges theories with an arbitrary gauge group. Regarding this latter line of research there is, however, some level of controversy as to whether it is possible to construct twisted gauge symmetries. In this lecture we address this issue from the point of view of canonically reparametrized noncommu- tative field theories.

77 Cohomological Approach to Diagonal Noncommutative Nonassociative Graded Algebras

Luis A. Wills-Toro, Juan D. Velez and Thomas Craven

We study finite abelian group graded algebras with no zero divisors on monomial in the generators. The algebra is generated by a set of algebra elements graded by a minimal set of generators of the grading group, and as many as them. There are functions q and r coding the noncommutativity and nonassociativity of the algebra. We study the cohomology of such q- and r-functions. We discover that the r-function coding nonassociativity has always trivial cohomology. Quaternions, octonions and sedenions are constructed in this manner and we study their noncommutativity and nonassociativity using cohomological tools. For deformed graded Lie algebras whose noncommutative nonassociative transformation parameters are governed by such algebras, even if the cohomology of the q-function is not trivial, there is a one to one correspondence with a plain (non-deformed) Lie algebra while maintaining their grading. We show then that in the case of colored or epsilon-Lie algebras, there is a transformation to plain Lie algebras that transform self-adjoint generators into self-adjoint generators, complementing the result of Scheunert.

78 Historical Talk

Did Copernicus Incorporate some Arab Innovations for Describing Planetary Motion into his Work Without Acknowledgement?

John Snygg

In 1957, a paper written by ibn al-Shatir (circa 1304 - 1375AD) was discovered in the Vatican archives which presented a variation of Ptolemy’s methods for describ- ing planetary motion. When this paper was shown to Copernican scholars, it was recognized that Copernicus used the same approaches without mention of any Arab source. Since then, it has become conventional wisdom among most historians of Islamic science that Copernicus did not reinvent these methods. In this talk, I will discuss the merits of these accusations against Copernicus.

79 List of Participants

Rafal Ablamowicz Fred Brackx Department of Mathematics Department of Mathematical Analysis Tennessee Technological University Ghent University Cookeville, TN, USA Ghent, Belgium [email protected] [email protected]

Thomas Batard Oliver Conradt Mathématiques, Image et Applications Section for Mathematics and Astronomy Université de La Rochelle Goetheanum La Rochelle, France Dornach, Switzerland [email protected] [email protected]

William Baylis Marek Czachor Department of Physics Theo. Phys. and Quantum Informatics University of Windsor Politechnika Gda´nska Windsor, ON, Canada Gdansky, Poland [email protected] [email protected]

Eduardo Bayro-Corrochano Cássius De Melo Dept. of Electrical Engineering and CS Instituto de Física Teórica CINVESTAV, Unidad Guadalajara UNESP Guadalajara, JAL, México São Paulo, SP, Brazil [email protected] [email protected]

Troy Brachey Hennie De Schepper Department of Mathematics Department of Mathematical Analysis Tennessee Technological University Ghent University Cookeville, TN, USA Ghent, Belgium [email protected] [email protected]

80 Nedim Degirmenci Klaus Gürlebeck Department of Mathematics Institute of Mathematics and Physics Anadolu University Bauhaus University Weimar Eski¸sehir, Turkey Weimar, Germany [email protected] [email protected]

Suleyman Demir Jacques Helmstetter Department of Physics Institut Fourier Anadolu University Université Grenoble I Eski¸sehir, Turkey Grenoble, France [email protected] [email protected]

Sirkka-Liisa Eriksson David Hestenes Department of Mathematics Department of Physics Tampere University of Technology Arizona State University Tampere, Finland Tempe, AZ, USA [email protected] [email protected]

Rita Fioresi Eckhard S. M. Hitzer Dipartimento di Matematica Department of Applied Physics Università degli Studi di Bologna University of Fukui Bologna, Italy Fukui, Japan [email protected] [email protected]

Ghislain Franssens Eduardo Hoefel Belgian Institute Departamento de Matemática for Space Aeronomy Universidade Federal do Paraná Brussels, Belgium Curitiba, PR, Brazil [email protected] [email protected]

81 Bernard Jancewicz Carlile Lavor Institute of Theoretical Physics Departmento de Matemática Aplicada Wroclaw University IMECC-UNICAMP Wroclaw, Poland Campinas, SP, Brazil [email protected] [email protected]

Marcos Jardim Julian Ławrynowicz Departamento de Matemática Department of Solid State Physics IMECC-UNICAMP University of Lodz Campinas, SP, Brazil Lodz, Poland [email protected] [email protected]

Lukas Krump Rafael Leão Mathematical Institute Departamento de Matemática Charles University in Prague Universidade Federal do Paraná Prague, Czech Republic Curitiba, PR, Brazil [email protected] [email protected]

Zhanna Kuznetsova Murat Limoncu Departamento de Matemática Department of Mathematics Universidade Federal de Juíz de Fora Anadolu University Juíz de Fora, ES, Brazil Eski¸sehir, Turkey [email protected] [email protected]

Anthony Lasenby Paul Loya Department of Physics Department of Mathematics University of Cambridge Binghamton University - SUNY Cambridge, England, UK Binghamton, NY, USA [email protected] [email protected]

82 Douglas Lundholm Artibano Micali Department of Mathematics Dép. des Sciences Mathématiques KTH - Royal Institute of Technology Université Montpellier II Stockholm, Sweden Montpellier, France [email protected] [email protected]

Marco A. Macías-Cedeño Igor Monteiro Tecnológico de Monterrey Departamento de Matemática Campus Santa Fé Universidade Federal do Rio Grande Ciudad de México, DF, México Rio Grande, RS, Brazil [email protected] [email protected]

Emilio Marmolejo-Olea Ricardo Mosna Instituto de Matematicas Departamento de Matemática Aplicada Unidad Cuernavaca – UNAM IMECC-UNICAMP Cuernavaca, MOR, Mexico Campinas, SP, Brazil [email protected] [email protected]

Mircea Martin Eduardo Notte-Cuello Department of Mathematics Departamento de Matemáticas Baker University Universidad de La Serena Baldwin City, KS, USA La Serena, Chile [email protected] [email protected]

Nolmar Melo Matej Pavšicˇ Departamento de Matemática Aplicada Department of Theoretical Physics IMECC-UNICAMP Josef Stefan Institute Campinas, SP, Brazil Ljubljana, Slovenia [email protected] [email protected]

83 Alessandro Perotti Dominic Rochon Dipartimento di Matematica Dép. mathématiques et d’informatique Università degli Studi di Trento Université du Québec Povo, Trento, Italy Trois-Rivières, QC, Canada [email protected] [email protected]

Bruto Max Pimentel Waldyr A. Rodrigues Jr. Instituto de Física Teórica Departamento de Matemática Aplicada UNESP IMECC-UNICAMP São Paulo, Brazil Campinas, SP, Brazil [email protected] [email protected]

Dimitrios Pinotsis Thalanayar Santhanam Department of Mathematics Department of Physics University of Reading Saint Louis University Reading, England, UK St. Louis, MO, USA [email protected] [email protected]

Rafael Reséndiz Bernd Schmeikal Departamento de Matematica Am Platzl 1 Universidad Autónoma Metropolitana A-4451 Garsten Ciudad de México, DF, México Austria [email protected] [email protected]

Roldão da Rocha Jr. Jon Selig CMCC MSFS Department Universidade Federal do ABC London South Bank University Santo André, SP, Brazil London, UK [email protected] [email protected]

84 Dalibor Smid George Stacey Staples Mathematical Institute Dept. of Mathematics and Statistics Charles University in Prague Southern Illinois Univ. Edwardsville Prague, Czech Republic Edwardsville, IL, USA [email protected] [email protected]

John Snygg Jorge Stolfi 433 Prospect Street Instituto de Computação East Orange, NJ 07017-3101, USA Universidade Estadual de Campinas [email protected] Campinas, SP, Brazil [email protected] [email protected]

Garret Sobczyk Douglas Sweetser Dpto. de Actuaría, Física y Matemáticas Quaternions.com Universidad de Las Américas Puebla 39 Drummer Road Puebla, Mexico Acton, MA 01720, USA [email protected] [email protected]

Fernando J. O. Souza Jürgen Tolksdorf Departamento de Matemática Max Planck Institut Universidade Federal de Pernambuco for Mathematics in the Sciences Recife, PE, Brazil Leipzig, Germany [email protected] [email protected]

Wolfgang Sprößig Francesco Toppan Institute of Applied Analysis Física Teórica TU Bergakademie Freiberg CBPF Freiberg, Germany Rio de Janeiro, RJ, Brazil [email protected] [email protected]

85 Sébastien Tremblay José David Vergara Dép. mathématiques et d’informatique Instituto de Ciencias Nucleares Université du Québec UNAM Trois-Rivières, QC, Canada Ciudad de México, DF, México [email protected] [email protected]

Alexandre Trovon Georges Weill Departamento de Matemática Department of Mathematics Universidade Federal do Paraná Cooper Union School of Engineering Curitiba, PR, Brazil New York, NY, USA [email protected] [email protected]

Jayme Vaz Jr. Luis A. Wills-Toro Departamento de Matemática Aplicada Dept. of Mathematics and Statistics IMECC-UNICAMP American University of Sharjah Campinas, SP, Brazil Sharjah, United Arab Emirates [email protected] [email protected]

Photography on page 3 by Antoninho Perri. Backcover photography captured by IKONOS-2 Satellite.

86 May 4, 1845 Exeter, England

March 3, 1879 Madeira, Portugal