Authors:Viktor Abramov Pages: 9 - 16 Abstract: Abstract We propose a notion of a super n-Lie algebra and construct a super n-Lie algebra with the help of a given binary super Lie algebra which is equipped with an analog of a supertrace. We apply this approach to the super Lie algebra of a Clifford algebra with even number of generators and making use of a matrix representation of this super Lie algebra given by a supermodule of spinors we construct a series of super 3-Lie algebras labeled by positive even integers. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0604-3 Issue No:Vol. 27, No. 1 (2017)

Authors:Alexandre Trovon; Osamu Suzuki Pages: 59 - 70 Abstract: Abstract In this paper we introduce the idea of Galois extension for a class of associative algebras and discuss binary and ternary Clifford algebras by such an algebraic construction. Nonion algebra is characterized by Galois extensions and a ternary structure is proposed for \({\mathfrak{su}(3)}\) leading to a duality for certain binary and ternary differential operators. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0565-6 Issue No:Vol. 27, No. 1 (2017)

Authors:I. Cação; M. I. Falcão; H. R. Malonek Pages: 71 - 85 Abstract: Abstract Recently, systems of Clifford algebra-valued orthogonal polynomials have been studied from different points of view. We prove in this paper that for their building blocks there exist some three-term recurrence relations, similar to that for orthogonal polynomials of one real variable. As a surprising byproduct of own interest we found out that the whole construction process of Clifford algebra-valued orthogonal polynomials via Gelfand–Tsetlin basis or otherwise relies only on one and the same basic Appell sequence of polynomials. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0596-z Issue No:Vol. 27, No. 1 (2017)

Authors:Sirkka-Liisa Eriksson; Heikki Orelma Pages: 99 - 110 Abstract: Abstract We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\, \,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\) . The function theory based on this metric is important, since in case \({k = n-1}\) , the metric is the hyperbolic metric of the Poincaré upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\) , calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\) -hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions ( \({k = n-1}\) ) and for k-hyperbolic harmonic functions in odd dimensional spaces. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0629-7 Issue No:Vol. 27, No. 1 (2017)

Authors:D. S. Shirokov Pages: 149 - 163 Abstract: Abstract In this paper we consider different operators acting on Clifford algebras. We consider Reynolds operator of Salingaros’ vee group. This operator “average” an action of Salingaros’ vee group on Clifford algebra. We consider conjugate action on Clifford algebra. We present a relation between these operators and projection operators onto fixed subspaces of Clifford algebras. Using method of averaging we present solutions of system of commutator equations. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0630-1 Issue No:Vol. 27, No. 1 (2017)

Authors:John Snygg Pages: 231 - 239 Abstract: Abstract This paper outlines how Clifford Algebra can be presented to high school or college students to deal with the symmetries of the Platonic solids. I concentrate on the cube but all five solids are discussed. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0607-0 Issue No:Vol. 27, No. 1 (2017)

Authors:A. Dargys; A. Acus Pages: 241 - 253 Abstract: Abstract A practical computational method to find the eigenvalues and eigenspinors of quantum mechanical Hamiltonian is presented. The method is based on reduction of the eigenvalue equation to well known geometrical algebra rotor equation and, therefore, allows to replace the usual det (H − E) = 0 quantization condition by much simple vector norm preserving requirement. In order to show how it works in practice a number of examples are worked out in Cl 3,0 (monolayer graphene and spin in the quantum well) and in Cl 3,1 (two coupled two-level atoms and bilayer graphene) algebras. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0549-6 Issue No:Vol. 27, No. 1 (2017)

Authors:R. T. Cavalcanti; R. da Rocha; J. M. Hoff da Silva Pages: 267 - 277 Abstract: Abstract It is shown that the meson algebra can be faced as the tensor product of Clifford algebras and, then, by constructing the DKP field by means of Elko spinors, we demonstrate that the symmetries of the so called very special relativity are inherited by the DKP field. PubDate: 2017-03-01 DOI: 10.1007/s00006-015-0563-8 Issue No:Vol. 27, No. 1 (2017)

Authors:Rafael Alves; Carlile Lavor Pages: 439 - 452 Abstract: Abstract The discretizable molecular distance geometry problem (DMDGP) is related to the determination of 3D protein structure using distance information detected by nuclear magnetic resonance (NMR) experiments. The chemistry of proteins and the NMR distance information allow us to define an atomic order \({v_{1},\ldots,v_{n}}\) such that the distances related to the pairs \({\{v_{i-3},v_{i}\},\{v_{i-2},v_{i}\},\{v_{i-1},v_{i}\}}\) , for \({i > 3}\) , are available, which implies that the search space can be represented by a tree. A DMDGP solution can be represented by a path from the root to a leaf node of this tree, found by an exact method, called branch-and-prune (BP). Because of uncertainty in NMR data, some of the distances related to the pairs \({\{v_{i-3},v_{i}\}}\) may not be precise values, being represented by intervals of real numbers \({[\underline{d}_{i-3,i},\overline{d}_{i-3,i}]}\) . In order to apply BP algorithm in this context, sample values from those intervals should be taken. The main problem of this approach is that if we sample many values, the search space increases drastically, and for small samples, no solution can be found. We explain how geometric algebra can be used to model uncertainties in the DMDGP, avoiding sample values from intervals \({[\underline{d}_{i-3,i},\overline{d}_{i-3,i}]}\) and eliminating the heuristic characteristics of BP when dealing with interval distances. PubDate: 2017-03-01 DOI: 10.1007/s00006-016-0653-2 Issue No:Vol. 27, No. 1 (2017)

Authors:Lei Dong; Lei Huang; Changpeng Shao; Yong Wen Pages: 475 - 489 Abstract: Abstract The Jordan forms of matrices that are the product of two skew-symmetric matrices over a field of characteristic \({\neq 2}\) have been a research topic in linear algebra since the early twentieth century. For such a matrix, its Jordan form is not necessarily real, nor does the matrix similarity transformation change the matrix into the Jordan form. In 3-D oriented projective geometry, orientation-preserving projective transformations are matrices of \({SL(4, \mathbb{R})}\) , and those matrices of \({SL(4, \mathbb{R})}\) that are the product of two skew-symmetric matrices are the generators of the group \({SL(4, \mathbb{R})}\) . The canonical forms of orientation-preserving projective transformations under the group action of \({SL(4, \mathbb{R})}\) -similarity transformations, called \({SL(4, \mathbb{R})}\) -Jordan forms, are more useful in geometric applications than complex-valued Jordan forms. In this paper, we find all the \({SL(4, \mathbb{R})}\) -Jordan forms of the matrices of \({SL(4, \mathbb{R})}\) that are the product of two skew-symmetric matrices, and divide them into six classes, so that each class has an unambiguous geometric interpretation in 3-D oriented projective geometry. We then consider the lifts of these transformations to SO(3, 3) by extending the action of \({SL(4, \mathbb{R})}\) from points to lines in space, so that in the vector space \({\mathbb{R}^{3, 3}}\) spanned by the Plücker coordinates of lines these projective transformations become special orthogonal transformations, and the six classes are lifted to six different rotations in 2-D planes of \({\mathbb{R}^{3, 3}}\) . PubDate: 2017-03-01 DOI: 10.1007/s00006-016-0701-y Issue No:Vol. 27, No. 1 (2017)

Authors:L. Campos-Macías; O. Carbajal-Espinosa; A. Loukianov; E. Bayro-Corrochano Pages: 581 - 597 Abstract: Abstract This paper describes a novel method for solving the inverse kinematics of a humanoid robot leg anthropomorphically configured with 6 degrees of freedom using conformal geometric algebra. We have used different geometric entities such as lines, planes, and spheres in order to achieve the desired position and orientation of the body and the foot, individually reconfiguring the amount of rotation for each joint. The proposed method can be used in a future work to design obstacle avoidance and self collision algorithms. The effectiveness of the proposed algorithm is proved via practical experiments. Results indicate that the proposed algorithm achieves the expected behavior. PubDate: 2017-03-01 DOI: 10.1007/s00006-016-0705-7 Issue No:Vol. 27, No. 1 (2017)

Authors:Gehová López-González; Nancy Arana-Daniel; Eduardo Bayro-Corrochano Pages: 647 - 660 Abstract: Abstract This work presents a parallelization method for the Clifford support vector machines, based in two characteristics of the Gaussian Kernel. The pure real-valued result and its commutativity allows us to separate the multivector data in its defining subspaces. These subspaces are independent from each other, so we can solve the problem using parallelism. The motivation is to present an easy approach that can be explained using the more common known concepts of complex numbers and quaternions, because in general there exists a lack of familiarity with geometric algebra. PubDate: 2017-03-01 DOI: 10.1007/s00006-016-0726-2 Issue No:Vol. 27, No. 1 (2017)

Authors:Abhijit Banerjee Abstract: Abstract We investigate the complete analytical solutions of quantum mechanical harmonic and isotonic oscillators formulated in the commutative ring of bicomplex numbers. We obtain the explicit closed form expressions for the excited eigenstates and corresponding energy eigenvalues associated with the harmonic and isotonic oscillator potentials by extending the formalism adopted in Banerjee (Ann Phys 377:493, 2017) recently to find the analytical closed form solutions for ground states. PubDate: 2017-03-16 DOI: 10.1007/s00006-017-0772-4

Authors:Stéphane Breuils; Vincent Nozick; Laurent Fuchs Abstract: Abstract This paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension (e.g. algebra of dimension 15). The method is compared with the main state of the art geometric algebra implementations, with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations. PubDate: 2017-03-16 DOI: 10.1007/s00006-017-0770-6

Authors:Hakan Simsek; Mustafa Özdemir Abstract: Abstract In this paper the generalization of the rotations on any lightcone in Minkowski 3-space \({\mathbb {R}}_{g}^{1,2}\) is given. The rotation motion on the lightcone is examined by means of a bilinear form and Lorentzian notions. We use the corresponding Rodrigues and Cayley formulas and benefit from the hyperbolic split quaternion product to obtain the corresponding rotation matrix. PubDate: 2017-03-16 DOI: 10.1007/s00006-017-0771-5

Authors:Ekaterina Pervova Abstract: Abstract We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals, which is not automatic in the diffeological context. We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved. PubDate: 2017-03-14 DOI: 10.1007/s00006-017-0769-z

Authors:Sergio Giardino Abstract: Abstract A quaternionic analog of the Aharonov–Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics. A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them. PubDate: 2017-03-07 DOI: 10.1007/s00006-017-0766-2

Authors:Waldyr Alves Rodrigues; Samuel A. Wainer Abstract: Abstract Using Clifford and Spin–Clifford formalisms we prove that the classical relativistic Hamilton Jacobi equation for a charged massive (and spinning) particle interacting with an external electromagnetic field is equivalent to Dirac–Hestenes equation satisfied by a class of spinor fields that we call classical spinor fields. These spinor fields are characterized by having the Takabayashi angle function constant (equal to 0 or \(\pi \) ). We also investigate a nonlinear Dirac–Hestenes like equation that comes from a class of generalized classical spinor fields. Finally, we show that a general Dirac–Hestenes equation (which is a representative in the Clifford bundle of the usual Dirac equation) gives a generalized Hamilton–Jacobi equation where the quantum potential satisfies a severe constraint and the “mass of the particle” becomes a variable. Our results can then eventually explain experimental discrepancies found between prediction for the de Broglie–Bohm theory and recent experiments. We briefly discuss de Broglie’s double solution theory in view of our results showing that it can be realized, at least in the case of spinning free particles.The paper contains several appendices where notation and proofs of some results of the text are presented. PubDate: 2017-03-07 DOI: 10.1007/s00006-017-0768-0

Authors:Mircea Martin Abstract: Abstract The goal of this article is to introduce a concept of Clifford structures on vector bundles as natural extensions of the standard complex and quaternionic structures, and to determine the derivations and linear connections on smooth Clifford vector bundles compatible with their Clifford structures. The basic object used to get such descriptions is an involution on the space of derivations of a Clifford vector bundle explicitly defined in terms of the specific Clifford structure. That involution is actually derived from an operation called the Clifford conjugation relative to a Clifford structure, which is defined in a purely algebraic setting as an involution on the space of derivations of a Euclidean Clifford algebra. Its definition essentially relies on the use and a complete description of the geometric concept of tangent Clifford structures of a Euclidean Clifford algebra. PubDate: 2017-03-03 DOI: 10.1007/s00006-017-0767-1