Authors:Peter Fletcher Abstract: In this paper we use matrix representations of quaternions and Clifford algebras and solve the same matrix equations in each case to find Daubechies quaternion and Clifford scaling filters. We use paraunitary completion of the polyphase matrix to find corresponding quaternion and Clifford wavelet filters. We then use the cascade algorithm on our filters to find quaternion and Clifford scaling and wavelet functions, which we illustrate using all possible projections onto two and three dimensions: to our knowledge, this is the first time that this has been done. We discuss the shapes of these functions and conclude with a consideration of what we could actually do with our filters. PubDate: 2018-06-05 DOI: 10.1007/s00006-018-0876-5 Issue No:Vol. 28, No. 3 (2018)

Authors:G. Dattoli; S. Licciardi; R. M. Pidatella; E. Sabia Abstract: In this paper we review the notion of hybrid complex numbers, recently introduced to provide a comprehensive conceptual and formal framework to deal with circular, hyperbolic and dual complex. We exploit the established isomorphism between complex numbers as abstract entities and as two dimensional matrices in order to derive the associated algebraic properties. Within such a respect we derive generalized forms of Euler exponential formula and explore the usefulness and relevance of operator ordering procedure of the Wei-Norman type. We also discuss the properties of dual numbers in terms of Pauli matrices. Finally we explore generalized forms of Dirac-like factorization, emerging from the properties of these numbers. PubDate: 2018-06-04 DOI: 10.1007/s00006-018-0870-y Issue No:Vol. 28, No. 3 (2018)

Authors:R. Lopes; R. da Rocha Abstract: The geometric product, defined by Graf on the space of differential forms, endows the sections of the exterior bundle by a structure that is necessary to construct a Clifford algebra. The Graf product is introduced and revisited with a suitable underlying framework that naturally encompasses a coframe in the cotangent bundle, besides the volume element centrality, the Hodge operator and the so called truncated subalgebra as well. PubDate: 2018-06-02 DOI: 10.1007/s00006-018-0875-6 Issue No:Vol. 28, No. 3 (2018)

Authors:Sebastian Bock Abstract: Recently, the classical orthogonal function systems of inner and outer monogenic Appell functions were used to find a new class of monogenic functions with (logarithmic) line singularities, which naturally extends the class of outer monogenic Appell functions. Based on these results, this article constructs another novel class of monogenic functions with line singularities, which now also relates the class of inner monogenic Appell functions and thus complements the previously known function classes. It is shown that a subset of the constructed functions are rational monogenic functions of the form \(\varvec{p}_{k}^{l}(\varvec{x})\,\varvec{q}^{l}(\overline{\varvec{\zeta }})^{-1}\) , where \(\varvec{p}_{k}^{l}(\varvec{x})\) and \(\varvec{q}^{l}(\overline{\varvec{\zeta }})\) are homogeneous polynomials in the respective variables. Finally, the classical and special classes of monogenic functions are brought together in a unified schematic representation and essential features of the individual function classes and their relationships to each other are shown. PubDate: 2018-06-02 DOI: 10.1007/s00006-018-0874-7 Issue No:Vol. 28, No. 3 (2018)

Authors:Pierre Anglès Abstract: There are claims in the literature on Clifford algebras that every Lie group can be represented as a spin group. In a letter, Pertti Lounesto emphasized, by explicit counter-examples, that this statement is false. This self-contained short paper intends to present a survey of publications of well-known scientists where explicitly developed counter-examples prove the importance of Lounesto’s letter. PubDate: 2018-05-31 DOI: 10.1007/s00006-018-0871-x Issue No:Vol. 28, No. 3 (2018)

Authors:R. A. Kycia; Z. Tabor; A. Woszczyna; D. Kabat; M. Tulik; Z. Latała Abstract: In the present paper a general setup for the determination of imperfect geometry of radiotherapeutic devices has been proposed that is based on a geometric algebra framework. To account for this imperfect geometry, two methods of a calibration were presented, consisting of determining for each angular position of a gantry a correction shift which must be applied to the origin of a laboratory frame of reference to place it along a radiation axis for this angular position. Closed form solutions for these corrections are provided. PubDate: 2018-05-24 DOI: 10.1007/s00006-018-0873-8 Issue No:Vol. 28, No. 3 (2018)

Authors:Dmitry Shirokov Abstract: We present a new class of covariantly constant solutions of the Yang–Mills equations. These solutions correspond to the solution of the field equation for the spin connection of the general form. PubDate: 2018-05-05 DOI: 10.1007/s00006-018-0868-5 Issue No:Vol. 28, No. 3 (2018)

Authors:Ovidiu Cristinel Stoica Abstract: A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra \(\mathbb {C}\ell _6\) , one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. The Dirac and Lorentz algebras appear naturally as subalgebras acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle \(\theta _W\) given by \(\sin ^2\theta _W=0.25\) . The model shares common ideas with previously known models, particularly with Chisholm and Farwell (Clifford (Geometric) algebras: with applications to physics, mathematics, and engineering. Birkhäuser Boston, Boston, pp 365–388, 1996), Trayling and Baylis (Clifford Algebras: applications to mathematics, physics, and engineering. Birkhäuser Boston, Boston, pp 547–558, 1996), and Furey (Standard Model Physics from an Algebra' Preprint arXiv:1611.09182, 2016). PubDate: 2018-05-05 DOI: 10.1007/s00006-018-0869-4 Issue No:Vol. 28, No. 3 (2018)

Authors:Adam Leon Kleppe; Olav Egeland Abstract: This paper presents a descriptor for course alignment of point clouds using conformal geometric algebra. The method is based on selecting keypoints depending on shape factors to identify distinct features of the object represented by the point cloud, and a descriptor is then calculated for each keypoint by fitting two spheres that describe the local curvature. The method for estimating the point correspondences is to a larger extent based on geometric arguments than the method of Kleppe et al. (IEEE Trans Autom Sci Eng, 2017), which results in improved performance. The accuracy of the curvature-based descriptor is validated in experiments, and is shown to compare favorably to state-of-the-art methods in an experiment on course alignment of industrial parts to be assembled with robots. PubDate: 2018-05-03 DOI: 10.1007/s00006-018-0864-9 Issue No:Vol. 28, No. 2 (2018)

Authors:Rafał Abłamowicz Abstract: Clifford algebras \(C \ell _{2,0}\) and \(C \ell _{1,1}\) are isomorphic simple algebras whose Salingaros vee groups belong to a class \(N_1\) . The algebras are isomorphic to the quotient algebra \(\mathbb {R}[D_8]/\mathcal {J}\) of the group algebra of the dihedral group \(D_8\) modulo an ideal \(\mathcal {J}=(1+\tau )\) where \(\tau \) is a central involution in \(D_8\) . Since all irreducible characters of \(D_8\) , including a single nonlinear character of degree 2, can be realized over \(\mathbb {R}\) , spinor representations of the Clifford algebras can be realized over \(\mathbb {R}\) and so \(C \ell _{2,0} \cong C \ell _{1,1} \cong \mathbb {R}(2)\) . Spinor modules in \(C \ell _{2,0}\) and \(C \ell _{1,1}\) are isomorphic to irreducible \(\mathbb {R}D_8\) -submodules of dimension 2 of the regular module \(\mathbb {R}D_8\) . As such, they are uniquely determined by the nonlinear character of degree 2. These results are generalized to the vee groups in classes \(N_{2k-1}\) and \(\Omega _{2k-1}\) ( \(1 \le k \le 4\) ). It is proven that each irreducible character of \(G_{p,q}\) in these classes can be realized over \(\mathbb {R}.\) Consequently, every nonlinear character of \(G_{p,q}\) uniquely determines a spinor module of \(C \ell _{p,q}\) which is faithful (resp. unfaithful) when \(G_{p,q}\) is in the class \(N_{2k-1}\) (resp. \(\Omega _{2k-1})\) . This paper is a continuation of [1]. PubDate: 2018-05-03 DOI: 10.1007/s00006-018-0867-6 Issue No:Vol. 28, No. 2 (2018)

Authors:Heinz Leutwiler Abstract: The classical theory of spherical harmonics on the unit sphere S is well-known. In an earlier paper, entitled “Modified Spherical Harmonics”, we dealt with a modification of this theory, adapted to the half-sphere \(S_{+}\) , in case of three dimensions. In the present paper we extend these results to the four-dimensional case. Although the results look quite similar, their proofs are not. In \(\mathbb {R}^4 =\left\{ (x,y,t,s) \right\} \) the Laplace equation \(\Delta h=0\) will be replaced by the equation \(s\Delta u+2\,\frac{\partial u}{\partial s}=0\) . Homogeneous polynomial solutions of this equation, if restricted to the half-sphere \(S_{+}=\left\{ (x,y,t,s): x^2 + y^2 + t^2 + s^2=1, s > 0 \right\} \) are called modified spherical harmonics. Endowed with a non-Euclidean scalar product on \(S_{+}\) , these functions behave like the classical spherical harmonics on the full sphere in \(\mathbb {R}^4\) . We shall give an explicit expression for the corresponding zonal harmonics and dwell on their connection with the Poisson-type kernel, adapted to the above differential equation. Finally we shall give an explicit orthonormal system of modified spherical harmonics. PubDate: 2018-04-27 DOI: 10.1007/s00006-018-0861-z Issue No:Vol. 28, No. 2 (2018)

Authors:Jacques Helmstetter Abstract: Let V be a vector space of finite dimension over a field K, and \(V^*\) the dual space; there is a canonical quadratic form on \(V\oplus V^*\) , and there is a well known group morphism from \(\mathrm {GL}(V)\) into \(\mathrm {O}(V\oplus V^{*})\) . This morphism can be factored through the Clifford-Lipschitz group \(\mathrm {GLip}(V \oplus V^{*})\) that lies over \(\mathrm {O}(V\oplus V^*)\) ; but when K is the field \(\mathbb {R}\) or \(\mathbb {C}\) , it is not possible to factor it through the smaller group \(\mathrm {Spin}(V\oplus V^*)\) ; this follows from a theorem published by H. Bass in 1974. It is worth recalling this theorem, and comparing it with an article published by Doran, Hestenes, Sommen and Van Acker in 1993, which asserts that every Lie group can be embedded in a spinorial group. The falseness of this assertion is explained at the end of this article. PubDate: 2018-04-27 DOI: 10.1007/s00006-018-0863-x Issue No:Vol. 28, No. 2 (2018)

Authors:Václav Zatloukal Abstract: We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local transformations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang–Mills interaction. Gauge field dynamics is not discussed in this article. Throughout, we employ the mathematical language of geometric algebra and calculus. PubDate: 2018-04-27 DOI: 10.1007/s00006-018-0865-8 Issue No:Vol. 28, No. 2 (2018)

Authors:Judith Vanegas; Franklin Vargas Abstract: The heat transfer problem in isotropic media has been studied extensively in Clifford analysis, but very little in the anisotropic case for this setting. As a first step in this way, we introduce in this work Dirac operators with weights belonging to the Clifford algebra \({\mathcal {A}}_n\) , which factor the second order elliptic differential operator \( {\tilde{\Delta }}_n= div (B \,\nabla ), \) where \(B \in \mathbb {R}^{n \times n}\) is a symmetric and positive definite matrix. For these weighted Dirac operators we construct fundamental solutions and get a Borel–Pompeiu and Cauchy integral formula. PubDate: 2018-04-26 DOI: 10.1007/s00006-018-0860-0 Issue No:Vol. 28, No. 2 (2018)

Authors:Rogério Serôdio; José Augusto Ferreira; José Vitória Abstract: The aim of this paper is to propose an iterative method to compute the dominant zero of a quaternionic unilateral polynomial. We prove that the method is convergent in the sense that it generates a sequence of quaternions that converges to the dominant zero of the polynomial. The idea subjacent to this method is the well known Sebastião e Silva’s method, proposed in “Sur une méthode d’approximation semblable à celle de Gräffe”, Portugaliae Mathematica, 1941, to approximate the dominant zero of complex polynomials. PubDate: 2018-04-26 DOI: 10.1007/s00006-018-0866-7 Issue No:Vol. 28, No. 2 (2018)

Authors:D. Alpay; A. Vajiac; M. B. Vajiac Abstract: This paper is an introduction of a new class of analytic functions defined on a ternary algebra, a three dimensional structure different from \({{\mathbb {C}}}\times {{\mathbb {R}}}\) , i.e. a real commutative algebra given by the linear span of \(\{1,\mathbf{e},\mathbf{e}^2\}\) , where \(\mathbf{e}\not \in {{\mathbb {C}}}\) is a generating unit. We define a single ternary conjugate and we build a new analytic function theory, different from previous approaches, on the basis of this single conjugation (akin to the quaternionic case). We give the solution to the Gleason problem which gives rise to Fueter-type variables and study the space of rational functions in this case. PubDate: 2018-04-26 DOI: 10.1007/s00006-018-0857-8 Issue No:Vol. 28, No. 2 (2018)

Authors:Rafael de Freitas Leão; Samuel Augusto Wainer Abstract: A beautiful solution to the problem of isometric immersions in \(\mathbb {R}^n\) using spinors was found by Bayard et al. (Pac J Math, arXiv:1505.02935v4 [math-ph], 2016). However to use spinors one must assume that the manifold carries a \(\text{ Spin }\) -structure and, especially for complex manifolds where is more natural to consider \(\text{ Spin }^{\mathbb {C}}\) -structures, this hypothesis is somewhat restrictive. In the present work we show how the above solution can be adapted to \(\text{ Spin }^{\mathbb {C}}\) -structures. PubDate: 2018-04-26 DOI: 10.1007/s00006-018-0856-9 Issue No:Vol. 28, No. 2 (2018)

Authors:Alberto Elduque; Adrián Rodrigo-Escudero Abstract: Some connections between quadratic forms over the field of two elements, Clifford algebras of quadratic forms over the real numbers, real graded division algebras, and twisted group algebras will be highlighted. This allows to revisit real Clifford algebras in terms of the Arf invariant of the associated quadratic forms over the field of two elements, and give new proofs of some classical results. PubDate: 2018-04-23 DOI: 10.1007/s00006-018-0862-y Issue No:Vol. 28, No. 2 (2018)

Authors:Lida Bentz; Tevian Dray Abstract: We classify the subalgebras of the split octonions, paying particular attention to the null subalgebras and their extensions. PubDate: 2018-04-23 DOI: 10.1007/s00006-018-0859-6 Issue No:Vol. 28, No. 2 (2018)

Authors:M. Ferreira; M. M. Rodrigues; N. Vieira Abstract: In this work we obtain the first and second fundamental solutions (FS) of the multidimensional time-fractional equation with Laplace or Dirac operators, where the two time-fractional derivatives of orders \(\alpha \in ]0,1]\) and \(\beta \in ]1,2]\) are in the Caputo sense. We obtain representations of the FS in terms of Hankel transform, double Mellin-Barnes integrals, and H-functions of two variables. As an application, the FS are used to solve Cauchy problems of Laplace and Dirac type. PubDate: 2018-04-23 DOI: 10.1007/s00006-018-0858-7 Issue No:Vol. 28, No. 2 (2018)