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Abstract: As an integral representation for holomorphic functions, Cauchy integral formula plays a significant role in the function theory of one complex variable and several complex variables. In this paper, using the idea of several complex analysis we construct the Cauchy kernel in universal Clifford analysis, which has generalized complex differential forms with universal Clifford basic vectors. We establish Cauchy–Pompeiu formula and Cauchy integral formula on the distinguished boundary with values in universal Clifford algebra. This work is the basis for studying the Cauchy-type integral and its boundary value problem in complex universal Clifford analysis. PubDate: 2021-10-12
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Abstract: In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with \(SL(2,\mathbb {H})\) , which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of \(SL(2,\mathbb {H})\) do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of SO(5, 1) is presented, providing a physical interpretation for the elements of \(SL(2,\mathbb {H})\) . Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by SU(4). PubDate: 2021-09-23
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Abstract: The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case \(n=4\) , the proofs for the case \(n=5\) and the case of arbitrary dimension n. The results can be used in symbolic computation. PubDate: 2021-09-22
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Abstract: Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Plücker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations for this product and its properties, and to formulas relating other blade products (scalar, inner, outer, etc., including those of Grassmann algebra) to angles between subspaces. Contractions are linked to an asymmetric angle, while commutators and anticommutators involve hyperbolic functions of the angle bivector, shedding new light on their properties. PubDate: 2021-09-21
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Abstract: Classical Segal–Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on \(\mathbb {R}^n\) with respect to a Gaussian measure, the space of monogenic square-integrable functions on \(\mathbb {R}^{n+1}\) with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of \(\mathbb {R}^n\) . PubDate: 2021-09-04
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Abstract: In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function. PubDate: 2021-09-03
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Abstract: Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function has zeon polynomial representations on associated equivalence classes of zeons. In this paper, zeros of polynomials over complex zeons are considered. Existing results for real zeon polynomials are extended to the complex case and new results are established. In particular, a fundamental theorem of zeon algebra is established for spectrally simple zeros of complex zeon polynomials, and an algorithm is presented that allows one to find spectrally simple zeros when they exist. As an application, inverses of zeon extensions of analytic functions are computed using polynomial methods. PubDate: 2021-08-20
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Abstract: This paper derives a linear, first-order, partial differential field equation (a Dirac-like equation) in the geometric calculus of the geometric algebra \({\mathcal {G}}_{4,1}\) that has free plane-wave solutions distinct from one another that correspond to the left and right chiral states of the electron and the neutrino. Besides the usual spacetime dependence of plane waves, the solutions have a multivector structure yielding a ladder of states with raising and lowering operators appropriate to electroweak theory and having an \(SU(2)_L\) relationship among the chiral electron and neutrino states. The required Dirac-like equation in \({\mathcal {G}}_{4,1}\) results from a systematic review of Dirac-like equations (i.e., first-order field equations whose solutions also satisfy the Klein–Gordon equation) in geometric algebras of lower dimension. PubDate: 2021-08-19
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Abstract: The k-Cauchy–Fueter operator is an Euclidean version of the helicity k/2 massless field equations on affine Minkowski space. In this article, a version of Schwarz lemma associated to the k-Cauchy–Fueter is established by applying Bochner–Martinelli formula. The constant in Schwarz lemma is sharper than the former results when \(n = 1,k = 1\) , the methods in Schwarz lemma may not only apply to the explicit dimension but also be valid for the general case which will generalize the results in Clifford analysis. PubDate: 2021-08-18
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Abstract: In this paper, we present the development of fractional bicomplex calculus in the Riemann–Liouville sense, based on the modification of the Cauchy–Riemann operator using the one-dimensional Riemann–Liouville derivative in each direction of the bicomplex basis. We introduce elementary functions such as analytic polynomials, exponential, trigonometric, and some properties of these functions. Furthermore, we present the fractional bicomplex Laplace operator connected with the fractional Cauchy–Riemann operator. PubDate: 2021-08-10
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Abstract: The main objective of this work is to show how to construct a ternary \({\mathbb {Z}}_3\) -graded Clifford algebra on two generators by using a group algebra of an extra-special 3-group G of order 27. The approach used is an extension of the method implemented to classify \({\mathbb {Z}}_2\) -graded Clifford algebras as images of group algebras of Salingaros 2-groups [2]. We will show how non-equivalent irreducible representations of the \({\mathbb {Z}}_3\) -graded Clifford algebra are determined by two distinct irreducible characters of G of degree 3. We comment on applying this approach to defining p-ary Clifford-like algebras on two generators and finding their irreducible representations on the basis of extra-special p-groups of order \(p^3\) for \(p > 3.\) Finally, we will comment on possibly using this approach to define p-ary Clifford-like algebras on three and more generators by using group central products and their group algebras. PubDate: 2021-07-30
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Abstract: On various occasions, when working with quaternionic linear spaces, there is a need to restrict them to their complex linear structure, then it becomes essential to understand whether the pre-existing internal product or norm in the quaternionic space will continue to be compatible with the complex structure of the new space obtained. There are other situations in which these types of questions arise, for example, if a linear space is originally complex but it turns out that it also admits the quaternionic structure. The objective of this work is to present the different options to change the linearities of some linear spaces and to analyze what happens with the pre-existing algebraic objects: to understand if they still work or if they induce some others that will be compatible with the new linear structure. PubDate: 2021-07-30
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Abstract: This paper extends approach of our recent paper together with Kähler to building special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media by means of different generalizations of the Cauchy–Riemann system with variable coefficients. A new class of three-dimensional solutions of the static Maxwell system in some special cylindrically layered media is obtained using class of exact solutions of the elliptic Euler–Poisson–Darboux equation in cylindrical coordinates. The principal invariants of the electric field gradient tensor within a wide range of meridional fields are described using a family of Vekua type systems in cylindrical coordinates. Analytic models of meridional electrostatic fields in accordance with different generalizations of the Cauchy–Riemann system with variable coefficients allow us to introduce the concept of \(\alpha \) -meridional mappings of the first and second kind depending on the values of a real parameter \(\alpha \) . In particular, in case \(\alpha =0\) , geometric properties of harmonic meridional mappings of the second kind are demonstrated explicitly within meridional fields in homogeneous media. PubDate: 2021-07-30
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Abstract: Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. An algebraist’s perspective on the many subgroups and subalgebras of Clifford algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions. PubDate: 2021-07-14
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Abstract: Conformal geometric algebra (CGA) is a framework that allows the representation of objects, such as points, planes and spheres, and deformations, such as translations, rotations and dilations as uniform vectors, called multivectors. In this work, we demonstrate the merits of multivector usage with a novel, integrated rigged character simulation framework based on CGA. In such a framework, and for the first time, one may perform real-time cuts and tears as well as drill holes on a rigged 3D model. These operations can be performed before and/or after model animation, while maintaining deformation topology. Moreover, our framework permits generation of intermediate keyframes on-the-fly based on user input, apart from the frames provided in the model data. We are motivated to use CGA as it is the lowest-dimension extension of dual-quaternion algebra that amends the shortcomings of the majority of existing animation and deformation techniques. Specifically, we no longer need to maintain objects of multiple algebras and constantly transmute between them, such as matrices, quaternions and dual-quaternions, and we can effortlessly apply dilations. Using such an all-in-one geometric framework allows for better maintenance and optimization and enables easier interpolation and application of all native deformations. Furthermore, we present these three novel algorithms in a single CGA representation which enables cutting, tearing and drilling of the input rigged model, where the output model can be further re-deformed in interactive frame rates. These close to real-time cut,tear and drill algorithms can enable a new suite of applications, especially under the scope of a medical VR simulation. PubDate: 2021-07-06
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Abstract: Recent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of \(H_3\rightarrow H_4\) in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the \({\mathrm {Pin}}\) and \({\mathrm {Spin}}\) covers. Using this connection with \(H_3\) via the induction theorem sheds light on geometric aspects of the \(H_4\) root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of \({\mathrm {Cl}}(3)\) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework. PubDate: 2021-06-28
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Abstract: The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as \(O(p+1,q+1)\) isomorphic to the nD conformal group with \(n=p+q.\) The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory. PubDate: 2021-06-23
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Abstract: Several topological methods have been used successfully in the study of the hypercomplex analysis; for example, in the theory of functions of several complex variables (Grauert et al., in: Encyclopaedia of mathematical science, vol. 74, Springer, 1991, Hirzebruch, in: Topological methods in algebraic geometry, Classics in Mathematics. Reprint of the 1978 Edition. Springer, Berlin, 1995, Krantz, in Function theory of several complex variables, 2nd edn. American Mathematical Society, Providence, 2001), in the Clifford analysis (Sabadini et al. in Adv. Appl. Clifford Algebras 24:1131–1143, 2014), and in the theory of slice regular functions (Colombo et al. in Math Nachr 285:949–958, 2012). Particularly, the fiber bundle is one of these topological subjects that had an intensive development in a number of papers (Bernstein and Philips in Sci Am 245(1):122–137, 1981, Bleecker, in: Guage Theory and Variational Principles. Dover Books on physics Dover Books on mathematics. Courier Corporation, North Chelmsford, 2005, Bredon in Topology and geometry, Springer, Berlin, 1913, Cohen in The topology of fiber bundles, Stanford University, Stanford, 1998, Hatcher in Algebraic-Topology, Cambridge University Press, Cambridge, 2002, Husemoller in fibre bundles, 3rd edn, Springer, Berlin, 1993, Steenrod in The topology of fibre bundles, Princeton University Press, Princeton, 1951, Walschap in Metric structures in differential geometry, Springer, New York, 2004, Weatherall in Synthese 193:2389–2425, 2016). The aim of this work is to show how the Splitting Lemma and the Representation Formula intrinsically determine a fiber bundle over the space of quaternionic slice regular functions and as a consequence, several properties of this function space are interpreted in terms of sections, pullbacks and isomorphism of fiber bundles. PubDate: 2021-06-21
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Abstract: Let \((g,~[-,-],~\omega )\) be a finite-dimensional complex \(\omega \) -Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra \({\mathrm{Der}}(g)\) and the automorphism group \({\mathrm{Aut}}(g)\) of \((g,~[-,-],~\omega )\) . We study \({\mathrm{Der}}^{\omega }(g)\) and \({\mathrm{Aut}}^{\omega }(g)\) , which are superalgebra of \({\mathrm{Der}}(g)\) and subgroup of \({\mathrm{Aut}}(g)\) , respectively. For any 3-dimensional or 4-dimensional complex \(\omega \) -Lie superalgebra g, we explicitly calculate \({\mathrm{Der}}(g)\) and \({\mathrm{Aut}}(g)\) , and obtain Jordan standard forms of elements in the two sets. We also study representation theory of \(\omega \) -Lie superalgebras and give a conclusion that all nontrivial non- \(\omega \) -Lie 3-dimensional and 4-dimensional \(\omega \) -Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional \(\omega \) -Lie superalgebra \(P_{2,k}(k\ne 0,-1)\) is 1-dimensional. PubDate: 2021-06-19
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Abstract: In this paper, we complement some recent results of L. Márki, J. Meyer, J. Szigeti and L. van Wyk, by investigating the constant-trace representations of a Clifford algebra \(C(V)\) of an arbitrary quadratic form \(q:V\rightarrow F\) (possibly degenerate) and we present some relevant applications. In particular, the existence of the polynomial identities of \(C(V)\) of particular form when the characteristic of the base field is zero is looked at. Furthermore, a lower bound is found on the minimal number t, such that \(C(V)\) can be embedded in a matrix ring of degree t, over some commutative F-algebra. Also, some results on the dimension of commutative subalgebras of \(C(V)\) are obtained. PubDate: 2021-06-11
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