Typographic error in Eq. (41c): the subscript.

We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra \({\mathcal {C}}\ell _{3,3}\) of the quadratic space \({\mathbb {R}}^{3,3}\). We show that this algebra describes in a unified way the operations of reflection, rotation (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using Hodge duality, we define an

We present GAALOPWeb for Matlab, a new easy to handle solution for Geometric Algebra implementations for Matlab. We demonstrate its usability for industrial applications based on a forward kinematics algorithm of a serial robot arm and illustrate it with the help of high run-time performance.

In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and a-points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative

A geometric algebra provides a single environment in which geometric entities can be represented and manipulated and in which transforms can be applied to these entities. A number of versions of geometric algebra have been proposed and the aim of the paper is to investigate one of these as it has a number of advantageous features. Points, lines and planes are presented naturally by element of grades

In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator \(D_s\), which is defined as the first-order \(\mathfrak {sp}(2n)\)-invariant differential operator acting on functions on \(\mathbb {R}^{2n}\) taking values in the metaplectic spinor representation.

Geometric Calculus is developed for curved-space treatments of General Relativity and comparison is made with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides

In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra \({\mathfrak {osp}}(1|2)\) in terms of a so-called radially deformed Dirac operator \({\mathbf {D}}\) depending on a deformation parameter c such that for \(c=0\) the classical Dirac operator is

We construct a discrete version of the plane wave solution to a discrete Dirac-Kähler equation in the Joyce form. A geometric discretisation scheme based on both forward and backward difference operators is used. The conditions under which a discrete plane wave solution satisfies a discrete Joyce equation are discussed.

We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to

We consider the factorization of differential operator of Klein–Gordon equation on the base of space-time algebra of sixteen-component sedeons. It is shown that generalized sedeonic wave equations can be used both to describe quantum particles and the force fields responsible for the interaction of particles. In particular, we discuss the first-order and second-order wave equations for sedeonic potentials

Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann and Clifford algebras and to the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families

The BiHom-version of infinitesimal bialgebra was studied in [22]. In this paper, the notion of covariant BiHom-bialgebra is introduced generalizing infinitesimal BiHom-bialgebra above. Especially, we give the characterizations of quasitriangular covariant BiHom-bialgebra, then we obtain the concepts of associative BiHom-Yang-Baxter pair and associative BiHom-Yang-Baxter equation. Furthermore, we present

In the classical Clifford analysis the Laplace operator is factorized by the Cauchy–Riemann operator \(\Delta =\overline{D}D\). The consequence is all components of a monogenic function are harmonic functions. In more general situation, suppose that we are given a linear partial differential equation. We wish to find a generalized Clifford algebra such that all components of a generalized monogenic

In a right quaternionic Hilbert space, for a bounded right linear operator, the Kato S-spectrum is introduced and studied to a certain extent. In particular, it is shown that the Kato S-spectrum is a non-empty compact subset of the S-spectrum and it contains the boundary of the S-spectrum. Using right-slice regular functions, local S-spectrum, at a point of a right quaternionic Hilbert space, and the

The article is devoted to a new type of octonion measures. The considered such measures are related with solutions of high order hyperbolic PDEs and related Markov processes. Their characteristic functionals are investigated. Cylindrical distributions of these measures are studied.

This paper deals with the construction of \(n=3 \text{ mod } 4\) Clifford algebra \(Cl_{n,0}\)-valued admissible shearlet transform using the shearlet group \(({\mathbb {R}}^{*} < imes {{\mathbb {R}}}^{n-1}) < imes {\mathbb {R}}^n\), a subgroup of affine group of \({\mathbb {R}}^n\). The admissibility conditions for a nonzero Clifford valued square integrable function have been obtained. Various properties

In this work, a novel methodology for the modelling and forecasting of time series using higher-dimensional networks in \(\mathbf{R }^{4,1}\) space is presented. Time series data is partitioned, transformed and mapped into five-dimensional conformal space as a network which we call the ‘Spatio-Temporal Ordinality Network’ (STON). These STONs are characterised using specific Clifford Algebraic multivector

In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which

In this paper, we first introduce the notion of BiHom–Lie superalgebras, which is a generalization of both BiHom–Lie algebras and Hom–Lie superalgebras. Also, we explore some general classes of BiHom–Lie admissible superalgebras and describe all these classes via G-BiHom-associative superalgebras, where G is a subgroup of the symmetric group \(S_{3}\). Finally, we obtain a method to construct the solutions

In this paper we extend the notion of the Brauer–Clifford group to the case of an \((S,{{\mathcal {G}}},H)\)-algebra, when H is a cocommutative Hopf algebra, \({{\mathcal {G}}}\) is a Lie algebra in the symmetric monoidal category of left H-modules, and S is a commutative algebra which is an H-module algebra, a \({\mathcal G}\)-module algebra and the H-action is compatible with the \({\mathcal {G}}\)-action

Using the concept of a transposition anti-involution in Clifford algebra \(C \, \ell _{1,1}\) and the isomorphisms \({\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),\) where \({\mathbb {H}}_s\) is the algebra of split quaternions and \(\text {Mat}(2,{\mathbb {R}})\) is the algebra of \(2 \times 2\) real matrices, one can find the Moore–Penrose inverse \(q^{+}\) of a non-zero

Let \({\mathcal {H}}\) be an oriented three-dimensional manifold and \({\mathbb {H}}_+={\mathbb {R}}_+\oplus {\mathcal {H}}\). The author introduces non-abelian vector valued Fourier transforms on \({\mathcal {H}}\) and Poisson integrals on \({\mathbb {H}}_+\). Through the boundary behaviour of Poisson integral, the author obtains the characterization of conjugate harmonic functions of Fueter type

Complex-valued Hopfield neural networks have been extended to 4-dimensional models using quaternions and commutative quaternions. The algebra of split quaternions is another 4-dimensional hypercomplex number system. In this work, a split quaternion-valued Hopfield neural network is defined. By the computer simulations using the twin-multistate activation function and projection rule, we evaluate the

This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decomposition with respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups

The paper provides an efficient method for obtaining powers and roots of dual complex \(2\times 2\) matrices based on a far reaching generalization of De Moivre’s formula. We also resolve the case of normal \(3\times 3\) and \(4\times 4\) matrices using polar decomposition and the direct sum structure of \(\mathfrak {so}_4\). The compact explicit expressions derived for rational powers formally extend

For a bounded right linear operators A, in a right quaternionic Hilbert space \(V_\mathbb {H}^R\), following the complex formalism, we study the Berberian extension \(A^\circ \), which is an extension of A in a right quaternionic Hilbert space obtained from \(V_\mathbb {H}^R\). In the complex setting, the important feature of the Berberian extension is that it converts approximate point spectrum of

Geometric algebra is a powerful mathematical framework that allows us to use geometric entities (encoded by blades) and orthogonal transformations (encoded by versors) as primitives and operate on them directly. In this work, we present a high-level C++ library for geometric algebra. By manipulating blades and versors decomposed as vectors under a tensor structure, our library achieves high performance

Remark 4.1 and Remark 4.5 in Section 4 will be true only if a is a power of two.

In this paper the Lorentz transformation, considered as the composition of a rotation and a Lorentz boost, is decomposed into a linear combination of two orthogonal transforms. In this way a two-term expression of the Lorentz transformation by means of quaternions is proposed. An analytical solution to the problem of finding eigenvectors is given. The conditions for the existence of eigenvectors are

From the beginning of David Hestenes rediscovery of geometric algebra in the 1960s, outermorphisms have been a cornerstone in the mathematical development of GA. Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. Over the last two decades, GA-based mathematical models

We present a hitherto unknown polar representation of complexified quaternions (also known as biquaternions), also applicable to complexified octonions. The complexified quaternion is factored into the product of two exponentials, one trigonometric or circular, and one hyperbolic. The trigonometric exponential is a real quaternion, the hyperbolic exponential has a real scalar part and imaginary vector

We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching

In this paper we derive a Cauchy integral representation formula for the solutions of the iterated sandwich equation \(\partial _{{\underline{x}}}^{2k-1}f\partial _{{\underline{x}}}=0\), where k is a positive integer and \(\partial _{{\underline{x}}}\) stands for the Dirac operator in the Euclidean space \({{\mathbb {R}}}^m\). We call these solutions \((2k-1)\)-infrapolymonogenic functions (or simply

In this paper, we investigate quaternionic analysis on the \((4n+1)\)-dimensional Heisenberg group. The tangential k-Cauchy–Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables, respectively. We give the Penrose integral formula for k-CF functions and establish the Bochner–Martinelli

In this work the dynamic model and the nonlinear control for a multi-copter have been developed using the geometric algebra framework specifically using the motor algebra \(G^+_{3,0,1}\). The kinematics for the aircraft model and the dynamics based on Newton-Euler formalism are presented. Block-control technique is applied to the multi-copter model which involves super twisting control and an estimator

Let V be a pseudo-Riemannian n-dimensional manifold or, more generally, let \((\xi ,Q)\) be a real fibre bundle whose base space is a paracompact space endowed with a non-degenerate quadratic form Q, (that is, with a structure group \({\mathrm {O}}(p,q),\)\(n=p+q\).) Let \(K_{p,q}\) denote the obstruction class for the existence of a \(\mathrm {Pin}(p,q)\)-spin structure on V or over \(\xi .\) Let

Evaluation of a robotic (serial or parallel) manipulator’s static and kinematics performance requires the solution of closed and complicated non-closed form systems of equations for both the forward and inverse problem types. In this work, a robotic system or manipulator is represented as a network whose motion generating kinematic pairs are represented as the network’s inter-connected nodes. Within

We investigate the 2D quaternion windowed linear canonical transform (QWLCT) in this paper. Firstly, we propose the new definition of the QWLCT, and then several important properties of newly defined QWLCT, such as bounded, shift, modulation, orthogonality relation, are derived based on the spectral representation of the quaternionic linear canonical transform (QLCT). Secondly, by the Heisenberg uncertainty

In this paper, we study composition operators on generalized Fock spaces of slice hyperholomorphic functions. More precisely, some relationship for composition operators between the slice hyperholomorphic setting and the classical setting is explored to characterize the boundedness and compactness of composition operators. Motivated by a question in Kumar (Adv Appl Clifford Algebras 27:1459–1477, 2017)

As a case of the Laplacian, the Helmholtz operator can be factorized using perturbed Dirac operators. In this article, we prove Schwarz lemmas for solutions of perturbed Dirac operators in vector space \(\mathbb {R}^{3}\) by the integral representation formulas. As an immediate consequence of the Schwarz lemma, one has Liouville’s theorem for monogenic functions in Clifford analysis.

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.