A Correction to this paper has been published: https://doi.org/10.1007/s10711-015-0119-z

In this paper, based on a proposed notion of generalized conjugate harmonic pairs in the framework of complex Clifford analysis, necessary and sufficient conditions for the solvability of inhomogeneous perturbed generalized Moisil–Teodorescu systems in higher dimensional Euclidean spaces are proved. As an application, we derive corresponding solvability conditions for the inhomogeneous Maxwell’s equations

The k-Cauchy–Fueter complex in quaternionic analysis was generalized to the Heisenberg group in a previous paper (Ren et al., in Adv Appl Clifford Algebras 30(2): Paper No. 20, 2020). But it is not known whether this differential complex is exact or not. In this paper, we apply the Penrose transform (the twistor method) to a double fibration of homogeneous spaces of \(\mathrm{SO}(2N,{\mathbb {C}})\)

The aim of this paper is to generalise the construction of n-ary Hom-Lie bracket by means of an \((n-2)\)-cochain of given Hom-Lie algebra to super case inducing n-Hom-Lie superalgebras. We study the notion of generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu and n-Hom-Lie superalgebras and their relation with generalized derivations and Rota-Baxter operators of Hom-Lie superalgebras

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras—subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and

Beltrami fields are complex vector fields \({\mathbf {F}}\) which satisfy the equation \({\text {curl}} {\mathbf {F}} + \lambda {\mathbf {F}}=0.\) Such fields appear in astrophysics, electromagnetics and plasma physics. We construct a complete system of solutions to the differential equation \((D+\lambda (x_3)+M^{\gamma (x_3){\mathbf {e}}_3})u=0\) for a complex quaternionic valued function u in a symmetric

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split

The theory of polynomial bases in one complex variable, as given by Whittaker and Cannon, was extended partially to the scope of Clifford analysis, where many results have been handled from different aspects. The emphasis of this paper is on introducing a certain type of bases of axially monogenic polynomials generated by the maximum modulus base. Determining the convergence properties of such bases

In para-Grassmann algebra a noncommutative, associative star product \(*\) (the Moyal product), which is a direct generalization of the star product in the algebra of Grassmann numbers is introduced. Isomorphism between the algebra of para-Grassmann numbers of order two equipped with the star product and the algebra of creation and annihilation operators \(a_{k}^{\pm }\) obeying the para-Fermi statistics

The CAM (Composition-Algebra based Methodology) (Wolk in Int J Mod Phys A 35:2050037–2050075, 2020; Pap Phys 9:090002–09007, 2017; Phys Scr 94:025301–025307, 2019; Adv Appl Clifford Algebras 27(4):3225–3234, 2017; J Appl Math Phys 6:1537–1538, 2018; Phys Scr 94:105301–105306, 2019; Adv Appl Clifford Algebras 30:4–14, 2020) [46,47,48,49,50,51,52] gauge model’s fiber bundle framework—previously shown

In this paper, we use the equivalence canonical form of four quaternion matrices to consider two systems of quaternion matrix equations. We derive some practical necessary and sufficient conditions for the existence of a solution to these two systems in terms of ranks of the quaternion matrices involved. We also give the general solutions to the systems when the solvability conditions are satisfied

We revisit the topic of two-state quantum systems using the Clifford Algebra in three dimensions \(Cl_3\). In this description, both the quantum states and Hermitian operators are written as elements of \(Cl_3\). By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system

In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real (“motion polynomials”), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition

In this paper, the study of linear differential equations involving one conventional and two nilpotent variables is started. This is a natural extension of the case of one involved nilpotent (para-Grassmann) variable studied earlier. In the case considered here, the two nilpotent variables are assumed to commute, hence they are generators of a (generalized) zeon algebra. Using the natural para-supercovariant

The aim of this paper is to generalise the construction of 3-BiHom-Lie superalgebras. We provide some properties that can be lifted to their \(T^{*}\)-extensions such as nilpotency, solvability and decomposition. We study representations, \(T_{\theta }\)-extensions and \(T^{*}_{\theta }\)-extensions of 3-BiHom-Lie superalgebras and prove the necessary and sufficient conditions for a 2n-dimensional

We show that if Projective Geometric Algebra (PGA), i.e. the geometric algebra with degenerate signature (n, 0, 1), is understood as a subalgebra of Conformal Geometric Algebra (CGA) in a mathematically correct sense, then flat primitives share the same representation in PGA and CGA. Particularly, we treat duality in PGA in the framework of CGA. This leads to unification of PGA and CGA primitives which

Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces

Because of the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\), it is possible to complexify the spacetime Clifford algebra \(C \ell _{1,3}({\mathbb {R}})\) by adding one additional timelike dimension to the Minkowski spacetime. In a recent work we showed how this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the new temporal

Conformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer graphics. In this paper we implement a simple ray tracer in CGA with a Blinn–Phong lighting model, before putting it to use to examine ray intersections with surfaces generated from the direct interpolation of

A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators

For functions \(f\in L^1\left( \mathbb {R}^2,\mathcal {H}\right) \) with the quaternion Fourier transform (QFT) \(\widehat{f}\) we give necessary and sufficient conditions in terms of \(\widehat{f}\) to ensure that f belongs either to one of the generalized Lipschitz classes \(H_{\alpha _1,\alpha _2}^m\) and \(h_{\alpha _1,\alpha _2}^m\) for \(0<\alpha _1,\alpha _2

We consider the technique of lifting frames to higher dimensions with the ridge idea that originally was introduced by Grafakos and Sansing. We pursue a novel approach with regard to a non-commutative setting, concretely the skew-field of quaternions. Moreover, we allow for splitting dimensions and for lifting with regard to multi-ridges. To this end, we introduce quaternionic Sobolev spaces and prove

Ternary Clifford algebras are an essential ingredient in a cubic factorization of the Laplacian and using a ternary Clifford analysis build on such spaces one obtains a Dirac-type operator D such that \(D^3=\Delta \). This paper is a continuation of the work of the authors in describing properties of generalized ternary Clifford algebras. Here we explore a blade decomposition and symmetries of these

In this article, we provide a complete classification of left octonionic modules (finite or infinite dimensions) in terms of new notions such as associative elements and conjugate associative elements. We give a simple approach to determine the irreducible left \(\mathbb {O}\)-modules by utilizing the Clifford algebra \(C\ell _{7}\). We find that every left \(\mathbb {O}\)-module has a basis in some

The Clifford Fourier transform (CFT) has been shown to be a crucial tool in the Clifford analysis. The purpose of this paper is to derive an analog of Titchmarsh’s theorems for the CFT for functions satisfying the Lipschitz and Dini–Lipschitz conditions in the space \(L^p(\mathbb {R}^{p,q},C\ell (p,q)), 1

In this present work, we generalize Titchmarsh’s theorem for the complex- or hypercomplex-valued functions. Firstly, we examine the order of magnitude of the windowed linear canonical transform (WLCT) of complex-valued functions that achieved certain Lipschitz conditions on \({\mathbb {R}}\). Secondly, we studied the order of magnitude of the 2-D continuous quaternion wavelet transform (CQWT) of certain

This paper deals with generalized axially symmetric potentials (GASP) which are solutions of a Weinstein-type equation in \(\mathbb {R}^3\) \(\dfrac{\partial ^2\Phi }{\partial x^2}+\dfrac{\partial ^2\Phi }{\partial y^2}+\dfrac{\partial ^2\Phi }{\partial z^2}+\dfrac{2(m+1)}{z}\dfrac{\partial \Phi }{\partial z}=0,\ m\in \mathbb {N}.\) GASP have been investigated by several generations of mathematicians

In this paper, the concept of the quaternionic adjoint matrix \(\chi _A\) of an elliptic biquaternion matrix A is introduced, which enable one to discuss the elliptic biquaternion problems through the quaternion ones. By this new concept, some fundamental problems, such as the right eigenvalues and eigenvectors, the singular value decomposition and the inverse can be investigated. Moreover, the least-squares

Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), González Cervantes

This paper explains that the definition of Moore–Penrose inverses in a given algebra A does not at all require any matrix representation of A. The pseudo-inverses y of a given element x of A (such that \(xyx=x\) and \(yxy=y\)) involve the right ideals complementary to xA and the left ideals complementary to Ax; the Moore–Penrose inverse corresponds to the complementary right and left ideals selected

Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: \(\min \Vert \mathbf {A}\mathbf {X}\mathbf {B}-\mathbf {C}\Vert _F\) subject to the constraints imposed to the right column space of \({\mathbf{A}}\) and the left row space of \({\mathbf{B}}\). Unique solution to the considered

This paper deals with the static Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} div(\Phi \overrightarrow{E})&{}=0,\\ \ curl\overrightarrow{E}&{}=0,\ (x_0,x_1,x_2)\in \mathbb {R}^3. \end{array} \right. \end{aligned}$$ The system is reformulated in quaternion analysis by Kravchenko in the form \(\mathcal {L}F=0\) with \(\mathcal {L}F=DF+F\alpha \). We consider special cases of the coefficient

In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space \(L^2({\mathbb {R}}^2,{\mathbb {H}})\), such as Parseval’s formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations

We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ in the upper half space \(\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}\). They are called \(\alpha \)-hyperbolic harmonic. An important result is that a function f is \(\alpha \)-hyperbolic

Unfortunately, there is the slight error in the affiliation of Dr. H. Banouh.

In this paper, we present a characterization of all linear fractional order partial differential operators with complex-valued coefficients that are associated to the generalized fractional Cauchy–Riemann operator in the Riemann–Liouville sense. To achieve our goal, we make use of the technique of an associated differential operator applied to the fractional case.

The special relativistic (space-time) Fourier transform (SFT) in Clifford algebra \(Cl_{(3,1)}\) of space-time, first introduced from a mathematical point of view in Hitzer (Adv Appl Clifford Algebras 17:497–517, 2007), extends the quaternionic Fourier transform to functions, fields and signals in space-time. The purpose of this paper is to advance the study of the SFT and investigate important properties

In this paper, we consider a version of the fractional Clifford–Fourier transform (FrCFT) and study its several properties and applications to partial differential equations in Clifford analysis. First, we give the definition of the FrCFT and its inverse transform in the form of integral. Then, we discuss the relationship between the FrCFT and the Clifford–Fourier transform (CFT) and give some properties

In this paper, we first define a two-sided Clifford Fourier transform(CFT) and its inverse transformation on \(L^{1}\) space. Then we study the differential of the two-sided CFT, the k-th power of \(F \{h\}\), Plancherel identity and time-frequency shift of the two-sided CFT. Finally we discuss the uncertainty principle of the two-sided CFT and give an application of the two-sided CFT to a partial

We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.

Unfortunately, the author names in the reference 9 was wrongly published in the original article and the reference should be

In this paper we continue our study on the density of the set of quaternionic polynomials in function spaces of slice regular functions on the unit ball by considering the case of the Bloch and Besov spaces of the first and of the second kind. Among the results we prove, we show some constructive methods based on the Taylor expansion and on the convolution polynomials. We also provide quantitative

In this paper we consider some basic properties of quaternionic measures.

This paper is concerned with properties for a class of degenerate elliptic equations in Clifford analysis. Here we obtain a direct proof of the existence and uniqueness for the Dirac equations by the method of variational integral. Also, we get the Poincaré inequalities for the case \(q<1\).

We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair \(h_1\), \(h_2\) of zonal harmonic functions such that \(f=h_1-\bar{x} h_2\). We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

In 1973, Kanzaki introduced a category \(\text {Quad}_2(K)\) made of generalized quadratic spaces (E, f, g) where E was a vector space of finite dimension over a field K, f a linear form on E, and g a quadratic form on E such that the bilinear form \((a,b)\longmapsto f(a)f(b)+g(a+b)-g(a)-g(b)\) was non-degenerate on E. There are direct sums and tensor products in \(\text {Quad}_2(K)\), and its objects

In this paper we present a new tool for Mathematica users, based on the new web-based geometric algebra algorithm optimizer (GAALOP). GAALOPWeb for Mathematica now supports the Mathematica user with an intuitive interface for the development, testing and visualization of geometric algebra algorithms, combining the geometric intuitiveness of geometric algebra with an efficient development of algorithms

In this paper we introduce generalized \((k_i)\)-monogenic functions in Clifford analysis. They are the general types of the k-hypermonogenic functions founded by Leutwiler and Eriksson. Each component of a generalized \((k_i)\)-monogenic function is a solution of a generalized Weinstein’s equation. We will construct \(2^n\) generalized Cauchy kernels and give an integral representation of the generalized

We obtain the octonionic Bergman kernel for half space in the octonionic analysis setting by two different methods. As a consequence, we unify the kernel forms in both complex analysis and hyper-complex analysis.

Solving boundary value problems with boundary element methods requires specific Green functions suited to the boundary conditions of the problem. Using vector algebra, one often needs to use a Green function for the Helmholtz equation whereas it is a solution of the perturbed Dirac equation that is required for solving electromagnetic problems using Clifford algebra. A wealth of different Green functions

The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8, \(\mathbb {C})\) of complex \(8 \times 8\) matrices. This algebra is well motivated by its close relation to the normed

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.