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.

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 such

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