We study the quaternionic Carleson measure, which provides an embedding of the slice regular Hardy space \({\mathcal {H}}^p({\mathbb {B}})\) into \(L^s({\mathbb {B}}, \text {d}\mu )\) with \(s>p.\) A new criterion is needed for a finite positive Borel measure to be an \(({\mathcal {H}}^p({\mathbb {B}}),s)\)-Carleson measure, given by the uniform integrability of slice Cauchy kernels. It turns out that

We describe Rota-Baxter operators on split octonions. It turns out that up to some transformations there exists exactly one such non-splitting operator over any field. We also obtain a description of all decompositions of split octonions over a quadratically closed field of characteristic different from 2 into a sum of two subalgebras, which describes the splitting Rota-Baxter operators. It completes

Following the idea of the fractional space-time Fourier transform, a linear canonical space-time transform for 16-dimensional space-time \(C\ell _{3,1}\)-valued signals is investigated in this paper. First, the definition of the proposed linear canonical space-time transform is given, and some related properties of this transform are obtained. Second, the convolution operator and the corresponding

We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic form in ordinary Clifford algebras. We present a natural realization of unitary Lie groups, which are important in physics and other applications, using only operations

It is easier to investigate phenomena in particle physics geometrically by exploring a real solution to the Dirac–Hestenes equation instead of a complex solution to the Dirac equation. The current research presents a formulation of the multidimensional Dirac–Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra \(\mathbb {C}\otimes C \hspace{-1.00006pt}\ell

This work presents a coordinate sphere bundle defined from theory of slice regular functions whose bundle projection and some real Banach spaces induce coordinate sphere bundles in which the quaternionic Banach modules of the slice regular functions of Bloch, Besov and Dirichlet are the base spaces. Finally, this work shows that Möbius invariant property of these quaternionic Banach modules defines

Holomorphic Cliffordian functions of order k are functions in the kernel of the differential operator \(\overline{\partial }\Delta ^k\). When \(\overline{\partial }\Delta ^k\) is applied to functions defined in the paravector space of some Clifford Algebra \(\mathbb {R}_m\) with an odd number of imaginary units, the Fueter–Sce construction establishes a critical index \(k=\frac{m-1}{2}\) (sometimes

This paper completely describes the form of all unital additive surjective maps, on the algebra of all bounded right linear operators acting on a two-sided quaternionic Banach space, that preserve any one of (left, right) invertibility, (left, right) zero divisors and (left, right) topological divisors of zero in both directions.

In this article, a Geometric Algebra (GA) and Geometric Calculus (GC) based exposition is carried out dealing with the formal characterization of sliding regimes for general Single-Input-Single-Output (SISO) nonlinear switched controlled Hamiltonian systems. Necessary and sufficient conditions for the local existence of a sliding regime on a given vector manifold are presented. Feedback controller

The paper is concerned with the definition, properties and uncertainty principles for the multi-dimensional quaternionic offset linear canonical transform. First, we define the multi-dimensional offset linear canonical transform based on matrices with symplectic structure. Then, we focus on the definition of the multi-dimensional quaternionic offset linear canonical transform and the corresponding

A monogenic function of two vector variables is a function annihilated by two Dirac operators. We give the explicit form of differential operators in the Dirac complex resolving two Dirac operators and prove its ellipticity directly. This opens the door to apply the method of several complex variables to investigate this kind of monogenic functions. We prove the Poincaré lemma for this complex, i.e

The paper deals with two second order elliptic systems of partial differential equations in Clifford analysis. They are of the form \({^\phi \!\underline{\partial }}f{^\psi \!\underline{\partial }}=0\) and \(f{^\phi \!\underline{\partial }}{^\psi \!\underline{\partial }}=0\), where \({^\phi \!\underline{\partial }}\) stands for the Dirac operator related to a structural set \(\phi \). Their solutions

Quaternion algebra \(\mathbb {H}\) is a noncommutative associative algebra, and recently quaternionic Fourier analysis has become the focus of an active research due to their potentials in signal analysis and color image processing. The problems related to quaternions are nontrivial and challenging due to noncommutativity of quaternion multiplication. This paper is devoted to establishing the framework

We consider square matrices over \(\mathbb {C}\) satisfying an identity relating their eigenvalues and the corresponding eigenvectors re-proved and discussed by Denton, Parker, Tao and Zhang, called the eigenvector-eigenvalue identity. We prove that for an eigenvalue \(\lambda \) of a given matrix, the identity holds if and only if the geometric multiplicity of \(\lambda \) equals its algebraic multiplicity

We consider an extension of Jackson calculus into higher dimensions and specifically into Clifford analysis for the case of commuting variables. In this case, Dirac is the operator of the first q-partial derivatives (or q-differences) \({_{q}}\mathbf {\mathcal {D}}= \sum _{i=1}^n e_i\,{_{q}}\partial _i\), where \({_{q}}\partial _i\) denotes the q-partial derivative with respect to \(x_i\). This Dirac

This paper presents a discussion on the branching problem that arises in the Weil representation of the exceptional Lie group of type \(G_2\). The focus is on its decomposition under the threefold cover of \(SL(2,\, {\mathbb {R}})\) associated with the short root of \(G_2\).

We construct the q-deformed Clifford algebra of \(\mathfrak {sl}_2\) and study its properties. This allows us to define the q-deformed noncommutative Weil algebra \(\mathcal {W}_q(\mathfrak {sl}_2)\) for \(U_q(\mathfrak {sl}_2)\) and the corresponding cubic Dirac operator \(D_q\). In the classical case this was done by Alekseev and Meinrenken in 2000. We show that the cubic Dirac operator \(D_q\) is

This paper presents the Geometric Algebra approach to the Wigner little group for photons using the Spacetime Algebra, incorporating a mirror-based view for physical interpretation. The shift from a point-based view to a mirror-based view is a modern movement that allows for a more intuitive representation of geometric and physical entities, with vectors and their higher-grade counterparts viewed as

In this article, we verify the boundedness of the Cauchy type integral operators under the generalized Hölder norm in Clifford analysis, which are called H-B theorems of the Cauchy integral operators in Clifford analysis. We first demonstrate the generalized 2P theorems and the generalized Muskhelishvili theorem in Clifford analysis by Du’s method derived from Du (J Math (PRC) 2(2):115–12, 1982) and

In this article we study some algebraic aspects of multicomplex numbers \({\mathbb {M}}_n\). For \(n\ge 2\) a canonical representation is defined in terms of the multiplication of \(n-1\) idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy \(\Lambda _n\), i.e. a composition of the n multicomplex conjugates \(\Lambda

In the present article, the research work of many years is summarized in an interim report. This concerns the connection between logic, space, time and matter. The author always had in mind two things, namely 1. The discovery/construction of an interface between matter and mind, and 2. some entry points for the topos view that concern graphs, grade rotations and contravariant involutions in geometric

The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The relatively recent field of Clifford Analysis lets us extend many results from Complex Analysis to higher dimensions. In this paper, I decompose the Cauchy-Riemann equations for a general Clifford algebra into grades using the Geometric Algebra formalism, and show that for

In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of \(AX-XB=C\), \(AXB-CX^{T}D=E\) and (anti)centrosymmetric solution of \(AXB-CYD=E\). And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations

An infinite-dimensional family of exact solutions of a three-dimensional biharmonic equation was constructed by the hypercomplex method.

In this paper, we investigate the eigenvalues of quaternion tensors under Einstein Product and their applications in color video processing. We present the Ger\(\check{s}\)gorin theorem for quaternion tensors. Furthermore, we have executed some experiments to validate the efficacy of our proposed theoretical framework and algorithms. Finally, we contemplate the application of this methodology in color

We compute and explore the full geometric product of two oriented points in conformal geometric algebra Cl(4, 1) of three-dimensional Euclidean space. We comment on the symmetry of the various components, and state for all expressions also a representation in terms of point pair center and radius vectors.

We are dedicated to addressing Riemann–Hilbert boundary value problems (RHBVPs) with variable coefficients, where the solutions are valued in the Clifford algebra of \(\mathbb {R}_{0,n}\), for biaxially monogenic functions defined in the biaxially symmetric domains of the Euclidean space \(\mathbb {R}^{n}\). Our research establishes the equivalence between RHBVPs for biaxially monogenic functions defined

In this paper, we extend Fueter’s theorem in hypercomplex function theory to encompass a class of pseudoanalytic functions associated with the main Vekua equation. This class includes Duffin’s \(\mu \)-regular functions as special cases, which correspond to the Yukawa equation. As the parameter \(\mu \rightarrow 0\), we recover the classical Fueter’s theorem.

The Plemelj-Sokhotski formulas, which deal with limiting values of the Bochner-Martinelli type integral, are powerful tools for analyzing boundary value problems. This article aims to study the boundary behavior of the Bochner-Martinelli type integral formula for the k-Cauchy-Fueter operator. Specifically, we consider the Plemelj-Sokhotski formulas, which will extend the corresponding results in the

We design several real representations of split quaternion matrices with the primary objective of establishing both necessary and sufficient conditions for the existence of solutions within a system of split quaternion matrix equations. This includes conditions for the general solution without any constraints, as well as \(X=\pm X^{\eta }\) solutions and \(\eta \)-(anti-)Hermitian solutions. Furthermore

This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in computer science

In this paper the Cayley–Laplace operator \(\Delta _{xu}\) is considered, a rotationally invariant differential operator which can be seen as a generalisation of the classical Laplace operator for functions depending on wedge variables \(X_{ab}\) (the minors of a matrix variable). We will show that the Bessel–Clifford function appears naturally in the framework of two-wedge variables, and explain how

A parametrization, given by the Euler angles, of Hermitian matrix generators of even and odd non-degenerate Clifford algebras is constructed by means of the Kronecker product of a parametrized version of Pauli matrices and by the identification of all possible anticommutation sets. The internal parametrization of the matrix generators allows a straightforward interpretation in terms of rotations, and

In this paper, we characterize all N-dimensional hypercomplex numbers having unital Archimedean f-algebra structure. We use matrix representation of hypercomplex numbers to define an order structure on the matrix spectra. We prove that the unique (up to isomorphism) unital Archimedean f-algebra of hypercomplex numbers of dimension \(N \ge 1\) is that with real and simple spectrum. We also show that

Building from ideas of hypercomplex analysis on the quaternionic unit ball, we introduce Hermitian, Riemannian and Kähler-like structures on the latter. These are built from the so-called regular Möbius transformations. Such geometric structures are shown to be natural generalizations of those from the complex setup. Our structures can be considered as more natural, from the hypercomplex viewpoint

In this paper, we are concerned with the problem of locating the zeros of polynomials and regular functions with quaternionic coefficients when their real and imaginary parts are restricted. The extended Schwarz’s lemma, the maximum modulus theorem, and the structure of the zero sets defined in the newly constructed theory of regular functions and polynomials of a quaternionic variable are used to

In this paper, we present two simple methods for constructing Beltrami fields. The first one consists of a composition of operators, including a quaternionic transmutation operator as well as the computation of formal powers for the function \(f(x)=e^{\textbf{i}\lambda x}\). For the second method, we generate Beltrami fields from harmonic functions, and using the intrinsic relation between the normal

Due to its potential application in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention. This paper addresses quaternionic subspace Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. We characterize subspace quaternionic Gabor frames in terms of quaternionic Zak transformation matrices. For an

We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are \(*\)-quantum spaces for the quantum orthogonal group \(\mathcal {O}(SO_q(3))\). We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of \(SO_q(3)\). The line bundles are associated to the quantum principal bundle via representations of SO(2)

In this paper we give an exposition of several recent results on local symmetries of real submanifolds in complex space, featuring new examples and important corollaries. Departing from Levi non-degenerate hypersurfaces, treated in the classical Chern–Moser theory, we explore three important classes of manifolds, which naturally extend the classical case. We start with quadratic models for real submanifolds

This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group \({\mathbb {S}}\), i.e., the unit 3-sphere in the quaternion

The theory of slice regular functions has been developed rapidly in the past few years, and most properties are given in slices at the early stage. In 2013, Colombo et al. introduced a non-constant coefficients differential operator to describe slice regular functions globally, and this brought the study of slice regular functions in a global sense. In this article, we introduce a slice Cauchy–Riemann

From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time.

Let $$\begin{aligned} \mathcal {D}_{16}=\left\{ Z\in \mathcal {M}_{1,2}(\mathfrak {C}^{c}):\;\begin{array}{lll} 1-\left\langle Z,Z \right\rangle +\left\langle Z^{\sharp },Z^{\sharp }\right\rangle>0,\\ 2-\left\langle Z,Z \right\rangle \; >0\end{array}\right\} \end{aligned}$$ be an exceptional domain of non-tube type and let \(\mathcal {U}_{\nu }\) and \(\mathcal {W}_{\nu }\) the associated generalized

In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion-valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff–Young inequality for QQPFT, which in particular sharpens the constant

In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate