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

Let \(X=Sp(1,n)/Sp(n)\) be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where n is greater than or equal to 1. The Fürstenberg boundary of X is denoted as \(\Sigma \). In this paper, we focus on the Plancherel formula on X associated with the Poisson transform of vector-valued \(L^2\)-space on \(\Sigma \). Through the Fourier-Jacobi transform and the Fourier-Poisson

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures

This paper investigates the distribution function and nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\)-valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for \( \mathbb{B}\mathbb{C}\)-valued functions, which characterizes valuable insights into the behavior and structure of \(\mathbb{B}\mathbb{C}\)-valued functions, allowing

In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform

We introduce new types of fractional generalized elliptic operators on a compact Riemannian manifold with Clifford bundle. The theory is applicable in well-defined differential geometry. The Connes-Moscovici theorem gives rise to dimension spectrum in terms of residues of zeta functions, applicable in the presence of multiple poles. We have discussed the problem of scalar fields over the unit co-sphere

In this paper, we introduce a new multidimensional fractional S transform \(S_{\phi ,\varvec{\alpha },\lambda }\) using a generalized fractional convolution \(\star _{\varvec{\alpha },\lambda }\) and a general window function \(\phi \) satisfying some admissibility condition. The value of \(S_{\phi ,\varvec{\alpha },\lambda }f\) is also written in the form of inner product of the input function f with

In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by Clifford cohomology. We show that Clifford cohomology controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of

The quaternion ambiguity function is an expansion of the standard ambiguity function using quaternion algebra. Various properties such as linearity, translation, modulation, Moyal’s formula and inversion identity are studied in detail. In addition, an interesting interaction between the quaternion ambiguity function and the quaternion Fourier transform is demonstrated. Based on these facts, we seek

For the right-sided multivariate continuous quaternion wavelet transform (CQWT), we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg’s inequality for the quaternion wavelet transform. Finally, we extend local uncertainty principle for a set of finite measure to CQWT.

Let \(\Gamma \) be a d-summable surface in \(\mathbb {R}^m\), i.e., the boundary of a Jordan domain in \( \mathbb {R}^m\), such that \(\int \nolimits _{0}^{1}N_{\Gamma }(\tau )\tau ^{d-1}\textrm{d}\tau <+\infty \), where \(N_{\Gamma }(\tau )\) is the number of balls of radius \(\tau \) needed to cover \(\Gamma \) and \(m-1\frac{d}{m}\), the operator \(S_\Gamma ^*\) transforms functions of the higher

In this paper, the octonion matrix equation \(AXB=C\) is studied based on semi-tensor product of matrices. Firstly, we propose the left real element representation and the right real element representation of octonion. Then we obtain the expression of the least squares Hermitian solution to the octonion matrix equation \(AXB=C\) by combining these representations with \(\mathcal {H}\)-representation

Let X be a two-sided quaternionic Banach space and let \(A, B, C: X \longrightarrow X\) be bounded right linear quaternionic operators such that \(ACA=ABA\). Let q be a non-zero quaternion. In this paper, we investigate the common properties of \((AC)^{2}-2Re(q)AC+|q|^2I\) and \((BA)^{2}-2Re(q)BA+|q|^2I\) where I stands for the identity operator on X. In particular, we show that $$\begin{aligned} \sigma

In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines

The aim of this paper is two-fold. First, we obtain various inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Sasakian space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion.

A null vector is an algebraic quantity with the property that its square is zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by \({{\mathcal {N}}}\). The rules of addition and multiplication in \({{\mathcal {N}}}\) are taken to be the same as those for real and complex square matrices. A distinct pair of null vectors is