Bone and Joint Research

Abstract 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) $${_{q}}\mathbf {\mathcal {D}}= \sum _{i=1}^n e_i\,{_{q}}\partial _i$$ q D = ∑ i = 1 n e i q ∂ i , where $${_{q}}\partial _i$$ q ∂ i denotes the q-partial derivative with respect to $$x_i$$ x i . This Dirac operator factorizes the q-deformed Laplace operator. Similar to the case of classical Clifford analysis, we then consider the q-deformed Euler and Gamma operators and their relations to each other. Nullsolutions of this q-Dirac equation are called q-monogenic. Using the Fischer decomposition, we can decompose the space of homogeneous polynomials into spaces of q-monogenic polynomials. Using the q-deformed Cauchy–Kovalevskaya extension theorem, we can construct q-monogenic functions. Overall, we show the analogies and the differences between classical Clifford and Jackson-Clifford analysis. In particular, q-monogenic functions need not be monogenic and vice versa.

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 invariant with respect to the $$U_q({\mathfrak {sl}}_2)$$ -action and $$*$$ -structures on  $$\mathcal {W}_q(\mathfrak {sl}_2)$$ , moreover, the square of  $$D_q$$ is central in  $$\mathcal {W}_q(\mathfrak {sl}_2)$$ . We compute the spectrum of the cubic element on finite-dimensional and Verma modules of  $$U_q(\mathfrak {sl}_2)$$ and the corresponding Dirac cohomology.

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 hyperplanes. This reinterpretation simplifies the implementation of homogeneous representations of geometric objects within the Spacetime Algebra and enables a relative view via projective geometry. Then, after utilizing the intrinsic properties of Geometric Algebra, the Wigner little group is seen to induce a projective geometric algebra as a subalgebra of the Spacetime Algebra. However, the dimension-agnostic nature of Geometric Algebra enables the generalization of induced subalgebras to $$(1+n)$$ -dimensional Minkowski geometric algebras, termed little photon algebras. The lightlike transformations (translations) in these little photon algebras are seen to leave invariant the (pseudo)canonical electromagetic field bivector. Geometrically, this corresponds to Lorentz transformations that do not change the intersection of the spacelike polarization hyperplane with the lightlike wavevector hyperplane while simultaneously not affecting the lightlike wavevector hyperplane. This provides for a framework that unifies the analysis of symmetries and substructures of point-based Geometric Algebra with mirror-based Geometric Algebra.

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 Lu (Boundary value problems of analytic functions. World Scientific, Singapore, 1993), which greatly refines the coefficients estimate of inequality in Du et al. (Acta Math Sci 29B(1):210–224, 2009) and Zhang (Complex Var Elliptic Equ 52(6):455–473, 2007). Then, we obtain the H-B theorems which extend and improve the corresponding results in Du et al. (2009) and Wang and Du (Z Anal Anwend, 2024).

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 $$\Lambda _n:=\dagger _1\cdots \dagger _n$$ , as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied in details, free $${\mathbb {M}}_n$$ -modules and their linear operators are considered and, finally, we develop Hilbert spaces on the multicomplex algebra.

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 Boolean lattices. In this part of the MiTopos theme I follow the historic approach to mathematical physics and remain with the Clifford algebra of the Minkowski space. It turns out that this interface is a basic morphogenetic structure inherent in both matter and thought. It resides in both oriented spaces and logic, and most surprisingly is closely linked to the symmetries of elementary particle physics.

Abstract 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 the Spacetime Algebra Cl(3, 1) these equations are the equations for a self-dual source free Maxwell field, and for a massless uncharged Spinor. This shows a deep link between fundamental physics and the Clifford geometry of Spacetime.

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 are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.

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 video compression, in which the reconstruction of an approximate original image is achieved by computing a limited number of the largest eigenvalues, yielding a favorable outcome. In summary, by utilizing block tensors in its iterations, this method converges more rapidly to the desired eigenvalues and eigentensors, which significantly reduces the time required for videos compression.

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 biaxially domains and RHBVPs for generalized analytic functions on the complex plane. We derive explicit solutions and conditions for solvability of RHBVPs for biaxially monogenic functions. Additionally, we explore related Schwarz problems and RHBVPs for biaxially meta-monogenic functions.

This paper presents an approach for extracting points from conic intersections by using the concept of pencils. This method is based on QC2GA—the two-dimensional version of QCGA (Quadric Conformal Geometric Algebra)—that is demonstrated to be equivalent to GAC (Geometric Algebra for Conics). A new interpretation of QC2GA and its objects based on pencils of conics and point space elements is presented, enabling the creation, constraining, and exploitation of pencils of conics. A Geometric Algebra method for computing the discriminants and center point of a conic will also be presented, enabling the proposition of an algorithm for extracting points from a conic intersection object.