# Advances in Applied Clifford Algebras

Online ISSN: 1661-4909
Print ISSN: 0188-7009
Recent publications
• Qian Huang
• Fuli He
• Min Ku
We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ($$\end{document}for short RHBVPs)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big )$$\end{document} for axially monogenic functions on Euclidean space Rn+1,n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n+1},n\in {\mathbb {N}}$$\end{document}. With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to (D-α)ϕ=0,α∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathcal {D}}-\alpha \big )\phi =0,\alpha \in {\mathbb {R}}$$\end{document}.

• Yun Shi
• Guangzhen Ren
The k-Cauchy–Fueter operator and the tangential k-Cauchy–Fueter operator are the quaternionic counterpart of Cauchy–Riemann operator and the tangential Cauchy–Riemann operator in the theory of several complex variables, respectively. In Wang (On the boundary complex of the k-Cauchy–Fueter complex, arXiv:2210.13656), Wang introduced the notion of right-type groups, which have the structure of nilpotent Lie groups of step-two, and many aspects of quaternionic analysis can be generalized to this kind of group. In this paper we generalize the right-type group to any step-two case, and introduce the generalization of Cauchy–Fueter operator on Hn×Rr.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^n\times {\mathbb {R}}^r.$$\end{document} Then we establish the Bochner–Martinelli type formula for tangential k-Cauchy–Fueter operator on stratified right-type groups.

We study the notions of centrally-extended higher ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-derivations and centrally-extended generalized higher ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-derivations. Both are shown to be additive in a ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-ring without nonzero central ideals. Also, we prove that in semiprime ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-rings with no nonzero central ideals, every centrally-extended (generalized) higher ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-derivation is a (generalized) higher ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-derivation.

Bicomplex convolutional neural networks (BCCNN) are a natural extension of the quaternion convolutional neural networks for the bicomplex case. As it happens with the quaternionic case, BCCNN has the capability of learning and modelling external dependencies that exist between neighbour features of an input vector and internal latent dependencies within the feature. This property arises from the fact that, under certain circumstances, it is possible to deal with the bicomplex number in a component-wise way. In this paper, we present a BCCNN, and we apply it to a classification task involving the colourized version of the well-known dataset MNIST. Besides the novelty of considering bicomplex numbers, our CNN considers an activation function a Bessel-type function. As we see, our results present better results compared with the one where the classical ReLU activation function is considered.

The fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath A⊗H4op,H4,ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( A\otimes H_{4}^{op}, H_{4}, \psi \right)$$\end{document}, where H4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{4}$$\end{document} is the Sweedler 4-dimensional Hopf algebra over a field k and A=Cl(α,β,γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=Cl(\alpha , \beta , \gamma )$$\end{document} is the Clifford algebra generated by two elements G, X with relations G2=α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^{2}=\alpha$$\end{document}, X2=β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{2}=\beta$$\end{document} and XG+GX=γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$XG+GX=\gamma$$\end{document}, (α,β,γ∈k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , \beta , \gamma \in k$$\end{document}) which becomes naturally an H4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{4}$$\end{document}-comodule algebra. We show that, when chark≠2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{char}\left( k \right) \ne 2,$$\end{document} this cowreath is always separable and h-separable as well.

The present study is the first of its kind which aims to analyze the Clifford-valued functions by introducing the notion of a two-sided Clifford-valued linear canonical transform in $L^2(\mathbb R^{n}, C\ell_{n})$, which not only embodies the classical Clifford Fourier transform, but also yields another new variant of Clifford transforms based on the fractional Clifford Fourier transform. To begin with, we study all fundamental properties of the proposed transform, including the inversion formula, translation and scaling covariances, Plancherel, and differentiation theorems. Subsequently, we introduce a novel Clifford-valued Mustard convolution associated with the proposed transform and express the proposed convolution in terms of a linear combination of eight standard convolutions.

Complete hom-Lie superalgebras are considered and some equivalent conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra are established. In particular, the relation between decomposition and completeness for a hom-Lie superalgebra is described. Moreover, some conditions that the linear space of αs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ^{s}$$\end{document}-derivations of a hom-Lie superalgebra to be complete and simply complete are obtained.

The Hayman Theorem of left-monogenic function in a Clifford Analysis Setting is established in this article. A few established conclusions regarding subharmonic functions in Euclidean half space are extended to Clifford half space.

For each quadratic form Q∈Quad(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\in \text{ Quad }(V)$$\end{document} on a vector space over a field K,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K},$$\end{document} we can define the Clifford algebra Cl(V,Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Cl}\,}}(V,Q)$$\end{document} as the quotient T(V)/I(Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{T}\,}}(V)/I(Q)$$\end{document} of the tensor algebra T(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{T}\,}}(V)$$\end{document} by the two-sided ideal generated by expressions of the form x⊗x-Q(x),x∈V.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\otimes x-Q(x),\, x\in V.$$\end{document} In the present paper we consider the whole family {Cl(V,Q):Q∈Quad(V)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{{{\,\textrm{Cl}\,}}(V,Q):\, Q\in \text{ Quad }(V)\}$$\end{document} in a geometric way as a Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document}-graded vector bundle over the base manifold Quad(V).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Quad }(V).$$\end{document} Bilinear forms F∈Bil(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in \text{ Bil }(V)$$\end{document} act on this bundle providing natural bijective linear mappings λ¯F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{\lambda }_F$$\end{document} between different Clifford algebras Cl(V,Q).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Cl}\,}}(V,Q).$$\end{document} Alternating (or antisymmetric) forms induce vertical automorphisms, which we propose to consider as ‘gauge transformations’. We develop here the formalism of Bourbaki, which generalizes the well known Chevalley’s isomorphism Cl(V,Q)→End(⋀(V))→⋀(V).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Cl}\,}}(V,Q)\rightarrow {{\,\textrm{End}\,}}(\bigwedge (V))\rightarrow \bigwedge (V).$$\end{document} In particular we realize the Clifford algebra twisting gauge transformations induced by antisymmetric bilinear forms as exponentials of contractions with elements of ⋀2(V∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigwedge ^2(V^*)$$\end{document} representing these forms. Throughout all this paper we intentionally avoid using the so far accepted term “Clifford algebra of a bilinear form” (known otherwise as “Quantum Clifford algebra”), which we consider as possibly misleading, as it does not represent any well defined mathematical object. Instead we show explicitly how any given Clifford algebra Cl(Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Cl}\,}}(Q)$$\end{document} can be naturally realized as acting via endomorphisms of any other Clifford algebra Cl(Q′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Cl}\,}}(Q')$$\end{document} if Q′=Q+QF,F∈Bil(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q'=Q+Q_F,\, F\in \text{ Bil }(V)$$\end{document} and QF(x)=F(x,x).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_F(x)=F(x,x).$$\end{document} Possible physical meaning of such transformations is also mentioned.

In this paper the mean ergodic theorem in bicomplex Banach modules is studied. Under appropriate conditions of boundness for the iterates compositions of a bicomplex linear and bounded operator on a bicomplex Banach module, and of weak compactness of the average sequence on the idempotent components, analogous to that of the classical case on Banach spaces, strong convergence of the mean sequence is achieved. Also, a result on ergodicity is given for bounded bicomplex strongly continuous semigroups in bicomplex Banach modules.

This paper presents a pedagogical introduction to the issue of how to implement Lorentz transformations in relativistic visualization. The most efficient approach is to use the even geometric algebra in 3+1 spacetime dimensions, or equivalently complex quaternions, which are fast, compact, and robust, and straightforward to compose, interpolate, and spline. The approach has been incorporated into the Black Hole Flight Simulator, an interactive general relativistic ray-tracing program developed by the author.

Spinors are central to physics: all matter (fermions) is made of spinors, and all forces arise from symmetries of spinors. It is common to consider the geometric (Clifford) algebra as the fundamental edifice from which spinors emerge. This paper advocates the alternative view that spinors are more fundamental than the geometric algebra. The algebra consisting of linear combinations of scalars, column spinors, row spinors, multivectors, and their various products, can be termed the supergeometric algebra. The inner product of a row spinor with a column spinor yields a scalar, while the outer product of a column spinor with a row spinor yields a multivector, in accordance with the Brauer–Weyl (Am J Math 57: 425–449, 1935, https://doi.org/10.2307/2371218) theorem. Prohibiting the product of a row spinor with a row spinor, or a column spinor with a column spinor, reproduces the exclusion principle. The fact that the index of a spinor is a bitcode is highlighted.

In this research, we look at problems in the theory of approximation of functions in real Clifford algebras. We prove analogues of direct and inverse approximation theorems in terms of best approximations of functions with bounded spectrum and the moduli of smoothness of all orders constructed by the generalized Steklov operators.

We consider spinorial fields in polar form to deduce their respective tensorial connection in various physical situations: we show that in some cases the tensorial connection is a useful tool, instead in other cases it arises as a necessary object. The comparative analysis of the different cases possessing a tensorial connection is done, investigating the analogies between space-time structures. Eventual comments on quantum field theory and specific spinors are given.

In several recent publications B-spline functions appeared with control points from abstract algebras, e.g. complex numbers, quaternions or Clifford algebras. In the context of constructions of Pythagorean hodograph curves, computations with these B-splines occur, mixing the components of the control points. In this paper we detect certain unifying patterns common to all these computations. We show that two essential components can be separated. The first one is the usual B-spline function squaring and integration, producing a new knot sequence and a new array of real coefficients for the control point computation. The second one is a special commutative multiplication which can be defined even in non-commutative algebras. We use this general Clifford algebra based approach to reconstruct some known results for the signatures (2, 0), (3, 0) and (2, 1) and add a new construction for the signature (3, 1). This last case is essential for the description of canal surfaces. It is shown that Clifford algebra is an especially suitable tool for the general description of B-spline curves with Pythagorean hodograph property. The presented unifying definition of PH B-splines is general and is not limited to any particular knot sequences or control points. In a certain sense, this paper can be considered as a continuation of the 2002 article by Choi et al. with regard to the B-splines.

In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator $$\mathcal S_a$$ with the decomposition process \begin{aligned} f=e_a\langle f, e_a\rangle +B_{a}*\mathcal S_a f, \end{aligned}where $$e_a$$ denotes the slice normalized Szegö kernel and $$B_a$$ the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka–Malmquist orthonormal system.

In every Clifford algebra Cl(V,Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Cl}(V,Q)$$\end{document} over a field, there is a Lip-schitz monoid (or semi-group) Lip(V,Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Lip}(V,Q)$$\end{document} that satisfies a lot of remarkable properties. In general, it is the multiplicative monoid generated by the vectors of the space V. Each of its two components is an irreducible algebraic submanifold. The present article is devoted to the equations that are satisifed by the coordinates of the elements of Lip(V,Q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Lip}(V,Q)$$\end{document}. The most remarkable property of these equations is their independence of the quadratic form Q. The theoretical knowledge involved in these equations is also concisely recalled.

As shared, collaborative, networked, virtual environments become increasingly popular, various challenges arise regarding the efficient transmission of model and scene transformation data over the network. As user immersion and real-time interactions heavily depend on VR stream synchronization, transmitting the entire data set does not seem a suitable approach, especially for sessions involving a large number of users. Session recording is another momentum-gaining feature of VR applications that also faces the same challenge. The selection of a suitable data format can reduce the occupied volume, while it may also allow effective replication of the VR session and optimized post-processing for analytics and deep-learning algorithms. In this work, we propose two algorithms that can be applied in the context of a networked multiplayer VR session, to efficiently transmit the displacement and orientation data from the users’ hand-based VR HMDs. Moreover, we present a novel method for effective VR recording of the data exchanged in such a session. Our algorithms, based on the use of dual-quaternions and multivectors, impact the network consumption rate and are highly effective in scenarios involving multiple users. By sending less data over the network and interpolating the in-between frames locally, we manage to obtain better visual results than current state-of-the-art methods. Lastly, we prove that, for recording purposes, storing less data and interpolating them on-demand yields a data set quantitatively close to the original one.

A novel invariant decomposition of diagonalizable $$n \times n$$ matrices into n commuting matrices is presented. This decomposition is subsequently used to split the fundamental representation of $$\mathfrak {su}({3})$$ Lie algebra elements into at most three commuting elements of $$\mathfrak {u}({3})$$. As a result, the exponential of an $$\mathfrak {su}({3})$$ Lie algebra element can be split into three commuting generalized Euler’s formulas, or conversely, a Lie group element can be factorized into at most three generalized Euler’s formulas. After the factorization has been performed, the logarithm follows immediately.

In this paper, we consider the actions of affine Yangian and $$W_{1+\infty }$$ algebra on three cases of symmetric functions. The first one is Schur functions of 2D Young diagrams. It is known that affine Yangian and $$W_{1+\infty }$$ algebra can be represented by 1 Boson field with center 1 in this case. The second case is the symmetric functions $$Y_\lambda ({\mathbf{p}})$$ of 2D Young diagrams which we defined. They become Jack polynomials when $$h_1=h, h_2=-h^{-1}$$. In this case affine Yangian and $$W_{1+\infty }$$ algebra can be represented by 1 Boson field with center $$-h_\epsilon /\sigma _3$$. The third case is 3-Jack polynomials of 3D Young diagrams who have at most N layers in z-axis direction. We show that in this case affine Yangian and $$W_{1+\infty }$$ algebra can be represented by N Boson field with center $$-h_\epsilon /\sigma _3$$. At each case, we define the Fermions $$\Gamma _m$$ and $$\Gamma _m^*$$ and use them to represent the $$W_{1+\infty }$$ algebra.

This paper addresses the study of the complexity of products in geometric algebra. More specifically, this paper focuses on both the number of operations required to compute a product, in a dedicated program for example, and the complexity to enumerate these operations. In practice, studies on time and memory costs of products in geometric algebra have been limited to the complexity in the worst case, where all the components of the multivector are considered. Standard usage of Geometric Algebra is far from this situation since multivectors are likely to be sparse and usually full homogeneous, i.e., having their non-zero terms over a single grade. We provide a complete computational study on the main Geometric Algebra products of two full homogeneous multivectors, that are outer, inner, and geometric products. We show tight bounds on the number of the arithmetic operations required for these products. We also show that some algorithms reach this number of arithmetic operations.

In this paper, we prove the sharp Pitt’s inequality for a generalized Clifford-Fourier transform which is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. As an application, the Beckner’s logarithmic uncertainty principle for the Clifford-Fourier transform is established.

In this paper, we first define supersymmetric Schur Q-functions and give their vertex operators realization. By means of the vertex operator, we obtain a series of non-linear partial differential equations of infinite order, called the super BKP hierarchy and the super BKP hierarchy governs the supersymmetric Schur Q-functions as the tau functions. Moreover, we prove that supersymmetric Schur Q-functions can be viewed as compound Schur Q-functions. This means that we can study the properties of supersymmetric Schur Q-functions according to Schur Q-functions, such as their applications in representation theory.

Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. In contrast to an ordinary graph, in which each edge connects exactly two vertices, an edge (or, “hyperedge”) can join any number of vertices in a hypergraph. Hypergraphs have been used for problems in biology, chemistry, image processing, wireless networks, and more. In the current work, zeon (“nil-Clifford”) and “idem-Clifford” graph-theoretic methods are generalized to hypergraphs. In particular, zeon and idem-Clifford methods are used to enumerate paths, trails, independent sets, cliques, and matchings in hypergraphs. An approach for finding minimum hypergraph transversals is developed, and zeon formulations of some open hypergraph problems are presented.

Most modern algorithms use convolutional neural networks to classify image data of different kinds. While this approach is a good method to differentiate between natural images of objects, big datasets are needed for the training process. Another drawback is the demand for high computational power. We introduce a new approach which involves classic feature vectors with structural information based on higher order Riesz transform. Following this way we create a framework specialized for texture data like images of rock cross-sections. The key advantages are faster computations and more versatile choices of the underlying machine learning tools while maintaining a comparable accuracy in comparison with state-of-the-art algorithms.

We generalize the space-time Fourier transform (SFT) (Hitzer in Adv Appl Clifford Algebras 17(3):497–517, 2007) to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3, 1)-valued signals over the domain of space-time (Minkowski space) $$\mathbb {R}^{3,1}.$$ We establish how it can be computed in terms of the SFT, and introduce its properties of multivector coefficient linearity, shift and modulation, inversion, Rayleigh (Parseval) energy theorem, partial derivative identities, a directional uncertainty principle and its specialization to coordinates. All important results are proven in full detail.

In this note, we first derive a Schödinger uncertainty relation for any pair of quaternionic observables and a mixed state. Then, the Wigner–Yanase skew information is introduced in the quaternion setting. Based on the skew information, we establish a new quantum uncertainty inequality for the non-Hermitian quaternionic observables.

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras $$\mathcal {G}_{p,q}$$ of vector space of dimension $$n=p+q$$. We present basis-free formulas for all characteristic polynomial coefficients in the cases $$n\le 6$$, alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $$n=4$$, and one of the formulas in the case $$n=5$$. We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases $$n\le 5$$. The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.

A number of new Lévy-Leblond type equations admitting four component spinor solutions have been proposed. The pair of linearized equations thus obtained in each case lead to Hamiltonians with characteristic features like L-S coupling and supersymmetry. The relevant momentum operators have often been understood in terms of Clifford algebraic bases producing Schrödinger Hamiltonians with L-S coupling. As for example, Hamiltonians representing Rashba effect or three dimensional harmonic oscillator have been constructed. Moreover the supersymmetric nature of one dimensional harmonic oscillator emerges naturally in this formulation.

This paper proposes to go beyond the Einstein General Relativity theory in a noncommutative geometric framework. As a first step, we rewrite the General Relativity theory intrinsically (coordinate-free formulation). As a second step, we rewrite the first and the second Bianchi identities, including torsion, within minimal algebraic hypotheses. Then, in order to extend the General Relativity theory for dealing with noncommutative scalar fields on a manifold $$\mathcal {M}$$, we are forced to adapt the definition of vector fields and connections. It leads to the consideration of one associative product p of vector fields and four derivations ($$\nabla$$: (vector, vector) to vector, $$\partial$$: (vector, scalar) to scalar, $$\delta$$: (scalar, vector) to scalar, $$\mathcal {C}$$: (scalar, scalar) to scalar). At last, the particular case where scalars are a Clifford numbers motivates future investigations towards a common writing of the Einstein field equations and the Dirac equation.

Let p and q be polynomials with degree 2 over an arbitrary field $$\mathbb {F}$$, and M be a square matrix over $$\mathbb {F}$$. Thanks to the study of an algebra that is deeply connected to quaternion algebras, we give a necessary and sufficient condition for M to split into $$A+B$$ for some pair (A, B) of square matrices over $$\mathbb {F}$$ such that $$p(A)=0$$ and $$q(B)=0$$, provided that no eigenvalue of M splits into the sum of a root of p and a root of q. Provided that $$p(0)q(0) \ne 0$$ and no eigenvalue of M is the product of a root of p with a root of q, we also give a necessary and sufficient condition for M to split into AB for some pair (A, B) of square matrices over $$\mathbb {F}$$ such that $$p(A)=0$$ and $$q(B)=0$$. In further articles, we will complete the study by lifting the assumptions on the eigenvalues of M.

In the present self-contained paper, we want, first, to construct a fundamental diagram, called (S.C), in homage to Carl Siegel and I. Satake that connects the following groups: $$\mathrm {SU}(m,m)$$, $$\mathrm {SO}^{*}(2m)$$, $$\mathrm {Sp}(2m,\mathbb {R}),$$ $$\mathrm {Sp}(4m,\mathbb {R})$$, $$\mathrm {SO}^{*}(4m)$$. Then, we define and study three Clifford algebras related to that diagram. First, we consider the morphism from $$\mathrm {Sp}(2m,\mathbb {R})$$ into $$\mathrm {SU}(m,m)$$, shown in the construction of the diagram (S.C.). Then, we define a Clifford algebra $$Cl^{m,m}$$, naturally associated with the group $$\mathrm {U}(m,m).$$ Let (E, b) be an m-dimensional skew-hermitian space over $$\mathbb {H}$$. For any $$x,y\in E,$$ write $$b(x,y)=h(x,y)+ja(x,y).$$ It is well known that h is a skew-hermitian complex form on $$\mathbb {E}_{2m}$$, the complex 2m-dimensional vector space underlying E, and a is a symmetric bilinear complex form on $$\mathbb {E}_{2m}$$. We proved previously in [4] that the special unitary group $$\mathrm {SU}(E,b)$$ of a skew-hermitian $$\mathbb {H}$$-right vector space (E, b), m-dimensional over $$\mathbb {H}$$, can be identified with the group $$\mathrm {SO}^{*}(2m)$$ defined by E. Cartan. We define a real Clifford algebra, namely $$Cl_{\mathbb {R}}^{*}(2m),$$ whose complexified algebra is $$C_{2m}^{+}(\mathbb {E}_{2m},a)$$, the even complex Clifford algebra associated with a. Both algebras are associated with the geometry of the skew-hermitian $$\mathbb {H}$$-space (E, b). Let $$V=(\mathbb {R}^{2m},\mathrm {Sp}(2m,\mathbb {R}))$$ be the standard model of a real symplectic space. We present some connections between the geometry of V and the algebras $$Cl^{m,m}$$, $$C_{2m}^{+}(\mathbb {E}_{2m},a)$$, $$Cl_{\mathbb {R}}^{*}(2m)$$. The last section wants to give a sketch of the prospects offered by these algebras for the study of the real conformal symplectic geometry. An appendix gives some indispensable recalls and some complements.

There is increasing demand for multi-level declassification of geographic vector field data in the big data era. Different from traditional encryption, declassification does not aim at making the original data unavailable through perturbation and transformation. During declassification process, the general geospatial features are usually retained but the detailed information is hidden from the perspective of data security. Furthermore, when faced with different levels of confidentiality, different levels of declassification are needed. In this paper, A declassification and reversion method with multi-level schemes is realized under the geometric algebra (GA) framework. In our method, the geographic vector field data is uniformly expressed as a GA object. Then, the declassification methods are proposed for vector field data with the rotor operator and perturbation operator. The declassification methods can progressively hide the detailed information of the vector field by vector rotating and vector perturbating. To make our method more unified and adaptive, a GA declassification operator is also constructed to realize the declassification computing of geographic vector field data. Our method is evaluated quantitatively by comparing the numerical and structure characterization of the declassification results with the original data. Divergence and curl calculating results are also compared to evaluate the reanalysis ability of the declassification results. Experiments have shown that our method can perform effective multi-level controls and has good randomness and a high degree of freedom in numerical and structure characteristics of geophysical vector field data. The method can well capture the application needs of geographic vector field data in data disclosure, secure transmission, encapsulation storage, and other aspects.

Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of spatial dimensions minus number of temporal dimensions). Real geometric algebras are periodic in s , but recursive in n . Recursion starts from the basis vectors of either the Euclidean plane or the Minkowskian plane. Although the two planes have different geometries, they have the same real geometric algebra. The direct product of the two planes yields Hestenes’ space-time algebra. Dimensions can be either open (for space-time) or closed (for the electroweak force). Their product yields the eight-fold way of the strong force. After eight dimensions, the pattern of real geometric algebras repeats. This yields a spontaneously expanding space-time lattice with the physics of the Standard Model at each node. Physics being the same at each node implies conservation laws by Noether’s theorem. Conservation laws are not pre-existent; rather, they are consequences of the uniformity of space-time, whose uniformity is a consequence of its recursive generation.

In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley–Wiener type, namely the Paley–Wiener spaces, the Hardy spaces on strips, and the Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels.

Sylvester-like matrix equations are encountered in many areas of control engineering and applied mathematics. In this paper, we construct some necessary and sufficient conditions for the system of Hermitian mixed type generalized Sylvester matrix equations to have a solution. The closed form formula to compute the general solution is also established when solvability conditions are satisfied. An algorithm and a numerical example are provided to validate our findings.

The hyperbolic Möbius transformations, which have been defined and proved to be hyperbolic conformal in Golberg and Luna-Elizarrarás (Math Methods Appl Sci 2020, https://doi.org/10.1002/mma.7109), are generalizations of Möbius transformations in complex space $${\mathbb {C}}(i)$$ and hyperbolic space $${\mathbb {D}}$$ to multidimensional hyperbolic space $${\mathbb {D}}^n$$. In this paper, we study the hyperbolic Möbius transformation in bicomplex space $${{\mathbb {B}}}{{\mathbb {C}}}$$ isomorphic to $${\mathbb {D}}^2$$ in detail, present a conjugacy classification according to the number of fixed points in $$SL(2,{{\mathbb {B}}}{{\mathbb {C}}})$$, and detailedly prove that the cross-ratio is invariant under hyperbolic Möbius transformations. Furthermore, the present paper generalizes the classical results, which have closed relation with fixed points and cross-ratios, to $${{\mathbb {B}}}{{\mathbb {C}}}$$ and may give new energy for the development of hyperbolic Möbius groups.

This paper presents a three-step program for extension of functions of complex analysis to the biquaternions by means of Cauchy’s integral formula: I. Investigate biquaternion bases consisting of roots of $$-1$$. A complex valued standard function (standardization factor) determines roots of $$-1$$. A root of $$-1$$ with a non-zero imaginary part, can uniquely determine a biquaternion ortho-standard basis. II. A single reference basis element determines two subspaces, one the span of scalars and the reference element, the other pure vector biquaternions orthogonal to the reference. The subspaces represent the distinct parts of the generalized Cayley-Dickson form. The Peirce decomposition projects into two subspaces: one is the span of the related idempotents and the other of the nilpotents. III. Using invertible elements in each of these subspaces, biquaternion functional extensions of holomorphic functions follow by Cauchy’s integral formula. Extensions retain analyticity in each biquaternion component. Cauchy integral formula uses separate idempotent and nilpotent representations of biquaternion reciprocals to define holomorphic function extensions. The Peirce projections allow extension to all viable biquaternions.

In the paper Geometric Algebra G6,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{6,3}$$\end{document} (Zamora-Esquivel in Adv Appl Clifford Algebras 24:493–514, 2014) a generalization of CGA was introduced. Now G6,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{6,3}$$\end{document} is called Quadric Geometric Algebra (QGA), and in which the use of axis aligned quadrics and their intersections were presented. In this paper the rotation of G6,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{6,3}$$\end{document} geometric entities is described, introducing new concepts like vanishing vectors, also the polar representation of quadric geometric entities is being used.

We reexamine the minimal coupling procedure in Hestenes’ geometric algebra formulation of the Dirac equation, where spinors are identified with even elements of the real Clifford algebra of spacetime. This point of view, as we argue, leads naturally to a non-Abelian generalisation of the electromagnetic gauge potential.

We introduce the Study variety of conformal kinematics and investigate some of its properties. The Study variety is a projective variety of dimension ten and degree twelve in real projective space of dimension 15, and it generalizes the well-known Study quadric model of rigid body kinematics. Despite its high dimension, co-dimension, and degree it is amenable to concrete calculations via conformal geometric algebra (CGA) associated to three-dimensional Euclidean space. Calculations are facilitated by a four quaternion representation which extends the dual quaternion description of rigid body kinematics. In particular, we study straight lines on the Study variety. It turns out that they are related to a class of one-parametric conformal motions introduced by Dorst in (Math Comput Sci 10:97–113, 2016, https://doi.org/10.1007/s11786-016-0250-8 ). Similar to rigid body kinematics, straight lines (that is, Dorst’s motions) are important for the decomposition of rational conformal motions into lower degree motions via the factorization of certain polynomials with coefficients in CGA.

In this article we introduce the notion of linear canonical wave packet transform in quater-nionic settings and we name it as the quaternionic linear canonical wave packet transform (QLCWPT). Firstly we establish the fundamental properties viz linearty, anti-linearty, parity, scaling, dilation and Parseval's identity. Furthermore, some key hormonic analysis results like energy conservation, inversion formula, characteristic of range and some bounds of QLCWPT are obtained. Towards the culmination of this paper, we established Heisenberg's uncertainty principle and logarithmic uncertainty principle associated with the proposed transform (QLCWPT).

In this study, we show that the quaternion product of quaternionic operator whose scalar part is a real parameter and vector part is a curve in R3 and a spherical striction curve represents a slant ruled surface in R3 if the vector part of the quaternionic operator is perpendicular to the position vector of the spherical striction curve. In R3, exploitting this operator, we define the slant ruled surface corresponding to the natural lift curve on the subset of the tangent bundle of unit 2-sphere, TM¯. Then, we classify q¯→-,h¯→- and a¯→- slant ruled surfaces. Furthermore, these surfaces can also be expressed with 2- parameter homothetic motions. Finally, we give the geometric interpretations of this operator with some examples.

The relations between Hilbert spaces over octonions and Hilbert spaces over Clifford algebras are discussed. It is shown that the category of Hilbert O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {O}$$\end{document}-bimodules is isomorphic to the category of Hilbert left Cℓ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\ell _{6}$$\end{document}-modules, and the category of Hilbert left O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {O}$$\end{document}-modules is isomorphic to the category of Hilbert left Cℓ7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\ell _{7}$$\end{document}-modules.

In this paper, we focus on the characterization of Lie algebras of fermionic, bosonic and parastatistic operators of spin particles. We provide a method to construct a Lie group structure for the quantum spin particles. We show the semi-simplicity of the Lie algebra for a quantum spin particle, and extend the results to the Lie group level. Besides, we perform the Iwasawa decomposition for spin particles at both the Lie algebra and the Lie group levels. Then, we give a general decomposition for spin particles. Finally, we investigate the coupling of angular momenta of spin half particles, and give a general construction for such a study.

Using the well known complex representation of the proper Lorentz group $$\mathrm {SO}_+(3,1)\cong \mathrm {PSL}(2,{\mathbb {C}})\cong \mathrm {SO}(3,{\mathbb {C}})$$ we study some Coriolis type effects in Special Relativity and Electromagnetism in close analogy with the more traditional kinematical treatment of the group of spatial rotations. Namely, we begin with the Clifford group of $${\mathbb {R}}^3$$ viewed as a complexification of $${\mathbb {H}}^\times$$ and consider the associated Maurer-Cartan form which yields a complex-valued analogue of the angular velocity characterizing the action of $$\mathrm {SO}(3)$$ in rigid body kinematics. It appears in a linear ODE’s written in biquaternion form or a Ricatti equation if one works with its projective version instead, providing a far richer structure compared to the real case without imposing serious technical obstruction at least in the decomposable setting, which is our main emphasis due to its importance in physics. There are several distinct terms in the non-commutative part of the so obtained connection describing well known effects named after Coriolis, Thomas, Hall and Sagnac. We also consider a restriction to the so-called Wigner little groups $$\mathrm {SO}(3)$$, $$\mathrm {SO}_+(2,1)$$ and $$\mathrm {E}(2)$$ discussing algebraic properties of the electromagnetic field. Some familiar constructions such as geometric phases, Hopf bundles and the Fubini-Study form appear naturally with this approach.

In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue’s dominated convergence theorem and some other results related to this theorem. Also we have proved the bicomplex version of Lebesgue-Radon-Nikodym theorem. Finally we have introduced the idea of hyperbolic version of invariant measure.

The quaternionic valued functions of a quaternionic variable, often referred to as slice regular functions, has been studied extensively due to the large number of generalized results of the theory of one complex variable. Recently, several global properties of these functions has been found such as a Borel–Pompieu formula and a Stokes’ Theorem from the study of a differential operator, see González-Cervantes and González-Campos (Complex Var Ellipt Equ 66(5):1–10, 2021, https://doi.org/10.1080/17476933.2020.1738410). The aim of this paper is to present a kind of quaternionic generalized slice regular functions that on slices coincide with pairs of complex generalized holomorphic functions associated to Vekua problems. The global properties of these functions are obtained from a perturbed global-type operator and among their local properties presented in this work are the versions of Splitting Lemma, Representation Theorem and a conformal property.

In this article, we show that the Cauchy integral formula for a monogenic function f:H⟶H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: {\mathbb {H}}\longrightarrow {\mathbb {H}}$$\end{document} for which Imf⊂C⊂H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Im } f\subset {\mathbb {C}}\subset {\mathbb {H}}$$\end{document} turns out to be the Bochner–Martinelli integral formula for an associated holomorphic functions g:C2⟶C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g: {\mathbb {C}}^2\longrightarrow {\mathbb {C}}$$\end{document}. To this end, we need to interpret the holomorphic self-mapping of C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^2$$\end{document} as a monogenic functions H→H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}\rightarrow {\mathbb {H}}$$\end{document} annihilated by a pair of Cauchy–Fueter type operators. We also need a concise version of the chain rule for quaternions as well as the explicit formulas among the various inner products in quaternions.

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