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

In this article, we introduce the notion of linear canonical wave packet transform in quaternionic settings and we name it the quaternionic linear canonical wave packet transform (QLCWPT). Firstly, we establish the fundamental properties, viz. linearity, anti-linearity, parity, scaling, dilation and Parseval’s identity. Furthermore, some key harmonic analysis results like energy conservation, inversion

In this paper, we establish Bernstein inequality, Jackson’s direct and inverse theorems for quaternion linear canonical transform using the functions with bounded spectrum.

The relations between Hilbert spaces over octonions and Hilbert spaces over Clifford algebras are discussed. It is shown that the category of Hilbert \(\mathbb {O}\)-bimodules is isomorphic to the category of Hilbert left \(C\ell _{6}\)-modules, and the category of Hilbert left \(\mathbb {O}\)-modules is isomorphic to the category of Hilbert left \(C\ell _{7}\)-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

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

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

This research aims to introduce and investigate the right and left W-MPCEP, W-CEPMP, and W-MPCEPMP generalized inverses for quaternion matrices. These generalized inverses are introduced as extensions of corresponding generalized inverses applicable to complex matrices. Some new characterizations and expressions of these inverses are presented. Determinantal representations of these new inverses are

In this article, we show that the Cauchy integral formula for a monogenic function \(f: {\mathbb {H}}\longrightarrow {\mathbb {H}}\) for which \(\text{ Im } f\subset {\mathbb {C}}\subset {\mathbb {H}}\) turns out to be the Bochner–Martinelli integral formula for an associated holomorphic functions \(g: {\mathbb {C}}^2\longrightarrow {\mathbb {C}}\). To this end, we need to interpret the holomorphic

In this paper we study the \(\mathfrak {sp}(2m)\)-invariant Dirac operator \(D_s\) which acts on symplectic spinors, from an orthogonal point of view. By this we mean that we will focus on the subalgebra \(\mathfrak {so}(m) \subset \mathfrak {sp}(2m)\), as this will allow us to derive branching rules for the space of 1-homogeneous polynomial solutions for the operator \(D_s\) (hence generalising the

We introduce a notion of n-Lie–Rinehart algebras as a generalization of Lie–Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation theory and describe a cohomology complex of n-Lie–Rinehart algebras. Furthermore, we investigate extension theory of n-Lie–Rinehart algebras by means of 2-cocycles. Finally, we introduce crossed modules

In this paper we systematically study the Riemann boundary value problems on the hyperplane for monogenic functions in Clifford analysis. The concept of the principal part of a sectionally regular function with the hyperplane as its jump surface is first introduced. Based on this concept the general forms of the Riemann boundary value problems on the hyperplane for monogenic functions are formulated

Clifford analysis has been the field of active research for several decades resulting in various methods to solve problems in pure and applied mathematics. However, the area of stochastic analysis has not been addressed in its full generality in the Clifford setting, since only a few contributions have been presented so far. Considering that the tools of stochastic analysis play an important role in

In the framework of higher-dimensional time-frequency analysis, we propose a novel Clifford-valued Stockwell transform for an effective and directional representation of Clifford-valued functions. The proposed transform rectifies the windowed Fourier and wavelet transformations by employing an angular, scalable and localized window, which offers directional flexibility in the multi-scale signal analysis

Lyapunov stability theory for smooth nonlinear autonomous dynamical systems is presented in terms of Geometric Algebra. The system is described by a smooth nonlinear state vector differential equation, driven by a vector field in \(\mathbb {R}^n\). The level sets of the scalar Lyapunov function candidate are assumed to be compact smooth vector manifolds in \(\mathbb {R}^n\). The level sets induce an

The Gauss–Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity

The Clifford algebraic formulation of the Duffin–Kemmer–Petiau (DKP) algebras is applied to recast the De Donder–Weyl Hamiltonian (DWH) theory as an algebraic description independent of the matrix representation of the DKP algebra. We show that the DWH equations for antisymmetric fields arise out of the action of the DKP algebra on certain invariant subspaces of the Clifford algebra which carry the

Multi-modal medical image fusion refers to the combination of patient area images obtained under diverse or identical imaging modalities, which improves the clinical applicability and provides more specific disease information for diagnosis. However, most of the existing image fusion algorithms usually divided color images into three channels of R, G, B for processing separately, which ignores the

The circuit analysis approach based on geometric algebra and \(\varvec{M}\), the power definition based on the geometric product between the voltage and the current multivectors, are used here to demonstrate the shortcomings of the traditional definition of the non-sinusoidal apparent power S. The shortcomings of S are illustrated in three ways. Firstly, by showing an example of how the norm of \(\varvec{M}\)

We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension 3. We in particular obtain a representation theorem for surfaces in \(\mathbb {L}(\kappa ,\tau )\) spaces. We then recover the Calabi correspondence between minimal surfaces in \(\mathbb {R}^3\) and maximal surfaces in \(\mathbb {R}_1^3\), and obtain a new Lawson type correspondence

In this study, we used the fact that unit circle for elliptic numbers is an ellipse to model motion of a planet around a star. For that purpose we first have given a standard derivation of elliptic orbits in Newtonian two-body problem. Then we translated the variables found in the coordinates where the origin is at the focus of the ellipse to elliptic number parameters where the origin is at the center

We study an operator related to the sub-Laplacian on the non-isotropic quaternionic Heisenberg group and construct the fundamental solution for this operator. For the isotropic case, we derive the closed form of this solution. The techniques we used can be applied to the standard Heisenberg group. We also give the connection between this operator and the Heisenberg sub-Laplacian.

The new applications of Clifford’s geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics

The notions of the MPCEP inverse and \(*\)CEPMP inverse are expanded to quaternion matrices and their determinantal representations are developed in terms of minors of appropriate matrices. Solvability of novel quaternion restricted matrix equations (QRME) is investigated and their solutions are obtained in terms of the corresponding MPCEP and \(*\)CEPMP inverses. Generalized Cramer’s rules for obtained

In this article, we prove the ellipticity of higher order conformally invariant differential operators in the higher spin spaces, where functions take values in certain irreducible representations of the spin group in the Euclidean space. We introduce the product formulas for these operators obtained in our previous work to greatly simplify our proofs. In both the bosonic cases and fermionic cases

Clifford-Legendre and Clifford–Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on m-dimensional euclidean space \({\mathbb R}^m\) and taking values in the associated Clifford algebra \({\mathbb R}_m\). New recurrence and Bonnet-type formulae for these polynomials are provided, and their Fourier transforms are computed. Explicit representations

On a symplectic manifold equipped with a symplectic connection and a metaplectic structure, we define two families of sequences of differential operators, called the symplectic twistor operators. We prove that if the connection is torsion-free and Weyl-flat, the sequences in these families form complexes.

We generalized the lower bound estimates for eigenvalues of the Dirac operator on spacelike hypersurfaces of Lorentzian manifolds obtained by Yongfa Chen in (Sci China Ser A Math 52(11):2459–2468, 2009) based on the constraint between the scalar curvature of the manifold, energy–momentum tensor and the mean curvature of the manifold. Afterwards, we examined the geometric data in the case of estimation

Let \(C \ell _{d}^{\frac{1}{3}}\) be the complex ternary Clifford algebra on d generators as defined in Cerejeiras and Vajiac (Adv Appl Clifford Algebras 31:13, 2021). The main objective of this work is to investigate algebraic structure of these algebras and present a new Structure Theorem. In particular, it is shown that when d is even, the algebras are central simple (CSA/\({\mathbb {C}}\)), and

This paper studies some systems of second order partial differential equations associated to a Dirac type operator \({^\varphi \!\underline{\partial }}\) in \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\), with respect to an arbitrary structural set \(\varphi \). We develop a method by which we can transform these general systems into those connected to the standard Dirac operator \(\underline{\partial

In this paper, we investigate and discuss in detail the structure and properties of product singular value decomposition for a quaternion tensor triplet under Einstein product (higher-order PSVD). As an application, we consider color video watermark processing with this higher-order PSVD.

The extensive computer-aided search applied in Bena et al. (The tadpole problem, 2020) to find the minimal charge sourced by the fluxes that stabilize all the (flux-stabilizable) moduli of a smooth K3 \(\times \) K3 compactification uses differential evolutionary algorithms supplemented by local searches. We present these algorithms in detail and show that they can also solve our minimization problem

As an analogue of the massless field equations in Euclidean space, we consider the so-called generalized Cauchy–Riemann equations introduced by E. Stein and G. Weiss. In the spin 1/2 case these equations reduce to the Dirac equation for spin 1/2 fields, which was thoroughly and intensively studied in Clifford analysis. For general spin it was recently shown that, in dimension 4, homogenous solutions

We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This

In this paper, we present Geometric Algebra as a powerful language to describe quantum operations using its geometric intuitiveness. Using the web-based GAALOPWeb, an online geometric algebra algorithm optimizer for computing with qubits, we describe new formulations for the NOT operation, as well as a strategy to describe the Z gate and especially the Hadamard operation both for one and multiple qubits

General spinors in polar form display a structure that is made up by real scalar degrees of freedom plus six components which can be recognized as Goldstone bosons: in the present paper we show that of all singular spinors, Weyl and Majorana spinors have no real degree of freedom and so that they can be interpreted as pure Goldstone states.

Typically, airborne laserscanning includes a laser mounted on an airplane or drone (its pulsed beam direction can scan in flight direction and perpendicular to it) an intertial positioning system of gyroscopes, and a global navigation satellite system. The data, relative orientation and relative distance of these three systems are combined in computing strips of ground surface point locations in an

We describe a possibility for geometric calculation of specific conics’ intersections in Geometric Algebra for Conics (GAC) using its operations that may be expressed as sums of products. The advantage is that no solver for a system of quadratic equations is needed and thus no numerical error is involved. We also describe specific conics connected to intersections of conics in a general mutual position

In this paper, we study the (anti-)self-duality equations \(*F\wedge F=\pm F\wedge F\) in the eight-dimensional Euclidean space. Using properties of the Clifford algebra \(Cl_{0,8}({\mathbb {R}})\), we find a new solution to these equations.

This work is intended as an attempt to extend the notion of bialgebra for Lie algebras to Leibniz algebras and also, the correspondence between the Leibniz bialgebras and its dual is investigated. Moreover, the coboundary Leibniz bialgebras, the classical r-matrices, and Yang–Baxter equations related to the Leibniz algebras are defined, and some examples are given. Finally, a method for the construction

In this paper, we study the quaternion matrix equation (1.1) by using the semi-tensor product of matrices. First, we propose a real vector representation of a quaternion matrix and study its properties. Then combining this real vector representation with semi-tensor product of matrices, the explicit expressions of the least squares solution, the least squares J-self-conjugate solution and the least

An Adinkra is a graph from the study of supersymmetry in particle physics, but it can be adapted to study Clifford algebra representations. The graph in this context is called a Cliffordinkra, and puts some standard ideas in Clifford algebra representations in a geometric and visual context. In the past few years there have been developments in Adinkras that have shown how they are connected to error

We propose a gauge model with the SU(2) symmetry, which describes a gravitational interaction of fundamental fermions (leptons and quarks) in the Minkowski space. In the Standard Model one uses a Dirac–Yang–Mills system of equations with U(2) gauge symmetry for electroweak interactions and with SU(3) gauge symmetry for QCD interactions. A key idea of the model is to use the Dirac-type equation (invented

We extend the special affine Fourier transform in the context of quaternion valued functions and study its properties including an uncertainty principle. The same transform is studied on a suitably constructed Boehmian space.

In this paper, we define and classify split quaternion operators. Then, we show that the split quaternion product of a split quaternion operator and a curve, which lies on Lorentzian unit sphere or on hyperbolic unit sphere, parametrizes a ruled surface in the 3-dimensional Minkowski space \(\mathbb {E}_{1}^{3}\) if the vector part of the operator is perpendicular to the position vector of the spherical

As an integral representation for holomorphic functions, Cauchy integral formula plays a significant role in the function theory of one complex variable and several complex variables. In this paper, using the idea of several complex analysis we construct the Cauchy kernel in universal Clifford analysis, which has generalized complex differential forms with universal Clifford basic vectors. We establish

In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with \(SL(2,\mathbb {H})\), which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation

Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Plücker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations

Classical Segal–Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on \(\mathbb {R}^n\) with respect to a Gaussian

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions

Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function