Showing papers in "Advances in Applied Clifford Algebras in 2020"

Quaternion Windowed Linear Canonical Transform of Two-Dimensional Signals

Abstract: We investigate the 2D quaternion windowed linear canonical transform (QWLCT) in this paper. Firstly, we propose the new definition of the QWLCT, and then several important properties of newly defined QWLCT, such as bounded, shift, modulation, orthogonality relation, are derived based on the spectral representation of the quaternionic linear canonical transform (QLCT). Secondly, by the Heisenberg uncertainty principle for the QLCT and the orthogonality relation property for the QWLCT, the Heisenberg uncertainty principle for the QWLCT is es- tablished. Finally, we give an example of the QWLCT.

A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics

Abstract: We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell’s Theorem in quantum mechanics, and its relation to the sphere $$S^7$$ and the exceptional group $$E_8$$, and proposed a new associative version of the division algebra normally thought to require the octonians. We show that what is being discussed by Christian is mathematically the same as our 1d up approach to 3d geometry, but that after the removal of some incorrect mathematical assertions, the results he proves in the first part of the paper, and bases the application to Bell’s Theorem on, amount to no more than the statement that the combination of two rotors from the Clifford Algebra Cl(4, 0) is also a rotor.

Polar Decomposition of Complexified Quaternions and Octonions

Abstract: We present a hitherto unknown polar representation of complexified quaternions (also known as biquaternions), also applicable to complexified octonions. The complexified quaternion is factored into the product of two exponentials, one trigonometric or circular, and one hyperbolic. The trigonometric exponential is a real quaternion, the hyperbolic exponential has a real scalar part and imaginary vector part. This factorisation is shown to be isomorphic to the polar decomposition of linear algebra.

Algebraic Properties of Bihyperbolic Numbers

Abstract: In this paper, we study the four-dimensional real algebra of bihyperbolic numbers. Under consideration of the spectral representation of the bihyperbolic numbers, we give a partial order of bihyperbolic numbers which allows us to obtain some relations in the ordered vector space of bihyperbolic numbers. Moreover, we state that the set of bihyperbolic numbers form a real Banach algebra with a new defined norm. We introduce conjugates, three hyperbolic valued moduli, real moduli, and multiplicative inverse of the bihyperbolic numbers. We give the concept of the absolute value of a bihyperbolic number which generalizes that of real numbers. Also, we represent the polar form of invertible bihyperbolic numbers.

Generalized Hadamard Product Bases of Special Monogenic Polynomials

Abstract: The notion of the generalized Hadamard product of bases in Clifford analysis is introduced. Certain convergence properties of such generalized Hadamard product bases in terms of those of their constituent bases are established. Unlike the complex case, some of these convergence properties may not be interchanged.

3-Hom-Lie Algebras Based on $$\sigma $$σ -Derivation and Involution

Abstract: We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using $$\sigma $$ -derivation and involution, and we obtain two 3-Hom-Lie algebras.

Sedeonic Equations in Field Theory

Abstract: We consider the factorization of differential operator of Klein–Gordon equation on the base of space-time algebra of sixteen-component sedeons It is shown that generalized sedeonic wave equations can be used both to describe quantum particles and the force fields responsible for the interaction of particles In particular, we discuss the first-order and second-order wave equations for sedeonic potentials describing fields with zero and nonzero mass of quantum We demonstrate the application of sedeonic space-time operators to describe quantum particles and in particular the calculation of energy spectrum of electron in an external magnetic field The relations between the sedeons and the spinors are discussed

Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

Abstract: We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair $$h_1$$ , $$h_2$$ of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$ . We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

Projection Factors and Generalized Real and Complex Pythagorean Theorems

Abstract: Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann and Clifford algebras and to the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families of subspaces. The complex Pythagorean theorems differ from the real ones in that measures are not squared, and this may have important implications for quantum theory. Projection factors of the complex line of a quantum state with the eigenspaces of an observable give the corresponding quantum probabilities. The complex Pythagorean theorem for lines corresponds to the condition of unit total probability, and may provide a way to solve the probability problem of Everettian quantum mechanics.

Quadratic Split Quaternion Polynomials: Factorization and Geometry.

Abstract: We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.

Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation

Abstract: The special relativistic (space-time) Fourier transform (SFT) in Clifford algebra $$Cl_{(3,1)}$$ of space-time, first introduced from a mathematical point of view in Hitzer (Adv Appl Clifford Algebras 17:497–517, 2007), extends the quaternionic Fourier transform to functions, fields and signals in space-time. The purpose of this paper is to advance the study of the SFT and investigate important properties such as continuity, Plancherel identity, Riemann–Lebesgue lemma, and to establish the associated Hausdorff–Young inequality. Moreover, using the observer related space-time split several uncertainty inequalities are established, including Heisenberg’s uncertainty principle and Hardy’s theorem.

Factorization and Generalized Roots of Dual Complex Matrices with Rodrigues’ Formula

Abstract: The paper provides an efficient method for obtaining powers and roots of dual complex $$2\times 2$$ matrices based on a far reaching generalization of De Moivre’s formula. We also resolve the case of normal $$3\times 3$$ and $$4\times 4$$ matrices using polar decomposition and the direct sum structure of $$\mathfrak {so}_4$$. The compact explicit expressions derived for rational powers formally extend (with loss of periodicity) to real, complex or even dual ones, which allows for defining some classes of transcendent functions of matrices in those cases without referring to infinite series or alternatively, obtain the sum of those series (explicit examples may be found in the text). Moreover, we suggest a factorization procedure for $$\mathrm {M}(n,{\mathbb {C}}[\varepsilon ])$$, $$n\le 4$$ based on polar decomposition and generalized Euler type procedures recently proposed by the author in the real case. Our approach uses dual biquaternions and their projective version referred to in the Euclidean setting as Rodrigues’ vectors. Restrictions to certain subalgebras yield interesting applications in various fields, such as screw geometry extensively used in classical mechanics and robotics, complex representations of the Lorentz group in relativity and electrodynamics, conformal mappings in computer vision, the physics of scattering processes and probably many others. Here we only provide brief comments on these subjects with several explicit examples to illustrate the method.

Double Hom-Associative Algebra and Double Hom–Lie Bialgebra

Abstract: Recently, Hom-structures have been widely investigated in literature. In this paper, we introduce the conceptions of double Hom-associative algebras and double Hom–Lie bialgebras, and give a necessary and sufficient condition for double Hom-associative algebras to be Hom-associative algebras. Meanwhile, we characterize a classical Hom–Yang–Baxter equation in terms of both Hom–Lie algebra morphisms and Hom–Lie coalgebra morphisms. Last but not least, we introduce the notion of double Hom–Lie bialgebras, and prove that double Hom-associative algebras are indeed quasi-triangular Hom–Lie bialgebras.

The Moore–Penrose Inverse and Singular Value Decomposition of Split Quaternions

Abstract: Using the concept of a transposition anti-involution in Clifford algebra $$C \, \ell _{1,1}$$ and the isomorphisms $${\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),$$ where $${\mathbb {H}}_s$$ is the algebra of split quaternions and $$\text {Mat}(2,{\mathbb {R}})$$ is the algebra of $$2 \times 2$$ real matrices, one can find the Moore–Penrose inverse $$q^{+}$$ of a non-zero non-invertible split quaternion q. In particular, using a well-known algorithm for finding the Moore-Penrose inverse $$Q^{+}$$ of a non-zero $$2 \times 2$$ matrix Q of rank 1, one can give four governing equations that the (unique) split quaternion $$q^{+}$$ corresponding to $$Q^{+}$$ must satisfy. We show how a dyadic expansion and a Singular Value Decomposition can be found for any split quaternion q, and we relate them to $$q^{+}$$ . Results presented in this paper may be useful in a plethora of recent applications of split quaternions.

The Tangential $$\varvec{k}$$k -Cauchy–Fueter Operator and $$\varvec{k}$$k -CF Functions Over the Heisenberg Group

Abstract: In this paper, we investigate quaternionic analysis on the $$(4n+1)$$-dimensional Heisenberg group. The tangential k-Cauchy–Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables, respectively. We give the Penrose integral formula for k-CF functions and establish the Bochner–Martinelli formula for the tangential k-Cauchy–Fueter operator. We also construct the tangential k-Cauchy–Fueter complex.

Spectrums of Functions Associated to the Fractional Clifford–Fourier Transform

Abstract: In this paper, we prove several versions of the real Paley–Wiener theorems for a fractional Clifford–Fourier transform which depends on two numerical parameters and paves the way in some sense for a functional calculus approach to generalizing the Fourier transform to Clifford analysis.

Split Quaternion-Valued Twin-Multistate Hopfield Neural Networks

Abstract: Complex-valued Hopfield neural networks have been extended to 4-dimensional models using quaternions and commutative quaternions. The algebra of split quaternions is another 4-dimensional hypercomplex number system. In this work, a split quaternion-valued Hopfield neural network is defined. By the computer simulations using the twin-multistate activation function and projection rule, we evaluate the noise tolerance.

The Unified Standard Model

Abstract: The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8, $$\mathbb {C})$$ of complex $$8 \times 8$$ matrices. This algebra is well motivated by its close relation to the normed division algebra of octonions. (Anti-)particle states are identified with basis elements of the vector space M(8, $$\mathbb {C})$$ . Gauge transformations are simply described by the algebra acting on itself. Our result shows that all particles and gauge structures of the Standard Model are contained in the tensor product of all four normed division algebras, with the quaternions providing the Lorentz representations. Interestingly, the space M(8, $$\mathbb {C})$$ contains two additional elements independent of the Standard Model particles, hinting at a minimal amount of new physics.

Generalized Fractional Cauchy–Riemann Operator Associated with the Fractional Cauchy–Riemann Operator

Abstract: In this paper, we present a characterization of all linear fractional order partial differential operators with complex-valued coefficients that are associated to the generalized fractional Cauchy–Riemann operator in the Riemann–Liouville sense. To achieve our goal, we make use of the technique of an associated differential operator applied to the fractional case.

BiHom–Lie Superalgebra Structures and BiHom–Yang–Baxter Equations

Abstract: In this paper, we first introduce the notion of BiHom–Lie superalgebras, which is a generalization of both BiHom–Lie algebras and Hom–Lie superalgebras. Also, we explore some general classes of BiHom–Lie admissible superalgebras and describe all these classes via G-BiHom-associative superalgebras, where G is a subgroup of the symmetric group $$S_{3}$$ . Finally, we obtain a method to construct the solutions of the BiHom–Yang–Baxter equation from BiHom–Lie superalgebras.

TbGAL: A Tensor-Based Library for Geometric Algebra

Abstract: Geometric algebra is a powerful mathematical framework that allows us to use geometric entities (encoded by blades) and orthogonal transformations (encoded by versors) as primitives and operate on them directly. In this work, we present a high-level C++ library for geometric algebra. By manipulating blades and versors decomposed as vectors under a tensor structure, our library achieves high performance even in high-dimensional spaces ($$\bigwedge \mathbb {R}^{n}$$ with $$n > 256$$) assuming (p, q, r) metric signatures with $$r = 0$$. Additionally, to keep the simplicity of use of our library, the implementation is ready to be used both as a C++ pure library and as a back-end to a Python environment. Such flexibility allows easy manipulation accordingly to the user’s experience, without impact on the performance.

Local Controllability of Snake Robots Based on CRA, Theory and Practice

Abstract: We introduce the transcript of direct and differential kinematics of a robotic snake in terms of Compass Ruler Algebra (CRA). We suppose that the robot moves on a planar surface. We provide the original and optimized code of elementary motions in GAALOP together with runtime comparisons.

Plancherel Theorems of Quaternion Hilbert Transforms Associated with Linear Canonical Transforms

Abstract: The Hilbert transform has wide applications in signal analysis. The quaternion Hilbert transforms associated with the linear canonical transform are recently used to form the quaternion analytic signal. In this paper, some properties of the 2D quaternion Hilbert transforms with the two-sided quaternion linear canonical transforms are investigated, such as the Plancherel theorems, the Parseval identities and the inversion formulas of the Hilbert transforms. In particular, we define the discrete generalized quaternion Hilbert transforms and use them for the color edge detection. The proposed edge detection methods are robust to noise and can simultaneously distinguish edges from the non-edge regions very successfully.

GAALOPWeb for Matlab: An Easy to Handle Solution for Industrial Geometric Algebra Implementations

Abstract: We present GAALOPWeb for Matlab, a new easy to handle solution for Geometric Algebra implementations for Matlab. We demonstrate its usability for industrial applications based on a forward kinematics algorithm of a serial robot arm and illustrate it with the help of high run-time performance.

Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

Abstract: We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ in the upper half space $$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$ . They are called $$\alpha $$ -hyperbolic harmonic. An important result is that a function f is $$\alpha $$ -hyperbolic harmonic if and only if the function $$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$ is the eigenfunction of the hyperbolic Laplace operator $$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$ corresponding to the eigenvalue $$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$ . This means that in case $$\alpha =n-2$$ , the $$n-2$$ -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincare upper half-space. We are presenting some connections of $$\alpha $$ -hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.

Spacetime Geometry with Geometric Calculus

Abstract: Geometric Calculus is developed for curved-space treatments of General Relativity and comparison is made with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides mathematical tools that streamline the formulation and simplify calculations. The formalism automatically includes spinors so the Dirac equation is incorporated in a geometrically natural way.

The Riemann Curvature Tensor and Higgs Scalar Field within CAM Theory

Abstract: The composition algebra based methodology (CAM) (Wolk in Pap Phys 9:090002, 2017, Phys Scr 94:025301, 2019, Adv Appl Clifford Algebras 27(4):3225, 2017, J Appl Math Phys 6:1537, 2018, Phys Scr 94:105301, 2019) has previously been shown to generate the pre-Higgs Standard Model Lagrangian. In this paper the symmetry of general covariance is incorporated into CAM. The Riemann curvature tensor thereby arises, from which gravity-field Lagrangians are constructed. A Higgs-like scalar field coupled to the spacetime metric tensor also manifests.

Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars

Abstract: We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.

Polynomial Approximation in Quaternionic Bloch and Besov Spaces

Abstract: In this paper we continue our study on the density of the set of quaternionic polynomials in function spaces of slice regular functions on the unit ball by considering the case of the Bloch and Besov spaces of the first and of the second kind. Among the results we prove, we show some constructive methods based on the Taylor expansion and on the convolution polynomials. We also provide quantitative estimates in terms of higher order moduli of smoothness and of the best approximation quantity. As a byproduct, we obtain two new results for complex Bloch and Besov spaces.

Determinantal Representations of the Solutions to Systems of Generalized Sylvester Equations

Abstract: In this paper, we consider three systems of coupled generalized Sylvester quaternion equations $$\begin{aligned} \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1} \\ A_{2}X_{2}-Y_{1}B_{2}=C_{2} \\ A_{3}X_{2}-Y_{2}B_{3}=C_{3} \end{array}, \right. \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1}\\ A_{2}Y_{1}-Y_{2}B_{2}=C_{2}\\ A_{3}Y_{2}-Y_{3}B_{3}=C_{3} \end{array}, \right. \end{aligned}$$and $$\begin{aligned} \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1}\\ A_{2}Y_{1}-Y_{2}B_{2}=C_{2} \\ A_{3}X_{2}-Y_{2}B_{3}=C_{3} \end{array}. \right. \end{aligned}$$We present new necessary and sufficient conditions for the solvability of each system, and derive the determinantal representations of the general solutions to the above systems by the row and column determinants of quaternion matrices.