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Abstract: Abstract 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 of n-Lie–Rinehart algebras to gain a better understanding of their third cohomology groups. PubDate: 2022-05-19

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Abstract: Abstract We study several theoretical aspects of both 2D and 3D intra multi-view geometry of calibrated cameras when all that they can reliably recognize is each other. Starting with minimal reconstructable configurations, we propose a method for obtaining the position-orientation structure of such camera ensembles, up to a global similarity. In the 3D setting we base our analysis on Rodrigues’ vector techniques familiar from mechanics and robotics. We also examine the average number of visible cameras and discuss some kinematic aspects of the problem. PubDate: 2022-04-29

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Abstract: Abstract 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. Then, the explicit expressions and explicit solvable conditions for solutions with any finite integer order at infinity are obtained. PubDate: 2022-04-20

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Abstract: Abstract This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our ideas, we include a brief introduction to the Lean language targeted at a reader with no prior experience with Lean or theorem provers in general. We formalize the multivectors as the quotient of the tensor algebra by a suitable relation, which provides the ring structure automatically, then go on to establish the universal property of the Clifford algebra. We show that this is quite different to the approach taken by existing formalizations of Geometric algebra in other theorem provers; most notably, our approach does not require a choice of basis. We go on to show how operations and structure such as involutions, versors, and the \(\mathbb {Z}_2\) -grading can be defined using the universal property alone, and how to recover an induction principle from the universal property suitable for proving statements about these definitions. We outline the steps needed to formalize the wedge product and \(\mathbb {N}\) -grading, and some of the gaps in mathlib that currently make this challenging. PubDate: 2022-04-18

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Abstract: Abstract 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 the study of objects, such as positive definite functions, reproducing kernels and partial differential equations, it is important to develop tools for the study of these objects in the context of Clifford analysis. Therefore, in this work-in-progress paper, we present further steps towards stochastic Clifford analysis by studying random variables, martingales, Brownian motion, and Itô formula in the Clifford setting, as well as their applications in Clifford analysis. PubDate: 2022-04-05

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Abstract: Abstract 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 associated global foliation of the state space. On any leaf of this foliation, a geometric subalgebra is naturally attached to the corresponding tangent vector space of the smooth vector manifold. The pseudoscalar (field) of this subalgebra completely characterizes the tangent space. Asymptotic stability of the system equilibria is described in terms of equilibria of, easily computable, rejection vector fields with respect to the pseudoscalar field. Nonexistence of invariant sets of the Lyapunov function directional derivative, along the defining vector field, are also tested using a simple tangency condition. Several illustrative examples are presented. PubDate: 2022-03-19

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Abstract: Abstract 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 in Clifford domains. The basic properties of the proposed transform such as inner product relation, reconstruction formula, and the range theorem are investigated using the machinery of operator theory and Clifford Fourier transforms. Moreover, several extensions of the well-known Heisenberg-type inequalities are derived for the proposed transform in the Clifford Fourier domain. We culminate our investigation by deriving the directional uncertainty principles for the Clifford-valued Stockwell transform. To validate the acquired results, illustrative examples are given. PubDate: 2022-03-19

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Abstract: Abstract 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 of the Gauss–Manin connection of pencils of hypersurfaces. As an application, we compute the periods of \(96\%\) of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard lattices and the endomorphism fields of their transcendental lattices. PubDate: 2022-02-22 DOI: 10.1007/s00006-022-01207-1

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Abstract: Abstract 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 representations of the fields. The matrix representation-free formula for the bracket associated with the DKP form of the DWH equations is also derived. This bracket satisfies a generalization of the standard properties of the Poisson bracket. PubDate: 2022-02-22 DOI: 10.1007/s00006-022-01202-6

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Abstract: Abstract 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 between CMC surfaces in \(\mathbb {R}_1^3\) and in the 3-dimensional pseudo-hyperbolic space \(\mathbb {H}_1^{3}.\) PubDate: 2022-02-21 DOI: 10.1007/s00006-022-01205-3

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Abstract: Abstract 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 of the ellipse. We noted that a similar argument may be used to model hyperbolic orbits in Newtonian gravity with hyperbolic numbers. However it seems that modelling parabolic orbits is not possible within the context of p-complex numbers. PubDate: 2022-02-21 DOI: 10.1007/s00006-022-01208-0

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Abstract: Abstract 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. PubDate: 2022-02-21 DOI: 10.1007/s00006-022-01206-2

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Abstract: Abstract 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 correlation between the channels and easily causes image information loss and blurring. This paper proposes a multi-modal color medical image fusion algorithm based on geometric algebra discrete cosine transform (GA-DCT). The GA-DCT algorithm combines the character of GA, which represents the multi-vector signal as a whole, can improve the quality of the fusion image and avoid a large number of complex operations related to encoding and decoding. Firstly, the source images are divided into several image blocks and expressed in GA multi-vector form; Secondly, we extend the traditional DCT to GA space and propose GA-DCT; Thirdly, we use GA-DCT to decompose the image to obtain AC and DC coefficients and finally a fusion algorithm are used to fuse the images. The experimental results show that the proposed algorithm can get clear and comprehensive fusion image, which also has great advantages under different compression ratios. PubDate: 2022-02-20 DOI: 10.1007/s00006-021-01197-6

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Abstract: Abstract 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 \(\varvec{M}\) contains S. Secondly, through six experiments that involve compliance with: Kirchhoff’s circuit laws, Tellegen’s theorem, the principle of conservation of energy, the equivalency of two terminal networks and the concept of reactive power compensation. Lastly, by showing how the use of S leads the current’s physical component power theory astray. The experiments show contradictions between the aforementioned circuit theory fundamentals and the results attained with S but a compelling harmony with the results attained with \(\varvec{M}\) . The evidence reveals two unprecedented discoveries: (1) that mathematical models aimed at explaining energy flow in non-sinusoidal circuits shouldn’t be based on the decomposition of S—as traditionally done— and, (2) the inappropriateness of extrapolating definitions from sinusoidal conditions to non-sinusoidal settings. PubDate: 2022-02-18 DOI: 10.1007/s00006-022-01200-8

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Abstract: Abstract 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 and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces. PubDate: 2022-02-14 DOI: 10.1007/s00006-021-01196-7

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Abstract: Abstract 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 solutions to new QRME are developed. A numerical example is presented to illustrate the derived results. PubDate: 2022-02-02 DOI: 10.1007/s00006-021-01192-x

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Abstract: Abstract 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, our product formula enables us to use similar methods as for the higher spin Laplace operator, exploiting the structure of the conformal Lie algebra and branching rules. PubDate: 2022-01-31 DOI: 10.1007/s00006-022-01198-z

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Abstract: Abstract 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 in terms of spherical monogenics and Jacobi polynomials are given, with consequences including the interlacing of zeros. In the case \(m=2\) we describe a degeneracy between the even- and odd-indexed polynomials. PubDate: 2022-01-31 DOI: 10.1007/s00006-021-01179-8

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Abstract: Abstract 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. PubDate: 2022-01-31 DOI: 10.1007/s00006-022-01199-y

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Abstract: Abstract 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 satisfies equality condition. PubDate: 2022-01-31 DOI: 10.1007/s00006-022-01201-7