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Abstract: Abstract In this paper, we are concerned with the problem of locating the zeros of polynomials and regular functions with quaternionic coefficients when their real and imaginary parts are restricted. The extended Schwarz’s lemma, the maximum modulus theorem, and the structure of the zero sets defined in the newly constructed theory of regular functions and polynomials of a quaternionic variable are used to deduce the bounds for the zeros of these polynomials and regular functions. Our findings generalise certain recently established results about the zero distribution for this subclass of regular functions. PubDate: 2024-08-07
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Abstract: Abstract In this paper, we present two simple methods for constructing Beltrami fields. The first one consists of a composition of operators, including a quaternionic transmutation operator as well as the computation of formal powers for the function \(f(x)=e^{\textbf{i}\lambda x}\) . For the second method, we generate Beltrami fields from harmonic functions, and using the intrinsic relation between the normal and tangential derivative, we solve an associated Neumann-type boundary value problem. PubDate: 2024-08-01
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Abstract: Abstract Due to its potential application in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention. This paper addresses quaternionic subspace Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. We characterize subspace quaternionic Gabor frames in terms of quaternionic Zak transformation matrices. For an arbitrary subspace Gabor frame, we give a parametric expression of its Gabor duals of type I and type II, and characterize the uniqueness Gabor duals of type I and type II. And as an application, given a Gabor frame for the whole space \(L^{2}({\mathbb {R}}^{2},\,{\mathbb {H}})\) , we give a parametric expression of its all Gabor duals, and derive its unique Gabor dual of type II. Some examples are also provided. PubDate: 2024-07-14
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Abstract: Abstract We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are \(*\) -quantum spaces for the quantum orthogonal group \(\mathcal {O}(SO_q(3))\) . We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of \(SO_q(3)\) . The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules \(\mathcal {E}_n\) of rank 1 and of degree computed to be an even integer \(-2n\) . The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For q real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra \({\mathcal {U}_{q^{1/2}}(sl_2)}\) which is dual to \(\mathcal {O}(SO_q(3))\) . PubDate: 2024-07-13
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Abstract: Abstract In this paper we give an exposition of several recent results on local symmetries of real submanifolds in complex space, featuring new examples and important corollaries. Departing from Levi non-degenerate hypersurfaces, treated in the classical Chern–Moser theory, we explore three important classes of manifolds, which naturally extend the classical case. We start with quadratic models for real submanifolds of higher codimension and review some recent striking results, which demonstrate that such higher codimension models may possess symmetries of arbitrarily high jet degree. This disproves the long held belief that the fundamental 2-jet determination results from Chern–Moser theory extend to this case. As a second case, we consider hypersurfaces with singular Levi form at a point, which are of finite multitype. This leads to the study of holomorphically nondegenerate polynomial models. We outline several results on their symmetry algebras including a characterization of models admitting nonlinear symmetries. In the third part we consider the class of structures with everywhere singular Levi forms that has received the most attention recently, namely everywhere 2-nondegenerate structures. We present a computation of their Catlin multitype and results on symmetry algebras of their weighted homogeneous (w.r.t. multitype) models. PubDate: 2024-07-05
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Abstract: Abstract Detecting the concavity and convexity of three-dimensional (3D) geometric objects is a well-established challenge in the realm of computer graphics. Serving as the cornerstone for various related graphics algorithms and operations, researchers have put forth numerous algorithms for discerning the concavity and convexity of such objects. The majority of existing methods primarily rely on Euclidean geometry, determining concavity and convexity by calculating the vertices of these objects. However, within the realm of Euclidean geometric space, there exists a lack of uniformity in the expression and calculation rules for geometric objects of differing dimensions. Consequently, distinct concavity and convexity detection algorithms must be tailored for geometric objects with varying dimensions. This approach inevitably results in heightened complexity and instability within the algorithmic structure. To address these aforementioned issues, this paper introduces geometric algebra theory into the domain of concavity and convexity detection within 3D spatial objects. With the algorithms devised in this study, it becomes feasible to detect concavity and convexity for geometric objects of varying dimensions, all based on a uniform set of criteria. In comparison to concavity-convexity detection algorithms grounded in Euclidean geometry, this research effectively streamlines the algorithmic structure. PubDate: 2024-07-02
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Abstract: Abstract This work studies the Clifford algebra approach to the density matrix. We discuss elementary examples of pure and mixed states by writing the density matrix as an element of the Clifford algebra of the three-dimensional space \(Cl_3\) . We also revisit the phenomenon of Larmor precession within the framework of Clifford algebra. Additionally, we discuss the geometrical interpretation of the so-called Clifford Density Element (CDE) for pure states in analogy to the Bloch sphere of conventional quantum theory. Finally, we discuss the dynamics of the CDE, which obeys an algebraic form of the Liouville von–Neumann equation. PubDate: 2024-06-29
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Abstract: Abstract This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group \({\mathbb {S}}\) , i.e., the unit 3-sphere in the quaternion algebra. Secondly, we reveal a new phenomenon: The compact convex set of such functions is not a Bauer simplex except when G is of exponent \(\le 2\) . In contrast, its complex counterpart is always a Bauer simplex, as is well known. We also present an integral representation for such functions as an application and some other minor results. PubDate: 2024-06-25
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Abstract: Abstract The aim of this paper is to analyze and prove different facts related with bicomplex Möbius transformations. Various algebraic and geometric results were obtained, using the decomposition of the bicomplex set as: \({{\mathbb {B}}}{{\mathbb {C}}}= {{\mathbb {D}}}+ \textbf{i}{{\mathbb {D}}}\) , and there were used actively both, hyperbolic and bicomplex, geometric objects. The basics of bicomplex Lobachevsky’s geometry are given. PubDate: 2024-06-24
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Abstract: Abstract The theory of slice regular functions has been developed rapidly in the past few years, and most properties are given in slices at the early stage. In 2013, Colombo et al. introduced a non-constant coefficients differential operator to describe slice regular functions globally, and this brought the study of slice regular functions in a global sense. In this article, we introduce a slice Cauchy–Riemann operator, which is motivated by the non-constant coefficients differential operator mentioned above. Then, A Borel–Pompeiu formula for this slice Cauchy–Riemann operator is discovered, which leads to a Cauchy integral formula for slice regular functions. A Plemelj integral formula for the slice Cauchy–Riemann operator is introduced, which gives rise to results on slice regular extension at the end. PubDate: 2024-06-24
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Abstract: Abstract From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time. PubDate: 2024-06-20
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Abstract: Abstract Let $$\begin{aligned} \mathcal {D}_{16}=\left\{ Z\in \mathcal {M}_{1,2}(\mathfrak {C}^{c}):\;\begin{array}{lll} 1-\left\langle Z,Z \right\rangle +\left\langle Z^{\sharp },Z^{\sharp }\right\rangle>0,\\ 2-\left\langle Z,Z \right\rangle \; >0\end{array}\right\} \end{aligned}$$ be an exceptional domain of non-tube type and let \(\mathcal {U}_{\nu }\) and \(\mathcal {W}_{\nu }\) the associated generalized Hua operators. In this paper, we determine the explicit formula of the action of the group \( E_{6(-14)}\) on \(\mathcal {D}_{16}\) . We characterized the \(L^{p}\) -range, \(1\le p < \infty \) of the generalized Poisson transform on the Shilov boundary of the domain \(\mathcal {D}_{16}\) . PubDate: 2024-06-13 DOI: 10.1007/s00006-024-01335-w
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Abstract: Abstract In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion-valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff–Young inequality for QQPFT, which in particular sharpens the constant in the inequality for the quaternion offset linear canonical transform (QOLCT). We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner–Ville distribution associated with QQPFT and obtain the Lieb’s uncertainty and entropy uncertainty principles for these three transforms. PubDate: 2024-06-11 DOI: 10.1007/s00006-024-01334-x
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Abstract: Abstract The aim of this paper is to study the properties of the Möbius addition \(\oplus \) under the action of the gyration operator gyr[a, b], and the relation between \((\sigma ,t)\) -translation defined by the Möbius addition and the generalized Laplace–Beltrami operator \(\Delta _{\sigma ,t} \) in the octonionic space. Despite the challenges posed by the non-associativity and non-commutativity of octonions, Möbius addition still exhibits many significant properties in the octonionic space, such as the left cancellation law and the gyrocommutative law. We introduce a novel approach to computing the Jacobian determinant of Möbius addition. Then, we discover that the gyration operator is closely related to the Jacobian matrix of Möbius addition. Importantly, we determine that the distinction between \(a\oplus x\) and \(x\oplus a \) is a specific orthogonal matrix factor. Finally, we demonstrate that the \((\sigma ,t)\) -translation is a unitary operator in \(L^2 \left( {\mathbb {B}^8_t,d\tau _{\sigma ,t} } \right) \) and it commutes with the generalized Laplace–Beltrami operator \(\Delta _{\sigma ,t} \) in the octonionic space. PubDate: 2024-06-08 DOI: 10.1007/s00006-024-01333-y
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Abstract: Abstract In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-s Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras. PubDate: 2024-06-03 DOI: 10.1007/s00006-024-01322-1
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Abstract: Abstract Let \(X=Sp(1,n)/Sp(n)\) be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where n is greater than or equal to 1. The Fürstenberg boundary of X is denoted as \(\Sigma \) . In this paper, we focus on the Plancherel formula on X associated with the Poisson transform of vector-valued \(L^2\) -space on \(\Sigma \) . Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of Sp(1, n) on \(L^2(X)\) . PubDate: 2024-05-28 DOI: 10.1007/s00006-024-01330-1
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Abstract: Abstract Multi-loop coupling mechanisms (MCMs) have been widely used in spacedeployable antennas. However, the mobility of MCMs is difficult to analyze due to their complicated structure and coupled limbs. This paper proposes a general method for calculating the mobility of MCMs using geometric algebra (GA). For the independent limbs in the MCM, the twist spaces are constructed by the join operator. For coupled limbs coupled with closed loops in the MCM, the equivalent limbs can be found by solving the analytical expressions of the twist space on each closed loop’s output link. Then, the twist spaces of the coupled limbs can be easily obtained. The twist space of the MCM’s output link is the intersection of all the limb twist spaces, which can be calculated by the meet operator. The proposed method provides a simplified way of analyzing the mobility of MCMs, and three typical MCMs are chosen to validate this method. The analytical mobility of the MCM’s output link can be obtained, and it naturally indicates both the number and the property of the degrees of freedom (DOFs). PubDate: 2024-05-27 DOI: 10.1007/s00006-024-01329-8
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Abstract: Abstract In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics. PubDate: 2024-05-27 DOI: 10.1007/s00006-024-01328-9
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Abstract: Abstract This paper investigates the distribution function and nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\) -valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for \( \mathbb{B}\mathbb{C}\) -valued functions, which characterizes valuable insights into the behavior and structure of \(\mathbb{B}\mathbb{C}\) -valued functions, allowing to analyze their properties and establish connections with other mathematical concepts. Next, the nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\) -valued functions with the hyperbolic norm are studied. By exploring the nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\) -valued functions, it is aimed to determine how the hyperbolic norm influences the rearrangement process and its impact on the function’s behavior and properties. PubDate: 2024-05-18 DOI: 10.1007/s00006-024-01327-w
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Abstract: Abstract In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform in several key aspects: it does not conform to the Plancherel theorem, alters the interplay between derivatives and multiplication, and the product of algebra elements does not necessarily maintain the magnitude relationships found in classical settings. To overcome these challenges, our approach involves constructing the Cayley–Dickson Fourier transform by sequentially applying classical Fourier transforms. A pivotal part of our strategy is the utilization of a theorem that facilitates the norm-preserving extension of linear operators between spaces \(L^p\) and \(L^q.\) Furthermore, our investigation brings new insights into the complexities surrounding the Beckner–Hirschman Entropic inequality in the context of non-associative algebras. PubDate: 2024-05-10 DOI: 10.1007/s00006-024-01326-x
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