Quaternion Algebra Research Articles

The reducible solution to a system of matrix equations over the Hamilton quaternion algebra

Reducible matrices are closely associated with the connection of directed graph and can be used in stochastic processes, biology and others. In this paper, we investigate the reducible solution to a system of matrix equations over the Hamilton quaternion algebra. We establish the necessary and sufficient conditions for the system to have a reducible solution and derive a formula of the general reducible solution of the system when it is solvable. Finally, we present a numerical example to illustrate the main results of this paper.

The two‐sided short‐time quaternionic offset linear canonical transform and associated convolution and correlation

In this paper, we introduce the two‐dimensional short‐time quaternion offset linear canonical transform (ST‐QOLCT), which is a generalization of the classical short‐time offset linear canonical transform (ST‐OLCT) in quaternion algebra setting. Several useful properties of the ST‐QOLCT are obtained from the properties of the ST‐QOLCT kernel. Based on the properties of the ST‐QOLCT and the convolution and correlation operators associated with QOLCT, we derive convolution and correlation theorems for the ST‐QOLCT. Finally, some potential applications of the ST‐QOLCT are introduced.

Mathematical Formulation of Arithmetic Surface (3, 5) Over Q

The primary arithmetic surface (3, 5) over Q is the focus of this work, which examines its construction and properties. Attaching the two symbols i,j where i2 = 3, j2 = 5, and ij = -ji yields a quaternion algebra over Q. This "example issue" involves the creation of a hyperbolic surface by linking the points (3, 5) over Q. If you want to demonstrate how to discover a Dirichlet domain, or how to generate a hyperbolic polygon, you may use this issue as an illustration. The list of generators that corresponds to the edge gluings is one possible motivation for seeking for a hyperbolic surface. The generators in this list have applications beyond only pure mathematics and science. Creating a co-compact Dirichlet domain in Hyperbolic 3-space is the focus of this article. Based on the concepts introduced in "Expository Note: An Arithmetic Surface," this article was written. We provide a brief summary of the material presented before constructing the Dirichlet domain in Hyperbolic 2-space associated with the quadratic form Q(x) = x2 +y2 -7z2. It is then shown that the new quadratic, Q(x) = w2 + x2 + y2 -7z2, also has a Dirchlet Domain in Hyperbolic 3-space.

МОДЕЛЬ РУХУ БЕЗПІЛОТНИХ ЛІТАЛЬНИХ АПАРАТІВ НА ОСНОВІ АЛГЕБРИ ДУАЛЬНИХ КВАТЕРНІОНІВ

The widespread use of unmanned aerial vehicles during warfare has intensified the problem of their management, especially when they are used in large groups. One of the main tasks is to ensure coordinated movement of the group's aircraft in space. Optimizing the movement of each device of the group in three-dimensional space is expedient to use mathematical models. The movement of any unmanned aerial vehicle can be presented as a combination of translational and rotational movements, and its speed as a combination of translational and rotational velocities. Previously, these movements were modeled separately using a system of differential equations or quaternions. In this article, a mathematical model of rotational and translational movements of an aircraft based on the algebra of dual quaternions is developed. Dual quaternions consisting of eight scalars are a compact representation of rigid transformations in space. Therefore, their properties determine the advantage in the course of motion simulation, as they reduce the amount of calculations. Thus, with the help of one dual quaternion, it is possible to provide both translational and rotational motions at once, and the operation of non-commutative multiplication of dual quaternions is used to simulate the movement. The model assumes that the real part of the dual quaternion determines the orientation of the UAV in space, and the dual part determines its position in three-dimensional space. In order to connect aircraft coordinate systems with the model, expressions for the transition from aircraft orientation angles (roll, yaw, and pitch) to dual quaternion parameters and vice versa are obtained. The functionality of the proposed model was confirmed using the developed software for modeling the coordinated movement of aircraft. The software is adapted for graphical display of a large number of aircraft in web browsers with WebGl support. Keywords: motion modeling; rotational and translational movement; unmanned aerial vehicles; quaternions; dual quaternions; algebra of quaternions.

Dual quaternion ambisonics array for six-degree-of-freedom acoustic representation

Spatial audio methods are gaining a growing interest due to the spread of immersive audio experiences and applications, such as virtual and augmented reality. For these purposes, 3D audio signals are often acquired through arrays of Ambisonics microphones, each comprising four capsules that decompose the sound field in spherical harmonics. In this paper, we propose a dual quaternion representation of the spatial sound field acquired through an array of two First Order Ambisonics (FOA) microphones. The audio signals are encapsulated in a dual quaternion that leverages quaternion algebra properties to exploit correlations among them. This augmented representation with 6 degrees of freedom (6DOF) involves a more accurate coverage of the sound field, resulting in a more precise sound localization and a more immersive audio experience. We evaluate our approach on a sound event localization and detection (SELD) benchmark. We show that our dual quaternion SELD model with temporal convolution blocks (DualQSELD-TCN) achieves better results with respect to real and quaternion-valued baselines thanks to our augmented representation of the sound field. Full code is available at: https://github.com/ispamm/DualQSELD-TCN.

Modified Hand–Eye Calibration Using Dual Quaternions

This paper presents a modified model for hand–eye calibration based on dual quaternion algebra. By using dual quaternions to represent the rotations and translations of a rigid body simultaneously in the task space, the formulation is elegant for the analysis of the hand–eye equation. The hand–eye transformation derived in this study is represented in a compact manner, which uses a combination of the dual part and the real part of the dual quaternion. Although the hand–eye equation can be solved by using six elements of a dual quaternion without using its scalar parts, the scaler numbers in both the real and dual parts of a dual quaternion contain part of the pose information. The originality is based on the derivation of the construct of the identification algorithm of external parameters of the camera by using all eight elements of a dual quaternion. Then, the data transformation between the cameras of the dual-arm hand–eye robot system is presented. The corresponding results demonstrate that the proposed hand–eye calibration algorithm can process measurement data with noise and can also improve the identification accuracy to verify its efficiency.

RKHS Representations for Augmented Quaternion Random Signals: Application to Detection Problems

The reproducing kernel Hilbert space (RKHS) methodology has shown to be a suitable tool for the resolution of a wide range of problems in statistical signal processing both in the real and complex domains. It relies on the idea of transforming the original functional data into an infinite series representation by projection onto an specific RKHS, which usually simplifies the statistical treatment without any loss of efficiency. Moreover, the advantages of quaternion algebra over real-valued three and four-dimensional vector algebra in the modelling of multidimensional data have been proven useful in much relatively recent research. This paper accordingly proposes a generic RKHS framework for the statistical analysis of augmented quaternion random vectors, which provide a complete description of their second order characteristics. It will allow us to exploit the full advantages of the RKHS theory in widely linear processing applications, such as signal detection. In particular, we address the detection of a quaternion signal disturbed by additive Gaussian noise and the discrimination between two quaternion Gaussian signals in continuous time.

Quaternion Attitude Control System of Highly Maneuverable Aircraft

In the era of rapid advancements in manned and unmanned aviation and robotics, there is a need for high-performance, robust attitude control of highly maneuverable fixed-wing aircraft, both manned and unmanned (UAVs). This paper presents an extension to research on spacecraft attitude control. The article extends existing concepts and applies them to the control problem of aircraft operating in Earth’s atmosphere. First, a general concept of quaternions is presented. Next, the attitude controller’s architecture is discussed. The controller synthesis is described using quaternion algebra. The quaternion-based attitude controller is then compared with a classical Euler-based attitude controller. The methodology for comparison and performance evaluation of both controllers is described. Lastly, the results of the simulations and a comparison of the two controllers are presented and discussed. The presented control scheme outperforms classical methods based on Euler angles, particularly at the aircraft’s high pitch and roll angles.

The structure of symmetric tensor powers of composition algebras

Let [Formula: see text] be a composition algebra which is either the Hamilton quaternion algebra [Formula: see text] or the Cayley octonion algebra [Formula: see text] over [Formula: see text]. In a previous work, the nth symmetric power [Formula: see text] of [Formula: see text] is shown to be a direct sum of central simple algebras, corresponding to the partitions of [Formula: see text] of length [Formula: see text], such that the component corresponding to the partition [Formula: see text] is isomorphic to the component [Formula: see text] of [Formula: see text] corresponding to the partition [Formula: see text] of [Formula: see text]. In this work, we study the building blocks [Formula: see text] of these decompositions. We show that the “local” structure of [Formula: see text], i.e. the complex-like subfields of [Formula: see text], determine both the complement of [Formula: see text] in [Formula: see text] and the trace map of [Formula: see text], induced from the trace map of [Formula: see text]. We also derive a recursive trace formula on the [Formula: see text]’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space [Formula: see text], which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra [Formula: see text] is described.

Theory of reduced biquaternion sparse representation and its applications

Traditional sparse representation models treat color image either represent color channels independently using the monochromatic model or concatenate color channels using the concatenation model. However, these two strategies cannot make full use of the correlation among the three color channels. In this paper, we propose a novel sparse representation model for color image based on reduced biquaternion. The advances of the proposed model are in two aspects: 1) compared with quaternion algebra, reduced biquaternion algebra is commutative algebra which has a low computation cost; 2) benefited from reduced biquaternion representation, the proposed model can treat the color image as a whole. We firstly extend the singular value decomposition (SVD) to the reduced biquaternion domain (RBSVD). The framework of the proposed model contains two major stages: first, reduced biquaternion orthogonal matching pursuit (RBOMP) is proposed in the stage of sparse coding; second, generalized K-means clustering for RBSVD (K-RBSVD) is proposed in the stage of dictionary training. Afterward, a new color image denoising algorithm is developed to demonstrate the effectiveness of the proposed model. The experimental results demonstrate that the proposed sparse representation model achieves state-of-the-art denoising performance in terms of both quantitative metrics and visual quality.

Some generalizations of Springer’s theorem in characteristic two

It is shown that an anisotropic [Formula: see text]-fold system of hermitian forms over a quaternion algebra in characteristic two remains anisotropic over all odd degree extensions of the base field. It is also shown that an anisotropic orthogonal involution on a central simple algebra of co-index 3 in characteristic two is anisotropic over any odd degree extension of the ground field.

Quaternion equations for hydrodynamic two-fluid model of vortex plasma

We discuss the application of quaternionic space-time algebra for the generalization of self-consistent equations describing the hydrodynamic two-fluid model of vortex plasma. It is shown that quaternionic formalism allows one to write the system of hydrodynamic equations in a compact form as one quaternion equation, which can be easy generalized to the case of damping plasma in an external electromagnetic field. As an illustration, we apply the proposed equations for the description of sound waves in electron–ion and electron–positron plasmas.

Exponential and logarithm of multivector in low-dimensional (n = p + q < 3) Clifford algebras

The aim of the paper is to give a uniform picture of complex, hyperbolic, and quaternion algebras from a perspective of the applied Clifford geometric algebra. Closed form expressions for a multivector exponential and logarithm are presented in real geometric algebras Clp;q when n = p + q = 1 (complex and hyperbolic numbers) and n = 2 (Hamilton, split, and conectorine quaternions). Starting from Cl0;1 and Cl1;0 algebras wherein square of a basis vector is either –1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras Cl0;2, Cl1;1, and Cl2;0. The sectors in the multivector coefficient space, where 2D logarithm exists are found. They are related with a square root of the multivector.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

In this paper, based on the slice regular quaternionic functions and spectral theory on the S-spectrum for quaternionic operators, firstly, the exponential growth bound of quaternionic operator semigroups is studied and the generalized Hille Lemma is obtained through introducing Bochner and Pettis integrals in quaternionic Banach space. Secondly, we study the quaternionic bisemigroups and the direct sum decomposition of quaternionic Banach space, then we introduce the notion of the quaternionic exponentially dichotomous operator and obtain its integral representation formula. It is crucial to note that the quaternionic operators we consider do not necessarily commute. Meanwhile, through establishing the S-spectral splitting theorem, the characterizations of quaternionic exponentially dichotomous operators are demonstrated. Moreover, the slice regular quaternionic bisemigroups generated by the quaternionic bisectorial operator are studied. Thirdly, by introducing the slice quaternionic Banach algebra and multiplicative quaternionic linear functionals, the perturbation invariance of exponential dichotomy for the quaternionic operators is explored. Under the noncommutative setting, our obtained results are essentially different from their analogues in the complex setting. As applications, we consider the Cauchy problems of quaternionic non-homogeneous evolution equations involving the quaternionic semigroup generators and exponentially dichotomous quaternionic operators. Additionally, the slice exponential dichotomy of quaternionic operators is characterized via the corresponding Cauchy problem.

Square integrals of the logarithmic derivatives of Selberg’s zeta functions in the critical strip

In our previous work (Y. Hashimoto, Selberg’s zeta function for the modular group in the critical strip, Math. Nachr. 294 (2021) 1899–1904, https://doi.org/10.1002/mana.202000268 ), we proposed an upper bound of the logarithmic derivative of Selberg’s zeta function for the modular group in the critical strip. This paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.

Distribution of toric periods of modular forms on definite quaternion algebras

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ and $\mathcal{O}$ an Eichler order in $D$ of square-free level. We study distribution of the toric periods of algebraic modular forms of level $\mathcal{O}$. We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on $\mathcal{O}$, we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic $L$-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients $\{a(n)\}_n$ of weight 3/2 modular forms, where $n$ ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them.

Explicit isomorphisms of quaternion algebras over quadratic global fields

Let L be a separable quadratic extension of either {mathbb {Q}} or {mathbb {F}}_q(t). We exhibit efficient algorithms for finding isomorphisms between quaternion algebras over L. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple L-algebra.

Continuous and discrete quaternion linear canonical wave packet transform

In this paper, we widen the domain of wave packet transform and linear canonical wave packet transform to the quaternion algebra. We prove certain properties of the quaternionic wave packet transform and quaternionic linear canonical wave packet transform. The reconstruction formula and Parseval’s relation for the introduced transforms are proved. Some examples of these transforms are discussed. We present an equivalent uncertainty principle as Heisenberg uncertainty principle for QLCWPT. We have introduced the discrete QLCWPT for application in image filtering. We have given the methodology of the application and then demonstrated it using an example.

О геометрической интерпретации кватернионов конусами

The geometric interpretation of quaternions is considered. The visualization complexity of quaternions is due to the fact that these objects have four independent parameters. A literature analysis shows that the problem of geometric interpretation of quaternions has not been completely solved to date.
 The first section provides general provisions on quaternions and the necessary notations. The second section describes the classical geometric interpretation of quaternions by arcs on a unit sphere. The third section describes a new geometric interpretation and its application to the problem of a vector finite rotation.
 The geometric interpretation of the quaternion as the surface of a right circular cone is presented. This representation allow demonstrating it as a holistic object in which the scalar and vector parts are interconnected, taking into account their modules and signs.
 For the considered normalized quaternion, it is easy to understanding an important entity, the quaternion versor: in general, it is a cone, which in the limiting case of a pure scalar quaternion transform into a sphere, and in the limiting case of a pure vector quaternion transform into an ordinary vector. This distinctive feature of the proposed geometric interpretation makes it possible, even when projected onto a plane, to clearly distinguish visualization of the quaternions with a nonzero scalar part from pure vector quaternions, which is difficult to do in the other known interpretations. The representation of quaternions by cones clearly demonstrates the need for a double quaternion product, when the vector is rotated around an arbitrary axis.
 Images of quaternions as cones, spheres and vectors can be useful in the study of quaternion algebra, which is currently finding increasing use in engineering.

Dynamics of Mobile Manipulators Using Dual Quaternion Algebra

Abstract This article presents two approaches to obtain the dynamical equations of mobile manipulators using dual quaternion algebra. The first one is based on a general recursive Newton–Euler formulation and uses twists and wrenches, which are propagated through high-level algebraic operations and works for any type of joints and arbitrary parameterizations. The second approach is based on Gauss’s Principle of Least Constraint (GPLC) and includes arbitrary equality constraints. In addition to showing the connections of GPLC with Gibbs–Appell and Kane’s equations, we use it to model a nonholonomic mobile manipulator. Our current formulations are more general than their counterparts in the state of the art, although GPLC is more computationally expensive, and simulation results show that they are as accurate as the classic recursive Newton–Euler algorithm.