Quaternion Algebra Research Articles

Fredholm theory in quaternionic Banach algebras

Muraleetharan and Thirulogasanthar (J. Math. Phys. 2018;59(10):103506, 27p.) introduced the concept of the Calkin S-spectrum of a bounded quaternionic linear operator. The study of this spectrum is established using Fredholm operator theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in a quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectrum. We also study the boundary S-spectrum. As an application, we investigate the Fredholm and Weyl S-spectra of bounded right quaternionic linear operators.

Commutative quaternion algebra and DSP fundamental properties: Quaternion convolution and Fourier transform

In recent years, quaternion algebra has been applied more and more in different applications. A primary motivation for investigating quaternion algebras is to extend the classical signal processing to quaternion signal processing theory. However, this algebra has non-commutative multiplication, leading to the uncertainty of such essential operations as the convolution and Fourier transform. As a result, it became difficult to process quaternion signals in the frequency domain. In this work, the arithmetic of quaternion numbers is analyzed using the Cayley-Dickson construction, but with an associative and commutative operation of multiplication. We call it the (2,2)-model of quaternions. This model's main properties of quaternions are described, including inverse, division, and conjugate operations. In the (2,2)-model, only two exponential functions with pure quaternion units exit, and therefore, only two types of the quaternion discrete Fourier transform (QDFT) can be defined. One of the crucial benefits of this model is that the unique definition of convolution will help reduce the complexity of quaternion convolution operation calculated using both types of QDFTs.

Jordan constants of quaternion algebras over number fields and simple abelian surfaces over fields of positive characteristic

AbstractWe compute and provide a detailed description on the Jordan constants of the multiplicative subgroup of quaternion algebras over number fields of small degree. As an application, we determine the Jordan constants of the multiplicative subgroup of the endomorphism algebras of simple abelian surfaces over fields of positive characteristic.

Quaternion algebras with derivations

We study derivations on quaternion algebras that stabilise quadratic subfields. Following the work of L. Juan and A. Magid [10] , we provide an explicit construction of a differential splitting field for a given differential quaternion algebra. We also examine the presence and impact of those derivations on quaternion algebras that admit new constants.

The spectral p-adic Jacquet–Langlands correspondence and a question of Serre

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

CPT Invariance of Quaternion Dirac equation

Abstract: In this paper the invariance of Quaternion Dirac equation under Lorentz Transformation, Charge conjugation, Parity transformation and Time reversal operation has been discussed successfully. The invariance under the combined operation of Charge conjugation, Parity and Time reversal (CPT) has also been discussed and expression for C, P, T and combined CPT operators have been obtained in terms of quaternions. Invariance condition for electric and magnetic field has also been obtained. It has been concluded that the Quaternion Dirac equation dominates over ordinary Dirac equation because of the advantage of algebra of quaternions.

Frequencies of astrophysical jets and gravitational strengths in the octonion spaces

This paper focuses on applying the octonions to explore the electromagnetic and gravitational equations in the presence of some material media, exploring the frequencies of astrophysical jets. Maxwell was the first to use the algebra of quaternions to describe the electromagnetic equations. This method encourages scholars in adopting the quaternions and octonions to study the physical properties of electromagnetic and gravitational fields, including the field strength, field source, linear momentum, angular momentum and so forth. In this paper, the field strength and angular momentum in the vacuum can be combined together to become one new physical quantity, that is, the composite field strength within the material media. Substituting the latter for the field strength in the vacuum will deduce the field equations within material media, including the electromagnetic and gravitational equations in the presence of some material media. In terms of the electromagnetic fields, the electromagnetic equations in the presence of some electromagnetic media are able to explore a few new physical properties of electromagnetic media. Especially, in case the magnetic flux density and magnetization intensity both fluctuate at a single frequency, their frequencies must be identical to each other within electromagnetic media. In some extreme cases, the electromagnetic equations within electromagnetic media will be degenerated into Maxwell’s equations in the vacuum. The above reveals that the electromagnetic equations within electromagnetic media are capable of extending the scope of application of electromagnetic theory. For the gravitational fields, there are some similar inferences and conclusions within gravitational media. Further they can be utilized to research the frequencies of astrophysical jets.

New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays.

During the past decades, many works on Hopf bifurcation of fractional-order neural networks are mainly concerned with real-valued and complex-valued cases. However, few publications involve the quaternion-valued neural networks which is a generalization of real-valued and complex-valued neural networks. In this present study, we explorate the Hopf bifurcation problem for fractional-order quaternion-valued neural networks involving leakage delays. Taking advantage of the Hamilton rule of quaternion algebra, we decompose the addressed fractional-order quaternion-valued delayed neural networks into the equivalent eight real valued networks. Then the delay-inspired bifurcation condition of the eight real valued networks are derived by making use of the stability criterion and bifurcation theory of fractional-order differential dynamical systems. The impact of leakage delay on the bifurcation behavior of the involved fractional-order quaternion-valued delayed neural networks has been revealed. Software simulations are implemented to support the effectiveness of the derived fruits of this study. The research supplements the work of Huang et al. (Neural Netw 117:67-93, 2019).

Equivalence between K-functionals and modulus of smoothness on the quaternion algebra

Equivalence between K-functionals and modulus of smoothness on the quaternion algebra

Kinematical modeling of circular and elliptical polarization of the polarized light via screw transformation

In this paper, it is shown that the motion of a polarized light wave in an optical fiber has a special meaning that relates the scalar and vector part of the dual quaternion to the kinematics concept of a screw. We describe the circular and elliptical polarization of a light wave according to how the electric field vector varies in a projection onto a plane perpendicular to the propagation direction using the Clifford algebra of the dual quaternion. Also, we determine the circular-Rytov curve (C R ) and elliptical-Rytov curve (E R ), which described the paths traced out by the tip of the electric field vector as it propagates in space, related to the polarization state of the electric field . The elliptical and circular polarization states are expressed by using the four Stokes parameters and their matrix form. Furthermore, some motivating examples are given and visualized their images with the help of the MAPLE program.

Clifford Algebra and Hypercomplex Number as well as Their Applications in Physics

The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2n-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.

Fast Quaternion Product Units for Learning Disentangled Representations in [Formula: see text

Real-world 3D structured data like point clouds and skeletons often can be represented as data in a 3D rotation group (denoted as [Formula: see text]). However, most existing neural networks are tailored for the data in the euclidean space, which makes the 3D rotation data not closed under their algebraic operations and leads to sub-optimal performance in 3D-related learning tasks. To resolve the issues caused by the above mismatching between data and model, we propose a novel non-real neuron model called quaternion product unit (QPU) to represent data on 3D rotation groups. The proposed QPU leverages quaternion algebra and the law of the 3D rotation group, representing 3D rotation data as quaternions and merging them via a weighted chain of Hamilton products. We demonstrate that the QPU mathematically maintains the [Formula: see text] structure of the 3D rotation data during the inference process and disentangles the 3D representations into "rotation-invariant" features and "rotation-equivariant" features, respectively. Moreover, we design a fast QPU to accelerate the computation of QPU. The fast QPU applies a tree-structured data indexing process, and accordingly, leverages the power of parallel computing, which reduces the computational complexity of QPU in a single thread from O(N) to O(logN). Taking the fast QPU as a basic module, we develop a series of quaternion neural networks (QNNs), including quaternion multi-layer perceptron (QMLP), quaternion message passing (QMP), and so on. In addition, we make the QNNs compatible with conventional real-valued neural networks and applicable for both skeletons and point clouds. Experiments on synthetic and real-world 3D tasks show that the QNNs based on our fast QPUs are superior to state-of-the-art real-valued models, especially in the scenarios requiring the robustness to random rotations. The code of this work is available at https://github.com/SuferQin/Fast-QPU.

Predicting Rigid Body Dynamics Using Dual Quaternion Recurrent Neural Networks With Quaternion Attention

We propose a novel neural network architecture based on dual quaternions which allow for a compact representation of information with a main focus on describing rigid body movements. After introducing the underlying dual quaternion math, we derive dual quaternion valued neural network layer which are generally applickable to all sorts of problems which can benefit from a mathematical description in dual quaternion space. To cover the dynamic behavior inherent to rigid body movements, we propose recurrent architectures in the neural network. To further model the rigid bodies interactions efficiently, we incorporate a novel attention mechanism employing dual quaternion algebra. The introduced architecture is trainable by means of gradient based algorithms. We apply our approach to a parcel prediction problem where a rigid body with an initial position, orientation, velocity and angular velocity moves through a fixed simulation environment which exhibits rich interactions between the parcel and the boundaries. There we can show that the dual quaternion valued models outperform their counterparts operation on the real numbers, confirming the successful introduction of an inductive bias through the usage of dual quaternion math. Furthermore, we used an advantageous custom data augmentation technique specifically tailored for the usage with our dual quaternion valued input data.

The η-Hermitian solutions to some systems of real quaternion matrix equations

Let Hmxn be the set of all m x n matrices over the real quaternion algebra. We call that A ? Hnxn is ?-Hermitian if A = A?* where A?* = -?A*?,? ? {i,j,k},i,j,k are the quaternion units. In this paper, we derive some solvability conditions and the general solution to a system of real quaternion matrix equations. As an application, we present some necessary and sufficient conditions for the existence of an ?-Hermitian solution to some systems of real quaternion matrix equations. We also give the expressions of the general ?-Hermitian solutions to these systems when they are solvable. Some numerical examples are given to illustrate the results of this paper.

Hyperquaternionic Representations of Conic Sections

In this paper, by means of an isomorphism, we express the Clifford algebra Cl5,3 as hyperquaternion algebra H ⊗ H ⊗ H ⊗ H (a four-fold tensor product of quaternion algebras) and we provide the hyperquaternionic approach to the inner product null space (IPNS) representation of conic sections.

Jordan semi-triple derivations and Jordan centralizers on generalized quaternion algebras

<abstract><p>In this paper, we investigate Jordan semi-triple derivations and Jordan centralizers on generalized quaternion algebras over the field of real numbers. We prove that every Jordan semi-triple derivation on generalized quaternion algebras over the field of real numbers is a derivation. Also, we show that every left (resp, right) Jordan centralizer on generalized quaternion algebras over the field of real numbers is a left (resp, right) centralizer.</p></abstract>

广义四元数代数的Jordan中心化子和Lie中心化子

广义四元数代数的Jordan中心化子和Lie中心化子

Elliptical kinematics of the accretive surface growth

The stresses within the soft tissue are not constant for some shell surfaces. They vary with position along the mantle edge. In this paper, we show that elliptical geometry is more convenient to describe this type of surface. Thus, we introduce the elliptical kinematics along an initial curve and construct some accretive surfaces with an elliptical cross-section. In fact, these surfaces are not only curves with an elliptical cross-sectional curve, but also the material points of the surface follow an elliptical trajectory during their formation. This situation can be easily explained through elliptical motion and elliptical quaternion algebra. Then, we investigate the relationship between velocity and eccentricity of the surfaces and compare it to the case of circular motion. Furthermore, we visualize some examples to support the theoretical results through the MAPLE program.

On Hermitian Solutions of the Generalized Quaternion Matrix Equation A X B + C X D = E

The paper deals with the matrix equation A X B + C X D = E over the generalized quaternions. By the tools of the real representation of a generalized quaternion matrix, Kronecker product as well as vec-operator, the paper derives the necessary and sufficient conditions for the existence of a Hermitian solution and gives the explicit general expression of the solution when it is solvable and provides a numerical example to test our results. The paper proposes a unificated algebraic technique for finding Hermitian solutions to the mentioned matrix equation over the generalized quaternions, which includes many important quaternion algebras, such as the Hamilton quaternions and the split quaternions.

Accurate Quaternion Polar Harmonic Transform for Color Image Analysis

Polar harmonic transforms (PHTs) have been applied in pattern recognition and image analysis. But the current computational framework of PHTs has two main demerits. First, some significant color information may be lost during color image processing in conventional methods because they are based on RGB decomposition or graying. Second, PHTs are influenced by geometric errors and numerical integration errors, which can be seen from image reconstruction errors. This paper presents a novel computational framework of quaternion polar harmonic transforms (QPHTs), namely, accurate QPHTs (AQPHTs). First, to holistically handle color images, quaternion-based PHTs are introduced by using the algebra of quaternions. Second, the Gaussian numerical integration is adopted for geometric and numerical error reduction. When compared with CNNs (convolutional neural networks)-based methods (i.e., VGG16) on the Oxford5K dataset, our AQPHT achieves better performance of scaling invariant representation. Moreover, when evaluated on standard image retrieval benchmarks, our AQPHT using smaller dimension of feature vector achieves comparable results with CNNs-based methods and outperforms the hand craft-based methods by 9.6% w.r.t mAP on the Holidays dataset.