Quaternion Field Research Articles

An extension of quaternionic metrics to octonions in a non-Riemannian spacetime: Including Yang–Mills-like fields within a division octonion algebra

In this paper, we develop an extended space-time geometry that includes an internal space built with division octonions with realization via Pauli matrices and Zorn (vectorial) matrices. Here, we extend a former field theory that used split octonion algebra to a division octonion algebra on a non-Riemannian manifold. The octonionic algebra is the last possible algebra allowed by the Hurwitz theorem. The interpretation of the “octonionic field” in this division octonionic algebra is not straightforward, however it behaves in a similar way to the quaternionic Yang–Mills fields. Former work suggests that this Yang–Mills-like octonionic field is associated with the field governing quarks within nucleons. In this work, we discover that the inclusion of octonionic fields in the geometry of an internal space necessarily excludes the quaternionic (Yang–Mills) fields in an extended non-Riemannian geometry. This is not what is expected from the standpoint of a “unified” field theory, which leads us to propose a different approach to Einstein’s unified field theory.

Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field

In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure.

Biquaternionic Analysis, Cyclic Quaternionic Fields, and Generalization of the Kerr–Penrose Theorem

Biquaternionic Analysis, Cyclic Quaternionic Fields, and Generalization of the Kerr–Penrose Theorem

A possibility of Klein paradox in quaternionic (3 + 1) frame

In light of the significance of non-commutative quaternionic algebra in modern physics, this study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing quaternionic wave function, we rewrite the Klein–Gordon equation in extended quaternionic form that includes scalar and the vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein–Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle’s kinetic energy is less than [Formula: see text]. Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero.

On the Uniqueness of Continuation for Polynomials of Harmonic Quaternion Fields

On the Uniqueness of Continuation for Polynomials of Harmonic Quaternion Fields

Fixed/Preassigned-Time Synchronization of Fully Quaternion-Valued Cohen–Grossberg Neural Networks with Generalized Time Delay

This article is concerned with fixed-time synchronization and preassigned-time synchronization of Cohen–Grossberg quaternion-valued neural networks with discontinuous activation functions and generalized time-varying delays. Firstly, a dynamic model of Cohen–Grossberg neural networks is introduced in the quaternion field, where the time delay successfully integrates discrete-time delay and proportional delay. Secondly, two types of discontinuous controllers employing the quaternion-valued signum function are designed. Without utilizing the conventional separation technique, by developing a direct analytical approach and using the theory of non-smooth analysis, several adequate criteria are derived to achieve fixed-time synchronization of Cohen–Grossberg neural networks and some more precise convergence times are estimated. To cater to practical requirements, preassigned-time synchronization is also addressed, which shows that the drive-slave networks reach synchronization within a specified time. Finally, two numerical simulations are presented to validate the effectiveness of the designed controllers and criteria.

A novel fixed-time error-monitoring neural network for solving dynamic quaternion-valued Sylvester equations

This paper addresses the dynamic quaternion-valued Sylvester equation (DQSE) using the quaternion real representation and the neural network method. To transform the Sylvester equation in the quaternion field into an equivalent equation in the real field, three different real representation modes for the quaternion are adopted by considering the non-commutativity of quaternion multiplication. Based on the equivalent Sylvester equation in the real field, a novel recurrent neural network model with an integral design formula is proposed to solve the DQSE. The proposed model, referred to as the fixed-time error-monitoring neural network (FTEMNN), achieves fixed-time convergence through the action of a state-of-the-art nonlinear activation function. The fixed-time convergence of the FTEMNN model is theoretically analyzed. Two examples are presented to verify the performance of the FTEMNN model with a specific focus on fixed-time convergence. Furthermore, the chattering phenomenon of the FTEMNN model is discussed, and a saturation function scheme is designed. Finally, the practical value of the FTEMNN model is demonstrated through its application to image fusion denoising.

Dirac–Proca–Maxwell equivalence of dyonic matter with quantum corrections for the quaternionic spinor fields

This research explores a novel approach to obtaining the dyonic Dirac–Proca–Maxwell fields within a single framework. Because Dirac fields are quantum fields for massive fermions, we construct quaternionic fields using a Dirac spinor basis to formulate quantum corrections for electric and magnetic field vectors, electromagnetic field equations, wave equations for scalar and vector potentials, continuity equations, and the propagation of electric and magnetic waves for massive dyons. The quaternionic form of the Dirac spinor fields is used to derive the quantum correction of the electromagnetic field energy of dyonic matter. It is demonstrated that the Maxwell equations for the classical electromagnetic field can be obtained from the quaternionic Dirac spinor fields by applying the quantum approximation condition. Therefore, the quaternionic quantum description of dyonic electromagnetic fields exhibits both wave and particle behaviour for the dynamics of matter particles.

A Comprehensive Review of Continuous-/Discontinuous-Time Fractional-Order Multidimensional Neural Networks.

The dynamical study of continuous-/discontinuous-time fractional-order neural networks (FONNs) has been thoroughly explored, and several publications have been made available. This study is designed to give an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time, including Hopfield NNs (HNNs), Cohen-Grossberg NNs, and bidirectional associative memory NNs, and similar models are considered in real (R), complex (C), quaternion (Q), and octonion (O) fields. Since, in practice, delays are unavoidable, theoretical findings from multidimensional FONNs with various types of delays are thoroughly evaluated. Some required and adequate stability and synchronization requirements are also mentioned for fractional-order NNs without delays.

Exponential control design on fixed-time synchronization of fully quaternion-valued memristive delayed neural networks without decomposition

This article mainly explores fixed-time (FXT) synchronization of fully quaternion-valued memristive neural networks (QVMNNs) with generalized delays. Firstly, based on the set-valued mapping theory and the nonsmooth approach, the discontinuity of quaternion-valued memristive connection weights is successfully solved. Next, by introducing the signum function, absolute-like norm and quadratic norm in the quaternion field, a direct non-decomposing method is put forward. Under this framework, several exponential-type controllers are designed, which have only one exponential term and no longer contain the traditional linear part and the power-law terms. Compared with designing four real-value controllers, the control schemes proposed in this article are simpler in form. Besides, several criteria of FXT synchronization and the upper bound of the setting time are deduced based on Lyapunov method and Taylor expansion. Lastly, the achieved synchronization results are confirmed via numerical examples.

Characteristic of Quaternion Algebra Over Fields

Quaternion is an extension of the complex number system. Quaternion are discovered by formulating 4 points in 4-dimensional vector space using the cross product between two standard vectors. Quaternion algebra over a field is a 4-dimensional vector space with bases and the elements of the algebra are members of the field. Each element in quaternion algebra has an inverse, despite the fact that the ring is not commutative. Based on this, the purpose of this study is to obtain the characteristics of split quaternion algebra and determine how it interacts with central simple algebra. The research method used in this paper is literature study on quaternion algebra, field and central simple algebra. The results of this study establish the equivalence of split quaternion algebra as well as the theorem relating central simple algebra and quaternion algebra. The conclusion obtained from this study is that split quaternion algebra has five different characteristics and quaternion algebra is a central simple algebra with dimensions less than equal to four.

Event-Triggered Quantized Quasisynchronization of Uncertain Quaternion-Valued Chaotic Neural Networks With Time-Varying Delay for Image Encryption.

In this article, a class of quaternion-valued master-slave neural networks (NNs) with time-varying delay and parameter uncertainties was first established by conducting the extension from real-valued chaotic NNs to the quaternion field. Then, based on logarithmic quantized output feedback, the quasisynchronization issue of the NNs was investigated via devising a neoteric dynamic event-triggered controller. In virtue of the classical Lyapunov method and a generalized Halanay inequality, not only corresponding synchronization criteria were obtained to realize the quasisynchronization of master-slave NNs but also a precise upper bound was provided. Moreover, Zeno behavior can be eliminated under the presented scheme in this article. The accuracy of the theoretical outcomes was demonstrated by means of Chua's circuit. Ultimately, some experimental results of pragmatic application in image encryption/decryption were exposed to substantiate the feasibility and efficacy of the current algorithm for the proposed quaternion-valued NNs.

Impulsive Destabilization Effect on Novel Existence of Solution and Global μ-Stability for MNNs in Quaternion Field

In this paper, a novel memristor-based non-delay Hopfield neural network with impulsive effects is designed in a quaternion field. Some special inequalities, differential inclusion, Hamilton rules and impulsive system theories are utilized in this manuscript to investigate potential solutions and obtain some sufficient criteria. In addition, through choosing proper μ(t) and impulsive points, the global μ-stability of the solution is discussed and some sufficient criteria are presented by special technologies. Then, from the obtained sufficient criteria of global μ-stability, other stability criteria including exponential stability and power stability can be easily derived. Finally, one numerical example is given to illustrate the feasibility and validity of the derived conclusions.

Image processing using the quantum quaternion Fourier transform

There is a growing interest in quantum image processing (QIP) that rose from the desire to exploit the properties of quantum computing to improve the performance of classical techniques and their applications. Since the introduction of a quaternion by Hamilton in 1843, quaternions had been used in a lot of applications. We show how the quantum states and quantum operators work in the field of quaternions and how we can use it to extend the quantum complex Fourier transform (QCFT) to the quantum quaternion Fourier transform (QQFT). We propose for the first time the novel algorithm for the QQFT. We explain the impossibility of the use of convolution in quantum image processing; we design filters in the quaternion Fourier spectrum and the space domain an edge detector and a quantum median filter.

Quaternion lattices and quaternion fields

Let Q_8 be the quaternion group of order 8 and {chi } its faithful irreducible character. Then {chi } can be realized over certain imaginary quadratic number fields K={mathbb Q}bigl (sqrt{-N}bigr ) but not over their rings of integers (Feit, Serre); here N is a positive square-free integer. We show that this happens precisely when {mathbb Q}bigl (sqrt{N}bigr ) but not {mathbb Q}bigl (sqrt{2}, sqrt{N}bigr ) can be embedded into a Q_8-field over the rationals (Galois with group Q_8) and N is not a sum of two integer squares. In particular, we get that {chi } cannot be integrally realized if N is (properly) divisible by some prime qequiv 7,({textrm{mod},}8).

System decomposition-based stability criteria for Takagi-Sugeno fuzzy uncertain stochastic delayed neural networks in quaternion field

<abstract><p>Stochastic disturbances often occur in real-world systems which can lead to undesirable system dynamics. Therefore, it is necessary to investigate stochastic disturbances in neural network modeling. As such, this paper examines the stability problem for Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks. By applying Takagi-Sugeno fuzzy models and stochastic analysis, we first consider a general form of Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks with time-varying delays. Then, by constructing suitable Lyapunov-Krasovskii functional, we present new delay-dependent robust and global asymptotic stability criteria for the considered networks. Furthermore, we present our results in terms of real-valued linear matrix inequalities that can be solved in MATLAB LMI toolbox. Finally, two numerical examples are presented with their simulations to demonstrate the validity of the theoretical analysis.</p></abstract>

Deformable Protein Shape Classification Based on Deep Learning, and the Fractional Fokker-Planck and Kähler-Dirac Equations.

The classification of deformable protein shapes, based solely on their macromolecular surfaces, is a challenging problem in protein-protein interaction prediction and protein design. Shape classification is made difficult by the fact that proteins are dynamic, flexible entities with high geometrical complexity. In this paper, we introduce a novel description for such deformable shapes. This description is based on the bifractional Fokker-Planck and Dirac-Kähler equations. These equations analyse and probe protein shapes in terms of a scalar, vectorial and non-commuting quaternionic field, allowing for a more comprehensive description of the protein shapes. An underlying non-Markovian Lévy random walk establishes geometrical relationships between distant regions while recalling previous analyses. Classification is performed with a multiobjective deep hierarchical pyramidal neural network, thus performing a multilevel analysis of the description. Our approach is applied to the SHREC'19 dataset for deformable protein shapes classification and to the SHREC'16 dataset for deformable partial shapes classification, demonstrating the effectiveness and generality of our approach.

S 3 -Slicer

Multi-axis motion introduces more degrees of freedom into the process of 3D printing to enable different objectives of fabrication by accumulating materials layers upon curved layers. An existing challenge is how to effectively generate the curved layers satisfying multiple objectives simultaneously. This paper presents a general slicing framework for achieving multiple fabrication objectives including support free, strength reinforcement and surface quality. These objectives are formulated as local printing directions varied in the volume of a solid, which are achieved by computing the rotation-driven deformation for the input model. The height field of a deformed model is mapped into a scalar field on its original shape, the isosurfaces of which give the curved layers of multi-axis 3D printing. The deformation can be effectively optimized with the help of quaternion fields to achieve the fabrication objectives. The effectiveness of our method has been verified on a variety of models.

Quaternionic fermionic field

The second quantization of the quaternionic fermionic field is undertaken using the real Hilbert space approach to quaternionic quantum mechanics ([Formula: see text]QM). The solution responds to an open problem of quaternionic quantum theory, and launches the basis to the development of the quaternionic interaction theory.

On the Quaternion Transformation and Field Equations in Curved Space-Time

In this paper, we use four-dimensional quaternionic algebra to describe space-time geodesics in curvature form. The transformation relations of a quaternionic variables are established with the help of basis-transformations of quaternion algebra. We deduce the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and the geodesic equation are also discussed to describe the quaternionic line element in curved space-time. We examine a quaternionic metric tensor equation for the Riemannian Christoffel curvature tensor. We present the quaternionic Einstein’s field-like equation, which indicates that quaternionic matter and geometry are equivalent. Relevance of the work:In recent decades, hypercomplex algebra, viz., quaternion and octonions, has been widely used to explain various branches of physics. In this way, we have investigated quaternionic transformations and field equations in curved space-time. The present novel work will help to explain the characteristics of the curved space-time universe in terms of quaternion algebra. It can also be used to describe quaternionic gravitational waves, the black hole formulation, and so on.