Narrow Results

The paper aims to explore the impact of composite operators containing a few physical quantities on the gravitational and electromagnetic fields, studying the influencing factors and physical properties of octonion field equations. Maxwell first utilized the quaternions and vector terminology to describe the electromagnetic fields. The octonions can be used to simultaneously describe the physical quantities of electromagnetic and gravitational fields, including the octonion field potential, field strength, field source, linear momentum, angular momentum, torque and force. In the octonion spaces, the field strength and quaternion operator are able to combine together to become one composite operator, making an important contribution to the field equations. Similarly, the field potential can also form some composite operators with the quaternion operator, and they have a certain impact on the field equations. Furthermore, other physical quantities can also be combined with the quaternion operator to form several composite operators. In these composite operators, multiple physical quantities can also have a certain impact on the field equations. In other words, the field strength does not occupy a unique central position, compared to other physical quantities in these composite operators. In the field theories described by the octonions, various field equations can be derived from the application of different composite operators. According to different composite operators, it is able to infer the field equations when the field strength or/and field potential make a certain contribution, and it is also possible to deduce several new field equations when the remaining physical quantities play a certain role. This further deepens the understanding of the physical properties of field equations.

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator-valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator-valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

Recently models of neural networks that can directly deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. Furthermore models of neural networks that can deal with quaternion numbers, which is the extension of complex numbers, have also been proposed. However they are all multilayer quaternion neural networks. This paper proposes models of fully connected recurrent quaternion neural networks, Hopfield-type quaternion neural networks. Since quaternion numbers are non-commutative on multiplication, some different models can be considered. We investigate dynamics of these proposed models from the point of view of the existence of an energy function and derive their conditions for existence.

Associative memory networks based on quaternionic Hopfield neural network are investigated in this paper. These networks are composed of quaternionic neurons, and input, output, threshold, and connection weights are represented in quaternions, which is a class of hypercomplex number systems. The energy function of the network and the Hebbian rule for embedding patterns are introduced. The stable states and their basins are explored for the networks with three neurons and four neurons. It is clarified that there exist at most 16 stable states, called multiplet components, as the degenerated stored patterns, and each of these states has its basin in the quaternionic networks.

We develop a quaternionic electrodynamics and show that it naturally supports the existence of magnetic monopoles. We obtained the field equations, the continuity equation, the electrodynamic force law, the Poynting vector, the energy conservation, and the stress-energy tensor. The formalism also enabled us to generalize the Dirac monopole and the charge quantization rule.

Super-Poincaré algebra in $D=6$ space–time dimensions has been studied in terms of quaternionic representation of Lorentz group. Starting the connection of quaternion Lorentz group with $SO(1,5)$ group, the $SL(2,ℍ)$ spinors for Dirac and Weyl representations of Poincaré group are described consistently to extend the Poincaré algebra to super-Poincaré algebra for $D=6$ space–time.

By applying the Hamilton’s quaternion algebra, we propose the generalized electromagnetic-fluid dynamics of dyons governed by the combination of the Dirac–Maxwell, Bernoulli and Navier–Stokes equations. The generalized quaternionic hydro-electromagnetic field of dyonic cold plasma consists of electrons and magnetic monopoles in which there exist dual-mass and dual-charge species in the presence of dyons. We construct the conservation of energy and conservation of momentum equations by equating the quaternionic scalar and vector parts for generalized hydro-electromagnetic field of dyonic cold plasma. We propose the quaternionic form of conservation of energy is related to the Bernoulli-like equation while the conservation of momentum is related to Navier–Stokes-like equation for dynamics of dyonic plasma fluid. Further, the continuity equation, i.e. the conservation of electric and magnetic charges with the dynamics of hydro-electric and hydro-magnetic flow of conducting cold plasma fluid is also analyzed. The quaternionic formalism for dyonic plasma wave emphasizes that there are two types of waves propagation, namely the Langmuir-like wave propagation due to electrons, and the ’t Hooft–Polyakov-like wave propagation due to magnetic monopoles.

We present a unified theory of charge and spin transport using quaternionic formalism. It is shown that both charge and spin currents can be combined together to form a quaternionic current. The scalar and vector part of quaternionic currents correspond to charge and spin currents, respectively. We formulate a unitarity condition on the scattering matrix for quaternionic current conservation. It is shown that in the presence of spin flip interactions, a weaker quaternionic unitarity condition implying charge flux conservation but spin flux nonconservation is valid. Using this unified theory, we find that spin currents are intrinsically nonlinear. Its implication for recent experimental observation of spin generation far away from the boundaries are discussed.

In this paper, an effective method based on the color quaternion wavelet transform (CQWT) for image forensics is proposed. Compared to discrete wavelet transform (DWT), the CQWT provides more information, such as the quaternion’s magnitude and phase measures, to discriminate between computer generated (CG) and photographic (PG) images. Meanwhile, we extend the classic Markov features into the quaternion domain to develop the quaternion Markov statistical features for color images. Experimental results show that the proposed scheme can achieve the classification rate of 92.70%, which is 6.89% higher than the classic Markov features.

The dynamics of a family of quadratic maps in the quaternion space is investigated. In particular, connectivity of the filled-in Julia sets is completely determined. It is shown that the connectedness locus of this family is not equal to what we call the quaternionic Mandelbrot set. Hyperbolic components will also be completely characterized.

The paper is inspired by a spectral decomposition and fractal eigenvectors for a class of piecewise linear maps due to Tasaki et al. [1994] and by an ad hoc explicit derivation of the Heisenberg uncertainty relation based on a Peano–Hilbert planar curve, due to El Nashie [1994]. It is also inspired by an elegant generalization by Zhang [2008] of the exact solution by Onsager [1944] to the problem of description of the Ising lattices [Ising, 1925]. This generalization involves, in particular, opening the knots by a rotation in a higher dimensional space and studying important commutators in the corresponding algebra. The investigations of Onsager and Zhang, involving quaternion matrices of order being a power of two, can be reformulated with the use of the "quaternionic" sequence of Jordan algebras implied by the fundamental paper of Jordan et al. [1934]. It is closely related to Heisenberg's approach to quantum theories, as summarized by him in his essay dedicated to Bohr on the occasion of Bohr's seventieth birthday (1955). We show that the Jordan structures are closely related to some types of fractals, in particular, fractals of the algebraic structure. Our study includes fractal renormalization and the renormalized Dirac operator, meromorphic Schauder basis and hyperfunctions on fractal boundaries, and a final discussion.

The structure of the unitary unit group of the group algebra 𝔽_2^kQ₈ is described as a Hamiltonian group.

In this paper, we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2 × 2 matrices with entries in a possibly non-commutative ring, for example, the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot.

The motion of binary star systems is re-examined in the presence of perturbations from the theory of general relativity. To handle the singularity of the Kepler problem, the equation of motion is regularized and linearized with quaternions. In this way first-order perturbation results are derived using the quaternion-based approach.

The paper aims to consider the strength gradient force as the dynamic of astrophysical jets, explaining the movement phenomena of astrophysical jets. J. C. Maxwell applied the quaternion analysis to describe the electromagnetic theory. This encourages others to adopt the complex quaternion and octonion to depict the electromagnetic and gravitational theories. In the complex octonion space, it is capable of deducing the field potential, field strength, field source, angular momentum, torque, force and so forth. As one component of the force, the strength gradient force relates to the gradient of the norm of field strength only, and is independent of not only the direction of field strength but also the mass and electric charge for the test particle. When the strength gradient force is considered as the thrust of the astrophysical jets, one can deduce some movement features of astrophysical jets, including the bipolarity, matter ingredient, precession, symmetric distribution, emitting, collimation, stability, continuing acceleration and so forth. The above results reveal that the strength gradient force is able to be applied to explain the main mechanical features of astrophysical jets, and is the competitive candidate of the dynamic of astrophysical jets.

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.

The quaternion Mandelbrot sets (abbreviated as M sets) on the mapping f : z ← z² + c with multiple critical points are constructed utilizing the cycle detecting method and the improved time escape algorithm. The topology structures and the fission evolutions of M sets are investigated, the boundaries and the centers of the stability regions are calculated, and the topology rules of the cycle orbits are discussed. The quaternion Julia sets with the parameter c selected from the M sets are constructed. It can be concluded that quaternion M sets have efficient information of the corresponding Julia sets. Experimental results demonstrate that the quaternion M sets with multiple critical points distinguish from that of zero critical point and the collection of the quaternion M sets with different critical points constitute the complete M sets on the mapping f : z ← z² + c.

A quaternion model for describing color image is proposed in order to evaluate its quality. Local variance distribution of luminance layer is calculated. Color information is taken into account by using quaternion matrix. The description method is a combination of luminance layer and color information. The angle between the singular value feature vectors of the quaternion matrices corresponding to the reference image and the distorted image is used to measure the structural similarity of the two color images. When the reference image and distorted images are of unequal size it can also assess their quality. Results from experiments show that the proposed method is better consistent with the human visual characteristics than MSE, PSNR and MSSIM. The resized distorted images can also be assessed rationally by this method.

This paper aims to present, in a unified manner, Cramer’s rule which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies Cramer’s rule for the system of v-quaternionic linear equations by means of a complex matrix representation of v-quaternion matrices, and gives an algebraic technique for solving the system of v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for Cramer’s rule in quaternionic and split quaternionic mechanics.

Let $F$ be a real-closed field and $D$ a $F$-division algebra. In this paper, we prove that if $D$ is algebraic over $F$ then the graph $\mathrm{\Gamma }({\mathrm{\text{M}}}_n(D))$ is connected and its diameter is at most $4$, for any $n\ge 3$. If in addition, the division ring $D$ is noncommutative, we also have the same results for $n=2$. As a corollary, we show that the diameter of the commuting graph of the matrix algebra of degree $n\ge 2$ over a generalized quaternion algebra ${ℍ}_F(a,b)$, where $F$ is a real-closed field, two elements $a,b\in F^{\ast }$, is also at most $4$. This fact is a strong improvement of the previous result by Akbari et al. asserting that this diameter is at most $6$ in case $F$ is the field $ℝ$ of real numbers.