Narrow Results

In this study, the electromagnetic fields are developed in the presence of both the electric and magnetic induction fields by quaternion algebra. In this sense, the polarization and magnetization effects, which are valid in the material media, gain much importance. Quaternions are one of the most convenient tools for representing electromagnetism with regard to having non-commutative but associative algebraic division ring. By defining the quaternion induction field, the quaternion source term has been obtained in basic and elegant notation for the first time. In addition, one type of Poynting theorem, named as the Minkowski form, has been presented including the permittivity and permeability constants by quaternions.

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance 1/n. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension n goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on more general Gaussian quaternionic random matrix models are also made.

In this paper, we obtain equations of circular surfaces by using unit quaternions and express these surfaces in terms of homothetic motions. Furthermore, we introduce new roller coaster surfaces constructed by the spherical indicatrices of a spatial curve in Euclidean $3$-space. Then, we express parametric equations of roller coaster surfaces by means of unit quaternions and orthogonal matrices corresponding to these quaternions. Moreover, we present some illustrated examples.

This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms.

We establish a few properties of eigenvalues and eigenvectors of the quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex Ginibre case (see [7, 11, 14]). We first recover a version of Kostlan’s theorem that was already at the heart of an argument by Rider [1], namely, that the set of the squared radii of the eigenvalues is distributed as a set of independent gamma variables. Our proof technique uses the De Bruijn identity and properties of Pfaffians; it also allows to prove that the high powers of these eigenvalues are independent. These results extend to any potential beyond the Gaussian case, as long as radial symmetry holds; this includes for instance truncations of quaternionic unitary matrices, products of quaternionic Ginibre matrices, and the quaternionic spherical ensemble.

We then study the eigenvectors of quaternionic Ginibre matrices. Angles between eigenvectors and the matrix of overlaps both exhibit some specific features that can be compared to the complex case. In particular, we compute the distribution and the limit of the diagonal overlap associated to an eigenvalue that is conditioned to be at the origin. This complements a recent study of overlaps in quaternionic ensembles by Akemann, Förster and Kieburg [1, 2].

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every $4$-dimensional (4D) real unital algebra ${𝒜}$ into an 8D real unital algebra denoted by $\mathrm{\text{FD}}({𝒜}).$ We also find the conditions on ${𝒜}$ under which $\mathrm{\text{FD}}({𝒜})$ is a division algebra. This covers the most classes of known $4$D real division algebras. The second process allows us to embed particular classes of $4$D RDAs into $8$D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic $8$D real division algebras whose derivation algebras contain $\mathrm{\text{su}}(2)$.

In this paper, we obtain a new version of Serret-Frenet formulae for quaternionic curve in ${ℝ}^4$ by using a method similar to the method given by Bharathi and Nagaraj and called it Type 2-Quaternionic Frame. Also, we give an application of this new type of the quaternionic frame by an example.

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.

In this paper, we discuss the fields described by Dirac wave equation written in Clifford algebra based on Macfalane quaternions. It is shown that the strengths of these fields are nonzero only in the area of sources and the interaction of such fields occurs by overlapping. We consider both the simple spherically symmetric models of sources, which demonstrate attractive and repulsive interaction, and more complicated core-shell systems, which provide the bound states formation.

We consider a generalized Vekua equation in biquaternionic formalism where the Cauchy-Riemann operator is replaced by the differential operator D of Dirac. For particular classes we construct differential operators of higher order which give a relation between the monogenic functions as solutions of Dw = 0 and the generalized pseudoanalytic functions as solutions of the generalized Vekua equation. This is done by considering a corresponding differential equation of second order. Using generating functions in the sense of L. Bers we can give further representations of such functions and we can obtain related pseudoanalytic functions of the second kind as solutions of another differential equation of first order.

In this study, we consider the Padovan Quaternions. Then we investigate the some new properties such as the summation formulas and binomial sum for these quaternions.

Kalman filter-based cooperative localization (CL) algorithms have been shown to significantly improve pose estimations within networks of vehicles but have relied predominantly on two-dimensional kinematic models of the member agents. An inherent deficiency of the commonly employed kinematic vehicle model is the ineffectiveness of CL with only relative position measurements. In this work, we present a singularity-free CL using the full three-dimensional (3D) nonlinear dynamic vehicle model suitable for decentralized control and navigation of heterogeneous networks. We develop the algorithm, present Monte Carlo simulation results with relative pose measurements, and assess the algorithm performance as the number of measurements increases. We further demonstrate that CL with only relative position measurements is effective when using the dynamic model and benefits from increasing number of measurements. We also evaluate the performance of CL with respect to measurement task distribution, which is important in cooperative control of autonomous vehicles.

A Dirac fermion is expressed by a four-component spinor, which is a combination of two quaternions and can be treated as an octonion. The octonion possesses the triality symmetry, which defines symmetry of fermion spinors and bosonic vector fields. The triality symmetry relates three sets of spinors and two sets of vectors, which are transformed among themselves via transformations G₂₃, G₁₂, G₁₃, G₁₂₃ and G₁₃₂. If the electromagnetic (EM) interaction is sensitive to the triality symmetry, i.e. EM probe selects one triality sector, EM signals from the five transformed world would not be detected and be treated as the dark matter. According to an astrophysical measurement, the ratio of the dark to ordinary matter in the universe as a whole is almost exactly 5. We expect quarks are insensitive to the triality, and triality will appear as three times larger flavor degrees of freedom in the lattice simulation.

A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space–time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism, the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to ${𝒩}=1$ supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.

The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z²+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.

Landsburg method of classifying mixed Nash equilibria for maximally entangled Eisert–Lewenstein–Wilkens (ELW) game is analyzed with special emphasis on symmetries inherent to the problem. Nash equilibria for the original ELW game are determined.

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher-dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I₂(h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group D_h of order 2h as a maximal subgroup. As a simple application, we demonstrate projections of the root and weight lattices of A₄ onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine W_a(A₄) can be decomposed into two orthogonal spaces whose point group is the dihedral group D₅ which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group W_a(A₃), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group W_a(A_n), n ≥ 4. We also note that the projection of the Voronoi cell of the root lattice of W_a(A₄) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields