Quaternionic Equation Research Articles

GEOMETRICAL HYPERCOMPLEX COUPLING BETWEEN ELECTRIC AND GRAVITATIONAL FIELDS

The present work shows a coupling of electrical and gravitational fields through Cauchy-Riemann conditions for quaternions present in a previous paper (1). It is also obtained an extended version of the Laplace-like equations for quaternions, now written in terms of both electric and gravitational fields.

Pexiderized functional equations for vector products and quaternions

The purpose of the present paper is to solve the pexiderized versions of functional equations investigated by B. Nyul and G. Nyul [2], raised by the connection between products of quaternions and products of three-dimensional vectors.

Algorithms for quaternion polynomial root-finding

In 1941 Niven pioneered root-finding for a quaternion polynomial P(x), proving the fundamental theorem of algebra (FTA) and proposing an algorithm, practical if the norm and trace of a solution are known. We present novel results on theory, algorithms and applications of quaternion root-finding. Firstly, we give a new proof of the FTA resulting in explicit formulas for both exact and approximate quaternion roots of P(x) in terms of exact and approximate complex roots of the real polynomial F(x)=P(x)P¯(x), where P¯(x) is the conjugate polynomial. In particular, if |F(c)|≤ϵ, then for a computable quaternion conjugate q of c, |P(q)|≤ϵ. Consequences of these include relevance of root-finding methods for complex polynomials, computation of bounds on zeros, and algebraic solution of special quaternion equations. Secondly, working directly in the quaternion space, we develop Newton and Halley methods and analyze their local behavior. Surprisingly, even for a quadratic quaternion polynomial Newton’s method may not converge locally. Finally, we derive an analogue of the Bernoulli method in the quaternion space for computing the dominant root in certain cases. This requires the development of an independent theory for the solution of quaternion homogeneous linear recurrence relations. These results also lay a foundation for quaternion polynomiography.

On a uniform estimate for the quaternionic Calabi problem

We establish a C 0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.

Bott-Duffin inverse over the quaternion skew field with applications

In this paper, we show some properties of the Bott-Duffin inverses \(A_{r_{ ( L_{1} ) }}^{ ( -1 ) }\) and \(A_{l_{ ( L_{2} ) }}^{ ( -1 ) }\) over the quaternion skew field. In particular, we establish the determinantal representations of these generalized inverses by the theory of the column and row determinants. Moreover, we derive some Cramer rules for the unique solution to some restricted linear quaternion equations. The findings of this paper extend some known results in the literature.

Functional equations for vector products and quaternions

In our paper we find all functions \({f : \mathbb {R} \times \mathbb {R}^{3} \rightarrow \mathbb {H}}\) and \({g : \mathbb {R}^{3} \rightarrow \mathbb {H}}\) satisfying \({f (r, {\bf v}) f (s, {\bf w}) = -\langle{\bf v},{\bf w}\rangle + f (rs, s{\bf v} + r{\bf w} + {\bf v} \times {\bf w})}\)\({(r, s \in \mathbb {R}, {\bf v},{\bf w} \in \mathbb {R}^{3})}\) , and \({g({\bf v})g({\bf w}) = -\langle{\bf v}, {\bf w}\rangle + g({\bf v} \times {\bf w})}\)\(({{\bf v},{\bf w} \in \mathbb {R}^{3}})\) . These functional equations were motivated by the well-known identities for vector products and quaternions, which can be obtained from the solutions f (r, (v1, v2, v3)) = r + v1i + v2j + v3k and g((v1 ,v2, v3)) = v1i + v2j + v3k.

The Quaternionic Equation ax + xb = c

The quaternionic equation ax + xb = c is studied in Section 3 of [2]. Unfortunately there is an error in Theorem 1 which states the main results (page 420). Here a correct theorem is stated. It is followed by a complete proof based on an alternative argument. And more information is given.

Quaternionic-valued ordinary differential equations II. Coinciding sectors

We give some sufficient conditions for the existence of at least one periodic solution of the quaternionic polynomial equations. In some cases we are able to prove uniqueness of periodic solutions inside some sectors of H and detect solutions heteroclinic to the periodic ones.

Hamiltonians and variational principles for Alfvén simple waves

The evolution equations for the magnetic field induction B with the wave phase for Alfvén simple waves are expressed as variational principles and in the Hamiltonian form. The evolution of B with the phase (which is a function of the space and time variables) depends on the generalized Frenet–Serret equations, in which the wave normal n (which is a function of the phase) is taken to be tangent to a curve X, in a 3D Cartesian geometry vector space. The physical variables (the gas density, fluid velocity, gas pressure and magnetic field induction) in the wave depend only on the phase. Three approaches are developed. One approach exploits the fact that the Frenet equations may be written as a 3D Hamiltonian system, which can be described using the Nambu bracket. It is shown that B as a function of the phase satisfies a modified version of the Frenet equations, and hence the magnetic field evolution equations can be expressed in the Hamiltonian form. A second approach develops an Euler–Poincaré variational formulation. A third approach uses the Frenet frame formulation, in which the hodograph of B moves on a sphere of constant radius and uses a stereographic projection transformation due to Darboux. The equations for the projected field components reduce to a complex Riccati equation. By using a Cole–Hopf transformation, the Riccati equation reduces to a linear second order differential equation for the new variable. A Hamiltonian formulation of the second order differential equation then allows the system to be written in the Hamiltonian form. Alignment dynamics equations for Alfvén simple waves give rise to a complex Riccati equation or, equivalently, to a quaternionic Riccati equation, which can be mapped onto the Riccati equation obtained by stereographic projection.

Determinantal representations of the generalized inverses $A_{T,S}^{(2)}$ over the quaternion skew field with applications

We first present some determinantal representations of one {1,5}-inverse of a quaternion matrix within the framework of a theory of the row and column determinants. As applications, we show some new explicit expressions of generalized inverses $A_{r_{T_{1}, S_{1}}}^{( 2)}$ , $A_{l_{T_{2},S_{2}}}^{(2)}$ and $A_{_{( T_{1},T_{2}) , ( S_{1},S_{2}) }}^{( 2) }$ over the quaternion skew field. Finally, we give the representations of the unique solution to some restricted left and right systems of quaternionic linear equations. The findings of this paper extend some known results in the literature.

Rotations and Screw Motion with Timelike Vector in 3-Dimensional Lorentzian Space

In the present paper we obtain the timelike Euler parameters of a Lorentzian orthogonal matrix in Lorentz space \({L^{3} = \mathbb{R}^{2,1}}\) by using Lorentzian matrix multiplication. Then, by using the timelike Euler parameters of a given rotation in a split quaternion formulation, we produce split quaternion equation of a rotation motion in L3. Moreover the components of a dual split quaternion are obtained by replacing the timelike L-Euler parameters with their split dual versions.

The Quaternionic Quantum Mechanics

A quaternionic wavefunction consisting of real and scalar functions is found to satisfy the quaternionic momentum eigenvalue equation. Each of these components are found to satisfy a generalized wave equation of the form $\frac{1}{c^2}\frac{\partial^2\psi_0}{\partial t^2} - \nabla^2\psi_0+2(\frac{m_0}{\hbar})\frac{\partial\psi_0}{\partial t}+(\frac{m_0c}{\hbar})^2\psi_0=0$. This reduces to the massless Klein-Gordon equation, if we replace $\frac{\partial}{\partial t}\to\frac{\partial}{\partial t}+\frac{m_0c^2}{\hbar}$. For a plane wave solution the angular frequency is complex and is given by $\vec{\omega}_\pm=i\frac{m_0c^2}{\hbar}\pm c\vec{k} $, where $\vec{k}$ is the propagation constant vector. This equation is in agreement with the Einstein energy-momentum formula. The spin of the particle is obtained from the interaction of the particle with the photon field.

Determinantal representations of the Moore–Penrose inverse over the quaternion skew field and corresponding Cramer's rules

Determinantal representation of the Moore–Penrose inverse over the quaternion skew field is obtained within the framework of a theory of the column and row determinants. Using the obtained analogues of the adjoint matrix, we get the Cramer rules for the least squares solution of left and right systems of quaternionic linear equations.

Schwarzschild solution of the generally covariant quaternionic field equations of Sachs

Sachs has derived quaternion field equations that fully exploit the underlying symmetry of the principle of general relativity, one in which the fundamental 10 component metric field is replaced by a 16 component four-vector quaternion. Instead of the 10 field equations of Einstein's tensor formulation, these equations are 16 in number corresponding to the 16 analytic parametric functions {\partial}x^{{\mu}'}/{\partial}x^{{\nu}} of the Einstein Lie Group. The difference from the Einstein equations is that these equations are not covariant with respect to reflections in space-time, as a consequence of their underlying quaternionic structure. These equations can be combined into a part that is even and a part that is odd with respect to spatial or temporal reflections. This paper constructs a four-vector quaternion solution of the quaternionic field equation of Sachs that corresponds to a spherically symmetric static metric. We show that the equations for this four-vector quaternion corresponding to a vacuum solution lead to differential equations that are identical to the corresponding Schwarzschild equations for the metric tensor components.

Linear Quaternionic Equations and Their Systems

The general linear quaternionic equation with one unknown and systems of linear quaternionic equations with two unknown are solved. Examples of equations and their systems are considered.

Quasi-Spherical and Multi-Quasi-Spherical Polynomial Quaternionic Equations: Introduction of the Notions and Some Examples

We introduce notions of quasi-spherical and multi-quasi-spherical polynomial quaternionic equations defined in terms of the shape of the set of the solutions of the equation. We establish that every quaternionic equation of the form $${\sum\limits_{\ell = 1}^m {a^{(\ell)}} x^2 b^{(\ell)} + \sum\limits_{p = 1}^q {c^{(p)}} xd^{(p)} + h = 0}$$is quasi-spherical. We get some sufficient conditions under which a quadratic quaternionic polynomial can be represented as a product of linear quaternionic polynomials (and thus the corresponding equation is multi-quasi-spherical).

Linear Manifolds in Sets of Solutions of Quaternionic Polynomial Equations of Several Types

We give complete description of possible shapes of the set of the solutions of any quaternionic equation of the form ax + xb = c. Moreover we study the set of the solutions of a quaternionic equation of the form ax 2 + x 2 b = c by the method of sections by hyperplanes perpendicular to the real axis; for every case where such section is an unbounded linear manifold a necessary and sufficient condition is found.

Multi-body dynamics simulation of geometrically exact Cosserat rods

In this paper, we present a viscoelastic rod model that is suitable for fast and accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (‘stiff’ dof), bending and torsion (‘soft’ dof). For inner dissipation, a consistent damping potential proposed by Antman is chosen. We parametrise the rotational dof by unit quaternions and directly use the quaternionic evolution differential equation for the discretisation of the Cosserat rod curvature.

Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n,ℍ), uniqueness (up to a constant) of a solution is proven, aas well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.

Quaternionic soliton equations from Hamiltonian curve flows in

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space . The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces M = G/H by viewing as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar–vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.