Related papers: Logarithmic Cycles of the Time: An Quaternionic Ap…
Trajectories in logarithmic potentials are investigated by taking as example the motion of an electron within a cylindrical capacitor. The solution of the equation of motion in plane polar coordinates, (r,{\phi}) is attained by forming a…
Denoting by $\mathbb{M}$ the complexification of the quaternionic algebra $\mathbb{H}$, we characterize the family of those $\mathbb{M}$-valued functions, defined on subsets of $\H$, whose values are actually quaternions, using an intrinsic…
Dual quaternion algebra and its application to robotics have gained considerable interest in the last two decades. Dual quaternions have great geometric appeal and easily capture physical phenomena inside an algebraic framework that is…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
We present fast and accurate ways to normalize two and three dimensional vectors and quaternions and compute their length. Our approach is an adaptation of ideas used in the linear algebra library LAPACK, and we believe that the…
A mathematical model is given for the occurrence of preferred orbits and orbital velocities in a Keplerian system. The result can be extended into energies and other properties of physical systems. The values given by the model fit closely…
We develop a relativistic free wave equation on the complexified quaternions, linear in the derivatives. Even if the wave functions are only one-component, we show that four independent solutions, corresponding to those of the Dirac…
More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. It is shown here that they…
We show that the classical algebra of quaternions is a commutative $\Z_2\times\Z_2\times\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.
Several completely integrable, indeed solvable, Hamiltonian many-body problems are exhibited, characterized by Newtonian equations of motion ("acceleration equal force"), with linear and cubic forces, in N-dimensional space (N being an…
Understanding the nature of time remains a key unsolved problem in science. Newton in the Principia asserted an absolute universal time that {\it `flows equably'}. Hamilton then proposed a mathematical unification of space and time within…
This article describes a method for constructing approximations to periodic solutions of dynamic Lorenz system with classical values of the system parameters. The author obtained a system of nonlinear algebraic equations in general form…
Besides the purely digital or analog interpretation of reality there is a third possibility which incorporates important aspects of both. This is the cyclic formulation of elementary systems, in which elementary particles are represented as…
The nature of the change in perspective that accompanies the proposal of a unified physical theory deriving from the single dimension of time is elaborated. On expressing a temporal interval in a multi-dimensional form, via a direct…
We discuss certain ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics. Far from being exhaustive, this article is intended above all to draw attention to these algebras,…
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of…
This article presents the exact solution of fixed points functions for the cycle of period four of the quadratic recurrence equations. The solution is demonstrated for the quadratic map and the logistic map. These recurrence equations,…
We present an alternative cyclic proof system for Peano arithmetic that could be simpler than the existing ones and well-adapted both for proof analysis and for automatizing inductive proof search. In addition, we will show how various…
A Lagrangian system with singularities is considered. The configuration space is a non-compact manifold that depends on time. A set of periodic solutions has been found.
In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [3] that only certain combinations of the…