Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics

Before we dive into the accessibility stream of nowadays indicatory applications of octonions to computer and other sciences and to quantum physics let us focus for a while on the crucially relevant events for todays revival on interest to nonassociativity. Our reflections keep wandering back to the $Brahmagupta$ $Fibonacc$ two square identity and then via the $Euler$ four square identity up to the $Degen$ $Ggraves$ $Cayley$ eight square identity. These glimpses of history incline and invite us to retell the story on how about one month after quaternions have been carved on the $Broughamian$ bridge octonions were discovered by $John$ $Thomas$ $Ggraves$, jurist and mathematician, a friend of $William$ $Rowan$ $Hamilton$. As for today we just mention en passant quaternionic and octonionic quantum mechanics, generalization of $Cauchy$ $Riemann$ equations for octonions and triality principle and $G_2$ group in spinor language in a descriptive way in order not to daunt non specialists. Relation to finite geometries is recalled and the links to the 7stones of seven sphere, seven imaginary octonions units in out of the $Plato$ cave reality applications are appointed . This way we are welcomed back to primary ideas of $Heisenberg$, $Wheeler$ and other distinguished fathers of quantum mechanics and quantum gravity foundations.Comment: 26 pages, 7 figure

On solutions of the matrix equations X−AXB=C and X−AXB=C

AbstractThis paper studies the solutions of complex matrix equations X−AXB=C and X−AXB=C, and obtains explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively

A Meshfree Method for Numerical Solution of Nonhomogeneous Time-Dependent Problems

We propose a new numerical meshfree scheme to solve time-dependent problems with variable coefficient governed by telegraph and wave equations which are more suitable than ordinary diffusion equations in modelling reaction diffusion for such branches of sciences. Finite difference method is adopted to deal with time variable and its derivative, and radial basis functions method is developed for spatial discretization. The results of numerical experiments are presented and are compared with analytical solutions to confirm the accuracy of our scheme

Numerical Method for Solving Nonhomogeneous Backward Heat Conduction Problem

In this paper we consider a numerical method for solving nonhomogeneous backward heat conduction problem. Coupled with the likewise Crank Nicolson scheme and an intermediate variable, the backward problem is transformed to a nonhomogeneous Helmholtz type problem; the unknown initial temperature can be obtained by solving this Helmholtz type problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several problems in both two and three dimensions. The results show that this numerical method can solve nonhomogeneous backward heat conduction problem effectively and precisely, even though the final temperature is disturbed by significant noise

A Numerical Method for 1D Time-Dependent Schrodinger Equation Using Radial Basis Functions

This paper proposes a new numerical method to solve the 1D time-dependent Schrodinger equations based on the finite difference scheme by means of multiquadrics (MQ) and inverse multiquadrics (IMQ) radial basis functions. The numerical examples are given to confirm the good accuracy of the proposed methods

The Method of Particular Solutions for Solving Inverse Problems of a Nonhomogeneous Convection-Diffusion Equation With Variable Coefficients

A new version of the method of particular solutions (MPS) has been proposed for solving inverse problems for nonhomogeneous convection-diffusion equations with variable coefficients (IPCD). Coupled with the time discretization and MPS, the proposed method is a truly meshless method which requires neither domain or boundary discretization. Even though the final temperature is almost undetectable or is disturbed by significant noise, the proposed method can still recover the initial temperature very well. The effectiveness of the proposed inverse scheme using radial basis functions is demonstrated by several examples in 2-D and 3-D

An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications

This paper, by means of complex representation of a quaternion matrix, discusses the consimilarity of quaternion matrices, and obtains a relation between consimilarity and similarity of quaternion matrices. It sets up an algebraic bridge between consimilarity and similarity, and turns the theory of consimilarity of quaternion matrices into that of ordinary similarity of complex matrices. This paper also gives algebraic methods for finding coneigenvalues and coneigenvectors of quaternion matrices by means of complex representation of a quaternion matrix

Algebraic Techniques for Canonical Forms and Applications in Split Quaternionic Mechanics

The algebra of split quaternions is a recently increasing topic in the study of theory and numerical computation in split quaternionic mechanics. This paper, by means of a real representation of a split quaternion matrix, studies the problem of canonical forms of a split quaternion matrix and derives algebraic techniques for finding the canonical forms of a split quaternion matrix. This paper also gives two applications for the right eigenvalue and diagonalization in split quaternionic mechanics