103,153 research outputs found
Quaternionic differential operators
Motivated by a quaternionic formulation of quantum mechanics, we discuss
quaternionic and complex linear differential equations. We touch only a few
aspects of the mathematical theory, namely the resolution of the second order
differential equations with constant coefficients. We overcome the problems
coming out from the loss of the fundamental theorem of the algebra for
quaternions and propose a practical method to solve quaternionic and complex
linear second order differential equations with constant coefficients. The
resolution of the complex linear Schrodinger equation, in presence of
quaternionic potentials, represents an interesting application of the
mathematical material discussed in this paper.Comment: 25 pages, AMS-Te
Solving simple quaternionic differential equations
The renewed interest in investigating quaternionic quantum mechanics, in
particular tunneling effects, and the recent results on quaternionic
differential operators motivate the study of resolution methods for
quaternionic differential equations. In this paper, by using the real matrix
representation of left/right acting quaternionic operators, we prove existence
and uniqueness for quaternionic initial value problems, discuss the reduction
of order for quaternionic homogeneous differential equations and extend to the
non-commutative case the method of variation of parameters. We also show that
the standard Wronskian cannot uniquely be extended to the quaternionic case.
Nevertheless, the absolute value of the complex Wronskian admits a
non-commutative extension for quaternionic functions of one real variable.
Linear dependence and independence of solutions of homogeneous (right) H-linear
differential equations is then related to this new functional. Our discussion
is, for simplicity, presented for quaternionic second order differential
equations. This involves no loss of generality. Definitions and results can be
readily extended to the n-order case.Comment: 9 pages, AMS-Te
Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system
The stability of the one-mode nonlinear solutions of the Fermi-Pasta-Ulam -
system is numerically investigated. No external perturbation is
considered for the one-mode exact analytical solutions, the only perturbation
being that introduced by computational errors in numerical integration of
motion equations. The threshold energy for the excitation of the other normal
modes and the dynamics of this excitation are studied as a function of the
parameter characterizing the nonlinearity, the energy density
and the number N of particles of the system. The achieved results confirm in
part previous results, obtained with a linear analysis of the problem of the
stability, and clarify the dynamics by which the one-mode exchanges energy with
the other modes with increasing energy density. In a range of energy density
near the threshold value and for various values of the number of particles N,
the nonlinear one-mode exchanges energy with the other linear modes for a very
short time, immediately recovering all its initial energy. This sort of
recurrence is very similar to Fermi recurrences, even if in the Fermi
recurrences the energy of the initially excited mode changes continuously and
only periodically recovers its initial value. A tentative explanation of this
intermittent behaviour, in terms of Floquet's theorem, is proposed.Comment: 37 pages, 41 figure
Right eigenvalue equation in quaternionic quantum mechanics
We study the right eigenvalue equation for quaternionic and complex linear
matrix operators defined in n-dimensional quaternionic vector spaces. For
quaternionic linear operators the eigenvalue spectrum consists of n complex
values. For these operators we give a necessary and sufficient condition for
the diagonalization of their quaternionic matrix representations. Our
discussion is also extended to complex linear operators, whose spectrum is
characterized by 2n complex eigenvalues. We show that a consistent analysis of
the eigenvalue problem for complex linear operators requires the choice of a
complex geometry in defining inner products. Finally, we introduce some
examples of the left eigenvalue equations and highlight the main difficulties
in their solution.Comment: 24 pages, AMS-Te
Quaternions and Special Relativity
We reformulate Special Relativity by a quaternionic algebra on reals. Using
{\em real linear quaternions}, we show that previous difficulties, concerning
the appropriate transformations on the space-time, may be overcome. This
implies that a complexified quaternionic version of Special Relativity is a
choice and not a necessity.Comment: 17 pages, latex, no figure
The General Linear Model and the Generalized Singular Value Decomposition; Some Examples
The general linear model with correlated error variables can be transformed by means of the generalized singular value decomposition to a very simple model (canonical form) where the least squares solution is obvious. The method works also if X and the covariance matrix of the error variables do not have full rank or are nearly rank deficient (rank-k approximation). By backtransformation one obtains the solution for the original model. In this paper we demonstrate the method with some examples
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