1,175 research outputs found
Natural FLRW metrics on the Lie group of nonzero quaternions
It is shown that the Lie group of invertible elements of the quaternion
algebra carries a family of natural closed Friedmann-Lemaitre-Robertson-Walker
metrics.Comment: A slightly more technical version of "Natural geometry of nonzero
quaternions" IJTP 46 (2) (2007) 251-25
Finding roots of polynomials over finite fields
We propose an improved algorithm for finding roots of polynomials over finite
fields. This makes possible significant speedup of the decoding process of
Bose-Chaudhuri-Hocquenghem, Reed-Solomon, and some other error-correcting
codes.Comment: 6 pages. IEEE Transactions on Communication
An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation
A semiclassical approximation approach based on the Maslov complex germ
method is considered in detail for the 1D nonlocal
Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak
diffusion. In terms of the semiclassical formalism developed, the original
nonlinear equation is reduced to an associated linear partial differential
equation and some algebraic equations for the coefficients of the linear
equation with a given accuracy of the asymptotic parameter. The solutions of
the nonlinear equation are constructed from the solutions of both the linear
equation and the algebraic equations. The solutions of the linear problem are
found with the use of symmetry operators. A countable family of the leading
terms of the semiclassical asymptotics is constructed in explicit form.
The semiclassical asymptotics are valid by construction in a finite time
interval. We construct asymptotics which are different from the semiclassical
ones and can describe evolution of the solutions of the
Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example
considered, an initial unimodal distribution becomes multimodal, which can be
treated as an example of a space structure.Comment: 28 pages, version accepted for publication in Int. J. Geom. Methods
Mod. Phy
The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation
The general construction of quasi-classically concentrated solutions to the
Hartree-type equation, based on the complex WKB-Maslov method, is presented.
The formal solutions of the Cauchy problem for this equation, asymptotic in
small parameter \h (\h\to0), are constructed with a power accuracy of
O(\h^{N/2}), where N is any natural number. In constructing the
quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations
(equations for middle or centered moments) is essentially used. The nonlinear
superposition principle has been formulated for the class of quasi-classically
concentrated solutions of the Hartree-type equations. The results obtained are
exemplified by the one-dimensional equation Hartree-type with a Gaussian
potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class:
Accelerator PhysicsComment: 36 pages, LaTeX-2
Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity
The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal
nonlinear drift term is reduced to a similar problem for the correspondent
linear equation. The relation between symmetry operators of the linear and
nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative
examples of the one-dimensional symmetry operators are presented.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
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