11,712 research outputs found
Discretized rotation has infinitely many periodic orbits
For a fixed k in (-2,2), the discretized rotation on Z^2 is defined by
(x,y)->(y,-[x+ky]). We prove that this dynamics has infinitely many periodic
orbits.Comment: Revised after referee reports, and added a quantitative statemen
Periodicity of certain piecewise affine planar maps
We determine periodic and aperiodic points of certain piecewise affine maps
in the Euclidean plane. Using these maps, we prove for
that all integer
sequences satisfying are periodic
A spectral solver for evolution problems with spatial S3-topology
We introduce a single patch collocation method in order to compute solutions
of initial value problems of partial differential equations whose spatial
domains are 3-spheres. Besides the main ideas, we discuss issues related to our
implementation and analyze numerical test applications. Our main interest lies
in cosmological solutions of Einstein's field equations. Motivated by this, we
also elaborate on problems of our approach for general tensorial evolution
equations when certain symmetries are assumed. We restrict to U(1)- and Gowdy
symmetry here.Comment: 29 pages, 11 figures, uses psfrag and hyperref, large parts rewritten
in order to match to the requirements of the journal, conclusions unchanged;
J. Comput. Phys. (2009
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure
Periodic Poisson Solver for Particle Tracking
A method is described to solve the Poisson problem for a three dimensional
source distribution that is periodic into one direction. Perpendicular to the
direction of periodicity a free space (or open) boundary is realized. In beam
physics, this approach allows to calculate the space charge field of a
continualized charged particle distribution with periodic pattern.
The method is based on a particle mesh approach with equidistant grid and
fast convolution with a Green's function. The periodic approach uses only one
period of the source distribution, but a periodic extension of the Green's
function.
The approach is numerically efficient and allows the investigation of
periodic- and pseudo-periodic structures with period lengths that are small
compared to the source dimensions, for instance of laser modulated beams or of
the evolution of micro bunch structures. Applications for laser modulated beams
are given.Comment: 33 pages, 22 figure
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