738 research outputs found
Constructing irreducible polynomials recursively with a reverse composition method
We suggest a construction of the minimal polynomial of
over from the minimal polynomial for all positive integers whose prime factors divide . The
computations of our construction are carried out in . The key
observation leading to our construction is that for holds
where and
is a primitive -th root of unity in . The
construction allows to construct a large number of irreducible polynomials over
of the same degree. Since different applications require
different properties, this large number allows the selection of the candidates
with the desired properties
Continuous symmetry reduction and return maps for high-dimensional flows
We present two continuous symmetry reduction methods for reducing
high-dimensional dissipative flows to local return maps. In the Hilbert
polynomial basis approach, the equivariant dynamics is rewritten in terms of
invariant coordinates. In the method of moving frames (or method of slices) the
state space is sliced locally in such a way that each group orbit of
symmetry-equivalent points is represented by a single point. In either
approach, numerical computations can be performed in the original state-space
representation, and the solutions are then projected onto the symmetry-reduced
state space. The two methods are illustrated by reduction of the complex Lorenz
system, a 5-dimensional dissipative flow with rotational symmetry. While the
Hilbert polynomial basis approach appears unfeasible for high-dimensional
flows, symmetry reduction by the method of moving frames offers hope.Comment: 32 pages, 7 figure
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