19,259 research outputs found
Quasi-periodic non-stationary solutions of 3D Euler equations for incompressible flow
A novel derivation of non-stationary solutions of 3D Euler equations for
incompressible inviscid flow is considered here. Such a solution is the product
of 2 separated parts: - one consisting of the spatial component and the other
being related to the time dependent part. Spatial part of a solution could be
determined if we substitute such a solution to the equations of motion
(equation of momentum) with the requirement of scale-similarity in regard to
the proper component of spatial velocity. So, the time-dependent part of
equations of momentum should depend on the time-parameter only. The main
result, which should be outlined, is that the governing (time-dependent)
ODE-system consist of 2 Riccati-type equations in regard to each other, which
has no solution in general case. But we obtain conditions when each component
of time-dependent part is proved to be determined by the proper elliptical
integral in regard to the time-parameter t, which is a generalization of the
class of inverse periodic functions.Comment: 17 pages, 8 figures; this article is accepted for publication in
JKSU
Division algebras of prime degree with infinite genus
The genus gen(D) of a finite-dimensional central division algebra D over a
field F is defined as the collection of classes [D'] in the Brauer group Br(F),
where D' is a central division F-algebra having the same maximal subfields as
D. For any prime p, we construct a division algebra of degree p with infinite
genus. Moreover, we show that there exists a field K such that there are
infinitely many nonisomorphic central division K-algebras of degree p, and any
two such algebras have the same genus.Comment: 4 page
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