3,843 research outputs found
Generalised polynomials and integer powers
We show that there does not exist a generalised polynomial which vanishes
precisely on the set of powers of two. In fact, if is and integer
and is a generalised polynomial such that
for all then there exists infinitely many , not divisible by , such that for some .
As a consequence, we obtain a complete characterisation of sequences which are
simultaneously automatic and generalised polynomial.Comment: 51 page
Algorithmic classification of noncorrelated binary pattern sequences
We show that it is possible to algorithmically verify if a given pattern
sequence is noncorrelated. As an application, we compute that there are exactly
noncorrelated binary pattern sequences of length . If we
restrict our attention to patterns that do not end with , we put
forward a sufficient condition for a pattern sequence to be noncorrelated. We
conjecture that this condition is also necessary, and verify this conjecture
for lengths .Comment: 20 page
On multiplicative automatic sequences
We show that any automatic multiplicative sequence either coincides with a
Dirichlet character or is identically zero when restricted to integers not
divisible by small primes. This answers a question of Bell, Bruin and Coons. A
similar result was obtained independently by Klurman and Kurlberg.Comment: 12 page
Automatic sequences as good weights for ergodic theorems
We study correlation estimates of automatic sequences (that is, sequences
computable by finite automata) with polynomial phases. As a consequence, we
provide a new class of good weights for classical and polynomial ergodic
theorems, not coming themselves from dynamical systems.
We show that automatic sequences are good weights in for polynomial
averages and totally ergodic systems. For totally balanced automatic sequences
(i.e., sequences converging to zero in mean along arithmetic progressions) the
pointwise weighted ergodic theorem in holds. Moreover, invertible
automatic sequences are good weights for the pointwise polynomial ergodic
theorem in , .Comment: 31 page
On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations
The dispersive effect of the Coriolis force for the stationary Navier-Stokes
equations is investigated. The effect is of a different nature than the one
shown for the non-stationary case by J. Y. Chemin, B. Desjardins, I. Gallagher
and E. Grenier. Existence of a unique solution is shown for arbitrary large
external force provided the Coriolis force is large enough. The analysis is
carried out in a new framework of the Fourier-Besov spaces. In addition to the
stationary case counterparts of several classical results for the
non-stationary Navier-Stokes problem have been proven
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