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2014-04-15 Nonlinear Seminar

Adam Nawrocki
Asymptotic behaviour of some $$ \mu $$-almost periodic functions

The function

\begin{displaymath}  
f(x)=\frac{1}{2+\cos x +\cos{(x\sqrt 2)}}
 \end{displaymath} (4)

is a classical example of an unbounded and continuous $$  \mu  $$-almost periodic function. For this function we have
\begin{displaymath} 
\lim_{x\to \infty}\frac{e^{-x}}{2+\cos x +\cos{(x\sqrt 2)}}=0.
 \end{displaymath} (5)

During this lecture we will consider if changing the number $$  \sqrt 2  $$ to another irrational number may have relevant impact on the behaviour of this function. We shall construct such a number $$  \alpha  $$ that the limit
\begin{displaymath} 
\lim_{x\to \infty}\frac{e^{-x}}{2+\cos x +\cos{(x\alpha)}}
 \end{displaymath} (6)

will not exist.
The lecture will be preceded by a short speech by Marcin Borkowski.

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