For questions concerned with irregularly periodic behavior, including but not limited to quasiperiodic functions, oscillations, and tiling.
Questions tagged [quasiperiodic-function]
31 questions
10
votes
1 answer
Is there a definition of a "pseudo period" for $f(x)=\sin(3x)+\sin(\pi x)$?
Sums of trigonometric functions may or may not be periodic functions; in particular, $\sin(ax)+\sin(bx)$ is periodic if $a/b$ is rational.
If we consider the function
\begin{equation}
f(x) = \sin(3x) + \sin(\pi x)
\end{equation}
it surely looks…
marco trevi
- 3,358
7
votes
1 answer
Does the cut-and-project method produce *the* Fibonacci chain?
The Fibonacci Chain is a one-dimensional quasicrystal, it is constructed using the following substitution rules
\begin{align}
S&\longrightarrow L\\
L&\longrightarrow LS\notag
\end{align}
which gives the following…
6
votes
1 answer
Quasiperiodic tiling of the hyperbolic plane?
Has anyone produced a quasiperiodic tiling of the hyperbolic plane?
Or is there a reason it cannot be done?
By quasiperiodic I mean that the structure is not strictly periodic (i.e. equal to itsef after translation) but that any arbitrary large…
Florian F
- 495
5
votes
3 answers
Asymptotics of Jacobi's third theta function.
For $z\in \mathbb{C}$ and $\tau \in i\mathbb{R}_{+}$ consider the function
$$ \theta_3(z;\tau)=\sum_{n\in\mathbb{Z}}\exp\left(2\pi i n z +\pi i \tau n^2\right) $$
This function satisfies the well known quasiperiodicity properties…
user385459
- 365
5
votes
1 answer
Almost periodic function vs quasi periodic function
I am doing some work regarding quasiperiodic functions but I am not able to figure out the difference between almost periodic and quasiperiodic functions. Can anyone let me know about it?
ashu sharma
- 51
4
votes
2 answers
Let $S$ be a discrete set of points on a line. Does there always exist a subset of $\mathbb{Z}^2$ whose projection on the line is $S$?
I am exploring projections of the integer lattice on a line $r$ passing through the origin.
I understand that the projection of a periodic 2D lattice on a line can be non periodic (a 1D quasicrystal, if I understand correctly).
I am trying to tackle…
marco trevi
- 3,358
4
votes
1 answer
Is there any research about a function with changing "period" like sin(1/x)?
I'm encountering a function with a "changing period". It has some sense of period but not exactly. For example: f(x) = cos(1/x). Generally, it has the form of f(x + g(x)) = f(x). I cannot find any information about it. Is there any research about…
Fengfeng
- 41
4
votes
1 answer
What give rise to the apparent quasiperiodicity of integer multiples of $\pi$ and $e$ plotted here?
Recently, I was investigating the following equation:
$$p\pi = qe, p,q \in \Bbb{N}$$
I then plot the sets $\{p\pi\}$ in black,$\{qe\}$ in yellow and obtained the following plot:
which apparently it has some kind of fringes that looks evenly spaced,…
Secret
- 2,420
3
votes
3 answers
is the number of digits in the decimal expansion of $2^x$ periodic?
I graphed the number of digits in the base $10$ expansion of the series $2^x$:
At first, it looks like a repeating pattern in the plot but when I overlay and shift a sequence on top of that graph, it exposes that it may not be periodic. This is not…
acacia
- 269
3
votes
2 answers
Whether Functions With A Periodic-Like Property Are Constant
Call a function $f(x)$ defined over $\mathbb R \setminus \{0\}$ “drunken periodic” if $$\forall k \in \mathbb Z \setminus \{0\}, q \in (0,1), f(kq)=f(q)$$Are all drunken periodic functions constant?
This question came from a math class I was…
Lieutenant Zipp
- 1,837
2
votes
0 answers
Defining boundary conditions on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$
I'm not a mathematician, so I apologise for any non-rigorous wordings.
In physics, we oftentimes take $\mathbb{R}$ and, as physicists say, we impose periodic boundary conditions $\psi(\phi)=\psi(\phi+2\pi)$ for any complex function $\psi$ (the…
TheQuantumMan
- 2,678
2
votes
0 answers
Is this conjecture by Hao Wang proven?
In this link, where aperiodic tilings are discussed, the author mentions the following statement
Interestingly, five years before Berger's proof, Hao Wang proved that
there would be an algorithm for deciding whether a given set of tiles
can tile…
2
votes
1 answer
histogramming phases between a periodic function and another periodic, quasiperiodic or almost-periodic function with irrational period relationship.
Background and motivation:
The Astronomy SE question How often does a full moon happen on the weekend? touches on issues I've always wondered about.
The current answer says 2/7 of full moons occur on weekends because
There is no exact alignment of…
uhoh
- 1,967
2
votes
0 answers
Layperson's explanation of Bochner (1926) Almost periodic function definition
I am trying to formulate a problem mathematically for which I have to mathematically formulate pseudo-periodic time series. According to this research paper, Almost-periodic and pseudo-periodic signals are similar.
I am not able to completely…
Arjun Chaudhary
- 121
2
votes
2 answers
Algebraic expression for the period of $\cos (\log (x))$?
This question relates only to $x \in \mathbb{R}^+$. The function $f(x) = \cos (\log (x))$ is clearly defined on the positive reals, with a monotonic decreasing period $p(x)$ which is defined at the limits of this range by
$$\underset{x\to…
Richard Burke
- 925