Questions tagged [quasiperiodic-function]
For questions concerned with irregularly periodic behavior, including but not limited to quasiperiodic functions, oscillations, and tiling.
31 questions
3 votes
1 answer
177 views
How can I show that this sequence is aperiodic and is not even eventually-periodic.
Let $\alpha$ be an irrational real number and let $f:\mathbf{N} \to \mathbf{N}$ be defined as $$ f(n) = \lceil \alpha (n+1) \rceil - \lceil \alpha n \rceil. $$ Here, as usual, $\lceil \cdot \rceil$ is ...
- 999
8 votes
1 answer
470 views
Can $\sum_{k=1}^{\infty}k \operatorname{sgn}(\cos(k))$ be normalized?
In this question, I considered a double-sequence $(x_k,y_k)$ whose first term was shown to be $x_k= k\operatorname{sgn}(\cos(k))$. I was interested when the quantity $X_n=\sum_{k=1}^n x_k$ or a ...
- 7,714
1 vote
0 answers
34 views
Is the limit of quasi periodic function a almost periodic in compact open topology?
I'm doing some research on the topic of quasi periodic function and almost periodic function. I already know that the limit of quasi periodic function is an almost periodic function in the sense of ...
- 11
1 vote
0 answers
31 views
set of orthogonal quasiperiodic functions
I hope you are well. Could you help me with the following? I want to build a set of quasi-periodic functions that are also orthonormal. This set will help me with a chaos problem. I tried with $\sin (\...
- 57
4 votes
2 answers
109 views
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, ...
- 3,378
0 votes
1 answer
80 views
Fitting trigonometric function
Let $y$ be a measured time series, and $\hat{y}(t) = A\cos(\omega t +\phi)$. How to find $A$, $\omega$, $\phi$ that minimizes $$\sum_{k = 0}^{N - 1}|\hat{y}(k) - y_k|^2$$ The amplitude spectrum looks ...
- 231
2 votes
0 answers
123 views
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+...
- 2,718
1 vote
1 answer
159 views
Quasi periodic entire function [duplicate]
This problem is exercise 6.1 from Silverman's Arithmetic of Elliptic Curves. 6.1 Let $\Lambda=\mathbb{Z}\omega_1$+$\mathbb{Z}\omega_2$ be a lattice.Suppose $\theta(z)$ is an entire function,with the ...
- 353
1 vote
1 answer
151 views
General solution to $f(cx) = \lambda f(x)$
Which functions satisfy $f(cx) = \lambda f(x)$ where $c, \lambda$ are any constants? The general solution for $c = \lambda$ is given in Steven Stadnicki's answer to this question: Is the function $\...
- 314
2 votes
0 answers
78 views
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 ...
2 votes
1 answer
136 views
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 ...
- 1,877
2 votes
0 answers
91 views
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 ...
- 121
0 votes
1 answer
125 views
Sinusoid with period as a linear function of time
I have tried many different methods but to no avail. I am trying to construct a sinusoidal function (e.g. sin(2πf*(x-a))) with a linearly changing period. This is my thought process: P(t) = a + bt f(t)...
- 101
7 votes
1 answer
545 views
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{...
4 votes
1 answer
108 views
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 ...
- 41