Mathematics

Questions tagged [quasiperiodic-function]

For questions concerned with irregularly periodic behavior, including but not limited to quasiperiodic functions, oscillations, and tiling.

31 questions
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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
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8
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1 answer

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 corresponding sum $Y_n=\sum_{k=1}^n y_k$ were equal…
Integrand
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7
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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 sequence \begin{align*}…
AccidentalTaylorExpansion
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6
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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
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5
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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
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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
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4
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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
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4
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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
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4
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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
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3
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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 the "ceiling" operator. I want to show that $f$…
fred
  • 999
3
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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
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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 taking…
Lieutenant Zipp
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2
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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
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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…
AccidentalTaylorExpansion
  • 1,926
2
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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
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