# Tagged Questions

Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.

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### Is $a_n = f(a + b c^n)$ when $a_0 = \ln(2)$ and $a_{n+1} = (e+1) a_n (a_n - 1)$?

Definition Let $a_n$ be a real sequence. Assume there exists a continuous real-periodic function $f(x)$ such that $f(n) = a_n$ And $f(x)$ has the period $t$ , where $t$ is An irrational real ...
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### Hot to show that system of nonlinear differential equations doesn't have periodic solutions?

Suppose we have nonlinear system of differential equations $$\frac{d\mathbf x}{dt} = \hat{A}(\mathbf x, \mu)\mathbf x$$ How to show that it has periodic solutions?
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### Non linear regression calculator

Are there any really good non linear regression calculators around the web? Or is something like matlab the best solution? I tried using excel and its solver tool, but it's complete garbage lol. ...
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### Periodic modular piecewise function

For every positive integer $n$, let $\text{mod}_5 (n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively ...
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### Proof for a periodic function

I have to solve the following exercise: The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. ...
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### Examples of real orbits with irrational period

Im looking for examples of periodic orbits on the real line given by piecewise analytic functions $g$. More specific at most 2 pieces. Im talking about integer iterations starting at $f(0)=0$ and with ...
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### Floquet Theory : zero equilibirum stability

Wondering if someone could explain exactly what it means when someone says "zero equilibirium is unstable because the floquet multiplier is greater than 1". I understand the FM part but I don't know ...
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### Does linear combination of periodic functions is periodic?

Does linear combination of periodic functions is periodic? I know that if $f$ is periodic and $g$ is periodic then $f+g$ is periodic. But what about the linear combination of two periodic functions??...
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### If $g(x):=f(x^2)$ is uniformly continuous, is $f$ a constant (assuming it's periodic)?

I know that $f:\mathbb{R}\to\mathbb{R}$ is continuous and periodic. How can I prove that $f$ is a constant whenever $g(x):=f(x^2)$ is uniformly continuous?
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### Are there entire functions that are unexpectedly periodic? [closed]

I wonder if there have ever been entire functions discovered/defined that turned out to be periodic and the periodicity was unexpected and surprising to mathematicians ?
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### What is the period of $f(x) = \sin^4(x)+ \cos^4(x)$?

This is an elementary problem but I'm just not getting the right answer. My reasoning is as follows: The period of $g(x) = \sin^4(x)$ is $\pi$ and that of $h(x) = \cos^4(x)$ is $π$ as well, so the ...
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### Are all solutions of $f_n (x)^{2n} + f_n ' (x)^{2n} = 1$ periodic?

Let $n$ be a strict positive integer and $x$ is a complex number. Define $f_n(x)$ as one of the solutions to $$f_n (x)^{2n} + f_n ' (x)^{2n} = 1$$ Where the derivative is with respect to $x$. Why ...
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### Prove that a continuous function on $\mathbb{R}$ which is periodic attains its extreme values.

Prove that a continuous function on $\mathbb{R}$ which is periodic attains its extreme values. I don't know how to start this proof, but I know I have to use the extreme value theorem, continuity ...
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### Are periodic functions stationary processes, e.g. y=sin(t)?

According to wikipedia, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when ...
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### Finding $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ periodicity

I need to find the period of $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ but couldn't find a nice method how? any ideas?
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### mathematical representation of a pure sinusoidal tone

Considering the case of a pure sinusoidal tone, e.g. the tuning A note at $440 \text{Hz}$, how can one mathematically represent the pressure wave resulting? For the sake of simplicity, I want to ...
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### if |f| is periodic then f is periodic [duplicate]

Decide whether the following statement about a function f: R -> R is true. If |f| is periodic, then f is periodic. Give a proof or counterexample.
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### How to make a function periodic?

I have a nice little equation here, $f\left(x\right)=\frac{4}{\pi ^2}\left(x+\frac{\pi }{2}\right)^2-1$, which ever so nicely approximates (with somewhat good accuracy), a period of the sine function, ...
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### Littlewood-Paley Decompositions and Periodic Besov Spaces

I'm currently working on some problems in $2$-dimensional periodic space, and it seems that the framework of Besov spaces will be useful to me. Since we're working in periodic space, we can consider ...
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### |f| is periodic implied f is periodic

I can think of an example where this wouldn't hold. Take 1,-1,1,-1,-1,1,-1,-1,-1,1,-1,-1,-1,-1. But I can also prove that the statement holds. Claim: $|f|$ is periodic then $f$ is periodic Proof: ...
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### Is $x_k=\sin(k)$ eventually/periodic?

A sequence is eventually periodic if we can drop a finite number of terms from the beginning and make it periodic. $$x_k=\sin(k)$$ I think this is periodic since the function is periodic and it ...
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### Prove or disprove: If $f$ is periodic, then $|f|$ is also periodic.

I started with the definition of periodic function and absolute value function. And I do it with discussing different cases of $x$ and $p$. But I got stuck with when $-p\leq x <0$ , I want to ...
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### References about functions of the following form $f(t)=f(t+P(t))$

Where I can find any references about real functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(t)=f(t+P(t)),\forall t\in \mathbb{R}$ ?
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### Why are the points of a period module necessarily isolated?

Given a non-constant meromorphic function $f(z)$ then how can we say that the absence of a finite accumulation point of our module $M$ implies that $f$ is constant? I understand that an accumulation ...
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### Integrals of a nonnegative, symmetric, periodic, continuous function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be such that $f$ is nonnegative, symmetric, $\ell$-periodic in both variables, zero exactly on the diagonal of its domain, and continuous. Specifically, $f$ has the ...
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### When talking about periods of a function why is it common to introduce a factor of $2$?

In many books I have read, when talking about periodic functions, they tend to write the period as $2\omega$. Why do we need the $2$? I haven't come across any working where the factor of $2$ is ...
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### proving periodic function f is uniformly continuous [duplicate]

So I'm new to uniform continuity and this is just an exercise that was scribbled on the board in my real analysis class. But it's is a tricky question for me: if $f:R\to R$ is periodic with period P (...
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### Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.

Let $f(x) = |x|$ for $x \in [-1,1]$ and extend $f$ to all of $\mathbb R$ by requiring $f(x+2)=f(x)$. Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$. This is easy to observe on a graph but my ...
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### How can $e^x$ be restated for small $x$?

Suppose I have the following equation: $$1=\frac{S_0}{\gamma}(1-e^{-\gamma T_n})+\lambda\int^{t_{n+1}}_{t_n}e^{-\gamma(t_{n+1}-\tau)}g(\tau)d\tau.$$ If I make two ...
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### Periodic boundary conditions: image summation vs periodic solution

Let's say we have a linear PDE and its solution is the field $\sigma(x,y)$. I want to use this field in a simulation, and in order to avoid boundary effects, I want to use Periodic Boundary Conditions....
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### Is there a natural, circular (toric) Gaussian density?

I am looking for density distributions over a circular set (think about $[-\pi, \pi[$ or $\Delta+[0, T[$ in general $\forall T \in \mathbb{R}^{+*}, \Delta \in \mathbb{R}$). Is there a density ...
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### Closed periodic curve

Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if \textbf{$\gamma$}(t+T)=\textbf{$\gamma$}...
### A function that is zero for every integer multiple of $k$
I'm new to this kind of question so it may be a trivial one but I can't find a general solution: I'm searching for a function that has a value of $0$ for every integer multiple of some fixed real $k$....