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

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1answer
64 views

Proving ODE is periodic if and only if p(t) is periodic

Consider the first-order nonautonomous equation $x′ = p(t)x$ where $p(t)$ is differentiable and periodic with period $T$. Prove that all solutions of this equation are periodic with period $T$ if and ...
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0answers
27 views

Is it true that a cycle with a period of 29 hours over 24 hours leads to a non-recurring pattern and how to prove it?

The default 'reset time' for Internet Information Services is 29 hours. The reason for this is that 'Wade [person on the team who developed the setting] suggested 29 hours for the simple reason ...
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1answer
35 views

Bound for function with constant /periodic second derivative

Consider a function $f : \mathbb{R}\to\mathbb{R}$ with $f''$ continuous and $f''(x)=f''(x+1)$ for all real numbers x. I need to show that there exists a real positive number $c$ such that $f(x)\leq ...
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1answer
25 views

Why are the Fourier's coefficient on $0,2\pi$ and $-\pi,\pi$ the same?

I was given the Following Fourier's coefficient and I was happy with it: $$\left\{ \begin{array}{ll} a_n(f)=\frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos\left(nx\right)\,\mathrm{d}x\\ b_n(f) = ...
5
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1answer
67 views

Periodic orbits on centre manifold

I am interested in periodic orbits of mechanical systems of second-order dynamics with no damping, i.e. governed by an equation of the type \begin{equation}(1)\quad \ddot x + f(x)=0 \end{equation} ...
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0answers
22 views

How to find out the Power of $x(t)$?

I am studying signals and system. I learned that \begin{align} P&=\lim_{L\to\infty} \frac 1{2L} \int_{-L}^{L} |x(t)|^2 dt\\ P&=\frac 1{T} \int_{<T>} |x(t)|^2 dt ~~~\mbox{, P could be ...
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0answers
35 views

Strong period of self-convolution of a strongly periodic cyclic function

Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all $j \in ...
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2answers
68 views

How to check the following system on existence of periodic solutions?

Let's have following system of DE: $$ \begin{cases} \dot{x} = y(1+x - y^{2}) \\ \dot{y} = x(1+y-x^{2})\end{cases}, \quad x, y \geqslant 0 $$ How to check whether this system contains periodic ...
2
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1answer
53 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
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1answer
31 views

Periodic product of sinusoids

(This is problem P-3.7 from the book 'Signal processing first') Let $x(t) = 2\cos(\omega_1t)\cos(\omega_2t) = \cos([\omega_1 + \omega2]t)+\cos([\omega_2 - \omega_1]t)$ where $0 < \omega_1 < ...
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1answer
95 views

Prove that 012345678910111213 etc is not a periodic sequence.

Prove that the sequence $012345678910111213...$ (all non-negative integers written one by one in natural order) is not periodic. I want to know the shortest and most elegant way to prove it. Can you ...
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0answers
21 views

Relationship between the integral of a periodic function on the unit circle and an infinite sum.

I am studying for my final and am stumped on this problem. Can someone give me hints or post a detailed solution? Suppose that $f$ is a continuous function on $\mathbb{R}$, with period $1$. Prove ...
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1answer
20 views

How to find period of a sum of periodic functions

I got this function: $$ x[n]=\sin(2*\pi*4/3*n) + \cos(2*\pi*5/2*n) $$ It is easy to see that period of the sin is 3/4 and the ...
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1answer
66 views

Prove that the Dirichlet function is a periodic function with no definite period

Prove that the Dirichlet function $f$ below is a periodic function with no definite period. $$f(x)=\begin{cases}1 ,& x \,\ \text{is rational} \\ 0 ,& x \,\ \text{is ...
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1answer
35 views

Change of scale of periodic function

Could someone show how to prove the following: If $f(x)$ has a period of $p$ show that $f(kx)$ has period of $p/k$.
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0answers
174 views

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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1answer
28 views

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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0answers
24 views

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. ...
0
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1answer
17 views

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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0answers
34 views

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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0answers
21 views

Getting sum of $2 \pi$ periodic function

I have a $2\pi$ periodic function which in the interval $[0,\pi]$ is $f(t) = \sin{\frac{t}{2}}$. I have to find the sum for $t \in \mathbb{R}$. But do I know anything about $f(t)$ outside of $t \in ...
1
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1answer
57 views

maximum Difference between two zeros

What is the maximum difference between the two consecutive zeros of the solutions of $y''+(1+x)y=0$ on $0\leq x<+\infty$? I have applied the Strum's comparison theorem (by comparison with ...
2
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1answer
32 views

Periodic Poincaré Inequality?

The classical periodic Poincaré inequality states that if $u\in H^1(\mathbb T^n)$ is such that $\displaystyle\int_{\mathbb T^n} u(x)\ dx=0$ then $$\|u\|_{L^2(\mathbb T^n)}^2\leq C_d \|\nabla ...
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0answers
15 views

Solve for a Fourier series problem

I am learning Fourier series and I have a problem which has me confused and would like to here others take on it. $$F(t)=\begin{cases}v_0 & \ \ 0 \le t\le T\\ 0 & \ \ T\le t \le \ ...
1
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1answer
28 views

Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$

Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$ satisfying the equation $f(x)=\text{sgn}(g(x)),$ is ...
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0answers
66 views

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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0answers
25 views

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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2answers
21 views

On Laplace transform of periodic functions

I recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: ...
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0answers
32 views

Suppose that a sequence {$a_n$} is eventually periodic. Prove that {$b_n$} is eventually periodic, given that they are in some relation.

To clarify, a sequence {$x_n$} is called eventually periodic if there exist positive integer $r$ and $p$ for which $x_{n+p}=x_n$ for all $n\geq r$. The relation they have is that, (i). ...
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2answers
53 views

Property of function $\varphi(x)=|x|$ on $\mathbb{R}$

Define $\varphi(x)=|x|$ on $[-1,1]$ and extend the definition of $\varphi(x)$ to all real $x$ by requiring that $\varphi(x+2)=\varphi(x).$ How do you prove that for any $s,t$ $$ ...
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0answers
37 views

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 ...
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1answer
94 views

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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27 views

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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1answer
29 views

How to find the period of this trigonometric function

$y$ = $|\sin x|$ I know the period is π by drawing the graph, but I can't prove it. Please use this method we have learnt for other functions. For example $y=\sin2x$ $\sin2x= \sin2(x+T)$ ...
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1answer
40 views

Integrating over discontinuities

I have the following integral: \begin{align} \int^T_0e^\tau I(\tau+t^0)d\tau. \end{align} In this integral, $I(t)$ is a function with period $T$. At each time $T, 2T, \ldots$, $I$ is increased by a ...
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1answer
40 views

If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic.

If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic. Is the converse true? Give a proof or counterexample.
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23 views

Understanding the relationship between phase and frequency in a particular equation

Context I have the following equation: \begin{align} a&=1-e^{-\omega t/x}+\frac{m}{\sqrt{1+x^2}}\Bigg[\sin(\alpha+\omega t + \varphi)-\sin(\alpha+\varphi)e^{-\omega t/x}\Bigg]. \end{align} This ...
1
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1answer
41 views

Formula for periodic sequence with increasing period

The first values of the sequence are: $\lt1, 0, 1, 1, 0, 0, -1, -1, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, ...
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3answers
140 views

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 ...
2
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1answer
72 views

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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1answer
54 views

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 ...
0
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2answers
42 views

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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1answer
21 views

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?
1
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1answer
37 views

Constant function and periodicity

If $f$ is constant function then every real number $p>0$ is its period. I was wondering is the converse true, that is: If every real number $p>0$ is period of a function $f : \mathbb R \to ...
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3answers
61 views

Function $f$ such that $f$ is non-periodic but $f(f(x))$ is?

Is there a "nice" example of a function $f$ such that $f(x)$ is non-periodic but the composition $f(f(x))$ is? By nice I mean that preferably it will be defined entirely on the domain $R$ and be ...
6
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1answer
77 views

Limit of $\int_E f(nx) dx$ for a $1$-periodic function $f$ on $[0,2\pi]$

Let $E$ be a measurable subset of $[0, 2\pi]$. Assume that $f \in C(\mathbb R)$ is $1$-periodic, i.e. $f(x + 1) = f(x)$. Compute $$\lim_{n\to\infty} \int_{E} f(nx) dx$$. Since $f$ is continuous on ...
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1answer
32 views

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 ...
0
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1answer
69 views

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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1answer
44 views

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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0answers
41 views

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 ...