Let $T$ be the tent map defined by $$T = \begin{cases} 2x \quad \text{ if } x \le 1/2 \\ 2(1-x) \quad \text{ if } x \ge 1/2\end{cases}$$ I am trying to prove that the periodic points of $T$ are dense in the unit interval, $I = [0,1]$. I know that all asymptotically periodic orbits of $T$ are eventually periodic, given that all fixed points are sources. However, I'm struggling in going about solving this given the tools I have. For context, this problem is problem 3.7 in Alligood's text. I was thinking that showing there are infinitely many periodic orbits is sufficient, but I'm unsure. I'm trying to find a more rigorous proof but am at a loss.
I need to find a period-3 orbit for a tent map defined by $$T(x)=\begin{cases}2x &amp; 0\le x\le0.5 \\[2ex] 2-2x &amp;0.5 &lt; x \le 1 \end{cases}$$  So, what is a fast and efficient way of finding it? I only know how to do it by guessing-and-checking. For example, I picked a random point, $x = \frac{2}{7}$ and found that $$T\left(\frac{2}{7}\right)=\frac{4}{7}\\[4ex]T\left(\frac{4}{7}\right)=\frac{6}{7}\\[4ex]T\left(\frac{6}{7}\right)=\frac{2}{7}$$ which is a period-3 orbit. Is there any better way of finding a period-3 orbit , more specifically, a period-3 point?
Where are packages stored installed with APT in Ubuntu 16.04? I know that packages installed with apt-get are stored in /var/cache/apt/archives.  I just installed mariadb server in Ubuntu 16.04 with following command:  sudo apt install mariadb-server   When I tried to find packages in /var/cache/apt/archives, there are no packages. Where are they stored if it's not in this directory? Is it unreasonable to assign a MySQL database to each user on my site? I'm creating a user-based website. For each user, I'll need a few MySQL tables to store different types of information (that is, userInfo, quotesSubmitted, and ratesSubmitted).  Is it a better idea to:  a) Create one database for the site (that is, "mySite") and then hundreds or thousands of tables inside this (that is, "userInfo_bob", "quotessubmitted_bob", "userInfo_shelly", and"quotesSubmitted_shelly")  or  b) Create hundreds or thousands of databases (that is, "Bob", "Shelly", etc.) and only a couple tables per database (that is, Inside of "Bob": userInfo, quotesSubmitted, ratesSubmitted, etc.)  Should I use one database, and many tables in that database, or many databases and few tables per database?    Edit:  The problem is that I need to keep track of who has rated what.  That means if a user has rated 300 quotes, I need to be able to know exactly which quotes the user has rated.  Maybe I should do this?  One table for quotes. One table to list users. One table to document ALL ratings that have been made (that is, Three columns: User, Quote, rating).  That seems reasonable. Is there any problem with that? How to add "Refresh" option to right click menu in Lubuntu? I have many users at the place I work, expressing the notion of switching to Ubuntu.  Most of them, for whatever weird reason wanting to know if its possible to have refresh added when you right click on the desktop.   Is that possible? What temperature can you attain with a solar furnace? Prove that the set of palindromes are not regular languages using the pumping lemma. Firstly I pick a language $xyz$ where $x = \epsilon$, $y = (abb)^{k}$, $z = (bba)^{k}$ where $|y| \ge$ the number of states in the automaton representing my language. Then $xyz = (abb)^k(bba)^k$ is a palindrome.  Now I split $y$ into $uvw$, where $v \ne \epsilon$ contains a state visited more  than once. If the language is regular then $xuv^iwz$ is within the language $\forall i \ge 0$.  I choose $i = 2$, then $uv^2w = (abb)^n$ where $n &gt; k$.  Then $xuv^2wz = (abb)^n(bba)^k$ where $n &gt; k$, which is not a palindrome.   Firstly, is my reasoning for this particular proof correct?  I'm unsure about how to properly apply the pumping lemma to this problem. And secondly the question asks me to state a proof for palindromes in general, whereas I only attempted to prove it for a single case. Is there an example I can choose which proves the conjecture for all palindromes? I assume not, seeing as I can't think of how that could be represented using regular expressions. How to disprove this fallacy that derivatives of $x^2$ and $x+x+x+\dots\quad(x\text{ times})$ are not same. Possible Duplicate:          \begin{align*} x^2 &amp;= \underbrace{x + x + x + \dots + x}_{x \text{ times}}, \\ \therefore \frac{\mathrm{d}}{\mathrm{d}x} (x^2) &amp;= \frac{\mathrm{d}}{\mathrm{d}x} (\underbrace{x + x + x + \dots + x}_{x \text{ times}}) \\ &amp;= \underbrace{1 + 1 + 1 + \dots + 1}_{x \text{ times}} \\ &amp;= x. \end{align*}  But we know that $$ \frac{\mathrm{d}}{\mathrm{d}x} (x^2) = 2x. $$  So what is the problem?  My take is that we cannot differentiate both sides because $\underbrace{{x+x+x+\cdots+x}}_{x \text{  times}}$ is not fixed and thus $1$ is not equal to $2$. how to avoid number repeation by using random class in c#? Wulfgar won't teach me Unrelenting Force's final word after Arngeir accepts the Horn of Jurgen Windcaller Incorrect whitespaces or list item size in Android SE app Noticed that sometimes when there are 10+ answers to a question - some answers look like they have \n\n...\n line breaks in the end, so the body of the answer has incorrect height.  It probably has something to do with ViewHolder or adapter not setting correct row height. Although as far as I know, it should do this according to text amount/length. Closing changes: [on hold], unclear, too broad, opinion-based, off-topic reasons, bye-bye to Too Localized How do I post text so that it is formatted as code?  What do I need to do so that my code shows up properly &mdash; not escaped or removed &mdash; when posted? And how to get the correct syntax highlighting?     For more information, see "" in the . How can I add a new user as sudoer using the command line? After I add a user using adduser, I can't see it via System > Administration > Users and Groups unless I log out and then log in again. Is that normal?  Also, can I set a newly added user as a sudoer or do I have to change that only after adding it? How can I do that via the shell?  Finally, can I delete the original user that was created upon initial installation of Ubuntu, or is this user somehow 'special'? I know  that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to be irrational or rational.  Could someone enlighten me? Thanks! Orientation on $\mathbb{CP}^2$ Why is lock(this) {...} bad? The  says that public class SomeObject {   public void SomeOperation()   {     lock(this)     {       //Access instance variables     }   } }  is &quot;a problem if the instance can be accessed publicly&quot;. I'm wondering why? Is it because the lock will be held longer than necessary? Or is there some more insidious reason? $f$ is entire and maps a rectangle to a rectangle, then $f$ is linear The problem is in the subject line, I have it for homework. $f$ is a complex valued function. For completeness:     Prove that if $f$ is an entire function and for some rectangle $R$, the image $f(R)$ is also a rectangle, then $f$ is linear.    The composition of linear maps is linear, so we can choose the two rectangles to have two edges coinciding with the real and imaginary axes, and their common vertex at the origin. So as a portion of the real line gets mapped to itself, we can take $f$ to be the analytic continuation of a real function. Then I'm stuck.  This question is confusing me. It's well known from the Riemann mapping theorem that there exists many holomorphic functions taking any rectangle to any other given rectangle. But if $f$ is linear, then the two rectangles must be similar. So the added restraint of $f$ being entire seems to be causing this hassle. Thoughts? Is there an off/disable control anywhere? I realize that the INFO button toggles it off, but it comes right back on as soon as you tap the shutter, and that it a problem in low light situations where the LCD keeps blasting me in the eye. How to enable the Edit button in Shutter? I am trying to enable the Edit button in Shutter. I installed the application from Ubuntu Software and are using Ubuntu 18.04.  After taking a screenshot the Edit button is disabled. And when hovering the mouse pointer over the button the text "Goo::Canvas/libgoo-canvas-perl needs to be installed for this feature" is shown.    I have tried to install the package libgoocanvas2-perl and restarted the computer, but the edit button is still not enabled:  sudo apt install libgoocanvas2-perl   That is the only package I can find.  Anyone that knows how to fix this? I keep going around in circles with this problem:  Given the functions $f:X\to Y$ and $g:Y\to Z$, if $g\circ f:X\to Z$ is bijective, then prove $f$ must be one-to-one and $g$ must be onto. I was able to prove this without difficulty. However the second part of the problem has me stumped:  a) Find functions $f,g$ where $g\circ f$ is bijective but $f$ fails to be onto.  b) Find functions $f,g$ where $g\circ f$ is bijective but $g$ fails to be one-to-one.  My attempt for (a): Let $f:\mathbb{R}\to \mathbb{R}^+$ be defined by $f(x)=e^x$ and $g:\mathbb{R}^+\to \mathbb{R}$ be defined by $g(x)=\ln(x)$. Then $g\circ f(x)=g(f(x))=g(e^x)=\ln(e^x)=x$ which is bijective, $g$ is onto and $f$ fails to be onto... or so I thought. Upon closer inspection $f$ $\underline{is}$ onto for the given codomain $\mathbb{R}^+$.  And now I'm stuck in a rut. Anyone out there have any hints? Why does the $\sum_{n>1}(\zeta(n)-1)=1?$ While I was looking at the values of the zeta function for the first natural numbers, I noticed that the sum of the values minus $1$, converge to $1$. Better put: $$\sum_{n=2}^{\infty} \left(\zeta(n)-1\right) = 1 $$ Furthermore, if you use only the even numbers for the zeta function, the sum will converge to $\frac{3}{4}$, or $$\sum_{n=1}^{\infty} \left(\zeta(2n)-1\right) = \frac{3}{4}$$  Leaving $$\sum_{n=2}^{\infty} \left(\zeta(2n-1)-1 \right)= \frac{1}{4}$$  This is probably common knowledge among mathematicians, but I couldn't find much about it on the internet. Is there a proof of this or perhaps even a simple explanation why this is so?