I am having trouble formalizing the answer to this question. Any help would be appreciated.     Is it true that for every continuous function $f:[0,2] \to \mathbb{R}$ such as $f(0)=f(2)$, $\exists x \in [0,1]:f(x)=f(x+1)$?   If the function is periodic of degree one this seems true to me. I tried proving by contradiction, supposing that $\forall x \in [0,1], f(x) \neq f(x+1)$, but I couldn't get any further.
Proof that if continuous function on $[0, 2]$ and $f(0) = f(2)$, then for some $x: f(x) = f(x - 1)$
Encrypting a folder I'm familiar with bitlocker, having been using it for a while, but I am looking at just encrypting a folder instead. But I just don't understand the options it's giving me.  When I right-click a folder > properties > advanced > [x] encrypt contents to secure data > OK &lt; apply > ...  I get:  () apply changes to this folder () apply changes to this folder, sub-folder and files.   This doesn't make sense to me at all, if I encrypt a folder then the whole folder, everything within it is encrypted, then why should I be given these options, how can I have a folder encrypted and files within that folder unencrypted, that seems stupid to me, it doesn't make sense at all.  Can anybody explain why these options exist and how can I simply encrypt a folder? Return vs. Enter? I noticed that my MacBook's keyboard has the word enter above the return key. I also noticed that on Mac OS X, there's a special symbol for return ↩, and another for enter ⌤. How would I send the enter key if my MacBook has no numpad?  Is there any difference between these two keys? Can I install Python 3.x and 2.x on the same Windows computer? Math contest integer triplet problem: $x + y + z + xy + yz + xz = xyz − 1$ Can any one help me with this?  Determine all integer triples $(x,y,z)$ such that $1 ≤ x ≤ y ≤ z$ and $x + y + z + xy + yz + xz = xyz − 1$.  I thought of Vieta's formula but don't let me lead you into a dead end. How do I recover files from an USB flash drive? Possible Duplicate:          I have a USB flash drive that claims it needs to be formatted every time I put it into my computer (Windows).  Yesterday the drive was healthy and had data on it.  The data is probalby still there.  Are there any free tools to restore the drive?  If not free, what tools are known to work in this situation? Imagine I have tons of RAM. Let's say 64GB. That's a lot for even gaming PCs. Now the default location of a pagefile in Windows is on the main OS drive, be it HDD or SSD, which are faster in general, but still not as fast as RAM.  Something tells me that disabling the pagefile on the hard drive or creating a virtual RAM drive and letting the pagefile be there could make Windows move all its virtual memory to RAM, and so increase the system's performance, but I'm not very knowledgeable in that area, so that might not be true at all.  I tried both, but I couldn't analyze the results to reach a definite conclusion with my knowledge level in memory things.  Would this work? If not, why? Change the URL in the browser without loading the new page using JavaScript How would I have a  action that may have some effects on the current page but would also change the URL in the browser so if the user hits reload or bookmark, then the new URL is used?    It would also be nice if the back button would reload the original URL.  I am trying to record JavaScript state in the URL. Find count of all combination of numbers whose sum is x I want to find the sum of all combination of numbers whose sum is x, for e.g.  when x = 3 f(x) = countOf(111,12,21,3) = 4 Make gnome-terminal show the command running as title I want the name of command running , e.g unzip to be visible through the title bar of gnome-terminal , but it seems to be impossible , if the application running doesn't set a title explicitly , even though I choose 'Replace initial title' option in the profile dialog. Recover files from hard drive that has been formatted thrice I have a situation.  I got married 6 months ago and stored some very important pictures on my laptop which was running Linux. Later I formatted the whole hard drive, installed Windows and a hacked version of Mac (just to play around) and I have been using the laptop for last 8 months or so after that.  Is there any software that can recover the files for me? I am happy to pay money! Prove that a graph with the same number of edges and vertices contains one cycle Below is an attempt at a proof of the following problem. Any feedback would be greatly appreciated. Thx!  Let $E$ be a set and $O_n = \{x: d(x, E) &lt; \frac{1}{n}\}$. Show  If $E$ is compact then $m(E) = lim_{n \rightarrow \infty} m(O_n)$.  This is false for $E$ closed and unbounded or $E$ open and bounded.   Note that all sets are real and measurable refers to Lebesgue measurable.  Suppose $E$ is compact. Then $m(E) &lt; \infty$. Also, since each $O_n$ is an open set of $\mathbb{R}^d$, $m(O_n) &lt; \infty$. If either of the following is true, we have that $lim_{n \rightarrow \infty}m(O_n)=m(E)$: $O_n \nearrow E$ $O_n \searrow E$ Now  $\bigcap_{n=1}^{\infty} O_n = \{ x:d(x,E) = 0\}$. That is, $x \in \bigcap_{n=1}^{\infty}O_n \iff \exists n&gt; \frac{1}{\epsilon},\ n \in \mathbb{N} \iff x\in \{x \mid d(x,E) =0\}$. Then we have that $\bigcap_{n=1}^{\infty}O_n = E$ and $O_n \searrow E$. Thus $ \ m(E) = lim_{n \rightarrow \infty} m(O_n)$. Suppose $E$ is closed and unbounded. Let $E = \{(x,y): y = 2 \}$ and $O_n = \{(x,y): d(x,E) &lt; \frac{1}{n}\}$. Since $E$ is a line in $\mathbb{R}^2$, $m(E) = 0$. Also, since the measure of a rectangle in $\mathbb{R}^d$ is its volume, $m(O_n) = \left| O_n \right| = \infty$, since the rectangle $O_n$ is unbounded. It is therefore apparent that $m(E) \not = lim_{n \rightarrow \infty} m(O_n)$. Suppose $E$ is open and bounded. Let $E=(0,1)$ and $O_n = \{x \mid d(x,E) &lt; \frac{1}{n}\}$. $\mathbb{R}-E$ is closed and $O_n \in \mathbb{R}-E$. Since $\mathbb{R}-E$ contains all its limit points $lim_{n \rightarrow \infty}O_n = O \in \mathbb{R}-E$. Thus $lim_{n \rightarrow \infty} m(O_n) \neq m(E)$. Restoring MySQL database from physical files Is it possible to restore a MySQL database from the physical database files. I have a directory that has the following file types:  client.frm client.MYD client.MYI    but for about 20 more tables.  I usually use mysqldump or a similar tool to get everything in 1 SQL file so what is the way to deal with these types of files? Trigger Setup Wizard at next boot using command-line I am looking for a way to use ADB to trigger Setup Wizard at next boot. I know I can run the following command and get it to pop up. However, I can't figure out what triggers it to start at boot.  adb shell am start -a android.intent.action.MAIN -n com.google.android.setupwizard/.SetupWizardTestActivity   The phone I am currently using is stock and I would like to keep it that way.   I've read the  question: Is there anything beyond subtle hints that Jack becomes the Face of Boe? Confusion with floating point numbers Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z})[X]$ is irreducible. (Hint: if it were reducible, it would either have a root or be of the form $g(X) \cdot h(x)$, where deg$g(X) = 2$ and deg$h(X) = 3$) Recall that $\bar{a}$ is shorthand for the coset $a + 2\mathbb{Z} \space $in$ \space (\mathbb{Z}/2\mathbb{Z})[X]$ (for any $a \in \mathbb{Z}$)  This question is worth $7$ marks out of a possible $75$, this is my solution..  $f(X) = X^5 + X^3 + \bar{1}$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$ iff it has no roots in $\mathbb{Z}/2\mathbb{Z}$. Since $f(\bar{0}) = \bar{1}$ and $f(\bar{1}) = \bar{1}$, it follows that $f(x)$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$.  Would this be a sufficient answer? am I missing something? Prior to doing some benchmarking work how would one free up the memory (RAM) that the Linux Kernel is consuming for its buffers and cache?    Note that this is mostly useful for benchmarking. Emptying the buffers and cache reduces performance! If you're here because you thought that freeing buffers and cache was a positive thing, go and read . The short story: free memory is unused memory is wasted memory. What is the variance of Y = AX where A is a matrix? Here $X,Y$ are vector of length $n$ and $A$ is  a $n\times n$ matrix. Suppose the covariance matrix $D(X)$ is known? Is there a bijection of the natural numbers which swaps $\frac{1}{n}$-summable subsets with $\frac{1}{\sqrt{n}}$-summable subsets? Let me start with a precise statement of the question.  For a subset $A\subseteq \mathbb{N}$ and a series of positive real numbers $\sum_{n=0}^\infty a_n$, I'll use the notation $\sum_A a_n$ as a shorthand for $\sum_{n\in A} a_n$.  Is there a bijection $f:\mathbb{N}\rightarrow \mathbb{N}$ with the property that for every $A\subseteq \mathbb{N}$, $\sum_A \frac{1}{n}$ converges iff $\sum_{f(A)} \frac{1}{\sqrt{n}}$ converges?  Background: Fix a series of positive terms $\sum_{n=0}^\infty a_n$.  Given a subset $A\subseteq \mathbb{N}$, call it $a_n$-small if $\sum_A a_n$ converges.  The following proposition is easy to prove:  Proposition:  The set $\{A\subseteq \mathbb{N}: A\text{ is } a_n\text{-small}\}\cup \{\mathbb{N}\}$ is a topology of closed sets on $\mathbb{N}$.  I will use the notation $(\mathbb{N},a_n)$ to refer to this topology.  Then one can ask how topological properties of $(\mathbb{N},a_n)$ are related to series properties of $\sum a_n$.  For example, I can show:  Proposition:  The following are equivalent.  $\sum a_n$ converges.  $(\mathbb{N},a_n)$ is discrete.  $(\mathbb{N},a_n)$ is disconnected.  $(\mathbb{N},a_n)$ is Hausdorff    And there are other nice things.  For example, $(\mathbb{N},a_n)$ is compact iff $(\mathbb{N},a_n)$ is cofinite iff $\liminf a_n &gt; 0$. With this language, my question can be reformulated as...  Are $\left(\mathbb{N}, \frac{1}{n}\right)$ and $\left(\mathbb{N}, \frac{1}{\sqrt{n}}\right)$ homeomorphic?  I have made very little progress on this.  Of course, the identity function $i:(\mathbb{N}, \frac{1}{\sqrt{n}})\rightarrow (\mathbb{N}, \frac{1}{n})$ is a continuous bijection, but the inverse map is not continuous.  Also, if there is such a bijection, there there is such a bijection with agrees with $i$ on any preassigned finite set. Edit  I thought I add in a slightly suprising (to me, at least) example when things work out to be homeomorphic. Begin with a convergent series $\sum c_n$. and divergent series $\sum a_n$ with $\lim a_n = 0$.  Create a new series $b_n$ using all the terms of $c_n$ and $a_n$ (in whatever order you wish).  Then $(\mathbb{N}, a_n)$ and $(\mathbb{N}, b_n)$ are homeomorphic.  The idea is that since $\lim a_n = 0$, there is a convergent infinite subseries $\sum_A a_n$.  Then we use a bijection which with $A$ and $A\cup \{\text{indices of }c_n\}$ to &quot;squeeze&quot; the $c_n$ in without changing the topology.  Of course, I am glossing over many details, but I can include them if desired.