Let $f : \mathbb{R} \to \mathbb{R}$ differentiable be such that $$f'(x)=f^2(x)$$ for all $x$ and $f(0)=0$. Show that $f(x)=0$ for all $x$.   My attempt:  Notice that $f'(x)=f^2(x)\geq 0$, thus $f$ is increasing. Since $f(0)=0, f(x)\geq0\; \forall x \geq 0$. Since $f$ is differentiable, $f$ is continous, therefore given $1&gt; \epsilon &gt;0$, there exists $\delta&lt;1$ such that for $x \in (-\delta,\delta)\implies f(x)&lt;\epsilon&lt;1.$ Take $x=\delta/2$. Then $f(x) = f'(c)\delta/2=f^2(c)\delta/2$. Then $f^2(c) \geq f(x)$. But since $c\in (-\delta,\delta)$, $0\leq f (c)&lt;1 \implies f^2(c) \leq f(c)\implies f(c)\geq f(x)$. But $c&lt;x$, therefore $f(c)\geq f(x)$ which means $f(c)=f(x)$. Therefore for all $y \in (c,x)$, $f(y)=f(x) \implies f'(y)=f^2(y)=0 \implies f(y)=0$. Where do I go from here?
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$ and $f'(x) = [f(x)]^{2}$, $\forall x \in \mathbb{R}$. Show that $f(x) = 0$,  $\forall x \in \mathbb{R} $.  I first (unsuccessfully) tried using the Mean Value Theorem, but this is in the Integrals chapter so the solution probably involves them. Can't really see where integrals come in here though.  What I've got so far:   (i) Since $f$ is differentiable, thus it is continuous and, hence,  integrable. Therefore $f^2$ is also integrable and as $f'=f^2$, $f'$ is too.   (ii) $f' \geq 0$, $\forall x \in \mathbb{R}$
For almost a year now, I have been in the uncomfortable position of having an idea.  However, there is one nice thing about this idea.  It makes a concrete, exact and relatively easy to test physical prediction.  The idea predicts that there is a 11,187 m/s (Earth’s escape velocity) aethereal wind directly into the surface of the Earth at its surface.  I believe it would be possible to test this by performing a vertical variation of modern versions of the Michelson-Morley experiment (MMX) with one arm pointed in the vertical direction. ()  In 2003, Müller et al. performed a normal (2 horizontal orthogonal arms) modern MMX using cryogenic optical resonators that found a “possible anisotropy of the speed of light c, (of) 2.6 +/- 1.7 parts in 10^15” (  )  In a brief conversation with Holger Müller a Professor at Berkeley and the lead author of that paper, he stated that to this knowledge no one had ever performed a variation of the experiment using a vertical arm.  He also mentioned that such an experiment would be complicated by the fact that gravity would slightly compress the length of the vertical bar making two equal length bars no longer equal in length.  “They haven't been done as far as i know. The problem is that any interesting physics signal would be hard to tell from a large signal from stretching of the arms under their own weight.” - H. Müller  I am interested in attempting to run this experiment myself.  To that end, I have the following questions:   Given current Physics understanding, is there any reason to expect that such a vertical variation of the MMX wouldn’t return the exact same results as all other MMXs, namely that there is no anisotropy in the speed of light? Given the complications mentioned by Professor Müller, are there reasonable methods available to overcome them?  Especially considering the size of the effect (c + 11,187m/s vs c) is substantially larger than the accuracy obtained in his and similar modern MMXs. What is the order of magnitude cost of such an experiment?  If I am to fund this personally, would such a project cost \$10,000?  \$100,000? \$1,000,000? more?   Any insight offered on this topic will be greatly appreciated.  EDIT: For what its worth, for people looking at this years later.  As I originally alluded to in a comment to [WetSavannaAnimal aka Rod Vance]'s much appreciated and comprehensive answer.  While, I've become increasingly comfortable with the idea that there is a flow of aether or spacetime or whatever you want to call it into the Earth at 11km/s; I strongly believe that a vertical MMX would return the same null result as the horizontal one.  And I believe it will do so because of length contraction; that length contraction exactly masks any anisotropy of the speed of light by ensuring that the roundtrip time of light in any direction is a constant; that length contraction ensures Lorentz co-variance. Filter users by country This was talked about a few podcasts ago. And they referenced some other site that does this.  Currently location is a string but adding a country dropdown (and for the usa a state drop down) could be a great addition to the site.  Then allow for user filtering by country / state. What is the difference between the memory usage report in System Monitor and the one by free? I am using System Monitor 2.28.0.  When I look in the Resources tab, I see a nice graph with for memory and swap.  Memory is about 60% 2.3 GiB of 3.8.  When I type the command free, I got :               total       used       free     shared    buffers     cached Mem:       3994908    3962396      32512          0     100852    1477032 -/+ buffers/cache:    2384512    1610396 Swap:      8000328      28468    7971860   cat /proc/meminfo | grep MemFree give  MemFree:           34536 kB   The situation has remained the save for several minute.  I started a lot of process with a script and the script is waiting for the free memory to get lower.  According to what I am seeing in the Process tab (or with top), the number in System Monitor seem a lot closed to the total of the memory of the process that the one reported by free.   Thanks When did Captain America learn what happened to Howard Stark? Near the end of Captain America: Civil War we learn that      Howard and Maria Stark were assassinated by the Winter Soldier   and Captain America admits that he already knew.  But how did he find out? I don't recall this being revealed in any of the previous movies, and I can't think of a reason why the characters who already knew would have gone out of their way to tell Steve about it. How to secure Elliptic Curve ElGamal encryption against known plaintext attacks? If I have an encoding function  $f(x)$ that maps a message $m$ to a point $P$ on a suitable Elliptic Curve $E$ . If I have the public key $Q$ of my recepient then I can encrypt the message as follows:  Choosing a random number $k &lt; n -1 $ where $n$ is the order of the curve E .  Calculating $C = [k] G $ where $G$ is the generator point of the curve E   Calculating $R = [k]Q $ where $Q$ is the recipient's public key   Now the message is encrypted by adding $P$ to $R$ to get          $C_e$ $( = R + P)$ which is sent along with $C$ to the recepient.  DECRYPTION:  Now for the recepient to decrypt my message they compute $R = [privatekey] C $ (where $C = [k]G $) and simply subtract $R$ from $C_e$ to get the encoded point $P$ back which is then decoded by an inverse function $f^{-1}(x)$ to recover $m$.  Now , two problems arise : If the attacker knows the plaintext (suppose if I follow a particular format of data while sending messages ) he could encode his guessed plaintext and subtract it from $C_e$ to recover $ R $ back ! . If I used that same $ R  $ to encrypt further "blocks" of my message then the security of the later parts of my message has been breached!   I realize that doing  something like a scalar multiplication over and over again for every block of data, would be a huge drawback in efficiency and speed.   Is there any way to compute a different $ R $ for every block of data, without using much resources and that too quickly ?   Does reusing the same $R$ for another message break security ?  NOTE: This question is for educational purposes only. It's for the sake of expanding my knowledge on Crypto. Integrate $\int \frac{25 - x^2}{x^4} \operatorname d\!x$ I think we use trig substitution for this. But I'm not sure where to even begin with this   $$\int \frac{25 - x^2}{x^4} \operatorname d\!x$$ What's everyone's opinion on closing an older question as a duplicate of a newer one, if the newer one is clearly the better resource? I'm looking at a special case right now, where the two were posted by the same person; the  was migrated to SU before a commenter pointed out it was a . Is it considered good practice to close the older as a duplicate of the newer?  More generally, is it ok to do that for separate questions posted by different people, if the older question went unanswered but the newer has a good answer?  (I should probably mention I saw , but I think this is separate) how do i change the color of typeface on photoshop? How do I undo 'git add' before commit? I mistakenly added files to Git using the command:  git add myfile.txt   I have not yet run git commit. Is there a way to undo this, so these files won't be included in the commit? "Same as I do" or "same as I have" She has the same book as I have.  Isn't it supposed to be She has the same book as I do?  But I often hear the former.  Why? What value to use to convert ADC value to voltage? While working on my project, I got to a point where I don't know the answer. I have a 10-bit ADC (the one incorporated in the ATMEGA328P) and it returns values up to 1023 (210 - 1).   The official documentation states that the equation to convert the values from the ADC to the corresponding voltage is:   $$ V = \frac{V_{in} \cdot ADC}{2^{10}} $$  But, if the values go up to 1023, then wouldn't it be better to replace the 210 by 210&nbsp;-&nbsp;1?  Thanks. I just replaced a broken light fixture in my closet. (The switch in the light fixture broke when I pulled too hard on the chain.)   When replacing the light fixture I noticed that there is only 30 V AC between the white and black wires. The ground wire is cut back and is not being used. By the way, the wiring in my house is all aluminum.   I know the new light fixture works. I checked it out with my multimeter, and I tried it in a different closet. But the light bulb won't turn on because there is only 30 V. (Even if I disconnect the light fixture completely there is still only 30 V.   It seems to me that it must be a problem with the wiring.  My attic does provide some (difficult) access to the wiring, but I am not sure where to start looking for the problem or what to look for. Can anyone offer some suggestions? How can I recursively clear the "Hidden" file flag on Windows Vista? A friend's computer was recently infected with a virus.  I hope the virus has been removed successfully. But, it appears that the virus set the "Hidden" flag on most directories and files on the system.   What's the most convenient way to clear this flag from selected directories and their contents? I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title).  I've already constructed two non-isomorphic total orderings. They are $$ 1 &lt; 2 &lt; 3 &lt;\ldots\\ 1 &gt; 2 &gt; 3 &gt;\ldots $$ where "$&lt;$" and "$&gt;$" are  strict total ordering relations.  Thanks in advance for your help. Find the center of the symmetry group $S_n$. Find the center of the symmetry group $S_n$.   Attempt:  By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\  g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is always the trivial permutation. Suppose $a$ is in $S_n$, but not equal to identity. Now we can imagine the permutation as bijective function that maps from $\{1,2,\dotsc,n\}$ to $\{1,2,\dotsc,n\}$.  So suppose $p$ is a permutation map. Then $p$ maps from a location $i$ to a location $j$. Take $p(i) = j$ where $i\neq j$. Let $k$ be in $\{1,2,\dotsc,n\}$, where $k$, $i$ and $j$ are all different elements. The cycle $r = (jk)$, then we will see if this commutes.  $rp(i) = rj$    Can someone please help me, I am stuck?  Thank you. Adding Basemaps from Google or Bing in QGIS ArcGIS Desktop has the option of using basemaps from ArcGIS Online. Does QGIS have any such options? How to run applications as root? I am having some strange issue with Kate and Kwrite. When I click on Open File, it crashes with segmentation fault.   I am a complete newbie to Linux, and I think the issue is that I am not running the application as root.   How do I run applications as root in Ubuntu? Is it bad practice to do this? What is the purpose of the whole root thing, where even though we need to use root so frequently, it is not utilized as default? What does the following for loop syntax mean in C? How to create a shortcut that executes an xdotool command to simulate a key press? I'm trying to simulate a media key press using a custom shortcut.  So far, I've been able to achieve the simulation I need using the command:  xdotool key XF86AudioPlay   It works perfectly, it pauses or starts the music player every time it is run.  The problem comes when trying to run it as a shortcut.  First I tried to run an alias with a custom shortcut, but it did not work.  As explained on :     The commands run by keyboard bindings are not parsed via a shell. Your best bet is to create a directory bin in your homedir.   Next I created the suggested script and named it simplay:  #!/bin/sh xdotool key XF86AudioPlay   It is located inside a bin folder which is part of the PATH, this works fine and can be run from any location.  When trying to create the Custom Shortcut, I was unable to get it to execute, the configuration is the following:  Name: test Command: simplay  Shortcut: Ctrl+Alt+R   After searching a bit more I found another way to execute the shortcut:  gnome-terminal -e simplay   I tried it on a terminal and noticed that when executing it it would rapidly open a new terminal window, execute the child process, pausing the music and immediately close the window.  I changed the command on the shortcut and when hitting it I noticed the same behavior opening a terminal and immediately being closed but this time the child process did not pause the music, which makes me think that the script has been called on all my previous attempts but it is somehow not being able to execute.  How can I solve this problem according to your knowledge? $M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences Let $0 \to M_1 \to M \to M_2 \to 0$ be an exact sequence of $A$-modules.  i) Prove: If $M_1$ and $M_2$ are finitely generated, then $M$ is too.  ii) Let $M$ and $N$ be sub-modules of an $A$-module $L$. Assume that $M+N$ and $M\cap N$  are finitely generated $A$-modules. Prove that $M$ and $N$ are finitely generated $A$-modules.  If have proved the first part, but i don't see how to do the second part. I tried to use the first part, so to find an exact sequence with $M$ in the middle and finitely generated modules in place of $M_1$ and $M_2$, but I can't find a surjective map $M \to P$ (for some f.g. $A$-module) in the sequence:  $$0 \to M\cap N \to M \to P \to 0$$  any hints/help would be appreciated.