Prove that $\sum_{t \vert n} d^3(t) = (\sum_{t \vert n}d(t))^2$ for all $n \in \mathbb{N}$
Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.
Did not receive site association bonus (+100 rep) I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need? When to use \par and when \\, \newline, or blank lines In this movie there are scientists who (spoilers ahead)       ...are creating a simulation. One scientist starts seeing odd looking characters but no one else sees them. He asks around and more and more characters are missing or don't know about the characters in the story he knows about.  He finds out that he is part of a simulation. He finds out that the characters only he remembers have been deleted. He also finds out that there are characters from his own simulation that have broken free of his simulation and are in his world.    Hints: I remember this being an older film (70's?); it might have been foreign film possibly with subtitles (not Hollywood); it might have been part of the Criterion collection; I'm pretty sure I streamed it from Hulu. Sorry, I can't remember more details. Can you release your zanpakuto if you are mute? So, this sounds like a stupid question I know but in the bleachverse 'names' are very important and you can only unleash the power of your zanpakuto if you can call out its name. Therefore, if a shinigami is mute, will they ever be able to achieve shikai/bankai? If so then how? Only one case that I know of is for shinigamis who have already achieved bankai to be able to release their shikais without a command.  Answers based on LN are welcome, but if you are citing from a dubious source or a source whose canon status can be question, please specifically mention it. I am mostly interested in an answer based on a  canon source. Are there any versions of t-SNE for streaming data? My understanding of  and the Barnes-Hut approximation is that all data points are required so that all force interactions can be calculated at the same time and each point can be adjusted in the 2d (or lower dimensional) map.  Are there any versions of t-sne that can efficiently deal with streaming data? So if my observations are arriving one at a time, it will find the best location on the 2d map to place the new observation, or continuously update all points on the 2d map to account for ht new observation.  Would this even make sense or does it go against the setup of t-sne. I was wondering if someone could critique my argument here. The problem is to find the probability where exactly 2 people in a room full of 23 people share the same birthday.  My argument is that there are 23 choose 2 ways times $\displaystyle \frac{1}{365^{2}}$ for 2 people to share the same birthday. But, we also have to consider the case involving 21 people who don't share the same birthday. This is just 365 permute 21 times $\displaystyle \frac{1}{365^{21}}$. To summarize:  $$\binom{23}{2} \frac{1}{365^2} \frac{1}{365^{21}} P\binom{365}{21}$$ For the ring $\Bbb{Z}[\sqrt{d}]=\{a+b\sqrt{d}\mid a,b \in \Bbb{Z}\}$ where $d$ is not divisible by the square of a prime, prove that the norm $N(a+b\sqrt{d})= |a^2-db^2|$ satisfies the assertions: $N(x)=0$ if and only if $x=0$; $N(xy)=N(x)N(y)$ for all $x$ and $y$; $x$ is a unit if and only if $N(x)=1$; and if $N(x)$ is prime, then $x$ is irreducible in $\Bbb{Z}[\sqrt{d}]$. English proverb for when a solution comes too late In Flemish we have a saying "Vijgen na pasen". Translated: "figs after Easter". It means a solution comes too late to be of any use.  What is the English equivalent for this?  Some googling gives me "Closing the barn door when the cow has bolted", but the explanation seems to point more towards the wrong solution for a problem. How do I change my Minecraft username? How to delete a non-empty directory in Terminal? How do I delete the following directory?  I typed:  rmdir lampp   This error comes up:  rmdir: failed to remove `lampp': Directory not empty   Is there a command to delete all the files in the directory and delete the directory folder? Does SQLDeveloper support executing scripts? I'm writing a shell script for Linux, using Bash, to translate any video-file into a MP4. For that, I'm using avconv with libvorbis for audio.   Inside my script, I have a question for the user :   read -p "- Audio Quality [scale from -2 to 10] ? "     if [ -n "$REPLY" ] ; then     ABITRATE="-aq $REPLY"     fi   My "ABITRATE" string goes into the final avconv command-line.  But I would like to give the user the opportunity to answer that question with a value in Kb (Kilobit), and translate it into the scale that libvorbis uses. The "scale from -2 to 10" is this :   Quality Kbit/s  Normalization -----------------------------  -2      ~32        y  -1      ~48        y   0      ~64        y   1      ~80        y   2      ~96        y   3     ~112        y   4     ~128        n   5     ~160        n   6     ~192        n   7     ~224        n   8     ~256        n   9     ~320        n  10     ~500        n   I would like to know how to check if my $REPLY is in a range of number. For example, I would like my script to do something like this :   if [ $REPLY is a number between 1 and 32 ] ; then   REPLY="-2" elif [ $REPLY is a number between 33 and 48 ] ; then   REPLY="-1" fi   Is this possible (I'm willing to say 'yes of course, shouldn't be hard' but I don't know the syntax to use) ? How do I prevent widow/orphan lines? How do I prevent a line from appearing by itself:  Orphan: at the bottom of the page, or Widow: at the top of the page? It is being surprisingly hard to achieve. I've tried so far:   Download file manager apps from Google Play. Unfortunately, "share" shares the files, not the file names. None of them had a option like copy file names Download backup apps. Unfortunately, none of them backup the file names of a folder.   Any idea to copy to the clipboard or share the file names of all the files of a folder? Prove $ \sum \frac{\cos n} { \sqrt n}$ converges The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra.     Recall that an infinite cyclic group is isomorphic to $\mathbb{Z}$. We wish to show that we do not have an isomorphism between $\mathbb{ZxZ\;and\;Z}$. Note that $\mathbb{ZxZ}$ is an infinite group (under addition of course). Now, in order for there to even be potential for an isomorphism, two spaces must have equal dimension. Since the $\mathbb{dim(\mathbb{ZxZ})}=2&gt;\mathbb{dim(\mathbb{Z})}=1$, we know that $\nexists$ an isomorphism between our spaces. Hence, $\mathbb{ZxZ}$ is not a cyclic group.   My question (besides a validity check): was there a better way to prove this? I just found this to be the easiest way.  EDIT Totally wrong with this one. Back to the cutting board. Rename all the files within a folder with prefix “Unix_” Rename all the files within a folder with prefix “Unix_” i.e. suppose a folder has two files a.txt and b.pdf than they both should be renamed from a single command to Unix_a.txt and Unix_b.pdf This is a problem from Rudin's Functional Analysis:  Show that there is a sequence of functions $\{ f_n \}$ in $X$ (the vector space of all complex functions) such that $f_n \to 0$ pointwise, but if any sequence $\{ \gamma_n \}$ tends to $\infty$, $\gamma_n f_n \not\to 0$.  The hint says to use the fact that the collections of all complex sequences converging to $0$ has the same cardinality as $[0,1]$. Honestly, this confused me more than it helped me.  My main issue is through the method of showing existence: I am not sure if I should attempt to construct an explicit sequence $f_n$, or if I need to appeal to other nonconstructive means such as Zorn's Lemma, etc. I have been trying to construct clever choices of sequences for a good bit now, but it always seems like I will be able to find a sequence that diverges slowly enough to allow that $\gamma_n f_n \to 0$.   Thanks! In my Java application, I want to run a batch file that calls "scons -Q implicit-deps-changed build\file_load_type export\file_load_type"  It seems that I can't even get my batch file to execute. I'm out of ideas.   This is what I have in Java:  Runtime.    getRuntime().    exec("build.bat", null, new File("."));   Previously, I had a Python Sconscript file that I wanted to run but since that didn't work I decided I would call the script via a batch file but that method has not been successful as of yet.