Show that $\sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}$
Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
Why does everyone do 9-overlord, not 10-overlord? Error/warning when using NSolve for simple equation Can't boot XP after Ubuntu Installation, how to fix? I installed Ubuntu and after the installation, i can't boot to XP. When i choose XP from the Grub menu, I got black screen for like 30 sec, and then it returns to the Grub menu.  Thanks! How to understand the memory usage and load average in linux server I am using a linux server which has 128GB of memory and 24 cores. I use top to see how much it is used. Its output is pasted at the end of the post. Here are two questions:  (1) I see that each of the running processes occupies a very small percentage of memory (%MEM no more than 0.2%, and most just 0.0%), but how the total memory is almost used as in the fourth line of output ("Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers")? The sum of used percentage of memory over all processes seems unlikely to achieve almost 100%, doesn't it?  (2) how to understand the load average on the first line ("load average: 14.04, 14.02, 14.00")?   Thanks and regards!  Edit:  Thanks!  I also really like to hear some rough numbers based on used percentage of memory to determine if a server is heavily loaded, since I once became the one who cramed the server without understanding the current load.  Is swap regarded as almost the same as memory? For example, when memory and swap are almost of same size, if the memory is almost running out but the swap is still largely free, may I just view it as if the used percentage of memory + swap is still not high and run other new processes?  How would you consider together CPU or memory (or memory + swap) usage? Do you become worried if either of them reaches too high or both?  Output of top：  $ top    top - 12:45:33 up 19 days, 23:11, 18 users,  load average: 14.04, 14.02, 14.00 Tasks: 484 total,  12 running, 472 sleeping,   0 stopped,   0 zombie Cpu(s): 36.7%us, 19.7%sy,  0.0%ni, 43.6%id,  0.0%wa,  0.0%hi,  0.0%si,  0.0%st Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers Swap: 63111312k total,   500556k used, 62610756k free, 124437752k cached    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND  6529 sanchez   18  -2 1075m 219m  13m S  100  0.2  13760:23 MATLAB 13210 timothy   18  -2 48336  37m 1216 R  100  0.0   3:56.75 absurdity 13888 timothy   18  -2 48336  37m 1204 R  100  0.0   2:04.89 absurdity 14542 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.34 absurdity 14544 timothy   18  -2  2888 2076  400 R  100  0.0   1:06.14 gatherData  6183 sanchez   18  -2 1133m 195m  13m S  100  0.2  13676:04 MATLAB  6795 sanchez   18  -2 1079m 210m  13m S  100  0.2  13734:26 MATLAB 10178 timothy   18  -2 48336  37m 1204 R  100  0.0  11:33.93 absurdity  12438 timothy   18  -2 48336  37m 1216 R  100  0.0   5:38.17 absurdity 13661 timothy   18  -2 48336  37m 1216 R  100  0.0   2:44.13 absurdity 14098 timothy   18  -2 48336  37m 1204 R  100  0.0   1:58.31 absurdity 14335 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.93 absurdity 14765 timothy   18  -2 48336  37m 1196 R   99  0.0   0:32.57 absurdity 13445 timothy   18  -2 48336  37m 1216 R   99  0.0   3:01.37 absurdity 28990 root      20   0     0    0    0 S    2  0.0  65:50.21 pdflush 12141 tim       18  -2 19380 1660 1024 R    1  0.0   0:04.04 top  1240 root      15  -5     0    0    0 S    0  0.0  16:07.11 kjournald  9019 root      20   0  296m 4460 2616 S    0  0.0  82:19.51 kdm_greet     1 root      20   0  4028  728  592 S    0  0.0   0:03.11 init     2 root      15  -5     0    0    0 S    0  0.0   0:00.00 kthreadd     3 root      RT  -5     0    0    0 S    0  0.0   0:01.01 migration/0     4 root      15  -5     0    0    0 S    0  0.0   0:08.13 ksoftirqd/0     5 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/0     6 root      RT  -5     0    0    0 S    0  0.0  17:27.31 migration/1     7 root      15  -5     0    0    0 S    0  0.0   0:01.21 ksoftirqd/1     8 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/1     9 root      RT  -5     0    0    0 S    0  0.0  10:02.56 migration/2    10 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/2    11 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/2    12 root      RT  -5     0    0    0 S    0  0.0   4:29.53 migration/3    13 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/3 Class method differences in Python: bound, unbound and static What is the difference between the following class methods?  Is it that one is static and the other is not?  class Test(object):   def method_one(self):     print "Called method_one"    def method_two():     print "Called method_two"  a_test = Test() a_test.method_one() a_test.method_two() Tool for straightening a bent metal bar of the gate? Someone's car hit our metal bar gate, so one of the rods is bent as shown in the photo. What kind of a tool can I use to bend this back, so it's straight? Thank you! I'm not too sure how to describe the effect that I'm going for or what it is even called.  Thus, I'm having some trouble finding any help through Google.    I want to accomplish something like this "SLOPED" text below, but have the bottoms of the letters follow the slope as well instead of being horizontal.    I have tried using the free transform tool, skewing, and using perspective, but I don't like the effect they are resulting in.  All of the tools that I just mentioned result in the appearance of the word being angled away from the viewer (perspective).  I want more of this style effect, whereas the letters are not skewed horizontally, but scaled evenly along a path that I lay out.  What is the name of this text effect?  Can anyone point me in the right direction to a tutorial or know how to accomplish this themselves?  I know that I could go through and do this letter by letter, but I was really hoping to find an easier method.  EDIT I have removed many elements of my design and quickly threw some sample text in it, but this should give you an idea of what I am trying to accomplish.  I want a mixture of the above and below effects. Accessing AWS S3 via a REST API (Apex callout with a Named Credential) Faster way of writing \mathbf An inequality about Stirling numbers For all positive integers $n,r,k$ that $n=rk$, how to prove the inequality $$(r-1)!^k\le \frac{c(n,k)}{S(n,k)}\le (n-k)!$$ $c(n,k)$ is unsigned Stirling numbers of the first kind and $S(n,k)$ is Stirling numbers of the second kind.  Any help would be appreciated! I’m struggling with getting \ifmmode to work correctly when used inside an array environment which is inside a displaymath environment.  Here is a minimal example:  \documentclass{article}  \newcommand{\signal}[1]{%   \ifmmode     1#1   \else     2$#1$   \fi }  \begin{document} \begin{displaymath}   \begin{array}{cc}     \signal{a} \signal{b} &amp; \signal{c} \signal{d} \\   \end{array} \end{displaymath} \end{document}   This example generates the following output:        Why doesn’t all invokations of \signal detect math-mode? Can we omit a duplicated preposition? Is on needed a second time in this sentence?        There is a bad effect either on human health or environment.    There is a bad effect either on human health or on environment. How to set the font for a \section title (and chapter etc) How do I specify the fonts, and general font size, for the titles, chapters and sections in my document  Update:  I can see how my question is confusing.  I use \section and \chapter etc. to organise my document, and I would like to change the way the section and chapter titles are displayed. cat files with directory Is there a command to show the directory/file name when cat files?  For example: assume two files f1.txt and f2.txt are under ./tmp  ./tmp/f1.txt ./tmp/f2.txt    Then when I do cat ./tmp/*.txt, only the content of files will be shown. But how to firstly show the file name, then the content?, e.g.:   (The needed command):  ./tmp/f1.txt:    This is from f1.txt  and so on  ./tmp/f1.txt:  This is from f2.txt ...   Is there a command to do it? (There seems to be no option for 'cat' to show the file names) How closely can a game legally resemble another? Most games build on other games' successes and many are downright clones. Where is the limit of imitation before legal issues come into play? Is it down to literary or graphic work like characters and storyline that cause legal problems, or can someone actually claim to own gameplay mechanics?  There are so many similar clone games out there that the rules are probably very slack or nonexistent, but I'd like to hear the views of more experienced developers / designers. Can a well-timed Twisted Image kill a player using Tree of Redemption? What happens if my opponent activates  and I cast  with the life-exchange ability on the stack?  Looking at the comp rules, these seem most relevant:     701.8a A spell or ability may instruct players to exchange something (for example, life totals or    control of two permanents) as part of   its resolution. When such a spell or ability resolves, if the   entire exchange can’t be completed, no part of the exchange occurs.      Example: If a spell attempts to exchange control of two target creatures but one of those              creatures is destroyed before the spell resolves, the spell does nothing to the other              creature.      112.7a Once activated or triggered, an ability exists on the stack independently of its source.    Destruction or removal of the source   after that time won’t affect the ability. Note that some    abilities   cause a source to do something (for example, “Prodigal Sorcerer deals   1 damage to    target creature or player”) rather than the ability   doing anything directly. In these cases, any    activated or triggered   ability that references information about the source because the   effect    needs to be divided checks that information when the ability   is put onto the stack. Otherwise, it    will check that information   when it resolves. In both instances, if the source is no longer in the   zone it’s expected to be in at that time, its last known information   is used. The source can still    perform the action even though it no   longer exists.   Is the entire exchange ability countered because the Tree is dead by the time it resolves? Or do we use the "last known state" (toughness 0), thereby killing its controller? How do I enable the "Universe" repository from the command line? How do I go about proving this?  If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs. $(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse? Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&amp;$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&amp;(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ as addition and multiplication, respectively. Do these operations form a field?  I have shown that every axiom of the field is satisfied except for the "existence of multiplicative inverses" axiom. This is where I need help. We either need to show that this axiom fails and so we don't have a field or that the axiom $aa^{-1}=1$ is satisfied for every $a\neq 0 \in \mathbb{R}$  The multiplicative identity in this problem is $(1,0)$.  Any solutions or hints are greatly appreciated. How can I attach a file to a question? I want to attach a code file to my question, but I am not able to find a link to do it. Is attaching files possible at all?