How to find out how Mathematica performed an integration?
Get a "step-by-step" evaluation in Mathematica
Can I safely charge my laptop with a non-standard, third-party charger? I have a Toshiba Satellite laptop. My charger has stopped working.  I have access to a Lenovo charger. Can I use this charger on my laptop? $Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$. Question is to prove that :     $Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$.   I see that $Q_8$ is isomorphic to subgroup of $S_8$ by left multiplication action.  Hint given was to prove that stabilizer of any point contains $\{\pm 1\}$.  To prove Cayley's theorem, stating any group is isomorphic to a subgroup of $S_n$ we take action of given group on a set $A$ having same cardinality.  with that motivation I want to check if there is a Isomorphism then there is a map from $G\times A \rightarrow A$. i.e., $G$ gives a permutation group $S_A$.  I tried in same manner. Suppose $Q_8$ is isomorphic to subgroup of $S_n$ with $n\leq 7$  then it should come from a group action of $Q_8$ on a set of cardinality atmost 7.   Suppose $Q_8$ acts on a set $A$ with possible cardinality at most 7.  $stab(a)=\{g\in Q_8 : g.a=g \forall a\in A \}$  $cl(a)=\{g.a : g\in Q_8\}$  I know number of elements in class of $a$ equals to index of stabilizer.  As $cl(a)=\{g.a : g\in Q_8\}\in A$ i.e., $cl(a)\subseteq A$ and as $|A|\leq 7$ i see that $|cl(a)|\leq 7$.  But, $|cl(a)|=|Q_8:stab(a)|$ for any element $a\in A$.  So, $|Q_8:stab(a)|=|cl(a)|\leq 7$ for all $a\in A$.   So, $stab(a)$ should be non trivial subgroup of $Q_8$ if not then $|Q_8:stab(a)|=8$  non trivial subgroup (proper) of $Q_8$ contains $\{\pm1\}$.  So, In the worst case, $\{\pm 1\}\subseteq stab(a)$ for all $a\in A$.  As $Ker(\eta)=\cap_{a\in A}stab(a)$ (where $\eta$ is the action of $Q_8$ on $A$)   we see that $\{\pm 1\}\subseteq Ket(\eta)$ which means that $Ker(\eta)$ is non trivial.  thus there is no isomorphism (coming from $\eta$) between $Q_8$ and any subgroup of $S_7$.  I would be thankful if someone can check whether my approach is correct or if there is any other simple possible way.  P.S : Usually what i do to see whether two groups are isomorphic or not is to check for cardinality, abelian property, no of elements with same order and so on. But I was having no idea when i fail in all these ways. With this Group actions i could see possibility for getting a precise conclusion on Isomorphisms.I would like to Thank Mr. Jyrki Lahtonen (a user of Math.SE) who made me to get used to Group actions.  P.S $2$: If any thing is wrong in my idea, it is entirely my fault, and if anything is correct in this whole credit should go to Mr. Jyrki Lahtonen I have a script that I need to execute on an NTFS partition. The script's permission is set to 600.  I attempted to modify the permissions by running chmod 755 script.sh, which doesn't report a failure or anything - but it also doesn't change the permissions on the file:  $ stat script.sh    File: `script.sh'   Size: 297070      Blocks: 584        IO Block: 4096   regular file Device: 811h/2065d  Inode: 35515       Links: 1 Access: (0600/-rw-------)  Uid: ( 1000/  xxxxxx)   Gid: ( 1000/  xxxxxx) Access: 2010-09-30 14:05:16.041621000 -0700 Modify: 2010-09-30 14:05:05.070157000 -0700 Change: 2010-09-30 14:05:05.070475000 -0700  $ chmod 755 script.sh $ stat script.sh    File: `script.sh'   Size: 297070      Blocks: 584        IO Block: 4096   regular file Device: 811h/2065d  Inode: 35515       Links: 1 Access: (0600/-rw-------)  Uid: ( 1000/  xxxxxx)   Gid: ( 1000/  xxxxxx) Access: 2010-09-30 14:05:16.041621000 -0700 Modify: 2010-09-30 14:05:05.070157000 -0700 Change: 2010-09-30 14:05:05.070475000 -0700   As you can see, it remains unchanged. Is there a program to mine bitcoins in ubuntu 13.10? How to get own process pid from the command prompt in Windows I'm trying to find a way to get my own PID from a command prompt (for later use in bat scripts).  So far the only useful way I found was to use getpids.exe from here : , but I'm looking for a command that's "built in" to Windows.  I'm looking for a "bullet proof" way. No assumptions about my process being the only cmd.exe or anything. Pros and Cons of a Decentralized Puppet Architecture We have around 300 RHEL servers that are currently connecting to a Puppetmaster server.  However, we have noticed some performance bottlenecks and it is the point of failure in our system.  I am fairly new to puppet in general and I am considering creating a decentralized puppet architecture instead of having Puppet clients connect to the Puppetmaster server.  Aside from what I would suspect to happen such as performance gain and lack of signing and exchanging SSL certs for new machines, what are other pros and cons to setting up a decentralized architecture? I am trying to write the Euclidean algorithm in the following way:  $A = \lfloor A \div B \rfloor \times B + (\text{remainder of}) \: A \div B $   Now is there any symbol I can use  to say "remainder of A $\div$ B"? I know that in the C programming language there is the operator % for modulus; is that a valid symbol in maths? Can I write A % B? Or is there some other way? How do I get a Realtek RTL8723BE wireless card to work? Book about classical mechanics I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.).  I have , but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics.  Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing. Suppose that $f$ is analytic on a close curve γ. Prove or disprove $\int_\gamma \overline{f(z)}f'(z)dz$ is purely imagine. Suppose that $f$ is analytic on a close curve γ. Prove or disprove   $$\int_\gamma \overline{f(z)}f'(z)dz$$  is purely imagine.  I know that $f$ is analytic on a close curve, then   $$\int_\gamma f(z) dz=0$$  I tried an example with $\gamma =e^{it}$ with $0\leq t\leq 2\pi$, I often get the real part of the integral equal zero.  I tried let $f=u+iv$ so $f'=u_x+iv_x$. Since $f=u+iv$, $\overline{f}=u-iv$  $$\overline{f(z)}f'(z)=uu_x+vv_x+i(uv_x-vu_x)$$  But this doesn't get me anywhere. Any help would be greatly appreciated. How to install Google Chrome I have a little problem with the placement of my picture. I want to insert it at the end of the introduction chapter but its place is not where I want.   I think the problem become from the:  \begin{figure} ... \end{figure}   Because when I insert the picture without using \begin{figure}... the picture is at the end of the chapter as I want.   Look the picture below. Quite often, the phrase "x for x's sake" is used in English, and so one could describe someone as being "argumentative for argument's sake" to describe someone who is arguing for the sake of arguing.  However, is there an adjective that means the same thing?  For example, it could be used in the context:     I don't want to be [X], but [argument...]   ... indicating that your argument is necessary and not intended to irritate. I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$   $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ holds.      I think the first step should look something like this:      $d = \gcd(a+b, a-b) = \gcd(2a, a-b)$   From here I tried to proceed with two cases. 1: $a-b$ is even, which leads to $\gcd(a+b, a-b) = 2$ 2: $a-b$ is odd, which leads to $\gcd(a+b, a-b) = 1$  My main problem I think is, that I do not know how I should include $\gcd(a,b) = 1$ in the proof.  Any help is appreciated. Thx in advance. Cherio Woltan Where are the necessary places to be appended with % to remove unwanted spaces? I have  experience where empty spaces cause unwanted effects. It is not easy to trace the cause of these unwanted effects. In order to eliminate any doubt, I often overuse % as follows.   \usepackage% [% left=3cm, right=3cm% ]% {geometry}   or  \newcommand{\mycommand}% {% This is my command.% }   I got an extreme example that will break what Leo Liu said     In fact, only spaces which would be   output have to be removed by comment.   The following code does not produce output, but we cannot remove the % and leave a blank line between two elements of a list below.  \newpsstyle{gridstyle} { gridwidth=0.4pt, % griddots=0 }   Shortly speaking, where are the necessary places to be appended with % to remove unwanted spaces? How unique is the php session id? I got the impression from various things that I've read that I should not rely on two users never getting the same sessionid. Isn't it a GUID? My problem: Prove that a set of vectors $S$ is linearly independent if and only if any finite subset of $S$ is linearly independent.  I tried like this:  Suppose S is LI.Then the vector $0$ cannot be expressed as a linear sum of all elements of $S$.  How it follows that a finite subset is also LI from this fact. I think $S$ can be finite or infinite.  This is a question from the book Linear Algebra - Friedberg et al. Usually, when we see two letters put together it is to define the pronunciation or to differentiate synonyms or just its foreign origins. But some words don't seem to have any reason to double up.     Why does vacuum have two of the letter U?  Why does aardvark have two of the letter A?  Why does llama have two of the letter L? What is the best solution for database connection pooling in python? I have developed some custom DAO-like classes to meet some very specialized requirements for my project that is a server-side process that does not run inside any kind of framework.    The solution works great except that every time a new request is made, I open a new connection via MySQLdb.connect.    What is the best "drop in" solution to switch this over to using connection pooling in python?  I am imagining something like the commons DBCP solution for Java.    The process is long running and has many threads that need to make requests, but not all at the same time... specifically they do quite a lot of work before brief bursts of writing out a chunk of their results.  Edited to add: After some more searching I found  which looks decent, but as I'm relatively new to python I guess I just want to make sure I'm not missing a more obvious/more idiomatic/better solution. If $n = p_1^{a_1} ... p_r^{a_r}$ the set of nilpotents of $\mathbb{Z}_n$ is $(p_1 p_2 ... p_{r-1} p_r)\Bbb Z_n$