Let $d=(m,n)$. Prove: $$φ (mn) = φ (m) φ (n)\frac{d}{φ (d)}$$ Can anyone help me with this proof?
The third formula on the  states that $$\varphi (mn) = \varphi (m) \varphi (n) \cdot \dfrac{d}{\varphi (d)} $$ where $d = \gcd(m,n)$.  How is this claim justified?  Would we have to use the Chinese Remainder Theorem, as they suggest for proving that $\varphi$ is multiplicative?
I have an object at a fixed position in 3D space. I would like the camera to circle around it while facing it. I want to path of the camera to be circular in 2D, meaning that it does not move up or down.  I need this to be pretty accurate. I don't want to manually set a keyframe every frame for position and rotation. I would like to do this maybe with an expression, not with manual keyframing. I am not sure how to go about this but I think it is going to involve Python as well as some trigonometry. Lost passport with Schengen visa My passport was stolen in Prague on my way to Vienna. I filed a police report and our embassy swiftly acted by giving me a temporary passport for my onward journey to Munich and Rome. Upon exit from Rome, the immigration ask for the Schengen visa, when I told him that I lost my passport, he did not ask for any document to support my claim, he just stamped my new passport with exit. I am not sure if on record, I am officially out of the Schengen area or not. I am scared that without the lost visa, it appears that I am still in the area and might reach the 90 day limit and will be denied when I re-apply as it would appear I broke the rule. What to do? Bake Simple Deform animation into mesh for Unity3D export? I'm trying to use a .blend file with a keyframe animation on the angle of a Simple Deform modifier in Unity3D. Unity3D however does not know how to handle modifiers and hence cannot play the animation.  How can I bake/save the animation to the .blend file so that Unity3D can play it? Whenever I create a document in Microsoft Word, it complains about a lot of my sentences being in passive voice. But, when I read that sentence aloud, it sounds fine to me. I am not sure if it is just me and will a statement in passive voice sound strange to a native speaker?  So, my question is, is it considered bad form to use passive voice generally? Or in some specific cases like written communications only?  Edit: If it is ok to use passive voice, then why does MS-Word complain? Find html element which id starts with Structuring game narrative. Handling NPC, Quest, General Dialogue. Is Json/XML the right way to go? Or any other solutions? Hello guys I'm try to make some games that have more to it than just simple high score scrolling shooters etc.  I'm trying to design my own systm of handling narrative. I want to be able to place NPC's in the game world, with scripts attached to them. Some of the NPC's would just have n random greetings. Others will be mission givers and will offer you a quest of sorts. (The quest could be go speak to Bob in the pub, or it might be bring me the Holy Grail and Crown Jewels, or something in between).  I've wracked my brain for a long while over this but sadly keep slipping back to the only way I know how, which is to manually code all the strings that people can say and have a horrible web of if statements and other such insanity to give me a rudimental dialogue system. Its not good enough even with just a few characters, but if I had dozens/hundreds of characters it would be even worse.  I've read lots about it already but I'm still very confused when I start trying on my own project. I've read that Json/XML might be the way to go and that I can add custom markups in the Json text to call methods in my code, but I dont even know if this is right and if so how I can do it.  The back of my brain is screaming at me that there is a simpler way to do it, but I've sat here for a very long time without managing to think of ideas.  Here is a diagram I drew which pretty much shows the flow of how I want my NPC to behave. If anyone can see this and then tell me how they might do something like this that would be fantastic!! (note: if Json is the correct advise, please give me a exact example of how exactly to format the Json in order to check the bools I mention in my flowchart - thanks!):    note: The circles denote an output from the data as Text for the dialogue system. The bools will need to be checked against by the data file to see which output to give (hope that makes sense and is possible) In Harry Potter, Do Spells Travel in a Straight Line? In the Harry Potter world, when a magical person casts a spell, sometimes they target a specific object, like when Harry has a wand only and needs his broom, he casts "Accio Firebolt" and his broom comes to him.  But a lot of the time people point their wands just in the direction of someone to cast a spell.  One example would be in a duel.  When they do this, are they casting a spell on a particular person or at that person?  It seems, from what I remember of the movies, that some spells travel in a straight line while others specifically target a person or object.  Do any spells travel in one direction, like a laser beam?  What kind of spells are directional (instead of focusing on a particular person or object)?  And can directional spells be deflected or blocked by something like a mirror or another physical object?  (By "directional," I could have said linear, but didn't want to start sounding mathematical or scientific.  In other words, do they travel in one direction, like a laser beam would, or will they change direction to hit the intended target?) On the usage of "the" article with acronyms and initialisms I've read  and  I know that  1. The article is often used with initials that are pronounced letter by letter (initialisms).  2. The article is often NOT used with initials that are pronounced as a word (acronyms).  I used 'often' because I'm not sure about possible exceptions.  In my field of study (Operations Research), initials are frequently used to represent optimization problems (e.g., the TSP). I discussed this point with my university supervisor who makes comments on my reports (English is not his native language). He responded that:  "What I learned from [...] was that whenever we use (specific) acronyms, we don't use 'the' when they are within a sentence, and we use 'the' when a sentence starts with them. For instance,  (i) The HCP is an NP-Complete problem.  (ii) We developed a new algorithm to solve HCP."  HCP stands for the Hamiltonian Cycle Problem. It is really confusing me because I do not know when to use and when not to use the article. For example, on , the author Jane Watson writes:     Use a definite article with an initialism if the spelled out term begins with “the” but is not covered in the initialism.   which implies that I should use the article with "HCP". What do you think? DEF file syntax error in visual studio 2012 Why does Beamer not change the itemize symbol when using \setbeamertemplate? This code:  \documentclass[10pt]{beamer} \setbeamertemplate{itemize item}[circle] \setbeamertemplate{itemize subitem}[triangle] \setbeamertemplate{itemize subsubitem}[square]   \begin{document} \begin{frame} \begin{itemize} \item first level     \begin{itemize}     \item second level         \begin{itemize}         \item third level         \end{itemize}     \end{itemize} \end{itemize} \end{frame} \end{document}    produces this output:    I expected the symbol of the first level item to be a circle, not a square. Is there an error in my code? The log file was too long to include in the question, so I . Piping for loop output prevents local variable modification I'm trying to write a simple bash function that takes, as its arguments, a number of files and/or directories. It should:   Fully qualify the filenames. Sort them. Remove duplicates. Print all that actually exist. Return the number of non-existent files.   I have a script that almost does what I want, but falls down on the sorting. The return value of the script as it stands is correct, but the output is not (unsorted and duplicates). If I uncomment the | sort -u statement as indicated, the output is correct but the return value is always 0.  N.B. Simpler solutions to solve the problem are welcome but the question is really about why is this occurring in the code I have. That is, why does adding the pipe seemingly stop the script incrementing the variable r?  Here's the script:  function uniqfile {     local r=0       for arg in "$@"     do           readlink -e "$arg" || (( ++r ))      done #| sort -u    ## remove that comment      return $r } Which close reason should I use for questions that do not demonstrate research effort? I regularily encounter questions on StackOverflow which are asked only because of the sole fact that the poster haven't made the effort to google for and the docs of a particular function/class/framework/whatever. My question is, to which category of the five proposed one should I vote when I vote to close the question? Is it: off-topic, not constructive, not a real question, exact dupe or too localized? I tend to be all for 'not a real question' - am I quite right? IntelliSense not working but it is enabled This problem is really proving to be a tricky one (and quite annoying).  In SQL Server Management Studio 2008, up until a few days ago my IntelliSense was working great.  Then all of a sudden it stopped.  The icon as it as enabled on the toolbar menu, and under Tools -> Options -> Text Editor -> T-SQL -> IntelliSense it says it is enabled there.  I have tried refeshing the IntelliSense cache with Ctrl-Shft-R but that doesn't work either.  Any ideas what happened to my IntelliSense and what I need to do to get it back? Why don't I have this option?    here's my disk layout  sudo fdisk -l  Disk /dev/sda: 640.1 GB, 640135028736 bytes 255 heads, 63 sectors/track, 77825 cylinders, total 1250263728 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x5b53cc54     Device Boot      Start         End      Blocks   Id  System /dev/sda1   *        2048      409599      203776    7  HPFS/NTFS/exFAT /dev/sda2          409600  1153767021   576678711    7  HPFS/NTFS/exFAT /dev/sda3      1216962560  1250050047    16543744    7  HPFS/NTFS/exFAT /dev/sda4      1250050048  1250261679      105816    c  W95 FAT32 (LBA)  Disk /dev/sdb: 4005 MB, 4005527552 bytes 32 heads, 63 sectors/track, 3880 cylinders, total 7823296 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x20d8782d     Device Boot      Start         End      Blocks   Id  System /dev/sdb1   *          63     7822079     3911008+   c  W95 FAT32 (LBA) Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$ I'm having issues figuring out how to approach this problem:     Conclude that if $ad-bc = \pm 1$, then   $$\gcd(x,y) = \gcd(ax +by, cx + dy)$$   The fact that $\gcd(x,y) = \gcd(x+ky, y)$ is a very special case of this exercise   I believe there is a property that if $a = bq_1 + 0$ where 0 is r, then b divides a, and $b=\gcd(a,b)$  Is that at all relevant in this case? Eigen Values Proof a.) Let A and B be $n$ x $n$ matrices. Prove that the matrix products   $AB$  and $BA$ have the same eigenvalues.      b.) Prove that every eigenvalue of a matrix A is also an eigenvalue of   its transpose $A^T$. Also, prove that if v is an eigenvector of A with   eigenvalue $\lambda$ and w is an eigenvector of $A^T$ with a different   eigenvalue $\mu \ne \lambda$, then v and w are orthogonal vectors with   respect to the dot product.   For a, I know that if their eigenvalues are the same then their eigenvectors must relate too.    For the first part of b, is it similar to proving that the $det(A) =det (A^T)$? And, I do not know how to do the second part. algebra - matrices and polynoms $T:\mathbb P_2[x]↦\mathbb P_2[x]$ s.t.:   $T(1)=3+2x+4x^2$, $T(x)=2+2x^2$, $T(x^2)=4+2x+3x^2$.   Is there base $B$ of $\mathbb P_2[x]$ that $[T]_B$= \begin{bmatrix}     2 &amp; 0 &amp; 0\\     -1 &amp; 1 &amp; 0\\     5 &amp; 0 &amp; 4\\  \end{bmatrix}  Please help. Thanks. Why does water pouring from a glass sometimes travel down the side of the glass? If you have a glass of water, say, three quarters full and you pour it at an angle of say, $45^{\circ}$ with respect to the the table, the water comes out of the glass and goes directly down towards the floor.  However, when the glass is more full, or even three quarters full and the 'angle of pouring' is far less with respect the table, when the water comes out of the glass, rather than going straight down it kind of stays stuck to the glass and travels down the outside of it.  Why does this happen? (I'm sure many of you have observed it yourself by accident and ended up making a mess). Hiding axis text in matplotlib plots Why do we use Root Mean Square (RMS) values when talking about AC voltage What makes it a good idea to use RMS rather than peak values of current and voltage when we talk about or compute with AC signals.