If you cast a spell with different effect options twice on the same creature, do multiple castings of the spell stack?
If you cast Blindness/Deafness as a 3rd level spell, can you target one creature with both effects?
What is the difference between local and global sequence alignments? There are a bunch of , and I don't want to get bogged down in the maths behind them as this not only between software but varies from software version to version.  There are two main divides in the programs; some use local alignments and others use global alignments. My question is threefold:   What are the fundamental differences between the two? What are the advantages and disadvantages of each? When should one use either a global or local sequence alignment? How do we prove that  $\operatorname{rank}(A) = \operatorname{rank}(AA^T) = \operatorname{rank}(A^TA)$ ?  Is it always true? Using a naked black hole as a mirror? The YouTube video is a good example how the gravity of this merging binary black holes bend light around themselves. Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity (for AB and BA not square)? I know that if A and B are square nxn matrices, then AB and BA have the same characteristic polynomial and thus the same eigenvalues (and same algebraïc multiplicity).   I'm wondering though if this can be generalized: if A is a nxm matrix and B a mxn matrix, then AB is a nxn matrix and BA a mxm matrix.  So my question is: will the eigenvalues of AB and BA, that differ from zero, have the same algebraïc multiplicity? I'm using OSX 10.9. I'm trying to set up Internet Sharing to connect my Android to my Mac directly over Wi-Fi. Every tutorial I've come across (including ) mentions checking a box for "AirPort" in a "To computers using" list. However, the option is simply not available on my machine:    I have a few related questions:   Why isn't the AirPort option available? Is there anything I can do to make it available? If there isn't, how can I work around it to finish setting up Internet Sharing so I can connect my Android and my Mac directly over Wi-Fi? Convergence of cos, sin, tan functions In Radian mode, continually pressing the $\cos$ function of a number causes the result to converge to $x=0.739085133$, a fixed point of $\cos(x)$. Repeating this behavior with the $\sin$ function causes the result to converge to $x=0$, a fixed point of $\sin(x)$.   What happens if this is done using the tan function? It seems that $x=0$ is a repelling fixed point with no convergence in this case.  What would be the best way to explain these results mathematically? Proof that the Euler characteristic is additive I'm reading through a set of notes which assumes that the Euler characteristic is additive, but doesn't give a proof, so I would like to understand why this is.  Let $A_n$ be a finitely generated Abelian group, non-zero for $0&lt;n&lt;k$. Consider the long exact sequence  $ \cdots \rightarrow A_n \rightarrow B_n \rightarrow C_n \rightarrow A_{n-1} \rightarrow B_{n-1} \rightarrow C_{n-1} \rightarrow \cdots $  where $B_n$, $C_n$ defined similarly to $A_n$.   I know that the Euler characteristic of $A$ is given by $\chi(A) = \sum_0^k (-1)^n \text{rank}(A_n)$, with similar definitions for $B$ and $C$. I want to prove that $\chi(B) = \chi(A) + \chi(C)$.  The notes say that for an exact sequence  $0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0$  of finitely generated Abelian groups $X,Y,Z$, we have the result rank($Y$) $=$ rank($X$) +  rank($Z$), and that the additivity of the Euler characteristic follows from this fact, but I can't see how. Need help identifying a comic book about a wizard I need help with identifying the comicbook, what follows is my best effort to gather whatever I remember from reading it years ago (circa year 2000, but I'm positive it was older than that):   The main character was a guy with white hair, who wanted to become powerful.  Through some training and a deal of some kind he was able to become a mage of sorts. The training was set on some other world, or possibly even parts were happening in different dimension. His powers might have had something with travelling between dimensions, but I'm not  100% sure. Also, I seem to recall there was a hard limit of how many times he could use them. Then followed a part where he was famous, but someone framed him into a scandalous plot (it was a sexual scandal, I think) using his clone or a doppelganger. At the end of the issue     the main character is seen trapped in some strange state, unable to move or escape using his powers (this might have to do something with the deal he made earlier). I remember the scene at the end as this: he is suspended in mid-air, his friends look at him in disbelief while some malevolent figure explains to them that he is now trapped forever.  His name was the title of the comic. The second and third parts were set on futuristic-looking planet.   Rings any bells? I'd like to read it again only I can't google it out. Resetting crontab file to default What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? Testing pointers for validity (C/C++) Is there any way to determine (programatically, of course) if a given pointer is "valid"? Checking for NULL is easy, but what about things like 0x00001234? When trying to dereference this kind of pointer an exception/crash occurs.  A cross-platform method is preferred, but platform-specific (for Windows and Linux) is also ok.  Update for clarification: The problem is not with stale/freed/uninitialized pointers; instead, I'm implementing an API that takes pointers from the caller (like a pointer to a string, a file handle, etc.). The caller can send (in purpose or by mistake) an invalid value as the pointer. How do I prevent a crash? Have you used any of the C++ interpreters (not compilers)? I am curious if anyone have used UnderC, Cint, Cling, Ch, or any other C++ interpreter and could share their experience. Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$   $\bf{My\; Try::}$ I have solved Using Direct formula::  $$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}$$  Now Put $n=7\;,$ We get   $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}\cdot \sin \frac{4\pi}{7}\cdot \sin \frac{5\pi}{7}\cdot \sin \frac{6\pi}{7}=\frac{7}{2^{7-1}}$$  So $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} =\frac{\sqrt{7}}{8}$$  Now my question is how can we solve it without using Direct Formula, Help me  Thanks Get steam to realize I have games already installed I uninstalled and re-installed steam. I didn't uninstall my games, and they still exist on my hard drive (I checked).  I have an SSD and a HDD, and to save space I install programs (including steam) to C:\Program Files (x86)\Steam drive (SSD) and games to D:\Games\SteamApps (HDD). How to I let steam know the programs are installed there? When I take the pth root, the bottom of the p comes very close to the \sqrt symbol.  In the New Century Schoolbook font (which I am using), the p actually intersects the \sqrt symbol, which looks even more terrible.  Is there a clean way to fix this?  \documentclass{article} \usepackage{fouriernc} % use the New Century Schoolbook font \begin{document} When I write $a^{(1/p)}$ as $\sqrt[p]{a}$, the bottom of the $p$ touches the top of the root symbol, which looks ugly. Even when the root symbol is larger, such as with $\sqrt[p]{\frac{1}{n}}$, it still looks bad (since the tail of the p almost hits that line). \end{document} Warning or confirmation on new tag creation In retrospect of this , I would like to ask the community how they feel about a warning or confirmation (or just special highlighting) on new tag creation.  Everybody hates stock "are you sure?" confirmation, but a lot of tags get created by mistake (and then used ad nauseam).   For example, I think I might have created the  tag by accident. Assuming the right tag was , there would be two tags for c# 4.0, until they are made synonyms. With a warning, some accidental tags could be prevented. Nearby objects won't move independently? In my cover letter I want to write that I am a graduate student in statistics. But I am not sure about the "in". So what is correct?     I am a graduate student in statistics.      I am a graduate student of statistics. Variance of the distribution of difference scores help  displays information about builtin commands. What is the source of help? Does it maintain a database for builtin commands, or does it read some files of each builtin commands (similar to the manpage of each utilities)?  Sometimes I find its information seems to expand that by --help  $ cd --help bash: cd: --: invalid option cd: usage: cd [-L|[-P [-e]]] [dir]  $ help cd cd: cd [-L|[-P [-e]]] [dir]     Change the shell working directory.      Change the current directory to DIR.  The default DIR is the value of the     HOME shell variable.      The variable CDPATH defines the search path for the directory containing     DIR.  Alternative directory names in CDPATH are separated by a colon (:).     A null directory name is the same as the current directory.  If DIR begins     with a slash (/), then CDPATH is not used.      If the directory is not found, and the shell option `cdable_vars' is set,     the word is assumed to be  a variable name.  If that variable has a value,     its value is used for DIR.      Options:         -L  force symbolic links to be followed         -P  use the physical directory structure without following symbolic         links         -e  if the -P option is supplied, and the current working directory         cannot be determined successfully, exit with a non-zero status      The default is to follow symbolic links, as if `-L' were specified.      Exit Status:     Returns 0 if the directory is changed, and if $PWD is set successfully when     -P is used; non-zero otherwise.   I thought help extract the help part of the executable, but for a  python script , it isn't  $ help ./pdf-merge.py bash: help: no help topics match `./pdf-merge.py'.  Try `help help' or `man -k ./pdf-merge.py' or `info ./pdf-merge.py'.  $ ./pdf-merge.py --help usage: pdf-merge.py [-h] [-v] [--ask] [--output OUTPUT] [--title TITLE]                     [--author AUTHOR] [--keyword KEYWORD] [--pdftk PDFTK]                     [--gs GS] [--pdfmarks PDFMARKS] [--unicode]                     PDF [PDF ...]  Merge PDFs preserving bookmarks. Thanks to Larry Cai for suggesting that Unicode be supported and for discussion about the `--pdfmarks` option.  positional arguments:   PDF                  an input PDF to merge  optional arguments:   -h, --help           show this help message and exit   -v, --version        show program's version number and exit   --ask                pause for manual pdfmark tweaking   --output OUTPUT      name of the output PDF   --title TITLE        title of output PDF   --author AUTHOR      author of output PDF   --keyword KEYWORD    keywords for the output PDF   --pdftk PDFTK        path to the pdftk executable   --gs GS              path to the gs (Ghostscript) executable   --pdfmarks PDFMARKS  path to pdfmarks file. If not given, a temporary file                        is used. If given and the file is missing, execution                        will stop after the file is created (before the                        Ghostscript run). If given and the file exists, no                        attempt will be make to use pdftk to generate the mark                        file (I assume your input file is what you want).   --unicode            instead of merging PDFs, convert PDF-formatted unicode                        strings. For example `--unicode '&lt;FEFF03B103B203B3&gt;'                        \u03b1\u03b2\u03b3`