count the numer of partial function between two sets with bijection
the numer of partial function between two sets in combinatoric way
Why isn't $\mathbb Z [\sqrt{pq}]$ a factorial domain This is exercise 8 on page 147 of Jacobson's Basic Algebra:     Let $p$ be a prime of the form $4n+1$ and let $q$ be a prime such that $ \left( \frac{q}{p} \right) =-1$. Show that $\mathbb Z[\sqrt{pq}]$ is not factorial.   Here, $ \left( \frac{q}{p} \right) =-1$ means that $\overline{q}$ is not a quadratic number in the field $\mathbb Z / (p)$.  In the special case when $p=5$ and $q=2$, $9 = 3 \cdot 3 = (\sqrt{10} +1 )(\sqrt{10 }-1)$, with $3$ and $\sqrt{10} +1$ or $\sqrt{10} - 1$ nonassociates. But I have no idea what can I do with the general case.  I know that primes of the form $4n+1$ can be written in the form $m^2 + n^2$ with $m,n \in \mathbb Z$. But when $q$ is odd and not of the form $4n+1$, $pq$ is not the sum of two quadratic numbers. How can I use the fact $\left( \frac{q}{p} \right) =-1$?  Thanks~ Electrostatic charge leakage What are the ways electrostatic charged objects leak charge in humid conditions?  Can airborne particles pick up charge by contact, then be repelled hence removing charge? If so would it be a significant factor? Configure long table caption Is there any program to provide a consistent interface across multiple archive types? At the moment, if I download a compressed file, it could be any of a .tar.gz archive, a tar.bz2 arhive, a .zip archive or a .gz archive. And each time I do so, I have to remember what the command line options for that program are.  Is there any CLI program where I can just go:  undocompression somefile.??  and let it figure out what format the archive is in? (overly long name used to avoid conflicting with any real program) I'm using the  package to add citations and a bibliography to my LaTeX document. I've noticed that only the first author plus "et al." is displayed for works with more than three authors. That's fine with me for in-text-citations, but I'd rather have the complete author information in the bibliography. How can I do that? Layers with same coordinates not lining up Can there be no 'a' between such and a noun? When it comes to losing weight, a quick fix to would be nice, but we   know there’s no such [a] thing. ()   It sounds like there isn’t ‘a’ in the news. So I wonder if it be right when there is no ‘a’ after ‘such.’ Would it be okay? Reference to undeclared entity exception while working with XML I am trying to set the innerxml of a xmldoc but get the exception: Reference to undeclared entity  XmlDocument xmldoc = new XmlDocument(); string text = "Hello, I am text &amp;alpha; &amp;nbsp; &amp;ndash; &amp;mdash;" xmldoc.InnerXml = "&lt;p&gt;" + text + "&lt;/p&gt;";   This throws the exception:      Reference to undeclared entity 'alpha'. Line 2, position 2..    How would I go about solving this problem? Say if I define a power series over some arbitrary field $F$ as   $$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$  Then can I say:  $$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i + j} $$ How can I change the umask for one command only? How can I interactively execute a command in Linux (zsh, if it matters) with a different umask from the default, for one command only? Perhaps a combination of commands combined in a single line? The new umask should apply only to that command and return to its default value for the next command entered.  Ideally the command would be agnostic to the default umask in force before its entered (in other words, the default umask doesn't have to specified). I know how much space is required to install Ubuntu.  How much space does a Ubuntu installation take up on the disk? Reapplying for a UK Standard Visitor visa after a refusal Should I use a comma in this sentence? I am trying to teach myself punctuation rules. I have this sentence:     The Disney version generally considered both racist and sexist portrayed the female characters in stereotypical roles and presented ethnically diverse characters with stereotypical features.   I think the sentence needs a comma after sexist and before portrayed. I also think it needs a comma after roles because of what I have identified as a coordinating conjunction. I.e.:     The Disney version generally considered both racist and sexist, portrayed the female characters in stereotypical roles, and presented ethnically diverse characters with stereotypical features.   Did I punctuate the sentence correctly? Is this a coordinating conjunction? What's the difference between these two sentences? And which one's grammatically correct? Unity: How do I revert Alt-tab behavior to switch between windows on the current workspace? Enabling Bias alt-tab sorting to prefer windows on the current viewport as described  made no difference. How can I get back the exact same behavior as 11.04, so that alt-tab only switches between windows on the current workspace?  Simply disabling the alt-tab and shift-alt-tab keybindings on the unity switcher seems to have helped, but it still switches workspaces on me sometimes.  For example, if I give a terminal window focus then press alt-tab, it switches to another terminal window on any workspace before trying to switch on the same workspace. Also, the Unity switcher still shows up when I alt-tab then hold alt even though I removed its alt-tab keybinding. The error: "mount: only root can do that" while mounting a partition I've installed Ubuntu 12.04 along with Windows 8.1. When I tried to open the C: drive in Ubuntu a window is popping out saying to run this command:  mount -t ntfs-3g -o remove_hiberfile /dev/sda2 /media/2A36EE3236EDFEA7   After running the command I'm getting the error mount: only root can do that.   What should I do? Modular arithmetic: division, fractions, solving linear congruences What do you call the feeling you get when you're satisfied to the point of nausea? So you're craving something, say a particular chocolate or sex with one very attractive person that you desire. Take the sexual encounter one for example. You sleep with the person so many times until you are fulfilled. But it gets somewhat "nauseating" to enjoy that thing again (the sex or the chocolate) because you've overindulged earlier.   Now what is that feeling called, the feeling of being overly "satisfied" or "quenched" to the point of nausea?  I wouldn't say what I'm looking for is satisfaction or quench. Is there a more specific term?  Example sentence:     "I've been eating T-bone steak all day yesterday. I had leftovers today, but I couldn't eat them - I'm just so ____ towards (or from) steak right now." Can you raise a number to an irrational exponent? The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. $\sqrt[y]{a^x}$. So what happens when you raise a number to an irrational number? Obviously it is not so simple to break it down like above. Does an irrational exponent still have a well formed meaning?  The only example that comes to mind is Euler's identity, but that seems likes a pretty exceptional case. What about in general? The square of an integer is congruent to 0 or 1 mod 4