How can we proof that if $p$ is prime and $k$ is integer number that $1&lt;k&lt;p-1$  then $(p-k)!(k-1)! \equiv (-1)^k\pmod p$
Here is a question from my number theory class.  Prove  $$(p-k)!(k-1)!\equiv (-1)^k \text{ mod p} $$ Help please!
Hard disk file system is shown as Raw file system; how can I access it? Plotting several 2d functions in a 3d graph I am trying to reproduce a similar figure to the following one. My first guess was to use pgfplots but I didn't find any example that I could use as a starting point. At the moment I am unsure if pgfplots is actually the right package or I should rely on something else e.g. TikZ.   Since I don't really know where to start, I would appreciate any piece of code that points me in the right direction.    EDIT  Here is some dummy data for the plots.  plot1   plot2   plot3   plot4 0       0       0       0 3.466   2.058   0       0 4.262   2.976   0.001   0 3.822   3.168   0.006   0.008 2.953   2.936   0.019   0.063 2.065   2.492   0.046   0.265 1.332   1.977   0.092   0.734 0.797   1.478   0.164   1.508 0.443   1.045   0.268   2.44 0.228   0.698   0.412   3.219 0.107   0.438   0.598   3.524 0.046   0.256   0.831   3.219 0.017   0.138   1.109   2.44 0.006   0.067   1.429   1.508 0.002   0.029   1.78    0.734 0       0.01    2.141   0.265 0       0.003   2.479   0.063 0       0.001   2.736   0.008 0       0       2.808   0 0       0       2.465   0 0       0       0       0 Watts P.M.P.O. and audio equipment power I am a beginner in electrical engineering, and to begin learning I decided to look into audio amplifiers. I have seen that the power of audio amplifiers is often described in the units of watts, watts RMS, and watts PMPO. So far, the only one of those units that I think I understand is watts. For instance, I just recently made . If I think: &quot;What is the power of my amplifier supposed to be?&quot;, then my thought process goes like this:  The transistors I used are the TIP120, which have a maximum rated power of 65W. Then, I decide that 60W is a safer maximum which would be reached if my amplifier runs 19V through one of my 6Ω speakers. Therefore, the power of my amplifier is 2x60W.  Then, I look at the datasheets of audio amplifier ICs, like the TPA3116 that says: &quot;50W into a 4Ω load at 21V&quot;, which makes me conclude that in that setting it'll output 14V maximum into the speaker, which I think makes sense. But then, I look at the datasheet of the TDA7294, and they talk about &quot;RMS Power&quot; and &quot;RMS Music Power&quot;, units which I couldn't find clear information about. Then, I look at home audio systems, like a Panasonic Stereo System I have which has &quot;9500 Watts P.M.P.O.&quot; written on the front. 9500W? That sounds stupidly high to me, but then I look again and it has 300W written on the back of it. If I'm correct, maybe a 1kW home stereo system could make sense, but a 2kW or greater one would not exist. So... Are my conclusions about  and the TPA3116 correct? Are my conclusions about audio equipment power correct? Are WattsPMPO, WattsRMS and RMSmusicpower, real useful units? The cardinality of the set of all finite subsets of an infinite set Let $X$ be an infinite set of cardinality $|X|$, and let $S$ be the set of all finite subests of $X$. How can we show that Card($S$)$=|X|$? Can anyone help, please? I go to change my birthdate on Roblox, but nothing shows up, the dropdowns are just blank with no Month, Day or Year:  It also shows me a pop up asking to put in the Missing Details to improve the experience, but no matter how many times I put in my age it doesn't save and asks it again. Can a wizard under the effects of Feign Death issue telepathic commands to their familiar? A wizard has a summoned familiar within 100 feet that is currently idle and has  (PHB, p. 240) cast on them by another PC. Would the wizard still be able to issue telepathic commands to their familiar?  Specifically, while Feign Death states that they appear dead and are incapacitated and blind, they aren't listed as being unconscious.   doesn't state that issuing telepathic commands requires an action (which you can't do while incapacitated), whereas it does state that seeing through your familiar's eyes requires an action as does dismissing it.  I was DMing a game where this occurred and I hastily ruled that the wizard was unable to issue commands, but I am second guessing myself after the fact. And knowing my PCs this situation is likely to happen again. Getting the starting shortcut in c# From io.Reader to string in Go Can't we compute $\lim_{n \to \infty}\frac{1+ \cdots +n}{n^2}=\lim_{n\to\infty}\frac{1}{n^2}+\cdots +\lim_{n\to\infty}\frac{n}{n^2}=0$? How do I prove $F(a)=F(a^2)?$ Let $E$ be an extension field of $F$. If $a \in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$.  let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and $k$ be the degree of the minimal polynomial $q(x)$ of $a^2$ over $F$.   Since $a^2 \in F(a)$, We have $F(a^2) \subset F(a)$, then $k\le n$  In order to prove the converse:  $q(a^2)=b_0+b_1a^2+b_2(a^2)^2\ldots+b_k(a^2)^k=0$   implies   $q(a)=b_0+b_1a^2+b_2a^4\ldots+b_ka^{2k}=0$ Then   $p(x)|q(x)$, because $p(x)$ is the minimal polynomial of $a$ over $F$.  If I prove that $n|2k$ we done, since $k$ is odd, we have $n|k$ and $n\le k$ and finally $n=k$.  So I almost finished the question I only need to know how to prove that $n|2k$ It should be only a detail, but I can't see, someone can help me please?  Thanks when EditText is empty and click calculateButton, app crashes Is it possible to allow usage for an app or program for a specific times on a Mac? I was trying to achieve this using a second account (which is the admin account) and Parental Controls, but it seemed this isn't possible to do so.  I know there are some programs which you can download that help you with this, (Cold Turkey, for example). But, it would be much easier if I could do this using only tools already available on macOS. Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics? What's the best way to communicate between view controllers? Being new to objective-c, cocoa, and iPhone dev in general, I have a strong desire to get the most out of the language and the frameworks.  One of the resources I'm using is Stanford's CS193P class notes that they have left on the web.  It includes lecture notes, assignments and sample code, and since the course was given by Apple dev's, I definitely consider it to be "from the horse's mouth".  Class Website:   Lecture 08 is related to an assignment to build a UINavigationController based app that has multiple UIViewControllers pushed onto the UINavigationController stack.  That's how the UINavigationController works.  That's logical.  However, there are some stern warnings in the slide about communicating between your UIViewControllers.  I'm going to quote from this serious of slides:   Page 16/51:     How Not To Share Data         Global Variables or singletons         This includes your application delegate      Direct dependencies make your code less reusable         And more difficult to debug &amp; test         Ok.  I'm down with that.  Don't blindly toss all your methods that will be used for communicating between the viewcontroller into your app delegate and reference the viewcontroller instances in the app delegate methods.  Fair 'nuff.  A bit further on, we get this slide telling us what we should do.  Page 18/51:     Best Practices for Data Flow         Figure out exactly what needs to be communicated   Define input parameters for your view controller   For communicating back up the hierarchy, use loose coupling         Define a generic interface for observers (like delegation)         This slide is then followed by what appears to be a place holder slide where the lecturer then apparently demonstrates the best practices using an example with the UIImagePickerController.  I wish the videos were available! :(  Ok, so... I'm afraid my objc-fu is not so strong.  I'm also a bit confused by the final line in the above quote.  I've been doing my fair share of googling about this and I found what appears to be a decent article talking about the various methods of Observing/Notification techniques:   Method #5 even indicates delegates as an method!  Except.... objects can only set one delegate at a time.  So when I have multiple viewcontroller communication, what am I to do?  Ok, that's the set up gang.  I know I can easily do my communication methods in the app delegate by reference's the multiple viewcontroller instances in my appdelegate but I want to do this sort of thing the right way.  Please help me "do the right thing" by answering the following questions:   When I am trying to push a new viewcontroller on the UINavigationController stack, who should be doing this push.  Which class/file in my code is the correct place? When I want to affect some piece of data (value of an iVar) in one of my UIViewControllers when I am in a different UIViewController, what is the "right" way to do this? Give that we can only have one delegate set at a time in an object, what would the implementation look like for when the lecturer says "Define a generic interface for observers (like delegation)".  A pseudocode example would be awfully helpful here if possible. SQL Server Dependencies Is there an easy way to chase down table/stored procedure/function dependencies in SQL Server 2005+?  I've inherited a giant application with lots of tables and even more stored procedures and functions that are long and interlinked.    At the end of the day is there a way to build a dependency tree?  Ideally what I'm looking for goes in both directions:  For a table/procedure - what depends ON it?:  Show me all the stored procedures that eventually reference it (ideally in a tree view such that sub procedures nest out to the bigger procedures that call them)  For a procedure - what does IT depend on?:  Show me all the procedures and tables that a given procedure will (or could) touch when running.  It seems this tool shouldn't be that hard to make and would be incredibly useful for DB maintenance generally.  Is anyone aware of such a thing?  If this doesn't exist, why the heck not?  The built-in functionality in Management Studio is nice but the information does not appear to be complete at all. What are Unity's keyboard and mouse shortcuts? In Unity, are there any pre-defined shortcuts, e.g. to open the dash or unhide the launcher? Resizing point on zoom out How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$ Find this sum     $$I=\sum_{n=0}^{\infty}\dfrac{1}{(3n+1)(3n+2)(3n+3)}$$   My try: let     $$f(x)=\sum_{n=0}^{\infty}\dfrac{x^{3n+3}}{(3n+1)(3n+2)(3n+3)},|x|\le 1$$   then we have $$f^{(3)}(x)=\sum_{n=0}^{\infty}x^{3n}=\dfrac{1}{1-x^3}$$ then we find the $f(x)$,Following is very ugly(can you someone can post your follow solution,) have other simple methods? Thank you very much. Printer "Filter failed" Upgraded to 18.04 from 16.04. I have a workplace printer connected through appsocket which stopped working.  Printer status is Idle-Filter failed.  Printing queue instantly goes to "stopped".  Right clicking on the printer shows "option finishings has value 3 and cannot be changed". I can't find any info on this error except one pdf to ignore the error.  The printer model Ricoh MP 2555 worked previously with the Ricoh MP C2800 driver. Now it works with neither 2800 or 2555.  The IP address of the printer pings fine. When I try and detect the printer on the network using IP address and I ask it to use the Ricoh MP 2555 driver, it asks for an additional driver which opens Ubuntu software (which never helps) and fails.  I tried changing drivers from "" by providing the ppd file. It still doesn't help. var/log/cups/error.log doesn't tell much, or I don't get it.  Hope someone knows about this error or some kind of a generic solution. How to determine if a string is a valid variable name?