Trying to find the inverse of the function $y=x!$ for domain restricted to $x&gt;0$. Does this function have an inverse for the given domain? If so, please include methods used for finding the inverse to help further my understanding of the solution. Thanks!
I'm trying to solve hard combinatorics that involve complicated factorials with large values.  In a simple case such as $8Pr = 336$, find the value of $r$, it is easy to say it equals to this: $$\frac{8!}{(8-r)!} = 336.$$  Then $(8-r)! = 336$ and by inspection, clearly $8-r = 5$ and $r = 3$.  Now this is all and good and I know an inverse function to a factorial doesn't exist as there is for functions like sin, cos and tan etc. but how would you possibly solve an equation that involves very large values compared to the above problem without the tedious guess and checking for right values.  Edit: For e.g. if you wanted to calculate a problem like this (it's simple I know but a good starting out problem) Let's say 10 colored marbles are placed in a row, what is the minimum number of colors needed to guarantee at least $10000$ different patterns? WITHOUT GUESS AND CHECKING  Any method or explanation is appreciated!
I've upgraded from 4.4.0 to 4.4.8 and then 4.5.2 on Ubuntu 16.04 LTS 64-bit, and it's still happening.   At 4.5.2 it's even more frequent, and mainly in Chrome (haven't tried firefox or opera just yet). Every time I open a new web page. Any way to fix that, or is it a bug?  00:02.0 VGA compatible controller: Intel Corporation HD Graphics 5500 (rev 09) 04:00.0 Display controller: Advanced Micro Devices, Inc. [AMD/ATI] Venus XTX [Radeon HD 8890M / R9 M275X/M375X] (rev 81)   Cheers. How do I refresh the package list? How to add custom C++ keywords to be recognized by Minted? I have defined several structures and types in a C++ project. I have to write some documentation for this code, and I use  to typeset my code.   I would like to be able to add some keywords to the minted database, so minted will also recognize these and give them the appropriate color. (I would like to have them colored in the same way a type, eg. double would be colored.) After having read various articles on this site about WWW or Non-WWW website setup, ,  and , and after having setup a correct redirection rule in .Htaccess to at least eliminate duplicate contents (currently the non-www solution (e.g. http://example.com) I have a feeling that my site rank less since I changed this.   My confusion was strengthened by these two opponents:     &amp;     Could the change of www.example.com >> http://example.com have a factual impact on ranking?  say either is better or worse. Are they holding information back?  I am confused fellow folks...  As much as I would like to separate facts from fiction, I am aware that not all the good valid information is written. Not on google webmaster site and perhaps not elsewhere. If so it seems out of date!  Does or does not a WWW domain impact SEO/rank better than its NON-WWW version? (Given that you have setup a proper redirect to avoid duplicates and go with one) Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$ Can I control voltage of USB data pins? I just bought a 5 volt dual relay module for my arduino.  But now that I have it I was wondering if there is any way that I could control it directly from USB.    The pinout of the module is GND  ONE  TWO  VOC This would make it really easy to just connect a USB cable to it. Assuming I did that is there any way to send 5 volt through the USB Data+ and Data-?  Thank you Fantasy book where a young prince becomes a mage I have filed my 2012 tax return jointly with my wife, along with my wife's ITIN application, back in late March.  In April, I got a letter for IRS ITIN department saying that we should send my wife's original passport to them. So we sent my wife's passport on May 15th. USPS tracking number confirmed that IRS received it on May 20th.  However, last week (end of June) we have received another letter from IRS ITIN department saying that our ITIN application has been rejected, rejected because we have not sent the original identification document that they requested!  Okay, let's put our tax return and ITIN aside. Where is my wife's passport? Right now my wife does not have her personal identification, nor proof of legal status in U.S. Could IRS have lost it?  I tried to call the IRS several times. One time I waited for 1 hour and 20 minutes to talk to a person, and that person said he has only 10 minutes to talk before he has to leave. He said they'll investigate and someone(?) will call us back soon, but never called back. I wrote them a letter, but no response. I tried calling several times more, but machine is saying 'sorry too many calls right now, we can't accept your call'.  What can I do? What happens with the force of gravity when the distance between two objects is 0? so I had my first approximation to the gravity equation  \begin{equation} F=GmM/r^2 \end{equation} and some questions arose that my teacher couldnt respond: if r approximates to 0 with mM being constant, then the force should tend to infinity, right? so that being true, then if we would travel to the center of the earth, we would be crushed by gravity? if so, how can there be anything right in the center of the earth? wouldnt anything there be compressed and destroyed by the infinite force?  having that in mind, my next question is as gravity force is measured from the center of an object, any part of the same object which is not at the center would be affected by the same object's gravity? if so, how can we differentiate between two objects that are not in a vacuum? my question originated thinking in us standing on the earth, in contact with it, are we, in a gravitational sense, one object? this lead to the last question, if gravity is a property of matter, then how applying the same logic as before, how does it work between two neutrons?(chose neutrons because i assumed it would be more difficult obtaining gravitational force if there was electrostatic forces from charges inbetween) do two newtrons in contact count as 1 gravitational object? and how thats its own gravity affect the newtron from center and out? Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$ So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group.   My thoughts are as follows:  Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is easily shown that $\alpha$ satisfies $\alpha^4-4\alpha^2+2=0$.  Define $f(x) := x^4-4x^2+2$. Then $f$ is irreducible over $\mathbb{Q}$ by Eisenstein with $p=2$.  So we have that $f$ is the irreducible polynomial for $\alpha$ over $\mathbb{Q}$.   Further $|\mathbb{Q}(\alpha):\mathbb{Q}|=4$.  For $\mathbb{Q}({\alpha})$ to be Galois, it must contain all roots of $f$.   Define $K=\mathbb{Q}(\alpha)$ for convenience.  Define $\alpha := \alpha_1$.  Since $f$ has only even powers, we know that $-\alpha := \alpha_2$ is a root, and therefore contained in $K$ since $K$ is a field.  We note that the other two roots are $\alpha_3=\sqrt{2-\sqrt{2}}$ and $\alpha_4=-\sqrt{2-\sqrt{2}}$.  So in order to show $K$ is Galois, it must be shown that $\alpha_3$ and $\alpha_4$ lie in $K$.  Now $\alpha_1^2=2+\sqrt2$ and so $\sqrt2 \in K$. Thus $-\sqrt2 \in K$ since $K$ is a field.  Can somebody explain why $\alpha_3$ and $\alpha_4$ lie in $K$?  Next we are to determine the Galois group of $K$.  Assuming $K$ is Galois, since it has degree $4$ over $\mathbb{Q}$ (shown earlier), we know that its Galois group has size $4$. There are only two groups of size $4$, namely $V_4$ and $C_4$, the Klein four group and the cyclic group of order $4$.  How do we determine which of these choice is in fact the Galois Group? I told my follower (Serana) to wait. But this is different. I told her to go somewhere and wait so it will activate the enemies. Well it did and I killed them. But Serana seems to be not following me anymore. I can't tell her to follow me. And she is constantly in a squatting position. The only thing I can do is tell her to wait somewhere else to move her. And I can't part ways because I'm doing a main quest and she has to be there. Can anybody give me a tip on what to do? Android app chokes on per-site meta profiles Steps to reproduce:    Go to a per-site meta. Find any post. Touch the user that posted it. See something like this (click to enlarge):           As you can see, it says "Stack Overflow" as the site, and shows my SO rep and badges, but it shows my Meta.Code Golf posts. Badge symbol in notification is of the site you're on, not where badge was earned Precision of Coulomb's law Up to which precision has the coulomb law proven to be true? I.e.  if you have two electrons in a vacuum chamber, 5 meters appart, have the third order terms been ruled out? Are there any theoretical limits to measure the precision ( Planck's constant?). Obviously there are practical limitations ( imperfect vacuum, cosmic rays, vacuum fluctuation). Still, does anyone know what was the smallest amount ever correctly predicted by that law?    Edit : Summary  On the high end of the energy spectrum a precision of 10^-16 has been shown ( 42 years ago )  For electron point charges at large distances the law might brake down due to practical reasons.  For moving particles QED gives a correction to the law: Determine $\text{Aut}(S)$ where $S = \mathbb{Q}[x]/(x^2)$ Here's the problem: if $S = \mathbb{Q}[x]/(x^2)$, compute the group  of automorphisms $\mbox{Aut}(S)$.   First off, I have the solution and I know it's isomorphic to the multiplicative group $\mathbb{Q}\backslash\{0\}$, so I'm trying to understand it. It uses what is called the substitution principle and the fact that the restriction of any endomorphism $\phi: \mathbb{Q}[x]\to \mathbb{Q}[x]$ to $\mathbb{Q}$ is the identity. It says that these two facts implies that the endomorphisms of $\mathbb{Q}[x]$ are in 1-1 correspondence with $h \in \mathbb{Q}[x]$, where the endomorphism corresponding to $h$ is $\phi_h : \mathbb{Q}[x] \to \mathbb{Q}[x]$ given by $\phi_h(f) = f(h)$ for $f \in \mathbb{Q}[x]$. I guess I just don't understand how it applied the substitution principle? It's probably obvious, but for some reason I am not getting it.  An explanation would be greatly appreciated! Or if you have a different solution, that'd be cool too. Thanks!! I want to upgrade the hard drive in my 2012 Macbook Pro to an SSD. I have the parts, but had a question about how to clone the contents of my current drive so I can install it onto the SSD.  I've seen tutorials where people suggest connecting the new SSD to the Macbook prior to installation and using a Carbon Copy Cloner to clone the drive, and then installing the SSD drive.  My question is, can I do a Time Machine backup of my current hard drive on an external hard drive, install the SSD, and then use the Time Machine backup from the external to transfer it onto the newly installed SSD? Is this the same as using the clone program above? How to get the Duke O' Death? I was watching some videos on YouTube showing how to get the Duke O' Death in GTA 5, however what I have seen is sort of unclear.  Some people say you must be a returning player on GTA 5 to get it (what does this mean too BTW?), some people don't say that. Some people say you have to be Trevor, some people don't. Some say the time of day matters, some don't. And apparently you can buy it from Warstock website (I can't).   Can someone please tell me how to get it?! GTA 5 came with my PS4 (I didn't play it on PS3) if that helps. Is there a way to input particular sections of source code using \lstinputlisting (from the listings package) in a flexible way?  It's possible to use the firstline and lastline options to input particular line ranges, but this isn't robust to changes to the source file.  For example, suppose that the input source file is:  int square(int x) {   return x * x; }  int cube(int x) {   return x * x * x; }   In my LaTeX document, I can include the cube procedure using the command \lstinputlisting[firstline=5,lastline=7]{source.c}.  But suppose I later decide to rewrite the cube procedure.  The input source file becomes:  int square(int x) {   return x * x; }  int cube(int x) {   int x2;   x2 = square(x);   return x * x2; }   Now my LaTeX document will not show the final two lines of cube because the listing uses only lines 5-7.  I could manually update the line range, but this is tedious and error-prone.  Is there a way to mark sections of source code with some kind of label so that the listing will be robust to these sorts of changes?  For instance, perhaps the input source file would be something like  int square(int x) {   return x * x; }  /**\beginlabel{lst:cube} **/ int cube(int x) {   int x2;   x2 = square(x);   return x * x2; } /**\endlabel{lst:cube} **/   so that I could use something like \lstinputlisting[lines=lst:cube]{source.c} to include the section that I want and remain robust to changes.  I am open to using other packages for source code listings (e.g., minted) if helpful. Proving $\sup B=\inf A$ Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Before I start, don't lecture me on the fact that FAT32 is old and hasn't been updated for nearly 20 years. Right now I have sorted that. I am trying to install Ubuntu Linux onto a 64GB flash drive and it wants me to format it to one of the EXT file systems.  The problem is if I put the flash drive into a Windows PC, I can't use it. I also can't install the required software onto every Windows PC that I will use. When I tried to install it on FAT32 I got an error and there is no option for NTFS.  Does anyone know how I could do this? Maybe a Windows based tool? Thanks in advance.