Why doesn't physics like infinity (or does it)? I was recently told that in physics, if your equation contains an infinity, it usually suggests that something is wrong. As an example, the infinity around black holes (also) tells us that general relativity isn't the full picture of gravity. How true is this? And what logic underlies this methodology of verifying a theory? Edit to suffice  The answers don't seem to explain why; there is no analytical or empirical grounding as to why 'limits are(n't) okay'.
When are infinities okay? I'm a little confused about how to handle infinities in physics. On the one hand, I always here things like infinities are not physically tenable answers. GR can't answer anything about the origin of the universe because of singularities. Throw away the radial solution that gives an infinite potential when solving the Laplace equation in spherical coordinates. But then there are other times that the infinities are okay. For instance, it is usually okay to set the reference point for your potential at infinity. There are certain times in David Tong's quantum field theory lecture notes where he seems to almost to hand wave away infinities. So, philosophically, physically, and mathematically when are infinities okay, and when are they not okay?
Writing to NTFS drive in Linux Why Did Anakin Skywalker Believe that Turning to the Dark Side of the Force Would Save Padmé? Just as the question asks: Why did Anakin come to believe that Padmé could be saved if he embraced the Dark side of the Force? Padmé clearly didn't believe it. Why was Anakin so driven to save Padmé through the dark side? Sometimes when I write, for example, "the boundaries of segments", I am suggested to use "segments' boundaries" or even "segment boundaries".   Or "the page's content" vs. "the content of the page" vs. "the page content". Then I would like to know is it always preferred to use phrases with apostrophe or adjective?  For example      In our terms, they are mainly textual elements within the page content, which serve as the starting point  of a data region (or a data region's starting point ?)  or a distinctive feature within it What to do if I can't get a visa to attend a conference that accepted my paper? My work got accepted into a very good category in a very good conference in my field. Due to complicated visa issues it will be very hard for me to attend that conference.   If I don't attend the conference they are certainly going to remove my work from the proceedings.  I am worried I won't be able to claim any sort of achievement if I don't attend this conference.  Can I still add it to my CV and specify that my work got accepted in this conference in a very good category? Possible Duplicate:          My reputation score has gone down, but the reputation tab in my profile does not indicate why. Is this a bug? What else could have caused it? How to add spaces at every X digit in a binary or hexadecimal number? I would like to format binary or hexadecimal numbers with a space separator at each 2 or 4 digit (or any other value like 8 digit or more).  A small picture is better than a long explanation, here is what I would like to do:    As you can see in this mwe, I use the fmtcount package with padzeroes:      \documentclass{article}     \usepackage[utf8]{inputenc}         \usepackage[T1]{fontenc}               \usepackage{fmtcount}     \begin{document}             Binary value : {\padzeroes[16]\binarynum{129}}$_{(2)}$ %             should display 0000 0000 1000 0001$_{(2)}$              Hexadecimal value: {\padzeroes[8]\Hexadecimalnum{65000}}$_{(16)}$ %             should display  00 00 FD E8$_{(2)}$     \end{document}   What I want to do is displaying numbers with various radix (2, 3, 7, 8, 10, 16) given a decimal number and format these numbers correctly with spaces and leading 0 (without having to perform the conversion by myself).  I already found some posts with similar questions, but none of them achieve this (they display number with thousand separators, like the numprint package, every 3 digits; or they do not make it automatic). Here are these posts :         EDIT two other linked questions:       I also prefer to use pdflatex. Graviational Lensing Data Can anyone provide me with (or point me to) real life gravitational lensing data that I can plug in to the first equation on this page (the standard lensing equation):   together with the luminous mass and distance to the galaxy(s).  I've looked for papers online and have not found any simple data that can be readily plugged in to the equation (I am not familiar with the details of how exactly the lensing is measured and have gotten lost in the mountains of data in the papers).  Thanks Boot failure : No DEFAULT or UI configuration directive found! I have a table:  table votes (     id,     user,     email,     address,     primary key(id), );   Now I want to make the columns user, email, address unique (together).  How do I do this in MySql?  Of course the example is just... an example. So please don't worry about the semantics. How do I get the Minecraft GUI scale to go back to Auto? I have been playing Minecraft, and I went into the settings to see what could be changed. I was in display and I saw the GUI scale was set to "Auto", which was fairly large. I cycled through the scales until I got back to auto, but the GUI scale was stuck at normal size and wouldn't allow me to make it large again. I know this isn't such a big problem, but I really like big font and this is a constant bother to me. This question is from [Number Theory George E. Andrews 1-1 #3].     Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$   This problem is driving me crazy. $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$  $(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can't figure it out    This is a similar problem in the book and I tried this method but it wasn't working out      $\quad$Thereom $\bf1$-$\bf2$: $\,\,\,\,$ If $\,x$ is any real number other than $1$, then $$\sum_{j=0}^{n-1}x^j=1+x+x^2+\ldots+x^{n-1}=\dfrac{x^n-1}{x-1}.$$    $\quad$Remark: $\displaystyle\sum_{j=0}^{n-1}A_j$ is shorthand for $A_0+A_1+A_2+\ldots+A_{n-1}.$     $\quad$Proof: Again we proceed by mathematical induction. If $n=1$ then $\displaystyle\sum_{j=0}^{1-1}x^j=x^0=1$ and $(x-1)/(x-1)=1$. Thus the theorem is true for $n=1$.  $\quad$ Assuming that $\displaystyle\sum_{j=0}^{k-1}x^j=(x^k-1)/(x-1)$, we find that $$ \eqalign{ \sum^{(k+1)-1}_ {j=0}x^j &amp; = \sum^{k-1}_ {j=0}x^j+x^k=\dfrac{x^k-1}{x-1}+x^k \\ &amp;= \dfrac{x^k-1+x^{k+1}-x^k}{x-1}\\ &amp;= \dfrac{x^{k+1}-1}{x-1}. }$$   Hence condition $(\rm ii)$ is fulfilled, and we have established the theorem.    $\quad$Corollary $\bf1$-$\bf1$: $\,\,$ If $\,m$ and $n$ are positive integers and if $m&gt;1$, then $n&lt;m^n.$ I want to crop (as in, take a subset of each frame, like when you crop a photo) a video in Windows Live Movie maker. Is there a way to do this, and failing that is there another free tool i can use to do that? System.FinalException: Record is read-only: Trigger.updateCompetitors: line 24 Find all functions $f$ such that $f:\mathbb{N}\rightarrow\  \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$   Let us plug in $n=1$  $f(f(1))+f(2)=3$  Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)$ can only take the values $1,2$. Now we divide the problem into cases.  Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. I can only give a "sort of recursive" way to define the function. Here it goes $$f:\mathbb{N}\rightarrow \mathbb{N}$$$$f(1)=1$$$$f(n)=n+1-f(f(n-1))$$  But this case is harder to deal with.Some help will be appreciated. Say, teamA score is 368 (369 to win), and teamB is batting at 368 (or 366). One run (or 3 runs) will make them win. The batsman hits the ball for a six, so score is 374 (or 372) which is more than the required 369. Even then the runs are counted for the team and for the batsman.  Now, on the last ball, suppose the batsman did not hit a six, but ran for a single : teamB will win. Can the batsmen continue to run until the ball is dead ?  Why will they want this ? (A) If batsman is at 98 and wants to reach a century. (B) If batsman has a career total of 9998 and wants to reach 1000 runs. (C) If the batsman has a very high record strike rate, and wants to remain at that record strike rate. (D) In the championship, this is the last quarterfinal, &amp; even winning will put teamB equal to teamC, and the better runrate at the end of the quarterfinals will enable them to move to the semifinals. (E) Many similar situations can happen.  So coming back to the question : Do the laws of cricket allow running on the last ball, even after winning ?  In most of the matches I have seen, after winning, the batsmen do not even look where the ball is and start celebrating the win. If the rules did not allow them to run more, then fine. If the rules did allow it, then has there been any game where they did run more ? Evaluating the integral $\mathop{\lim}\limits_{n \to \infty} \int_{-1}^{1} f(t)\cos^{2}(nt) \ dt $ Given that $f\colon [-1,1] \to \mathbb{R}$ is a continuous function such that $ \int_{-1}^{1} f(t) \ dt =1$, how do I evaluate the limit of this integral: $$\lim_{n \to \infty} \int_{-1}^{1} f(t) \cos^{2}{nt} \,dt$$   What I did was to write $\cos^{2}{nt} = \frac{1+\cos{2nt}}{2}$ and substitute it in the integral so that I can make use of the given hypothesis of $\int_{-1}^{1} f(t) \ dt =1$. So the integral becomes,   \begin{align*} \int_{-1}^{1} f(t)\cos^{2}{nt} \ dt = \int_{-1}^{1} f(t) \biggl[\frac{1+\cos{2nt}}{2}\biggr] \ dt  &amp;  \\  \hspace{3cm} = \frac{1}{2}\int_{-1}^{1}f(t) \ dt + \int_{-1}^{1} \frac{f(t)\cos{2nt}}{2} \ dt \end{align*}  But I don't really know how I can evaluate the second integral and also I can't realize as to why that integral condition on $f$ has been assumed. Moreover without assuming that condition on $f$ is it possible to evaluate this integral? If yes, then what would the answer be? Force newlines with cat wildcard printing I want to use cat with wildcards in bash in order to print multiple small files (every file is one sentence) to standard output. However, the separate file contents are not separated by a newline, which I'd like to for ease of reading.  How can I add a file delimiter of some sort to this command? I am afraid that listing doesn't support javascript.  \begin{lstlisting}[language=javascript] ... \end{lstlisting}   What am I doing wrong? Multiple cases in switch statement Is there a way to fall through multiple case statements without stating case value: repeatedly?  I know this works:  switch (value) {    case 1:    case 2:    case 3:       // Do some stuff       break;    case 4:    case 5:    case 6:       // Do some different stuff       break;    default:        // Default stuff       break; }   but I'd like to do something like this:  switch (value) {    case 1,2,3:       // Do something       break;    case 4,5,6:       // Do something       break;    default:       // Do the Default       break; }   Is this syntax I'm thinking of from a different language, or am I missing something? How do I add "citations" at the end of the document as done here?