Evaluating $\int_{-\infty}^{\infty} e^{-{(x-1)}^2} dx$
How can I evaluate $\int_{-\infty}^{\infty} e^{-x^2} dx$ without using polar coordinates
Why not make  but for questions too?  I love that everything is well though behind SO but I can't see no reason as to why great questions can't be prized with "Badges by subject". How to run/use popcorn-time? I got popcorn time installed using this link:    But I have no clue how to use or run Popcorn time.  Do I open my torrent client and download a movie and then it gets going? or do I have to open it? If I have to open it, how do I do that, I opened the dash and tried clicking on it but nothing happened. Please help me!! :D Why non-measurable sets exist? What Is the reason behind the fact that $\Bbb R$ is not a complete measure space? Is it only due to the cardinality of  $\Bbb R$? Or other structures on  $\Bbb R$ also play a role? Can one make every subset of  $\Bbb R$ trivially measurable by assigning them to a number of a complete ordered field of the cardinality $\aleph_2$ ? Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$ I want to solve this system of equations, I have been out of Maths for a long!!  $$\sqrt{x}+y=7$$  $$\sqrt{y}+x=11$$  Just wondering easiest step to find values for $x$ and $y$ from the above equations? How can I change (or remove / reset ) my name on a site in Stack Exchange network? How can I customize the Ubuntu installer? Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $ Given $$\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$$ evaluate:   $$\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx. $$   I can find that  $$\left(ax+\frac{b}{x}\right)^2 = a^2x^2+2ab+\frac{b^2}{x^2}$$  therefore: $$\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx = e^{2ab}\int_0^{\infty}e^{-\left(ax+\frac{b}{x}\right)^2}dx$$  but I can't find any clue then. How to find the magnetic field due to a revolving electron of hydrogen atom in first orbit Is a transit visa required to pass through Bangkok Airport? WOL - Wake On Lan over internet `du -sh` reports different used size than `df -h` Possible Duplicates:             df -h says that I have 494G used on /var:  Filesystem            Size  Used Avail Use% Mounted on /dev/sda3             564G  494G   41G  93% /var   But du -sh /var says it's 350G  What makes them different? Disable cache for some images The client can do several actions per frame, for instance, requesting a movement, shooting etc. Should i send a packet to the server for each action the client performed during the frame, or should i stack them in an unique packet and send it at the end of the frame?  EDIT: i'm using Jmonkeyengine and the provided network API SpiderMonkey. I am trying to remember the title of a children's SF book I once read, maybe 20 to 25 odd years ago.   I don't have a great number of details (and some of what I recall may be completely wrong), but the details (such as they are):   A war between two alien groups ended, and some technology that should have been destroyed was instead imprinted into a human baby. A few years later, both groups find this out and attempt to retrieve it. The main characters are the child, and an alien that I believe was called "Yacob" (or similar) who was sent to protect the child/retrieve the technology, and the antagonists were the other group of aliens that had previously lost the war and were trying to get the technology to start it up again (again, I don't remember the exact details, but I think it was a fairly simple black &amp; white/good vs. bad scenario, being a children's book and all). There was a scene in which Yacob/the alien gives the child blue food, who complains that it's blue, and the alien states "close your eyes and it's just food". The alien had a space shuttle/some other piece of tech disguised as a van. I vaguely recall a scene in which it is under attack and it flies or something. How do I dynamically invoke a class method in PHP? How can I dynamically invoke a class method in PHP? The class method is not static. It appears that   call_user_func(...)   only works with static functions?  Thanks. Imagine a black hole that is fast-approaching its final exponential throws of Hawking evaporation. Presumably, at all points in this end process there will remain a region that identifiably remains &quot;the black hole&quot; until the the very end, as opposed to huge swarm of fundamental particles that is being radiated out from it. As the mass of the black hole descends to that of individual particles, it would seem entirely feasible that the very last fermionic Hawking radiation event available to the almost-deceased black hole could leave it with an unbalanced charge, e.g. -1, and an unbalanced spin, say 1/2. It would also have some kind of mass of course, but that aspect of the final residue could be fine-tuned to any specific value by photon emissions of arbitrary frequencies. After photon emission mass trimming, the resulting black hole residuum would reach a point where it is no longer be able to evaporate into any known particle, because there is no longer any lower-mass option available to it for removing the -1 charge and 1/2 spin. The black hole residuum will at that point be stuck, so to speak, stuck with exact charge, spin, and mass features of an electron. And so my question: Is it an electron? And if so, by equivalence, is every electrons in the universe really just a particular type of black hole that cannot evaporate any further due to the constraints of charge and spin conservation? And if so, why are charge and spin so uniquely combined in such black hole remnants, so that e.g. a remnant of -1 charge and zero spin is not permitted, at least not commonly, and the mass is forced to a very specific associated level? Is there anything in the current understanding of general relativity that would explain such a curious set of restrictions on evaporation? The full generalization of this idea would of course be that all forms of black hole evaporation are ultimately constrained in ways that correspond exactly to the Standard Model, with free fundamental particles like electrons being the only stable end states of the evaporation process. The proton would be a fascinating example of an evaporation that remains incomplete in a more profound way, with the three quarks remaining incapable of isolated existence within spacetime. The strong force, from that perspective, would in some odd sense have to be a curious unbalanced remnant of those same deeper constraints on the overall gravitational evaporation process. This may all be tautological, too! That is, since Hawking radiation is guided by the particles possible, the constraints I just mentioned may be built-in and thus entirely trivial in nature. However, something deeper in the way they work together would seem... plausible, at least? If an electron is an unbalanced black hole, then the particles given off would also be black holes, and the overall process would be not one of just particle emission, but of how black holes split at low masses. Splitting with constraints imposed by the structure of spacetime itself would be a rather different way of looking at black hole evaporation, I suspect. (final note: This is just a passing thought that I've mulled over now and then through the years. Asking it was inspired by  by Ben Crowell. I should add that I doubt very seriously that that my wild speculations above have anything to do with Wheeler's concept of geons, though.) How is it possible for the Root Name Servers to handle all DNS requests? I was reading about DNS some days ago and learned how the requests are processed. If you surf to www.example.com, then a request will go to the Root Name Servers to see who owns that .com address, then another request will go to another, more local, DNS server to see who owns the example.com address and so on.  How is it technically possible that the 13 Root Name Servers can handle all requests done by earth's billions of Internet users simultaneosly without beeing ddos:ed? Why every polynomial over the algebraic numbers $F$ splits over $F$? I read that if $F$ is the field of algebraic numbers over $\mathbb{Q}$, then every polynomial in $F[x]$ splits over $F$. That's awesome! Nevertheless, I don't fully understand why it is true. Can you throw some ideas about why this is true? Proving at least 99 pairwise sums of the reals are non-negative Suppose, one hundred real numbers are given and their sum is 0.Then how can I prove that at least 99 of the pairwise sums of these hundred numbers are non-negative?  I tried this: Let the real numbers be $a_i$,$1\leq i\leq100$. By the given condition, $a_1+a_2+\dots +a_{100}=0$.Suppose for the sake of contradiction, that at most 98 of the pairwise sums of these given reals are non-negative.I just tried to examine the case when exactly 98 of them are non-negative. I labelled the sums $S_1$,$S_2,\dots S_{4950}$,WLOG assume that sums $S_1,S_2,\dots S_{4852}$ are negative, the rest are non-negative. $$S_1+S_2+\dots +S_{4950}=99(\sum_{i=1}^na_i)=0$$ Case I: $S_{4853}+a_{4854}+\dots +a_{4950}=0$.That simply means that $S_1+S_2+\dots +S_{4852}=0$.But, by the given condition, $S_1+S_2+\dots +S_{4852}&lt;0$,a contradiction.After that, I am not able to make any progress.  Can anyone please suggest a more efficient method to solve this problem.I feel this approach will involve enormous case work. Is a closed subset of a compact set (which is a subset of a metric space $M$) compact? Is there a way to prove this using sequential compactness instead of open cover definitions?  My first gut reaction was that the fact was obvious since we can show that the closed subset $[a,b]$ is compact in $\mathbb R$.