If $f=g$ a.e. and $f$ and $g$ are continuous, then $f=g$ every where. Let $f,g:\mathbb R\longrightarrow \mathbb R$ two continuous functions. Is it true that if $f=g$ a.e. then $f=g$ every where ?  I would say yes. My argument would be the following one : Let $$\Omega =\{x\mid f(x)\neq g(x)\}.$$  So let $x\in \Omega $. Is it true that there is a sequence in $\mathbb R\backslash \Omega $ that converge to $x$ ? If yes, the claim is proved. But if yes, How can I prove this claim ? If no, how can I do ?   Does this result can be adapted to an abstracted measurable space $(X,\mu)$ ?
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere? I wanna prove that   "if $f: \mathbb{R}^n \to \mathbb{R}$ is continuous and satisfies $f=0$ almost everywhere (in the sense of Lebesgue measure), then, $f=0$ everywhere."  I am confident that the statement is true, but stuck with the proof. Also, is the statement true if the domain $\mathbb{R}^n$ is restricted to $\Omega \subseteq \mathbb{R}^n$ that contains a neighborhood of the origin "$0$"?
Recently, I have noticed that on most applications I have on Ubuntu 16.04, when I load the application, the window is displayed with a strange multi-coloured border (see the images below). If I maximise the window though, the border disappears and the window is displayed as expected.  This started happening for no apparent reason. If I reboot my machine, things return to normal at first, but after a while the problem comes back.  The issue occurs on most applications, including Nautilus, Software Centre, Terminal, System Settings, etc... I also get the same border appearing when I right click on the desktop, with the border surrounding the pop-up menu.  However, I have noticed that it does not happen on Chrome, but it does happen on other applications which did not come with Ubuntu by default.  Any ideas on what is causing the problem? Thanks! Coding for an ordered covariate I am performing ordinal regression, I have 5 response categories and several predictors both continuous and categorical. I would like to add a predictor which is categorical but ordered (1, 2, 3, 4). I don't think it would be appropriate to apply the usual dummy coding for unordered categorical predictors, but when I searched for how to code this I did not find much information.  In  "linear coding" or "assuming linearity of the predictor effect" is mentioned, but without further details.  Does it mean I just use my ordered values as they are, i.e. use them as a continuous variable? How can I remap a keyboard key? One of my laptop's keys has fallen off. Is there any way I can remap another key to serve as that key?  I'll 'sacrifice' the other key because I never use it. Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly:   A priori, we are dealing with an irreducible complex representation $(V,\rho)$ of the Clifford algebra of signature $(p,q)$, i.e. generalized Dirac spinors. That Majorana spinors exist means abstractly that there is a  on $V$, i.e. a conjugate-linear map $\phi : V\to V$ with $\phi^2 = \mathrm{id}_V$ that commutes at least with the $\mathfrak{so}(p,q)$ action.  Every single source I can find for Majorana spinors uses operations like the transpose, complex conjugation and Hermitian adjoint on the $\Gamma$-matrices to obtain matrices acting on the same space. This is abstractly wrong, the transpose acts on the dual, the complex conjugation on the conjugate, and the Hermitian adjoint needs an inner product we have no reason for choosing. Of course, since $V$ is finite-dimensional, one can pick a basis and define the uncanonical isomorphisms to its dual and its conjugate, but I find this inelegant, particularly since the standard derivations require us to make a particular such choice with respect to the signs the $\Gamma$-matrices have under e.g. ${}^\dagger$. Finally, the Majorana spinors are usually defined by some equation involving an unnatural and arbitrary-looking product of $\Gamma$-matrices, which varies from source to source according to different sign conventions and sign choices made in the course of the derivation.  It's inelegant because the rest of the theory of spinors can be developed without making such uncanonical choices. Both the uniqueness of the dimension of the irreducible Dirac representations (there are two of them in odd dimensions) and the existence of the Weyl spinors in even dimensions can be derived purely from the abstract properties of the Clifford algebra, no choices made, no transpose, adjoint or conjugate occuring. The (admittedly slightly subjective question) is: Is there a way to show in which dimensions Majorana spinors exist that neither requires an uncanonical choice of basis nor arbitrary choices of signs?  Some partial results:   In even dimensions, the Dirac representation is necessarily self-conjugate since it is the only irreducible representation of the Clifford algebra, so all that is left to show is that a conjugate-linear $\mathfrak{so}$-equivariant map on it squares to $\mathrm{id}_V$ and not to $-\mathrm{id}_V$. However, I can't seem to exhibit any particular equivariant map on it that one could simply check for its square. In odd dimensions, one first needs to figure out whether the two inequivalent Dirac representations are conjugate to each other or self-conjugate.     As further motivation that a clear proof using only canonical properties of the Clifford algebra itself is required, consider the confusing and contradictory claims in the literature:   Polchinski, "String Theory", Vol. 2, p.434: $\mathrm{SO}(d-1,1)$ has Majoranas for $d = 0,1,2,3,4 \mod 8$, corresponding to a special case of $p-q = d-1-1 = 6,7,0,1,2 \mod 8$. Fecko, "Differential Geometry and Lie Groups for Physicists", pp. 651: $\mathrm{Cliff}(p,q)$ has Majoranas for $p-q = 0,2\mod 8$.  This clearly conflicts with Polchinski's claims e.g. for $d=3$. Figueroa-O'Farrill, , pp. 18: We have Majoranas for $p-q = 0,6,7\mod 8$ and "symplectic Majoranas" for $p-q = 2,3,4\mod 8$.   Note that these results conflict simply by the number of possible $p-q$ regardless of whether I've correctly taken care of the differing conventions of whether $p$ or $q$ denotes timelike dimensions. Can I add new methods to the String class in Java? How to create a Windows 7 installation USB from OS X? I'm trying to create a Windows 7 installation USB (for use on a PC) from OS X.  I've tried following these instructions from a :   Insert the USB flash drive and run the command diskutil list to find out the disk name, we'll use /dev/disk1 as an example Now unmount the disk using diskutil unmountDisk /dev/disk1 dd if=/path/to/Win7.iso of=/dev/disk1 bs=8192   But the instructions don't work. When I insert the created USB drive into my PC and try to boot from it, I get this message:     Reboot and Select proper Boot device or Insert Boot Media in selected Boot device and press a key   What am I missing? what is/are the purpose(s) of inline? Quotient ring of Gaussian integers A very basic ring theory question, which I am not able to solve. How does one show that    $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$, if $a,b$ are relatively prime.   My attempt was to define a map, $\varphi:\mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ and show that the kernel is the ideal generated by $\langle{3-i\rangle}$. But I couldn't think of such a map. Anyhow, any ideas would be helpful. Array List count to users Why can't we use double pointer to represent two dimensional arrays? Number agreement between subject and object I am looking for a proof possibly using the sandwich theorem and/or Bernoulli's inequality for proving the following statement:  If $x&gt;0$ then $x^{1/n}$ tends to $1$. I have dual boot configured with Windows 7 (32bit) and Ubuntu 12.04(64bit). I installed Ubuntu after windows, now my windows is corrupt due to some virus and I need to reinstall it. Currently when I start my PC, grub loader comes up and I need to select Ubuntu or Windows.   I was wondering how can I install windows again and still have my Ubuntu install? Is there a way to do this?  I do not want to install Ubuntu again and lose my settings and data there.   Thanks, Ketan When is double-quoting necessary? How to superimpose LaTeX on a picture? I know how to add a picture to my LaTeX file, e.g. \includegraphics[width=7cm]{curve.pdf}. What I'd now like to do is place some latex symbols onto the picture. Like a \gamma to name a curve. I do not want to do that in the graphics editing program e.g. because I might want to rename that curve later and don't want to repeat the editing process.  I know I can do a \put in a picture environment. But this environment forces me to specify its size at the beginning. That would be cumbersome since I'd have to go open the graphics file and do some calculations. Is there a way to get the dimensions (or aspect ratio) of a graphics file within LaTeX, so I might use it there? But even if I could do these calculations automatically, I still wouldn't like the redundancy of specifying the width for \includegraphics again, but I could live with that.  Also, is it possible to specify the size in cm to the picture environment? I just tried and got some errors.  So basically what I am looking for is a solution as simple as possible which:  a) lets me specify a width or a height and a file, and displays the image scaled while keeping its aspect ratio and then   b) lets me put LaTeX stuff inside the image in relative coordinates (percentages or 0...1), so I won't have to redo anything should I decide that the image needs to be a bit larger or smaller.  (b is kind of like setting \unitlength to the whole width=height for the 'picture' environment. But of course most images are not square and I also don't appreciate the redundancy (but could live with it).) Ok, so I have made a .tmx file for my first level, but I am unsure how I would load it into my game. Currently I am iterating through an internal array to set the blocks, but I need to know how to load the .tmx file into the code.  Here is my World.java:  import java.awt.Graphics; import java.awt.Graphics2D; import java.awt.Image; import javax.imageio.ImageIO; import java.io.*;  class World {     Image ground;     Block[] blocks = new Block[46];     int map[][] = {             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},             {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},         };     World() {         int enc = -1;         for (int i = 0; i &lt; map.length; i++) {             for (int j = 0; j &lt; map[i].length; j++) {                 if (map[i][j] == 1) {                     enc++;                     blocks[enc] = new Block(j * 32, i * 32, 32, 32, 1);                 } else if (map[i][j] == 2) {                     enc++;                     blocks[enc] = new Block(j * 32, i * 32, 32, 32, 2);                 }             }         }     } } "To your left" versus "on your left" Is it possible to enter Russia by train 1-1.5 hours before the visa validity period starts? A friend of mine is taking the Train from Finland to Russia. She has now paid and sent the Tourist visa application, with the requested validity being from 3 to 16 August.  Now she's considering entering the evening before, so I would like to ask the following:  The Train she wants to take arrives at Vyborg (first Russian station) on 2 August at 22:22, then St Petersburg at 23:27. The Russian border guards board the train at Vyborg and do the checks while the Train is travelling between the stations, so sometime between 22:30-23:00  If the visa validity starts on 3 August, and she is checked at 22:30-23:00 the evening before, what would happen? Are they very strict about the starting date if it's within 1-1.5 hours, and how bad would the consequences be?  Although they do the check while travelling between Vyborg and St Petersburg, could they simply wait until arrival in St Petersburg, then hold her in custody for 30 minutes and then stamp her in? Solve a linear system with more variables than equations When exactly can a caster stop concentrating on a spell?