How to write this system in latex?
How to look up a symbol or identify a letter from a math alphabet or other character?
How can we prove that for every pair $N \in \mathbb{N}$, and natural number $\beta\in [2, \infty)$ there exists a unique set of integers $x_i \in [0, \beta -1]$, $k\in [0,\infty)$ such that:  $$N = \sum_{i=0}^{k}\beta^ix_{i}, \quad $$  It's a question if every positive integer has its unique representation in every base $\beta$. Is the validity of above statement somehow related to Horner's rule? Adding script conditionally If you have two envelopes, and ... Suppose you're given two envelopes. Both envelopes have money in them, and you're told that one envelope has twice as much money as the other. Suppose you pick one of the envelopes. Should you switch to the other one?  Intuitively, you don't know anything about either envelope, so it'd be ridiculous to say that you should switch to the other envelope to maximize your expected money.   However, consider this argument. Let $x$ be the amount of money in the envelope you picked. If $y$ is the amount of money in the other envelope, then the expected value equals  $$E(y) = \frac{1}{2}\left(\frac{1}{2}x\right) + \frac{1}{2}\left(2x\right) = \frac{5}{4} x$$  But $5x/4 &gt; x$, so you should switch!    The Wikipedia article says that $x$ stands for two different things, so this reasoning doesn't work. I say this is not a valid resolution.   Consider opening up the envelope that you pick, and finding $\$10$ inside. Then you can run the expected value calculation to get $$E(y) = \frac{1}{2} \cdot \$5+\frac{1}{2} \cdot \$20 = \$12.50$$  This means that if you open one of the envelopes and find $\$10$, you should switch to the other envelope. The $\$10$ doesn't stand for two different things, it literally just means $\$10$.   But you don't have to open up the envelope to run this calculation, you can just imagine what's inside, and run the calculation based on that. This is what "Let $x$ be the amount in the envelope" means. The problem with the argument is not that $x$ stands for two different things.   So what is the problem?    Previous questions on stack exchange have given the resolution that I just said I wasn't satisfied by, so please don't mark this as a duplicate. I want a different resolution, or a more satisfying explanation of why $x$ does stand for two different things.  Apparently there is still research being published about this problem - maybe it isn't so obvious?    I think there's something subtle wrong with the premise. Because there's no uniform probability distribution on $\mathbb{R}$, statements like "random real number" are not well-defined. Likewise, I think "one envelope has twice as much money as the other" assumes some probability distribution on $\mathbb{R}$, and perhaps our expected value calculation assumes that this distribution is uniform, which it cannot be ... Square roots -- positive and negative It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.  I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified? How do I safely investigate a USB stick found in the parking lot at work? I work at an embedded software company. This morning I found a USB stick in the parking lot in front of the building. With all the stories of "dropped USB stick attacks" in mind, I'm obviously not going to just plug it into my laptop. OTOH, I'm curious to know whether this was actually an attempt to compromise our systems, or it's really just an innocent case of somebody accidentally losing a USB stick. How do I safely inspect the USB stick without risking exposure?  I'm worried not just about malware and crafted file system images; there's also stuff like power surge attacks:.  EDIT: Many of the answers seem to assume I want to keep the drive and use it afterwards. I have no interest in that at all, I know USB sticks are cheap, and that it wouldn't be mine to keep anyway. I only want to know whether this was indeed a semi-targeted attack, partly out of curiosity whether this actually happens in real life and not just in security papers, but also so that I could warn my coworkers.  I want to know how I would figure out whether the stick contains malware. And that's not just a matter of looking at the drive contents and seeing a suspicious autorun.inf or a carefully crafted corrupt file system - I very much also want a way to inspect the firmware. I sort-of expected that there were tools for extracting that and comparing to known-good or known-bad binaries. Why aren't search engines indexing my content? This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it? Can the Universe be deterministic and still allow free will? Possible Duplicate:             As asked in the title, can a Universe which is deterministic at its fundamental level still allow free will? Why does the CAN bus use a 120 ohm resistor as the terminating resistor and not any other value? I know the reasons for using terminating resistors on a  and how important it is.  But why 120&nbsp;ohm? How did this value come up? Is there any specific reason to use 120&nbsp;ohm? Identifying a resistor with a weird band combination Prove that : $a_n>0$, Then, $\sum_{n=1}^\infty a_n$ converges iff $\sum_{n=1}^\infty \sin(a_n)$ converges. I am given the problem: Let $a_n&gt;0$, prove $\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty \sin(a_n)$ converges.  We have this problem in a homework, and I don't believe it can be true. The statement being biconditional implies that when $\sum_{n=1}^{\infty} \sin(a_n)$ converges, so too does $\sum_{n=1}^\infty a_n$, but if we take $a_n=\pi n$ or even $a_n=\pi$, then $\sum_{n=1}^\infty \sin(a_n)$ converges while $\sum_{n=1}^\infty a_n$ diverges. Is my thinking incorrect? How to make a teleport command that does not work in the Nether? I have a vanilla server 1.8.7 that uses a book to tp players to spawn or to their home (an invisible armor stand with their uuid connected to it). Problem is, if they set home (place their armor stand) in the overworld and go to the nether and try to tp home (activated using a trigger), it sends them to the same co-ordinates in the nether, and most likely kills them.   I did some digging, and there seems to be a dimension parameter, but I can't quite figure out the syntax or how to get it to work.  I updated the command block to:  /tp @a[score_tpHome_min=1,Dimension=0] 68 68 68   I was hoping it would only tp the player when they were in the overworld (Dimension 0), but it seems to be ignoring that parameter in the nether and still teleporting to those co-ordinates in the nether.   I also tried:  /tp @a[score_tpHome_min=1,Dimension:0] 68 68 68   Which is the way other people were using the parameter in other ways, but that gave an error. Rough Smart objects problem in Photoshop I am using Adobe CC(2015) latest version and the behaviour of the smart objects isn't normal. When I paste a vector from Illustrator to Photoshop, the vector looks low quality or rough.   ORIGINAL  IN PHOTOSHOP   However, if I click over the smart object layer(in Photoshop)and this is opened in Illustrator, the vectors looks as the original one.   VECTOR OPENED IN ILLUSTRATOR FROM PHOTOSHOP SMART OBJECT   How can I solve this? Playstation Plus for PS4 on multiple Console accounts? Let $\mathbb F$ be a field and $\mathbb  K $ be an extension field of $\mathbb F$  such that $\mathbb K$ is algebraically closed.   Let $\mathbb L$ be the field of all elements of $\mathbb K$ which are algebraic over $\mathbb F$. Then $\mathbb L_{|\mathbb F}$ is an algebraic extension.   My question is : Is $\mathbb L$ algebraically closed ?  I am trying to prove the existence of algebraic closure, so please don't assume that every field has an algebraic closure. This bug was found on a couch in . The couch, and the two beds in the room show no signs of other bugs. Is this a bed bug that I should worry about or something harmless? It was slightly larger than a common tick. $\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$ I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to  $\mathbb  Z_5$.   The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \}  \to  \mathbb N$ defined as $N(a+bi)=a^2+b^2$, then I can express any element $z$ in $\mathbb Z[i]$ as $z=(1+2i)q+r$ with $N(r)=1,2,3,4$ or $r=0$.   If I define $f:\mathbb Z[i] \to \mathbb  Z_5$ as $f(x=(1+2i)q_x+r_x)=\overline{r_x}$, then it is clear that $f(x)=0$ if and only if $x \in  \langle 1+2i \rangle$. The problem with this function is that it doesn't satisfy $f(x+y)=f(x)+f(y)$ and it is not surjective for if $r_x=f(x)=3 (5)$, then if $r_x=a+bi$, we have $a^2+b^2=3$, which is absurd.  I don't know what else to do, any suggestions would be appreciated. How does tangzhong (water roux) make bread softer? As I understand it, tangzhong in relation to bread making is an Asian technique in which a portion of the flour in the recipe is pre-cooked with water to make a roux, before being cooled and added to the rest of the ingredients. It is supposed to make the resulting bread very soft, often for days longer than non-tangzhong breads. I thought maybe it works by binding the proteins so gluten development is restricted, but I'd have thought that would inhibit rise, which seems not to be an issue.  Disclaimer: I have not yet made a tangzhong bread, but I believe I have eaten a few... My first attempt is currently doing a bulk rise.  How does tangzhong work exactly? How does it actually change the dough to achieve the distinct softness for which it is so famous? Command not found $ (dollar sign) Any command that starts with $ returns  command not found   How can I fix this?  I am  using 16.10 Ubuntu. Mac-like alt-tab replacement for Windows 7? Is there an alt-tab replacement for Windows 7 that works in the same manner as command-tab and command-tick (tilde) on OS X?  Requirements:  1. Completely keyboard-shortcut-based, no mouse action should be necessary.  2. 2 keyboard shortcuts with the following behavior:  a.) alt+tab:       cycles through open applications (NOT windows... applications, meaning even if i have 10 firefox windows open, I only want to see firefox once in the list that comes up when i hit this shortcut)  b.) alt+`:     cycles through WINDOWS that are instances of the currently active application (meaning if firefox is my active window and i hit shortcut2, it will cycle through those 10 firefox windows, while ignoring any other windows from ANY other application).    Update 08:40 5/21: I tried out eidylon's suggestion, TopDesk.  It doesn't provide any of the functionality described here.  Other apps I've already tried:   VistaSwitcher Switcher Dexpot (awesome app, still using, but doesn't do what I want in this case... also looked for plugins but couldn't find one that does it) UltraMon Nvidia nView   update: Unfortunately I still haven't found a solution for this.  update: made this a community wiki in case anyone is ever able to find a solution for this. Can I answer my own questions, even if I knew the answer before asking?