Using enumerate environment in a figure caption
Using the itemize environment inside a caption
USB WiFi dongle device that can emulate a virtual mass storage device for a home video player Is there a USB WiFi dongle device that can:   Emulate a virtual mass storage device for a home video player But in fact shows the files from a network-shared folder? Proof verification: column rank = row rank $\newcommand{\span}{\operatorname{span}}\newcommand{\rank}{\operatorname{rank}}$I would like to prove that for every martix, column rank = row rank Let $A\in M_{m\times n}$ be some matrix. fix $R_A$ = vector space of the rows of $A$ , $C_A$ = vector space of the columns of $A$  $$\rank (R_A) = dim \ (\span \{R_1, R_2, \ldots, R_m\})$$ $$\rank (C_A) = dim \ (\span \{C_1, C_2, \ldots, C_n\})$$  let $x$ be some vector. $$x\in \operatorname{Null}(A) \Leftrightarrow \forall i  \ \ (1 \leq i \leq m) : \langle x,R_i\rangle = 0$$ (to be clear - I'm referring to the inner product of $x$ with each row of $A$) Using the rank nullity theorem, $\dim \operatorname{Null}(A) =  n - \rank(R_A)$ as $n$ =  number of columns. what I would like to do is to say: $$\dim \operatorname{Null}(A) =  n - \rank(R_A)$$ $$\dim \operatorname{Null}(A) = n - \rank(C_A)$$ therefore,  $\dim \operatorname{Null}(A) = n - \rank(R_A)= n - \rank(C_A)  \Longrightarrow \rank(R_A) = \rank(C_A) $ it that false? is using the rank- nullity  theorem this way is cheating? or just doesn't prove what needs to be proven formally? What's the difference between postfix/smtp and postfix/smtpd A bit confused when reading through the logs of my smtp/mail server, I'm turning this questions to you guys.   In the maillog files I'm seeing entries for postfix/smtp as well as postfix/smtpd. I can also see that there are possible settings for smtp as well as smtpd in the file main.cf  Trying to find some reasonable explanation through google search seemed to be impossible, so here is my question to you guys.  My server was kind of compromised and the spam mails where going through postfix/smtp and not smtpd. Please help me to understand. Thanks in advance! Rotate a long bar in space and get close to (or even beyond) the speed of light $c$ Proof for two sequences producing the maximum value when sorted I am trying to improve my discrete Math skills by doing some proof exercises. But I am struggling to understand how to start proving the following hypothesis:  $$ S = {x}_1{y}_1 + {x}_2{y}_2 + \ldots + {x}_n{y}_n $$ Where ${x}_1,{x}_2\ldots{x}_n$ and ${y}_1,{y}_2\ldots{y}_n $ are ordering of two different sequences of positive real numbers, each containing n elements. S takes its maximum value when ${x}_1,{x}_2\ldots{x}_n$ and ${y}_1,{y}_2\ldots{y}_n $ are sorted in non decreasing order.  I was thinking that I should maybe use a proof by contradiction. Where the negation of the hypothesis is: S takes it minimum value over all ordering of the two sequences when both sequences are sorted into non increasing order.  However I am not sure my negation of the hypothesis is correct or how proceed from there. I do not want the complete proof because I am trying to learn how to do it myself but if some one can tell me how to make a start on this I would be grateful. What is the physical meaning of commutators in quantum mechanics? This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well covered in textbooks, so I would like to know of various perspectives and explanations that I can use when teaching.  The question comes up naturally in what is usually students' second course in quantum physics / quantum mechanics. At that stage one is fairly comfortable with the concept of wavefunctions and with the Schrödinger equation, and has had some limited exposure to operators. One common case, for example, is to explain that some operators commute and that this means the corresponding observables are 'compatible' and that there exists a mutual eigenbasis; the commutation relation is usually expressed as $[A,B]=0$ but no more is said about that object.  This naturally leaves students wondering     what is, exactly, the physical significance of the object $[A,B]$ itself?   and this is not an easy question. I would like answers to address this directly, ideally at a variety of levels of abstraction and required background. Note also that I'm much more interested in the object $[A,B]$ itself than what the consequences and interpretations are when it is zero, as those are far easier and explored in much more depth in most resources.    One reason this is a hard question (and that commutators are such confusing objects for students) is that they serve a variety of purposes, with only thin connecting threads between them (at least as seen from the bottom-up perspective).   Commutation relations are usually expressed in the form $[A,B]=0$ even though, a priori, there appears to be little motivation for the introduction of such terminology. A lot of stock is placed behind the canonical commutation relation $[x,p]=i \hbar$, though it is not always clear what it means.  (In my view, the fundamental principle that this encodes is essentially de Broglie's relation $\lambda=h/p$; this is made rigorous by the  but that's quite a bit to expect a student to grasp at a first go.) From this there is a natural extension to the Heisenberg Uncertainty Principle, which in its general form includes a commutator (and an anticommutator, to make things worse). Canonically-conjugate pairs of observables are often introduced, and this is often aided by observations on commutators. (On the other hand, the energy-time and angle-angular momentum conjugacy relations cannot be expressed in terms of commutators, making things even fuzzier.) Commutators are used very frequently, for example, when studying the angular momentum algebra of quantum mechanics. It is clear they play a big role in encoding symmetries in quantum mechanics but it is hardly made clear how and why, and particularly why the combination $AB-BA$ should be important for symmetry considerations.  This becomes even more important in more rigorous treatments of quantum mechanics, where the specifics of the Hilbert space become less important and the algebra of observable operators takes centre stage. The commutator is the central operation of that algebra, but again it's not very clear why that combination should be special. An analogy is occasionally made to the Poisson brackets of hamiltonian mechanics, but this hardly helps - Poisson brackets are equally mysterious. This also ties the commutator in with time evolution, both on the classical side and via the Heisenberg equation of motion.   I can't think of any more at the moment but they are a huge number of opposing directions which can make everything very confusing, and there is rarely a uniting thread. So: what are commutators, exactly, and why are they so important? Can "Wow!" be a sentence? I washed and dried my passport by mistake, do I need a new one? Around two years ago I washed and dried my passport by mistake at the laundry room. My ID, my old US student visa and my stamps from european countries are still readable but the pages are not "flat" anymore and the front cover is kind of erasing (some parts of the Brazil symbol is gone).   In April I'll travel to US for a conference, I'll need to get a new US visa (B1/tourism visa) but I'm worried if my passport is not "good to go" anymore because it was "washed and dried". Apparently there is an electronic chip in it and I don't know if it is still working.  Should I get a new passport or can I save some money and time and use this one? It expires only in 2019. Open Finder window from current Terminal location? If I am in a specific path in a Terminal window, how can I open that same window in a new Finder window?   Note: This is the opposite of . Tags are not showing up on the right side of the screen This is on Stack Overflow. On Meta I'm able to see what the tags are on the right side of the screen. In fact, I think yesterday or some point earlier today I saw tags on the right side of the screen (I can't screenshot or remember correctly).  It's right above the asked and active heading.  I know this is minor but what happened to it? It might be useful for browsing instead of having to look at the bottom of the question post.    Found an image on Google and it shows the tags on the right side of the screen: CreateService get error: ERROR_INVALID_ADDRESS (0x000001e7) Can not update Bash on Debian 6.0 (Squeeze) I can not update  on a Debian 6.0 (Squeeze) server to get rid of the discovered vulnerability:  bash --version GNU bash, version 4.1.5(1)-release (x86_64-pc-linux-gnu)  apt-get update apt-get install bash Reading package lists... Done Building dependency tree Reading state information... Done bash is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 7 not upgraded.   Can I use Squeeze-LTS for this server just to update Bash? After one week I will be on another server, so I will not make any other updates.  uname -m x86_64  lsb_release -a No LSB modules are available. Distributor ID: Debian Description:    Debian GNU/Linux 6.0.5 (squeeze) Release:        6.0.5 Codename:       squeeze Why is my variable local in one 'while read' loop, but not in another seemingly similar loop? Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$? Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H: hg \not \in H$?  I'm new to abstract algebra and this would appear to be a useful result. However, by definition, a subgroup is nearly closed, and I don't think this directly implies that for any element $g$ from $G$ not in $H$, the composition $gh \not \in H$.  Is this in fact a true statement? When I create a UV sphere and add the subdivision surface modifier, the following wrinkle appears.    How can I avoid it?  I've read the article  but there seems to be no clear solution.  UPDATE 1  I've used the word 'apply' in the first sentence of the initial version of this question. It's misleading since I want to preserve the modifier without 'applying' it. Rather, what I meant is 'adding' the modifier.  UPDATE 2  I followed josh sanfelici's solution, but still there is an artifact at the pole.  Here is the screen: (A rendered view with a sun lamp)    Maybe it's the limitation of the general-purpose subdivision modifier.  BTW, when the cast modifier is added after the subdivision modifier, the sphere becomes perfect (even for my original wrinkled sphere).  But since I want to edit the resulting sphere (for example, scaling X, Y, etc) while modifier is added but not applied, it's not usable. What to do when the correct answer to a question changes over time? My question concerns . I asked the question, and at the time selected the correct answer. However, time passes and libraries get updated, and now a different answer is actually correct.   I should probably select this new answer as the correct answer, but I feel bad for the person that will lose 15 reputation points even though is answer was the a good answer at that given point in time.  Any suggestions? Good day. I'm not too sure how I should go about asking my question, so if it is in anyway confusing, feel free to edit it.   In my Group Theory course we defined and showed a few properties of the commutator subgroup $[G,G]$ of a group $G$. And then we stated and proved a lemma in lecture that I'm having some difficulty wrapping my head around. I'll state it here for clarity.  Lemma a) The commutator subgroup $[G,G]$ of $G$ is a normal subgroup of $G$. b) The quotient group $G / [G,G]$ is the largest abelian quotient of G in the sense that every homomorphism $\varphi : G \to A$ where $A$ is an an abelian group, induces a unique homomorphism $\bar \varphi : G/[G,G] \to A$ so that $\varphi =\bar \varphi \circ \pi$ where $\pi : G \to G/[G,G]$ is the quotient homomorphism.  Now, it's not the proof of this lemma that I'm having trouble with, it's the italicized text that is messing with me. I completely understand what it is that the italicized text is saying, I'm just not too sure why it is that the property gives us the "largeness" that is claimed. I can see that the above forces $G$ to "pass through" $G/[G,G]$ before it "reaches" $A$ (I apologize for the casual terminology), so I can sort of see why $G/[G,G]$ should be the largest quotient, in a non-rigorous sort of way.  Overall I'm very confused. Perhaps I'm missing something very simple. Any help or nudges in the right direction would be much appreciated.   As a side note I did try a search on Google and here on Stack Exchange and couldn't find anything that answered my question. So if I missed something obvious, then my apologies. Thank you in advance. What is this agglomeration of pink cells I found attached to a stone? Singleton: How should it be used The blackboard bold font in the AMSFonts package only has capital letters.  I sometimes wish to use a blackboard bold "1", for which I can use \usepackage{bbold}. But this changes the entire blackboard bold font, and I like the original AMSFonts versions of the capital letters better.  Is there a simple way for me to get \mathbb{1} from one package and the blackboard bold capitals from another?