The phase velocity can be faster than light. Some argue that the phase velocity doesn't convey information, but this doesn't convince me.   We can emit a wave of a single one frequency. Then it will move in the space, but the group velocity will be 0. We can convey information in the following way: 2Hz means 1, 1 Hz means 0. We emit single-frequency wave for a second. Then we don't emit anything for a second. And so on.
I understand that information cannot be transmitted at a velocity greater than speed of light. I think of this in terms of the radio broadcast: the station sends out carrier frequencies $\omega_c$ but the actual content is carried in the modulated wave, the sidebands $\omega_c\pm\omega_m$. The modulation envelop has its group velocity and this is the speed at which information is being transmitted. We also know, for example, that x-rays in glass have a phase velocity which is greater than the speed of light in vacuum.   My question is, what exactly is it that is travelling faster than the speed of light?    EDIT: I know it couldn't be anything physical as its superluminal. My problem is, what is it that has absolutely no information content yet we associate a velocity with it.
Correct interpretation of a "double-negative" in American Pop Culture (i.e. "I ain't afraid of no ghost!") Two question on harmonic function I've been struggling with this all day today. I imagine it's not very hard, but my algebra skills are terrible. So, how can I show that if $m&gt;n$ and $a$ is a positive integer, then $$a^{2^n}+1 \mid a^{2^m}-1.$$ I just can't get a coherent picture of what I'm supposed to do. If this too (see yesterday's question) boils down to the Unique Factorization Theorem, I would be very grateful for any intuition on how to think about this! I have a set of coordinate points. The points are distributed in such a way that some points are clustered around a specific area. There is an option to find out mean coordinates(s) of the whole point layer. I would like to know if there is any way in QGIS to find out the mean coordinates of the points from a specific area from a point layer. lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $ I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n  $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n  ) = \limsup a_n \limsup b_n  $$ Help please, it´s not a homework I want to learn. "Try Ubuntu" option is not dislayed I'm going to mount my 47" LCD above my fireplace. I'm not looking for arguments against this decision. I'm looking for wiring solutions. The diagram below should help explain the situation.   _________________________________________________ |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | | p |   | m | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    #################    |     |     | |     |     |    ##             ##    |     |     | |     |     |    ##             ##    |     |     | |    P|     |    ##             ##  m |     |P C  | |_____|_____|____##_____________##____|_____|_____|    | = stud # = fireplace brick P = existing power outlet C = existing cable outlet p = new power outlet m = new media outlet   Essentially, there is an existing power outlet on either side of the fireplace and a cable outlet on the far right side of the fireplace. I'm intending to put my non-TV media equipment on the right side of the wall and the television above the fireplace. This means I need two things above the fireplace: power and HDMI. I don't want to just run an HDMI cable in the wall. I'd prefer to have some sort of conduit to simplify replacing the cable or having a second (with one of these on each end: ). I only need to drill through two studs that are 6" apart above the right side of the fireplace in order to access the vertical space where I want the other end. For the power, I need to drill through two 6"-spaced studs above the left-side of the fireplace, but then another 2 full-height studs.  There are a couple of caveats:   This is an externally facing wall, which means there's insulation. I'm assuming this will make things more difficult. I'd like to refrain from cutting more holes in the wall than is necessary for the outlets (to reduce drywall damage/repair).   I've considered a couple of options:   Cut the openings for the outlets above the fireplace, and use a flexible drill bit to drill through the studs above the fireplace. This minimizes dry-wall damage, but how difficult is this with insulation in the wall, how can I feed the conduit through those holes, and how would I drill through the full-height studs on the left for power? Cut a wide section of dry-wall (4" high?) from the desired media outlet to the right until reaching the gap above the desired lower outlet. This results in some dry-wall damage, but would allow more control for drilling holes in the studs and running conduit. This still doesn't solve the problem of getting through the full-height stud for the power. For the power, I would need to extend the cut-out two studs further.   What should I do? My son is almost 3 (33 months) and for the past few months, he has been holding his poop in, to the point that he won't go for many days a time. When this started, the doctor advised us to give him Miralax to make his stool softer so it comes out more easily. Even with the Miralax, he still managed to hold it for two days at a time sometimes, but at least it was coming out at all, so we continued with it.  Anyway, we've been giving him the Miralax for months now, and now he's (supposed to be) going to preschool in September, which of course requires that he be potty trained. We don't want to hold him back since he's developmentally ready otherwise, so we started potty training a few weeks ago.  He picked up the potty training really quickly – for peeing – but he still holds his poop in no matter what. If he's sitting on the potty, as soon as he's about to do it, he stands up and just keeps holding it in (and once he starts to stand, if we even so much as ask him to sit back down, he screams and cries).  We have tried everything to convince him (or distract him enough) to sit and poop: reading books, singing songs, playing games, playing with toys, offering him snacks/treats (graham crackers), giving him stickers, etc. Nothing works.  As of today, it has been a week since the last time he pooped, with the exception of a little bit today and a little bit yesterday (both of which ended up all over the bathroom floor since he won't sit down). The doctor says now we should give him an enema to force it all to come out, but even so, that won't solve anything going forward since he still won't go in his diaper or on the potty in the first place (what are we going to do, give him an enema every week, indefinitely?).  The issue isn't that he's scared of the potty itself, since (a) he was holding it before we started potty training, and (b) he knows exactly what he's supposed to do, since he has no problem with peeing (he sits down himself, points it out afterward, knows that it gets emptied into the big potty, etc.).  It's that he just won't poop, ever – not in his diaper, not on the potty – and the only reason why anything comes out at all is because we've been giving him the Miralax every day (and even then, these aren't "full" poops, just whatever happens to come out because it's soft enough that he's unable to hold it all in).  I'm at a complete loss as to why this even started, and what if anything we can do that would actually help. Is it a behavioral problem? Medical? Both?? Help!    UPDATE: I just wanted to update this question since a lot has changed since I originally posted it. We discovered there was a specific situation where he would go in his diaper voluntarily: in our bedroom closet with the door closed. So we knew then that the withholding was intentional, since it was now established that he could go if he wanted to.  So we let him continue going in the closet and, over time, found that he usually went around the same time each day. Based on that, we then had him sit on the potty when we knew it was most likely he would need to go. We also managed to find two things that distracted him enough to sit (blowing bubbles and playing with his toy laptop).  By doing this, we were able to get him to go on the potty without really realizing it (so he didn't try to fight it), and then made a big deal out of it every time he did. Over time, that helped establish that that's what he was supposed to do, and now he does so voluntarily. He hasn't pooped in his diaper in probably 3 weeks by now.  I wouldn't say he's "fully" potty trained, in that an adult still needs to go with him and distract him (and occasionally remind him when he needs to go in general), but finally we made progress. Can ordered list produce result that looks like 1.1, 1.2, 1.3 (instead of just 1, 2, 3, ...) with css? Counting in a rectangular grid with obstacles I was asked to find the no. of ways to get from (0,0) to (8,8) with the 4 points of the shaded box unpassable, aka I can't use the 4 points namely (3,3)(4,4)(3,4)(4,3) while on my way to (8,8). Right and up movements.  I know it is something like a pascal triangle and worked out the solution of 4050 ways.  May I know if there are any faster means using permutation formula?  Thank you in advance. I am struggling to find such languages $L_{1}$, $L_{2}$, and $L_{3}$ such that  $$ L_{1} \subseteq L_{2} \subseteq L_{3} $$ where $L_{1}, L_{3} \notin  \mathbb{R}$ and $L_{2} \in  \mathbb{R}$.  I know they exist, I need help finding them. How to find historical stock price for a de-listed or defunct company? With services like Google Finance or Yahoo Finance it is very convenient to find the historical stock price metric of a listed company, but if the company is not listed, I can't find the data.  For example, I want to see the price metrics for Macromedia, which had been listed in 1993 on NASDAQ (MACR), then acquired by Adobe in 2005. How could I find its historical price metrics? Before running a ridge regression model, do I need to preform variable selection? I am currently constructing a model that uses last year's departmental information to predict employee churn for the current year.  I have 55 features and 318 departments in my data set.  A good portion of my independent variables are correlated, and because of this, I believe that performing a ridge regression on my data will lead to optimal predictions when I bring the model into production.  I have studied ridge regression and understand that the lambda coefficient computed for a given predictor can minimize the effect that predictor has on the model to next to nothing.  Does this mean that performing a ridge regression means I don't have to bother with variable selection? If I do need to perform a variable selection technique, would implementing a stepwise regression and then using those selected variables in my ridge regression be a valid approach at variable selection? I already posted this question on stack exchange but was informed that stack exchange was the better platform to ask statistical questions.  I am sorry for the confusion. For which Natural $n\ge2: \phi(n)=n/2$ I have installed Ubuntu 13.04 on my desktop (runs like a charm). Then I used Crouton to install Ubuntu 12. 04 on my chromebook samsung (ARM). It also works quite well.  Here comes my question. Can we have a true Ubuntu (preferably 13.04) on my ARM chromebook?  It seems that the Crouton/Ubuntu is not 100% the real thing.  The best option would be to have the full Ubuntu 13.04 and remove the Chrome OS. Can't execute .out files, getting permission denied I have written a C++ program and complied it to produce a.out file. However, whenever I try to run it, I get Permission Denied. I read that we can use sudo, but I can't quite get it to work. I use something like, sudo "./a.out" but that too doesn't work.  Edit:  Here is the message I get when I try "./a.out".  bash: ./a.out: Permission denied Can a magus with a high BAB cast two spells or attack and cast, with their full attack action? Suppose you are a level 10 Magus or something similar. (Something with a decent BAB progression and spellcasting abilities) When you hit +6/+1, can you cast a spell, then attack, vice-versa, or simply case two spells? Everywhere I look I can't get a clear answer. right now   I have setup /etc/fstab like this:  //192.168.5.167/H /mnt/ssd cifs credentials=/root/.smbreds,noauto,x- systemd.automount 0 0   I want to set it up like first network should be connected and then fstab mount should get mounted. Tony's Private life in Spider-Man: Homecoming I really don't know how to do this in terms of spoilers, but here goes...  After watching the end of Homecoming, it appears that       Tony and Pepper are very much back together.   Is there any in-universe explanation for this change from the way the universe seemed to be in Civil War? I have the following table:     \documentclass[a4paper,preprint,aps,                    unsortedaddress]{revtex4}     \usepackage[top=2.5cm,bottom=2.5cm,                 left=3.5cm,right=2.5cm]{geometry}     \usepackage{longtable,array,booktabs}      \ifdefined\mc        \renewcommand{\mc}{\multicolumn}     \else        \newcommand{\mc}{\multicolumn}     \fi      \begin{document}           \setlength\LTleft{-3.0cm}         \setlength\LTright{-2.0cm}          \scriptsize      \begin{longtable}{ l r @{--} p{1.3cm} &gt;{\centering}p{2.6cm}                         *{4}{p{1.25cm}} p{2.1cm}}                    %% headers and footers      \caption{Table} \\      \hline\hline     \endfirsthead      \mc{9}{c}{\tablename\ \thetable\ -- \emph{Continued from previous page}} \\      \hline     aaaaaaaaaaaaaaaaaaaaaaaa &amp; \mc{2}{c}{bbb--ccc} &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     \hline     \endhead      \hline     \mc{9}{r}{\emph{Continued on next page}} \\     \endfoot      \hline\hline     \endlastfoot      %% body of table          aaaaaaaaaaaaaaaaaaaaaaaa &amp; \mc{2}{c}{bbb--ccc} &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     aaaaaaaaaaaaaaaaaaaaaaaa &amp; bbb &amp; ccc &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     aaaaaaaaaaaaaaaaaaaaaaaa &amp; bbb &amp; ccc &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     aaaaaaaaaaaaaaaaaaaaaaaa &amp; bbb &amp; ccc &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\      \end{longtable}     \end{document}   which I would like to shift to the left by setting \setlength\LTleft{-3.0cm}  and \setlength\LTright{-2.0cm}. Furthermore, I have to apply \scriptsize for the table. However, none of these commands are working with revtex4. Could you please provide a solution to this problem? (This question is the same as in , but I cannot sign up with that username.)  If it is not possible, an alternative solution would also be useful for me. For this case, I attached the first page of my real table.