In Peskin and Schroder's Quantum field theory text (page 28, the passage below Eq. 2.53), it asserts that for a spacelike interval $(x-y)^2&lt;0$, one can perform a Lorentz transformation taking $(x-y)\rightarrow -(x-y).$ This is argued by noting that both the points $(x-y)$ and $-(x-y)$ lies on the spacelike hyperboloid.   The specific Lorentz transformation matrix $\Lambda^{\mu}{}_{{\nu}}$ that acting on $(x-y)^\mu\equiv \Delta x^\mu$ takes it to $-\Delta x^\mu$ is given by $\Lambda=diag(-1,-1,-1,-1)$. Is that correct?   For $(\Delta x)^2\equiv \Delta x_\mu\Delta x^\mu&gt;0$, why the Lorentz transformation $\Lambda=diag(-1,-1,-1,-1)$ will not take $\Delta x^\mu$ to $-\Delta x^\mu$?
Related post   In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory.  Consider commutator $$ [ \phi(x), \phi(y) ] = D(x-y) - D(y-x) \tag{2.53} $$ where $D(x-y)$ is the two-point correlation function,  $$D(x-y):= \langle 0 | \phi(x) \phi(y) | 0 \rangle  = \int \frac{d^3 p}{ (2\pi)^3} \frac{1}{ 2E_{\mathbf{p}}} e^{-ip(x-y)}\tag{2.50}$$  P&amp;S argued that each term in the right-hand-side of (2.53) is Lorentz invariant, since $$\int \frac{d^3p }{ (2\pi)^3} \frac{1}{2E_{\mathbf{p}}} = \int \frac{ d^4 p }{ (2\pi)^4} (2\pi) \delta(p^2-m^2)|_{p^0&gt;0} \tag{2.40}$$ is Lorentz invariant.  Since there exists a continuous Lorentz transformation in the spacelike interval $(x-y)^2&lt;0 $ such that $(x-y) \rightarrow - (x-y) $ and $D(y-x)=D(x-y)$, (2.53) equals zero in the spacelike interval. In timelike interval, since such continuous Lorentz transformation does not exist, (2.53) is non-zero in general.  My question is, consider a non-continuous Lorentz transmation in the timelike interval, $PT$, namely time reversal times parity transformation. I can also let  $(x-y) \rightarrow - (x-y) $. Why (2.53) in the timelike interval is non-zero?  I guess $PT$ will let (2.40) go to $p^0&lt;0$ branch. But I am not sure if it breaks the Lorentz invariant of (2.40) and (2.50).
Is "shined" correct? If so, is "he shined X on the tree" also correct? Recently, I overhead a former professor of mine use the word shined, a word that makes me grammatically uncomfortable.  She used it as following: "Then, after we shined a light on the other ball, what happened?..."  My first question is: Is using shined mainstream, over what sounds better to my ear, shone?  If so, is there any historical flip-flopping between the two usages?  Secondly, if shined is in fact legit, can it be used transitively?  Saying something like,      The light shined all throughout the night.   while odd, doesn't sound as bad to my ear as,     He shined the light on the ball throughout the night.   The latter sounds quite bad to my ear.  Indeed, à propos the original question, can shone &mdash; the better sounding alternative to my ear &mdash; be used transitively? If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$. How to create posters using LaTeX Please don't throw this one out as a duplicate of  I'm not asking about the plurality of the noun immediately following those two words - I'm interested in exactly why the example below is problematic (context: I have several children, one named John)...     1: One of my children's name is John.   I can't see any obvious reason why singular name there is completely unacceptable, but it certainly doesn't sit well with me. It seems to turn on whether one of my children can be treated collectively as the "subject" of the Saxon genitive 's, but I don't see a problem with...     2: It's actually one of my children's, but you can sleep in this bed.   I realise that in practice we'd normally pluralise names in the first example. But of course, this implies that one of attaches to my children's names, which isn't really the same construction.  So - is there some kind of "rule, principle" debarring #1 above? Or is it just "one of those things"?    (Apologies if my later switch from brothers to children invalidates any comments or answers.) How can I shorten my command line (bash) prompt? Currently it is:   michael@Castle2012-Ubuntu-laptop01:~/Dropnot/webs/rails_v3/linker/spec/controllers$   Outside of renaming my machine and directory structure...  How could I make it be something more like:  michael:controllers$ One frequent aspect of pdflatex output that could be improved is to what extent one can select text output from the generated pdf-file and copy-and-paste it. This is essentially an issue of having the right output encoding. There are at least two important aspects to this:   In the ideal case, Unicode would be the output encoding. Ligature glyphs (such as ﬁ) are ideally de-ligated in the output (here: "fi").   It seems that output encodings are specified through the fontenc package. What is the range of possible output encodings, and most importantly: Is there a way to specify a Unicode output encoding and one that deals with ligatures in the intended way?  (Note: It is important to distinguish output from input encodings. How to use different input encodings in LaTeX has been documented widely. My understanding is that this is best handled by the inputenx package, as it supersedes the inputenc package.)    Quick guide to the solutions: The information in the answers and answer threads is a bit distributed, so here the short summary: one approach is to use \input glyphtounicode together with \pdfgentounicode=1; the other approach is to use cmap/mmap.  Addendum: Sometimes one will need to load the package accsupp and enclose one's macro definition in \BeginAccSupp{method=hex,unicode,ActualText=&lt;codepoint&gt;}[...]\EndAccSupp{} to generate a specific code point. See the caveat about non-BMP code points  and the fix (starting from accsupp version v0.4, 2012/11/18) , which provides the new unichar package option. Given an array of n Objects, let's say it is an array of strings, and it has the following values:  foo[0] = "a"; foo[1] = "cc"; foo[2] = "a"; foo[3] = "dd";   What do I have to do to delete/remove all the strings/objects equal to "a" in the array? "Give a hand" vs "Help" Currently, I'm working in an international company with many non-native speakers (so am I) but with pretty good level of English, I suppose.  I have noticed that my colleagues usually use phrase "give a hand" when asking for  help or even talking about it and seems to be avoiding the strong word "help".  It looks a bit confusing to me especially after I made people laugh when I said  "I will help you" in one of our conversations, don't remember exact context though...  So could you please describe the difference between them? And is there something wrong or hidden meaning with help-based phrase? How to export data files using specific number format Adverb-adjective compound premodifier with or without hyphen? Remarkably low condensation temperature  or  Remarkably-low condensation temperature?  The focus of remarkable is that it is such a low temperature. We are having an office disagreement and any help is much appreciated. I am very confused about the probability formula $E[\min(X,k)]=\int_0^kxf(x)dx+k(1-F(k))$ I understand the proof of this first formula.  I am confused about where he got the first E[min(X, k)] formula. It seems to have just come out of nowhere. And also, can someone explain why the two formulas are correct please? Convergence in Distribution in Order Statistics Let $X_1, X_2, \ldots$ be iid from Exp$(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. (a) Find the limiting distribution of $M_n = Y_1 - \theta\ln(n)$ and $T_n = nY_n$, where $Y_1$ and $Y_n$ represent the first and last order statistics (among $n$ observations) respectively. Answer: I have obtained the CDF of X to be: $$F_x(x) = 1 - e^{-\frac{x}{\theta}}$$ the CDF of $Y_1$ to be: $$F_{Y_1} = 1 = e^{-\frac{xn}{\theta}}$$ the CDF of $M_n$ to be: $$F_{M_n}(x) = \left[1 - \frac{1}{n}e^{-\frac{x}{\theta}}\right]^n$$ with limiting distribution $$e^{-e^{-\frac{x}{\theta}}}$$ the CDF of $T_n$ to be: $$F_{T_n}(x) = 1 - e^{-\frac{x}{\theta}}$$ which is also the limiting distribution. (b) Show that $M_n$ and $T_n$ are asymptotically independent. Answer: I've tried finding the joint PDF for ordered statistics by using the equation $$g_{ij}(y_i, y_j) = \frac{n!}{(i - 1)!(j - i - 1)!(n - j)!}[F(y_i)]^{i - 1}[F(y_j) - F(y_i)]^{j - i - 1}[1 - F(y_j)]^{n - j}f(y_i)f(y_j)$$ however, it is a mess based on my CDFs. Any suggestions on a different approach? 12.04.3 can start only after I press "resume" in rescue mode (every boot) - problem with Nvidia driver I've tried to install maybe ten Linux distros on my new PC, and every time I have problems with video drivers: black screen or visual artefacts.  I was able to install only Ubuntu 12.04.3 64-bit, but also with problems: black screen on startup (sometimes all was normal when I used Live mode before install, sometimes I was not able to load Live Mode - Ubuntu says that I have problems with video and gives me a choice with four elements, one of them "use safe graphics mode", but I even can't choose this - the system is frozen).   In tty I installed nvidia-current, but it didn't help me. Every boot I need to press ESC and enter rescue mode and just press Resume and after this the system boots normally. If I try to do a normal restart I get a black screen after Ubuntu logo again.  I tried switching nvidia driver in the options - I have version 319-updates and version 319 [Recommended], but neither works.  I have Palit GeForce GTX650, all updates. 'Some index files failed to download' while updating Several years ago I heard of something called dummy subjects in high school. It was then stated that, for example, it is a dummy subject when it starts many instances of sentences, e.g.        It is rainy today.   It is hard to imagine how hard is to live alone.   It is not my university degree that matters here.     My question is if we can use it with plural subjects? For example in       If some think that in Mahmoud Dowlatabadi’s works it is only social themes that matter and the writer is ignoring fictional arrays and artistic techniques, at least, endings of his stories show wrongness of this idea.     My question deals with the italicized part. If this usage is OK, can we also write they are social themes that matter? I mean can they also play the role of a dummy subject? A common technique after training, validating and testing the Machine Learning model of preference is to use the complete dataset, including the testing subset, to train a final model to deploy it on, e.g. a product.      My question is: Is it always for the best to do so? What if the   performance actually deteriorates?   For example, let us assume a case where the model scores around 65% in classifying the testing subset. This could mean that either the model is trained insufficiently OR that the testing subset consists of outliers. In the latter case, training the final model with them would decrease its performance and you find out only after deploying it.   Re-phrasing my initial question:     If you had a one-time demonstration of a model, such as deploying it   on embedded electronics on-board an expensive rocket experiment, would you trust a   model that has been re-trained with the test subset in the final step without being   re-tested on its new performance? Will Area 51 be supported in the app? Will  be supported on the Android App in the future? I visit A51 rather frequently and I would like to see it in the App. Chatroom owners can star their own messages I was able to star my own message by pinning it and unpinning it.  Pinning a message adds a star, even to your own messages, but when the pin is removed, the star remains. XTerm only display part of some unicodes and Urxvt displays nothing In my case, (U)XTerm does display some unicodes, but only partially. For exmaple: for letters are: 𝒫φϕ𝒟. Here is a screenshot of UXTerm and Urxvt respectively   It seems that the problem is related to the letter space or letter length. By the way, everything is fine with gnome-terminal.  My current .Xresources file is: Xft.dpi:       120 Xft.antialias: true Xft.hinting:   true Xft.rgba:      rgb Xft.autohint:  false Xft.hintstyle: hintslight Xft.lcdfilter: lcddefault  XTerm*faceName:          Fixed XTerm*faceSize:          14 XTerm*reverseVideo:      false XTerm*selectToClipboard: true  UXTerm.vt100*faceName:          Fixed UXTerm.vt100*faceSize:          14 UXTerm.vt100*reverseVideo:      false UXTerm*selectToClipboard: true  URxvt.font:            xft:Fixed:pixelsize=18  URxvt.depth:                      32 URxvt.background:                 #222D31 URxvt*scrollBar:                  false URxvt*mouseWheelScrollPage:       false URxvt*cursorBlink:                true URxvt*background:                 black URxvt*foreground:                 grey URxvt*saveLines:                  5000  How to make XTerm and Urxvt to display those characters correctly? Is it possible? I am considering buying my first DSLR.  How do I know whether I need a 18-200 lens, 17-85, or 18-55? Or something else?