How can I prove that the following transformation is canonical:  $\begin{cases}\overline{q}_i=\dfrac{q_i}{Q} \\ \overline{p}_i=Qp_i-2Pq_i \end{cases},\ i\in\overline{1,n}$  where $Q=\sum_{i=1}^n q^2_i$ and $P=\sum_{i=1}^n p_i q_i$?  I struggled a lot trying to obtain that the jacobian matrix is symplectic, but it looks too ugly. Is there any easier method the suits to this problem?
We had a couple of examples where we were supposed to calculate the  (CT), but we never actually talked about a condition that decides whether a transformation is a canonical one or not.   Let me give you an example: We had the transformation: $$P=q \cdot \cot(p), \qquad Q=\ln \left(\frac{\sin(p)}{q}\right).$$  How do I see whether this transformation is a canonical one or not?   You don't have to carry out the full calculation, but maybe you can give me a hint what I need to show here?
"Stack Overflow requires external JavaScript from another domain" message What's the difference between JavaScript and Java? What's the difference between JavaScript and Java? I was doing a very simple search:  grep -R Milledgeville ~/Documents   And after some time this error appeared:  grep: memory exhausted   How can I avoid this?  I have 10GB of RAM on my system and few applications running, so I am really surprised a simple grep runs out of memory. ~/Documents is about 100GB and contains all kinds of files.  grep -RI might not have this problem, but I want to search in binary files too. First time using bevel tool, doesn't bevel evenly. What am i doing wrong? First off I'll let you know I'm a total noob at blender I've used 3ds Max up until this point, but the company I'm working for at the moment needs me to use blender so I'm trying to make the transition over.  Literally all I'm making at the moment is a simple book and cannot understand how the bevel tool isn't working similar to how the chamfer tool would in 3ds Max.  I want a nice curved edge for the spine of the book however when I press control + B and try to bevel it bevels the edge unevenly with the new bevel pushing forward.    Is there a way I can get it to bevel evenly like it does in Max? I tried to install ffmpeg in trusty/Ubuntu 14.04 and got the following message:  $sudo apt-get install ffmpeg  Reading package lists... Done Building dependency tree        Reading state information... Done Package ffmpeg is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'ffmpeg' has no installation candidate   Why isn't the package available? Why do we use the simple past but not the present or future in the following expressions:  Don't you think it's time we went a little further Don't you think it's time we ate Don't you think it's time we went home I've browsed the forums and found a number of posts that have addressed this issue, but none of the solutions seem to work for me. I have the following script that I just copied from the bibtex home page to get familiar with it. Instead of the citation number I get a question mark. I compile using Latex+Bibtex+Latex+Latex+PDFLatex+ViewPDF just as has been previously suggested and the problem persists.   \documentclass[11pt]{article} \usepackage{cite}  \begin{document}  \title{My Article} \author{Nobody Jr.} \date{Today} \maketitle  Blablabla said Nobody ~\cite{Nobody06}.  \bibliography{mybib} \bibliographystyle{plain} \end{document}   My bibliography (Bib.bbl)  @misc{ Nobody06, author = "Nobody Jr", title = "My Article", year = "2006" }    Looking at previous posts one thing that is concerning is that my .bbl looks empty as shown below. Further, I don't have a .blg  \begin{thebibliography}{}  \end{thebibliography} What is the difference between an "API," a "framework," an "IDE," and an "engine?" Normality of dependent variable = normality of residuals? This issue seems to rear its ugly head all the time, and I'm trying to decapitate it for my own understanding of statistics (and sanity!).   The assumptions of general linear models (t-test, ANOVA, regression etc.) include the "assumption of normality", but I have found this is rarely described clearly.   I often come across statistics textbooks / manuals / etc. simply stating that the "assumption of normality" applies to each group (i.e., categorical X variables), and we should we examining departures from normality for each group.   Questions:     does the assumption refer to the values of Y or the residuals of Y?  for a particular group, is it possible to have a strongly non-normal distribution of Y values (e.g., skewed) BUT an approximately (or at least more normal) distribution of residuals of Y?   Other sources describe that the assumption pertains to the residuals of the model (in cases where there are groups, e.g. t-tests / ANOVA), and we should be examining departures of normality of these residuals (i.e., only one Q-Q plot/test to run).  does normality of residuals for the model imply normality of residuals for the groups? In other words, should we just examine the model residuals (contrary to instructions in many texts)?  To put this in a context, consider this hypothetical example:   I want to compare tree height (Y) between two populations (X).   In one population the distribution of Y is strongly right-skewed (i.e., most trees short, very few tall), while the other is virtually normal  Height is higher overall in the normally distributed population (suggesting there may be a 'real' difference). Transformation of the data does not substantially improve the distribution of the first    population.  Firstly, is it valid to compare the groups given the radically different height distributions? How do I approach the "assumption of normality" here? Recall height in one population is not normally distributed. Do I examine residuals for both populations separately OR residuals for the model (t-test)?     Please refer to questions by number in replies, experience has shown me people get lost or sidetracked easily (especially me!). Keep in mind I am not a statistician; though I have a reasonably conceptual (i.e., not technical!) understanding of statistics.  P.S., I have searched the archives and read the following threads which have not cemented my understanding: Childrens fantasy, blind monk leading boy who can control the elements Need hint for $\lim_{x\to 0} \frac{(x+1)^\frac{1}{x}-e}{x}$ Please give me some hints or solution for this limit. I had it on my exam and I'm curious how to solve it. $$ \lim_{x\to 0} \frac{(x+1)^\frac{1}{x}-e}{x} $$ How often did Gandalf return to Hobbiton after the events of The Hobbit? It has been a while since I read the Lord of the Rings novels (don't have a copy at the moment) but I seem to recall he went to Hobbiton at least once, as the young hobbits were always excited to see him.   Did he travel there often? I'm one of those exceptions that deviates from the norm: I run CyanogenMod without any Google apps.   Is it possible to download the Stack Exchange Android app .apk directly? Python as a statistics workbench Why current in series circuit is the same? I have read in the internet that the charges do not have any other path to go and they must go through the same in a series circuit,hence the current is same.  It was quite convincing but what confused me was: "A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. Resistors act to reduce current flow..."(according to the Wikipedia). This means that the resistors slow down the rate of flow of charges. By definition, electric current is the rate of flow of charges. Then must not the current be reduced in a resistor even when the amount of charge is same? $\exp(A)$ is orthogonal iff $A$ is skew symmetric Let $A \in M_n(\mathbb R)$ and $\exp(A)=\sum_{i=0}^{\infty}\frac{A^i}{i!}$. How to prove $\exp(A)$ is an orthogonal matrix iff $A^t=-A$?  I can get that $\exp(A)$ is an orthogonal matrix iff $\exp(A^t)=\exp(-A)$, but how to get $A^t=-A$ from it?  What's more, is $\exp(A)$ injective or not? Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$? I believe I see that $a_n = 2^n(a_0+1) - 1$ but am somewhat uncertain where to proceed afterwards. I am a complete beginner at number theory and would appreciate it if someone could point me in the right direction--surely there is some obvious argument I am missing. Wrong posts number on Review section Lately it's happening to me this:  After completing one or more "First Posts" reviews, I go back to the Review Section main page:   And the post counter shows that there are still posts to review. I click the link to review them, but there is nothing.  Refreshing the page doesn't solve the issue: it shows the wrong number again. The title says it all.  If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?  So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?  How should I answer his question? Apex Array Slice Sometimes I like to use a Queueable to perform work on a list of Ids or sObjects over the heaver Batchable Interface.   However, to do this, we need a very solid implementation of an Array slice method.  Most other programming languages have a native slice method, but Apex does not.  While I have written my own version of slice and it's working fine, I would be interested in hearing about other implementations of slice - the more efficient the better.  My version is in this Utilities class on github   It's modelled after the Javascript version described on the mozilla site   I would expect that the usage and output be similar to this:  String[] arr = ['zero', 'one', 'two', 'three']; String[] sliced = Utilities.slice(arr,1, 3);  System.debug(arr);      // ['zero', 'one', 'two', 'three'] System.debug(sliced);   // ['one', 'two']