This questions does not relate to how to use two passports, but which passport number to put down when purchasing an international ticket online. I have Dutch and SA citizenship..  Which passport number do I give for the ticket - SA or Dutch?  So far I had conflicting answers. My understanding is that you give the passport number of the country that you are leaving and returning to?
I am a citizen of two different countries, and have two passports.  How should I use my passports when traveling?
Prove $\lim\limits_{n\to\infty}n^{-1/(p+1)}(1^{1^p}\cdot 2^{2^p}\cdots n^{n^p})^{n^{-p-1}}=e^{-1/(p+1)^2}$ Prove that for real number $p\geq  0$ we have   $$\lim_{n\to\infty}\frac{(1^{1^p} \cdot2^{2^p}\cdots n^{n^p})^\frac{1}{n^{p+1}}}{n^\frac{1}{(p+1)}} = \exp\left(\dfrac{-1}{(p+1)^2}\right).$$ Note: This was posted when I was starting out C#. With 2014 knowledge, I can truly say that auto-properties are among the best things that ever happened to the C# language.  I am used to create my properties in C# using a private and a public field:  private string title; public string Title {     get { return title;  }     set { title = value;  } }   Now, with  3.0, we got auto-properties:  public string Title { get; set; }   I know this is more a philosophical/subjective questions, but is there any reason to use these auto-properties except from saving five lines of code for each field? My personal gripe is that those properties are hiding stuff from me, and I am not a big fan of black magic.  In fact, the hidden private field does not even show up in the debugger, which is OK given the fact that the get/set functions do nothing. But when I want to actually implement some getter/setter logic, I have to use the private/public pair anyway.  I see the benefit that I save a lot of code (one vs six lines) without losing the ability to change the getter/setter logic later, but then again I can already do that by simply declaring a public field "Public string Title" without the need of the { get; set; } block, thus even saving more code.  So, what am I missing here? Why would anyone actually want to use auto-properties? The new GNU Octave 4.0.1 was released March 23, 2016.  I would like to find some standard way to install it in El Capitan (mine OS X 10.11.3) because it has significantly better support in some functions than Matlab, discussed in the thread .  I do brew search octave and get  homebrew/science/octave    Caskroom/cask/octave       Caskroom/cask/xoctave       where one of the first two should be a correct one. The project's homepage  for OS X is down  about installing Octave on OS X systems.  There is also ruby's way of installing Octave described in the thread  but for Yosemite.  I would like to find some standard way of getting Octave for OS X.   I do successfully brew update and brew upgrade before the commands. My Homebrew is 0.9.5 (git revision 80d1; last commit 2016-03-24).  . I run brew info octave but I get  Error: No available formula with the name "octave"     How can you install Octave in some standard way for El Capitan? Insert eps graphics with graphicx (file not found) How can I add a newline in the text of a label in WPF such as the following?  &lt;Label&gt;Lorem    ipsum&lt;/Label&gt; Comparing two logit or probit curves using a single parameter I've conducted a psychological experiment on the same subject, under two different condition. For each condition I've collected the number of correct and wrong answer for each stimulus (number of trials per stimulus = 10; number of stimulus = 15). From the data collected I fitted a psychometric curve (probit or logit curve), and now I want to compare the results.  Consider subject 1: His answers are collected in the following matrices  **COND_1**     correct cnt   Y  N   1   0 10   2   0 10   3   0 10   4   0 10   5   0 10   6   0 10   7   1  9   8   4  6   9   5  5   10  7  3   11  8  2   12  7  3   13 10  0   14 10  0   15 10  0  **COND_2**     correct cnt   Y  N   1   0 10   2   2  8   3   2  8   4   3  7   5   2  8   6   4  6   7   8  2   8  10  0   9   8  2   10 10  0   11  9  1   12 10  0   13 10  0   14 10  0   15 10  0  COND_1.1 &lt;- matrix(c(0, 2, 2, 3, 2, 4, 8, 10, 8, 10, 9, 10, 10, 10, 10, 10, 8, 8, 7, 8, 6, 2, 0, 2, 0, 1, 0, 0, 0, 0), byrow=F, ncol=2)  COND_1.2 &lt;- matrix(c(0, 0, 0, 0, 0, 0, 1, 4, 5, 7, 8, 7, 10, 10, 10, 10, 10, 10, 10, 10, 10, 9, 6, 5, 3, 2, 3, 0, 0, 0), byrow=F, ncol=2)  cnt &lt;- seq(from=0, to=1.4, by=0.1)  ddprob.1.1 &lt;- glm(COND_1.1 ~ cnt, family = binomial(link = "probit"))  ddprob.2.1 &lt;- glm(COND_2.1 ~ cnt, family = binomial(link = "probit"))     Observing the plot, you can surely say that performance under condition 2 is better then the performance under condition 1. In fact if you compute some parameters like AUC, p25, p50, p75 (the value of the stimulus at which the number of correct answers are 25%, 50% and 75% each) [threshold parameters], you'll see that:  COND_1      COND_2 AUC    &lt;    AUC p25    &gt;    p25 p50    &gt;    p50 p75    &gt;    p75     Now consider subject 2:  COND_1     correct cnt   Y  N   1   0 10   2   0 10   3   0 10   4   0 10   5   3  7   6   3  7   7   4  6   8   5  5   9   9  1   10  9  1   11 10  0   12 10  0   13 10  0   14 10  0   15 10  0  COND_2     correct cnt   Y  N   1   0 10   2   1  9   3   0 10   4   4  6   5   2  8   6   6  4   7   4  6   8   7  3   9   4  6   10  7  3   11  7  3   12 10  0   13  9  1   14 10  0   15 10  0   COND_1.2 &lt;- matrix(c(0, 0, 0, 0, 3, 3, 4, 5, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 7, 7, 6, 5, 1, 1, 0, 0, 0, 0, 0), byrow=F, ncol=2) COND_2.2 &lt;- matrix(c(0, 1, 0, 4, 2, 6, 4, 7, 4, 7, 7, 10, 9, 10, 10, 10, 9, 10, 6, 8, 4, 6, 3, 6, 3, 3, 0, 1, 0, 0), byrow=F, ncol=2) cnt &lt;- seq(from=0, to=1.4, by=0.1)  ddprob.1.1 &lt;- glm(COND_1.2 ~ cnt, family = binomial(link = "probit")) ddprob.2.1 &lt;- glm(COND_2.2 ~ cnt, family = binomial(link = "probit"))   The plotted curves are shown in the image below.    Can you say what performance is better? Under condition 1 or under condition 2? Extrapolating AUC, p25, p50, p75, you have:  COND_1      COND_2 AUC     &gt;   AUC p25     &gt;   p25 p50     =   p50 p75     &lt;   p75   So my question is: Is there a method to say that a generic curve (logit, probit or whatelse) is HIGHER then another? Is there a method to compare (a single number of "performance") that describe the differences of the two curves? My example shows that AUC, p25, p50, p75 are not good parameters. I'd like to compute a single numeric parameter for each curves, to make a simple t-paired test, extrapolated from the distribution seen. I have 16GB RAM. Do I need 32GB swap? I read many places that the rule of thumb for swap space is to double the amount of physical RAM. However, 32 GB does seem a LOT. Do I need that much? Do I need it at all with this high amount of physical RAM? How to create a new object instance from a Type One may not always know the Type of an object at compile-time, but may need to create an instance of the Type.    How do you get a new object instance from a Type? What is spin as it relates to subatomic particles? I often hear about subatomic particles having a property called "spin" but also that it doesn't actually relate to spinning about an axis like you would think. Which particles have spin? What does spin mean if not an actual spinning motion? Are film capacitors affected by DC bias? is there a quick way to sort the items of a select element? Or I have to resort to writing javascript?  Please any ideas.  &lt;select size="4" name="lstALL" multiple="multiple" id="lstALL" tabindex="12" style="font-size:XX-Small;height:95%;width:100%;"&gt; &lt;option value="0"&gt; XXX&lt;/option&gt; &lt;option value="1203"&gt;ABC&lt;/option&gt; &lt;option value="1013"&gt;MMM&lt;/option&gt; &lt;/select&gt; I uploaded a pre-print of an article on arXiv 1 year ago. The article was accepted in SCI journal and currently available online. However, I found a typo mistake in one equation and want to add more discussion related one figure (On researchgate I am receiving a lot of queries regarding that figure). Should i upload a new version of pre-print on arXiv? (How citations will work in this case? as arxiv version is more advanced compare to Journal version) OrI may upload a short document on arXiv as an errata/correction with extra discussion?Or I may address errata and discussion in online page of journal? Actually this journal allows the registered users to comment and start discussion related to published article. Is there a platform-agnostic and filesystem-agnostic method to obtain the full path of the directory from where a program is running using C/C++? Not to be confused with the current working directory. (Please don't suggest libraries unless they're standard ones like clib or STL.)  (If there's no platform/filesystem-agnostic method, suggestions that work in Windows and Linux for specific filesystems are welcome too.) Adding 3 electron spins I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers needed are total spin and total z-component.  But how do you add three 1/2-spins? It should yield 8 different eigenstates. Which quantum numbers do you need to characterise the 8 states?  It is not as easy as using C-G-coefficients and the usual quantum numbers as for the total momentum the doubly degenerate 1/2 state and quadruple degenerate 3/2 state can describe only 6 or the 8 states. You will need an additional quantum number for the degeneracy.  So how do you get the result?  (I actually tried out myself with a large 8x8 matrix. The total spin 1/2 is each doubly degenerate. For the additional quantum number I chose the cyclic permutation. Spin 1/2 states are neither symmetric nor antisymmetric. But what is the usual way to derive this?)  EDIT: For reference I'm adding my results for up to 4 spins from some time ago:   If you recall the basics of quantum mechanics with matrices it is actually a straightforward matrix diagonalization and requires no specialized knowledge. However, you still need to find an additional operator which breaks degeneracy. I chose the cyclic permutation, which seems to do the job. Please refer to the below answer, since I haven't checked all details. Beamer, Babel and Tikz conflict Automatically define macros in LyX documents I'm using LyX all the time and over the last 2 years I've accumulated some very handy macros for my lecture notes. As it is today, every time I start a new document, I copy and paste the macros from one of my other documents. Is it possible, somehow, to automatically load macros for all files?  (I asked this on SO a while back and got no answers. I hope in this new home I'll get a response). How do you square an ideal? For some of you, this question is going to seem extremely basic.  I think I understand what an ideal is. Such as $\langle 2, 1 + \sqrt{-5} \rangle$, it consists of all numbers in this ring of the form $2a + (1 + \sqrt{-5})b$.  But then what is $\langle 2, 1 + \sqrt{-5} \rangle^2$? My first thought was $\langle 4, -4 + 2 \sqrt{-5} \rangle$, but that seems wrong somehow. I also had something in the back of my mind saying $\langle 2 \rangle$, but I'm not sure about that one either.  Then I thought about trying to figure out $\langle 2, 1 + \sqrt{-5} \rangle \langle 2, 1 + \sqrt{-5} \rangle$ when I realized I don't actually understand how to multiply ideals to begin with.  I'm only using $\langle 2, 1 + \sqrt{-5} \rangle$ as an example (though that does draw in one question identified as similar that looks much more relevant than all the questions identified as "Questions that may already have your answer"). In a principal ideal domain, would it be correct to think that $\langle a \rangle \langle b \rangle = \langle ab \rangle$?  Any help would be much appreciated. I want to know where the Metro applications (Mail, People, Messages, ...) executable file path is.  Also, how can I run Mail application when my PC starts? How to self study Linear Algebra I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is indeed the wrong forum for the type of question I'm about to ask, do please say so).  I am a 16 year old guy who is passionate about physics and as a result wants to increase his knowledge in mathematics, the language of physics. I've read and heard a lot about Linear Algebra and how crucial it is to physics and I am deeply motivated to self study this intriguing part of mathematics. However, I have limited knowledge of maths. I (for example) know basic algebra, trig, diff/int calc and some analytical geometry, but I wouldn't say I master these subjects past the high school curriculum. Now my question is: Would you kind people say I am able to self study Linear algebra or is it too tough and/or does it require too much of a math background? And are there any good books out there for BEGINNERS in LA? I found this (free) e-book called: Elementary Linear Algebra by Kenneth Kuttler and another one called 'Linear Algebra: Theory and its applications' also by Kuttler? Are these any good? Or would you recommend other books? If you guys have any tips regarding books for LA but also tips in general, please, I'd appreciate them! When I try to connect to Google Play, it would say "download Google play service". After I downloaded it, it says that it is "not supported by your device, please contact the manufacturer".