Say we have the following problem ():      Bobo the amoeba has a 25%, 25%, and 50% chance of producing 0, 1, or 2   offspring, respectively. Each of Bobo’s descendants also have the same   probabilities. What is the probability that Bobo’s lineage dies out?   You can probably the probability of extinction $p$ as  $$p = \frac{1}{2}p^2 + \frac{1}{4}p + \frac{1}{4}$$   Which if you solve leads to solutions 1 and .5. That said, why is the "correct" answer 1/2?  There is a discussion of a similar problem on some of the comments on this thread (), but I don't really understand the reasoning, especially since in this problem we don't have a root above the value of 1.   Thoughts?
I was asked this question during an interview for a trading position with a proprietary trading firm. I would very much like to know the answer to this question and the intuition behind it.  Amoeba Question:  A population of amoebas starts with 1. After 1 period that amoeba can divide into 1, 2, 3, or 0 (it can die) with equal probability. What is the probability that the entire population dies out eventually?
Product of quotient maps and quotient space that is not Hausdorff I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff. How to set the default program for opening files without an extension in Windows? How to set the default program for opening files without an extension in Windows? Is there a way to forget node names between TikZ pictures? I have a document with several TikZ pictures in it.  I generally use simple names for the nodes such as:  \node (A) at .....  \node (B) at .....   and so forth.  Because of this I have name clashes between the various pictures.  That's generally not a problem.  However I find myself in a position where I may or may not have defined a certain node within a given picture.  So I use the following snippet to test whether the node has been defined:  \makeatletter \long\def\ifnodedefined#1#2#3{%%   \@ifundefined{pgf@sh@ns@#1}{#3}{#2}} \makeatother   Unfortunately this fails if I've already created a node by the same name in another picture.  Is there a way to scope the definitions of nodes or is there a way to undefine nodes once you no longer need them?  P.S.  I'm aware that I could embed each picture in a standalone file and import it using \includegraphics.  Unfortunately, this approach will not work for the document I'm creating.  MWE  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc}  \makeatletter \long\def\ifnodedefined#1#2#3{%%   \@ifundefined{pgf@sh@ns@#1}{#3}{#2}} \makeatother  \setlength{\parindent}{0pt} \pagestyle{empty} \begin{document}   There's no node \textbf{D} here, so I draw a triangle:\par \begin{tikzpicture}    \coordinate (A) at (0,0);   \coordinate (B) at (1,0);   \coordinate (C) at (1,1); %  \coordinate (D) at (0,1);    \ifnodedefined{D}     {  \draw (A) foreach \myn in {B,C,D} {  -- (\myn) } --cycle;  }     {  \draw (A) foreach \myn in {B,C}   {  -- (\myn) } --cycle;  }  \end{tikzpicture}     There is a node \textbf{D} here, so I draw a square:\par \begin{tikzpicture}    \coordinate (A) at (0,0);   \coordinate (B) at (1,0);   \coordinate (C) at (1,1);   \coordinate (D) at (0,1);    \ifnodedefined{D}     {  \draw (A) foreach \myn in {B,C,D} {  -- (\myn) } --cycle;  }     {  \draw (A) foreach \myn in {B,C}   {  -- (\myn) } --cycle;  }  \end{tikzpicture}      There isn't supposed to be a node \textbf{D} here.  I'm expecting a triangle, but get a square.\par \begin{tikzpicture}    \coordinate (A) at (0,0);   \coordinate (B) at (1,0);   \coordinate (C) at (1,1); %  \coordinate (D) at (0,1);    \ifnodedefined{D}     {  \draw (A) foreach \myn in {B,C,D} {  -- (\myn) } --cycle;  }     {  \draw (A) foreach \myn in {B,C}   {  -- (\myn) } --cycle;  }  \end{tikzpicture}  \end{document}   produces: I just upgraded my texlive distribution and can no longer insert space in table rows using \\[x pt] syntax when colortbl is loaded.  It seems that recent changes to array.sty might be the problem as this is the only package to change recently.  Can anyone suggest a workaround?  MWE  \listfiles \documentclass{article} \usepackage{colortbl} \begin{document} \begin{tabular}{l}   \hline   No space below with \texttt{colortbl.sty} loaded. \\[10mm]   \hline \end{tabular} \end{document}   File versions in use:   article.cls    2014/09/29 v1.4h Standard LaTeX document class   size10.clo    2014/09/29 v1.4h Standard LaTeX file (size option) colortbl.sty    2012/02/13 v1.0a Color table columns (DPC)    array.sty    2018/04/07 v2.4g Tabular extension package (FMi)    color.sty    2016/07/10 v1.1e Standard LaTeX Color (DPC)    color.cfg    2016/01/02 v1.6 sample color configuration   pdftex.def    2018/01/08 v1.0l Graphics/color driver for pdftex I don't know how to disconnect a face from the rest of the mesh in edit mode.  This isn't like pressing P to separate the selection into a completely different object. I just want to select a face and disconnect/rip it away from the rest of the mesh but keep it as a part of the original object.  One way to do it would be to duplicate the face with ShiftD and then delete the original face. But I suspect there is a "native" command for disconnecting selected faces and I would like to learn it. Were any of the Elves in Tolkien twins? Thinking about the idea of Elves having low numbers overall, got me to wondering: as far as Tolkien's writing, could elves ever have twins?  As a "yes" answer, I need an example of twins (any of JRRT work). They must be full on Elves, not half-Men. As a "no" answer, has to be an explicit statement in his work stating Elves ALWAYS had one child at a time. Anything else is a "maybe, we don't know". What does static/dynamic BVH do? In the render properties panel, under the performance rollout there is a setting for static vs. dynamic BVH.  I am simply wondering what this setting does.  From the :     The raytracing acceleration structure used is a bounding volume hierarchy... This code includes spatial splits to make it more competitive with kd-trees.   Unfortunately, though I know quite a bit about cycles, this is totally Greek to me.  I couldn't really find much else useful on the web other than it may have some impact on animations only.  Can anybody explain this better (maybe starting with BVH in general)? If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful?  Is there a better way of calculating these probabilities rather than simply summing together the combinations of success and failure probabilities? In FreeBSD and also in Linux, how can I get the numerical chmod value of a file? For example, 644 instead of -rw-r--r--? I need an automatic way for a Bash script. I once had the task of determining the following performance parameters from inside a running application:   Total virtual memory available Virtual memory currently used Virtual memory currently used by my process Total RAM available RAM currently used RAM currently used by my process % CPU currently used % CPU currently used by my process   The code had to run on Windows and Linux. Even though this seems to be a standard task, finding the necessary information in the manuals (WIN32 API, GNU docs) as well as on the Internet took me several days, because there's so much incomplete/incorrect/outdated information on this topic to be found out there.  In order to save others from going through the same trouble, I thought it would be a good idea to collect all the scattered information plus what I found by trial and error here in one place. Libraries module. Where to extract? I have a LAMP server with following configuration.  CPU : Intel(R) Xeon(R) CPU   RAM : 32 GB  HDD : 80 GB  Running on virtual environment.  All the things in the server are running smoothly. But I have noticed that the RAM usage is the server is increasing each day without any reason.   This is the current memory usage on my server.  # free -g              total       used       free     shared    buffers     cached Mem:            31         13         17          0          0         10 -/+ buffers/cache:          1         29 Swap:            2          0          2   And you can see the last 8 weeks memory usage below.   And i have executed the following command to find how much memory each process is consuming.  # ps -eo size,pid,user,command --sort -size | awk '{ hr=$1/1024 ; printf("%13.2f Mb ",hr) } { for ( x=4 ; x&lt;=NF ; x++ ) { printf("%s ",$x) } print "" }'          0.00 Mb COMMAND        2253.62 Mb /usr/libexec/mysqld --basedir=/usr --datadir=/var/lib/mysql --user=mysql --pid-file=/var/run/mysqld/mysqld.pid --skip-external-locking --socket=/var/lib/mysql/mysql.sock        1171.66 Mb java -jar /etc/vmagent/bin/VMAgent.jar         138.27 Mb /sbin/audispd         127.75 Mb /usr/sbin/named -u named -c /etc/named.conf -u named -t /var/named/run-root          74.29 Mb auditd          73.03 Mb automount          43.14 Mb /usr/bin/spamd --username=popuser --daemonize --nouser-config --helper-home-dir=/var/qmail --max-children 5 --create-prefs --virtual-config-dir=/var/qmail/mailnames/%d/%l/.spamassassin --pidfile=/var/run/spamd/spamd_full.pid --socketpath=/tmp/spamd_full.sock          43.14 Mb spamd child          43.14 Mb spamd child          37.13 Mb /usr/sbin/httpd          37.02 Mb /usr/sbin/httpd          36.52 Mb /usr/sbin/httpd          36.46 Mb /usr/sbin/httpd          36.23 Mb /usr/sbin/httpd ................   For troubleshooting , i have restarted the MySQL and APACHE servers. But the result was same. Restarting the entire server will fix the issue, but i want to know what is the real issue.  Let me know if you have any idea about this issue.  Thanks,  Arun S What is a Windows Handle? How to install pygments on windows 7 When I run LaTeX file its gives error as  "! Package minted Error: You must have pygmentize installed to use this package", you must have pygments installed. when I go through minted documentation minted itself required pygments, so Tex friend How do I install this pygment, I don't know the procedure to install. Please guid me or any link pygement.exe file How to get a list of appendices? How to get a list of appendices like the list of figures and list of tables? How to set it on a separate page, like the list of figures and list of tables? Using JQuery and Prototype in the same page Several of my pages use both JQuery and Protoype. Since I upgraded to version 1.3 of JQuery this appears to be causing problems, because both libraries define a function named '$'.  JQuery provides a function noConflict() which relinquishes control of $ to other libraries that may be using it. So it seems like I need to go through all my pages that look like this:  &lt;head&gt;           &lt;script type="text/javascript" src="/obp/js/prototype.js"&gt;&lt;/script&gt;     &lt;script type="text/javascript" src="/obp/js/jquery.js"&gt;&lt;/script&gt; &lt;/head&gt;   and change them to look like this:  &lt;head&gt;           &lt;script type="text/javascript" src="/obp/js/prototype.js"&gt;&lt;/script&gt;     &lt;script type="text/javascript" src="/obp/js/jquery.js"&gt;&lt;/script&gt;     &lt;script type="text/javascript"&gt;         jQuery.noConflict();         var $j = jQuery;     &lt;/script&gt; &lt;/head&gt;   I should then be able to use '$' for Prototype and '$j' (or 'jQuery') for JQuery. I'm not entirely happy about duplicating these 2 lines of code in every relevant page, and expect that at some point somebody is likely to forget to add them to a new page. I'd prefer to be able to do the following    Create a file jquery-noconflict.js which "includes" jquery.js and the 2 lines of code shown above Import jquery-noconflict.js (instead of jquery.js) in all my JSP/HTML pages   However, I'm not sure if it's possible to include one JS file in another, in the manner I've described? Of course an alternate solution is simply to add the 2 lines of code above to jquery.js directly, but if I do that I'll need to remember to do it every time I upgrade JQuery. Why did the Jedi want to bring balance to the Force? During the Prequel trilogy, the Jedi wanted to bring balance to the Force, in accordance with a prophecy.  There seemed to be a general agreement that this was a 'good thing', but a disagreement with the assertion that Anakin was the one to do it.  Why did the Jedi Council believe it was good to bring balance to the Force?  The Republic was mighty (in theory), the Council was established as THE source of Force users, the Sith were thought extinct, and the number of Dark Jedi was apparently very, very low.  It seemed, for all intents and purposes, as if the Force was strongly imbalanced in favor of the Light side...so why did the Council want to 'balance' it? Both sides differentiation expression with integral I have a problem with derivation of some function. This is related to computing the Poincare Map of Logistic Population model with Periodic harvesting. This function is:  $\phi(t,x_0) = x_0 + \int_{0}^{t} f(s,\phi(s,x_0)) ds $,  where $\phi(t,x_0)$ is a function: $\mathbb{R}  \times \mathbb{R} \rightarrow \mathbb{R}$.  I'm trying to differentiate  equation of this function both sides  with respect to $x_{0}$. In book from this equation comes from the result of this differentiation is:  $\frac{\partial \phi}{\partial x_0} (t,x_0) = 1 + \int_0^{t} \frac{\partial f}{\partial x_0} (s, \phi(s,x_0)) \cdot \frac{\partial \phi}{\partial x_0} (s, x_0)ds$.  I can't understand expression under the integral. In this book it is explained that it is clear from chain rule.Could someone explain to me this transformation in easy way? I will be grateful for your help Best regards Which pronoun to refer to "person"? It? He or She? They? "When he found his seat on the plane, Sam recognized the person who was sitting in the seat next to his. It? She? was a woman he knew."  Which is more common/natural? Isn't 'it' more correct – grammatically speaking – than 'she', since 'a person' can be a 'he' or a 'she', and obviously you cannot use the singular 'they' here? Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: