I'm trying to learn how to produce math diagrams like this in Latex. I haven't had any luck finding stuff online because I don't really know what to search for. Any help?
I've just started using TikZ and need to create a simple triangle commutative diagram like this one.    The closest I've been able to get is this.   How do I get the h arrow to go from A2 to A3? Everything I've tried just causes errors.  Here's my current code:  \begin{tikzcd}[column sep=small] A \arrow{r}{i}  \arrow{rd}{g}    &amp; A3 \arrow{d}{h} \\     &amp; A2 \end{tikzcd}
Can't change font size for GD imagestring() For $X$ and $Y$ random variables; $X$ follows the uniform distribution.  (1): if $Y=-\log X$  (2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}.  Why is this so? Intuitively statement (2) make sense to me. But i'd like a mathematical proof.  -Probably wrong working:  (1) seems to imply $\exp(-Y) = X$ (which is like saying $X$ is exponentially distributed, which is a contradiction, since its actually uniform!); or is it incorrect for me to do this since $X$ and $Y$ are random variables?  Ultimately how do I prove (2)?  Thanks The default PS1 variable on my machine (Kubuntu 13.10) is this:  \[\e]0;\u@\h: \w\a\]${debian_chroot:+($debian_chroot)}\u@\h:\w\$   I'm looking for a reference for how the $PS1 variable works that, at a minimum, allows me to understand the above PS1 variable. This is my first question.   Let $a_1, a_2,\ldots, a_k$ be natural numbers $\leq n$ with $m$ prime factors.  Let $p_1, p_2, \ldots, p_r$ be the prime numbers $\leq n$.   Let $$C_{m,n} = \frac{\sum_{i=1}^{k}a_i}{\sum_{i=1}^{r}p_i}$$ Does the following limit exist when $m\geq 2$? Obviously, when $m=1$,  $C_{1,n}=1$.  $$\lim_{n\to \infty} C_{m,n}$$  I have some guesses but not defendable enough.  My best,  The FM. Static posts page with home.php For my website, I am using a static home page page Home (template front-page.php) and for posts- page News (template home.php)  I need to show all the post titles and exceprts on that page News (templ. home.php), but also show the original title and the_content from that page News itself.  This code below while statement shows all the posts, which is good, but the top the_title also shows the title from the first post, not from the News page itself.  How to achieve what I need?  &lt;?php get_header(); ?&gt;      &lt;?php     the_title();           the_content();     ?&gt;  &lt;?php if(have_posts()) while(have_posts()) : the_post(); ?&gt;          &lt;?php         the_title();         ?&gt;          &lt;?php         the_content();         ?&gt;      &lt;?php     endwhile;     ?&gt;    &lt;?php get_footer(); ?&gt; Explain what is a transistor in plain English! Javascript Remoting - sending sObject as argument There is a  being thrown out of a flask, literally hovering out of the container  My question is how detailed a physical explanation of this phenomenon we have?  Question: More precisely, if we used the best available mechanical physics simulator software, and we tried to simulate this behavior, would the software be able to reproduce the effect or not? My guess is not but I would like to know why not.  Or do we have to content ourselves only with handwaved explanations? I am from the school of thought that you don't understand physics until you have a reasonable certainty that you can reproduce the physics in a computer. addplot3[surf] creates lines instead of a surface I have been trying to create a surface plot but I am unable to get it to work. Any help would be greatly appreciated. I've tried using surf which only produces lines instead of a surface (Fig. 1). Note: Although I was originally able to produce output, thesurf plot no longer compiles: "Package tikz Error". I've tried compiling with both pdflatex and lualatex with -shell-escape flag.    Figure 1  I also tried to use the patch style plot. This produces gibberish (Fig. 2)    Figure 2  MWE:  \documentclass{article} \usepackage{pgfplots} \usepgfplotslibrary{external} \usepgfplotslibrary{patchplots} \tikzexternalize  \begin{document} \begin{figure}[t]   \centering    \tikzsetnextfilename{surface_plot}   \begin{tikzpicture}     \begin{axis} [       height=12cm,       width=15cm,       grid=major,       grid style={dotted},       colormap/jet,       mesh/ordering=x varies     ]       %\addplot3[patch,patch type=biquadratic,shader=faceted interp,patch refines=3] coordinates {       \addplot3[surf] coordinates {  (-6,-6,446.788)  (-5,-6,423.361)  (-4,-6,413.264)  (-3,-6,402.349)  (-2,-6,388.948)  (-1,-6,391.747)  (0,-6,386.874)  (1,-6,423.778)  (2,-6, 439.19)  (3,-6,449.074)  (4,-6,462.155)  (5,-6,484.399)  (6,-6,505.412) (-6,-5, 423.48)  (-5,-5,392.382)  (-4,-5,406.696)  (-3,-5,392.592)  (-2,-5,371.158)  (-1,-5,384.142)  (0,-5,394.666)  (1,-5,408.105)  (2,-5, 422.51)  (3,-5,420.173)  (4,-5,445.816)  (5,-5,464.731)  (6,-5,485.349) (-6,-4,402.646)  (-5,-4,371.879)  (-4,-4,376.239)  (-3,-4,396.874)  (-2,-4,382.761)  (-1,-4,386.665)  (0,-4,397.148)  (1,-4,410.875)  (2,-4,425.998)  (3,-4,385.143)  (4,-4,420.458)  (5,-4,438.432)  (6,-4,    458) (-6,-3,423.219)  (-5,-3,370.582)  (-4,-3,464.786)  (-3,-3,443.921)  (-2,-3,425.196)  (-1,-3,427.138)  (0,-3,438.002)  (1,-3, 453.97)  (2,-3,473.021)  (3,-3,418.583)  (4,-3,435.887)  (5,-3,414.993)  (6,-3,475.148) (-6,-2,401.995)  (-5,-2,385.612)  (-4,-2,438.956)  (-3,-2,418.583)  (-2,-2,400.093)  (-1,-2,400.773)  (0,-2, 410.58)  (1,-2,426.024)  (2,-2, 444.92)  (3,-2,397.131)  (4,-2,413.739)  (5,-2,431.741)  (6,-2,451.267) (-6,-1,382.978)  (-5,-1,335.274)  (-4,-1,416.598)  (-3,-1,396.718)  (-2,-1,378.348)  (-1,-1,377.455)  (0,-1,32.0089)  (1,-1,400.683)  (2,-1,419.918)  (3,-1,377.371)  (4,-1,393.286)  (5,-1,374.648)  (6,-1,429.077) (-6, 0,365.902)  (-5, 0,350.965)  (-4, 0,397.527)  (-3, 0, 378.36)  (-2, 0,360.454)  (-1, 0,358.447)  (0, 0,    363)  (1, 0,381.147)  (2, 0,401.177)  (3, 0,362.155)  (4, 0,378.174)  (5, 0,395.513)  (6, 0, 414.34) (-6, 1,372.511)  (-5, 1,326.945)  (-4, 1,407.215)  (-3, 1,388.646)  (-2, 1,371.945)  (-1, 1,371.003)  (0, 1,380.027)  (1, 1,396.414)  (2, 1,416.602)  (3, 1,374.897)  (4, 1,391.276)  (5, 1, 373.32)  (6, 1,428.284) (-6, 2,391.149)  (-5, 2,376.191)  (-4, 2,406.941)  (-3, 2,369.633)  (-2, 2,393.583)  (-1, 2,394.184)  (0, 2,404.263)  (1, 2,420.588)  (2, 2,  440.5)  (3, 2,416.631)  (4, 2,374.948)  (5, 2,391.855)  (6, 2,418.511) (-6, 3,375.967)  (-5, 3,361.698)  (-4, 3,357.421)  (-3, 3,368.273)  (-2, 3,377.106)  (-1, 3,378.664)  (0, 3,369.175)  (1, 3, 382.42)  (2, 3,397.821)  (3, 3,377.942)  (4, 3,384.944)  (5, 3,387.862)  (6, 3,424.429) (-6, 4,396.478)  (-5, 4,381.554)  (-4, 4,376.162)  (-3, 4,388.843)  (-2, 4,375.881)  (-1, 4,378.384)  (0, 4, 355.14)  (1, 4,403.654)  (2, 4,419.779)  (3, 4,398.599)  (4, 4,407.113)  (5, 4,426.715)  (6, 4,448.245) (-6, 5,409.364)  (-5, 5,380.492)  (-4, 5,387.843)  (-3, 5,360.034)  (-2, 5, 362.23)  (-1, 5,364.909)  (0, 5,358.321)  (1, 5,363.226)  (2, 5,379.325)  (3, 5,397.152)  (4, 5,410.086)  (5, 5,433.909)  (6, 5,460.548) (-6, 6, 423.99)  (-5, 6,408.967)  (-4, 6,400.475)  (-3, 6,372.922)  (-2, 6,371.116)  (-1, 6, 373.92)  (0, 6,384.778)  (1, 6,  398.8)  (2, 6, 415.62)  (3, 6,420.127)  (4, 6,436.064)  (5, 6,461.957)  (6, 6,491.123)        };     \end{axis}   \end{tikzpicture}      \end{figure} \end{document}   Thank you so much. Applying Zorn's Lemma to natural numbers Zorn's Lemma states that if every chain $C$ in a partially ordered set $X$ has an upper bound then there is at least one maximal element in $X$.  Let $n\in \mathbb{N}$, $C_n \subset \mathbb{N}$ be the set containing the natural numbers $\{1,\dots,n\}$. Then if $k\le n$, $C_k\subset C_n$, so $C_n$ is a chain. Also, $C_n$ has an upper bound, such as $n+1$. But what is the maximal element of $\mathbb{N}$?  There's something I don't get in Zorn's Lemma statement. Mouse cursor flickering and disappearing Python does not find packages installed without Anaconda I'm trying to install kivy for python 3 using apt-get:      sudo apt-get install python3-kivy   And it is installed in      /usr/lib/python3/dist-packages   But, when I tried to load it in python, I get a ModuleNotFoundError      Python 3.6.3 |Anaconda custom (64-bit)| (default, Oct 13 2017, 12:02:49)      [GCC 7.2.0] on linux     Type "help", "copyright", "credits" or "license" for more information.     &gt;&gt;&gt; import kivy     Traceback (most recent call last):       File "&lt;stdin&gt;", line 1, in &lt;module&gt;     ModuleNotFoundError: No module named 'kivy'   When I check the system path from python, I only get packages in Anaconda:      &gt;&gt;&gt; import sys     &gt;&gt;&gt; print('\n'.join(sys.path))      /home/santiago/anaconda3/lib/python36.zip     /home/santiago/anaconda3/lib/python3.6     /home/santiago/anaconda3/lib/python3.6/lib-dynload     /home/santiago/anaconda3/lib/python3.6/site-packages     /home/santiago/anaconda3/lib/python3.6/site-packages/torchvision-0.2.0-py3.6.egg   Is there any way to include the '/usr/lib/python3' path to Anaconda?  Thanks. Why Was It 'Essential' That Voldemort Kill Harry Potter? ‘So the boy ... the boy must die?’ asked Snape, quite calmly.   ‘And Voldemort himself must do it, Severus. That is essential.’   Deathly Hallows - page 551 - UK Hardcover - chapter 33, The Prince's Tale   Dumbledore tells Snape it is 'essential' that Voldemort be the one to kill Harry, I'm assuming in order for the piece of Voldemort's soul in Harry to be properly killed as well.  But why?   Hermione destroyed the Hufflepuff cup; Ron destroyed the Slytherin locket; Neville killed Nagini; Harry destroyed the diadem and Tom Riddle's diary. So the Horcruxes were not immune to destruction at the hand of someone other than Voldemort. And as it ended up, Harry himself wasn't even a Horcrux, but rather just an unknowing host to a parasitic bit of Voldemort's soul.   Why was it 'essential' that Voldemort be the one to kill Harry in order for the piece of Voldemort's soul to die? ToList()-- does it create a new list? I've googled and also searched the answers here but couldn't find my problem so here goes:  I know how to bevel a curve using another object and when I try it with a normal circle it's fine. But I'm trying to use this flat oval shape to bevel a larger curve of the same shape and it's not following the curve at the correct rotation. I've tried everything, rotating the bevel object, clicking random options in the object data menu.. I can't seem to budge it from looking how it is in the screenshot. In the screenshot you can see the curve I've failed to bevel correctly and also the bevel object in the bottom right.  Basically, I want the flat part of the oval to be above and below the curve, not on the sides as it is. I hope that makes sense. Anyone know what I need to do? Pluralization rule for "five-year-old children", "20 pound note", "10 mile run" Editing a starred post in chat breaks display for user who starred it Steps to reproduce:   User A posts a message User B stars it, the star next to the post appears yellow User A edits the post The display of the star in User B's view becomes black, as if they haven't starred it   This is the worst bug ever in the history of StackExchange and makes the entire network completely unusable for me. I'm going to Yahoo! Answers until this is fixed. Let $A$ be a $m \times n$ matrix with entries from some field $F$. Define the determinant rank of $A$ to be the largest possible size of a nonzero minor, i.e. the size of the largest invertible square submatrix of $A$. It is true that the determinant rank is equal to the rank of a matrix, which we define to be the dimension of the row/column space.  It's not difficult to see that $\text{rank} \geq \text{determinant rank}$. If some submatrix of $A$ is invertible, then its columns/rows are linearly indepedent, which implies that the corresponding rows/columns of $A$ are also linearly indepedent.  Is there a nice proof for the converse? How can I swap or replace multiple strings in code at the same time? Why did JJ Abrams recreate the Star Trek Timeline? In the 2009 version of Star Trek, Nero and the Romulans come back through time via Blackhole and kill Kirk's dad and the original Star Trek timeline is altered  creating a whole new reality.  Why would Abrams not just stick to the timeline given by Roddenberry instead of altering it?