Will string theory ever be provable?
What experiment would disprove string theory?
How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$? For my proof I distinguished the two possible cases which derive from $7 \mid a^2+b^2$:   Case one: $7\mid a^2$ and $7 \mid b^2$ Case two (which (I think) is not possible): $7$ does not divide $a^2$ and $7$ does not divide $b^2$, but their sum.  I've shown that case $1$ implies $7\mid a$ and $7\mid b$, so I just have show that case 2 isn't possible - I'll be happy for any input.  MfG, Karl Execute command in remote active terminal Install App Image such as beaker browser and pin it to dock (favorites) in Ubuntu 20.04? Please let me know how to change the size of the camera view. The red dotted line of the square is too small for me. tikz relative coordinates Suppose one defines coordinates a and b. One may draw a line connecting these two coordinates (\draw (a) -- (b)). Suppose one desires to draw a line that is shifted upward 1 unit relative to a and b. A first guess is the command \draw (a){}+(0,1) -- (b){}+(0,1). This command, however, connects the point a+(0,1) to the point b, presumably because addition is subsequent to the draw command. How does one obtain the desired result (drawing from a+(0,1) to b+(0,1)?  MWE:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \begin{tikzpicture}  \draw[fill] (0,0) circle (2pt) coordinate (a);  \draw[fill] (5,0) circle (2pt) coordinate (b);  \draw (a) -- (b);  \draw (a){}+(0,1) -- (b){}+(0,1); \end{tikzpicture} \end{document}     Edit: How does one draw coordinates at a point shifted relative to the arithmetic average of the coordinates of two points (something akin to \node at ($(((s1)+(s2))/2)+(0,1)$)). Am wondering if somewhere there is a list of industry "minimums" for browser support. My company release two "portals" over the past 6 months and did very little to support IE8 users, who at this time, according to my G Analytics makes up about 14% of all IE visitors. Amazingly enough IE7 is 25%...  Figured support for IE7 was out the window already, but figured 14% was significant enough to warrant some support.  Any arguments to support my side of wanting to support IE8?  Any arguments to NOT support IE8? Is there a way to programmatically change the screen resolution or enable/disable multiple monitors in Windows XP?  For example to change from 1024x768 with one monitor to 1280x104 on two monitors?  I would be most interested in a win32 function to do this but anything that can be tied to a windows shortcut would suffice. Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$ I need to proof this following equality :  $$\binom{m+n}{m}=\sum_{i=0}^m \left(\binom{m}{i}\binom{n}{i}\right)$$  This is what I did combinatoric proof:  Left : subset with $m$ members from $m+n$  Right : $\binom{m}{i} \rightarrow$ The number of subsets in set $m$ could also be written as $2^m$ AND $\binom{n}{i} \rightarrow$ the number of subsets with $m$ members from set $n$  In case $ n \leq m \implies \binom{n}{i}=2^n$  So this is what I came up with I still can't see why those expression are equal.  Any ideas? Thanks. Intuitive explanation of convergence in distribution and convergence in probability What is the intuitive difference between a random variable converging in probability versus a random variable converging in distribution?  I've read numerous definitions and mathematical equations, but that does not really help. (Please keep in mind, I am undergraduate student studying econometrics.)  How can a random variable converge to a single number, but also converge to a distribution? I have been given the task to rewrite a function as a sum of an even and an odd function. But when I went to analyze the parts of the original function I noticed that one part was neither even nor odd.   The original function was: $g(x) = (x + 1) / (x^2 - 3x + 4$)  After analyzing I realized that when using f(-x) to determine symmetry it produced a 'neither' result. How would I go about writing a function that is neither even nor odd as an even function. (It would have to be an even function as the (x+1) part of the original function is odd and since I need to write it as a sum of even and odd, the denominator would need to be even. Create Generic method constraining T to an Enum Why isn’t the definite article used in "The commission commanded that work [...] should cease"? In this sentence:     The commission commanded that work on the building should cease.   Why isn't the used before the word work? It seems that the work is specified by on the building.  Another example is,     Construction of the building will start next week.   I understand construction here is something like putting up the building, and so the construction isn’t general in meaning. But the definite article isn’t used here, either.  I’ve asked about this several times, but none gave me a good answer. "Startup Applications" not working Is $\le$ defined by $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2 \land𝑦_1 \ge 𝑦_2$ a linear order? I need to tell if the relation $\le$ defined by relationship $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2$ $\land$ $𝑦_1 \ge 𝑦_2$ linear order?   I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help! How to set up beamer to grey out past items but hide future items? Pagination returns 404 after page 20 Here's a fragment from Jack London's Star Rover:     Wordsworth knew. He was neither seer nor prophet, but just ordinary man like you or any man. What he knew you know, any man knows. But he most aptly stated it in his passage that begins "Not in utter nakedness, not in entire forgetfulness.   To the best of my — admittedly sketchy — knowledge of grammar, the second sentence should read neither a seer nor (a) prophet, but just an ordinary man. Even if we accept the seer and the prophet as "poetic entities" ("father and son," "robber and robbed," "man to man," or whatever), logic would still dictate that ordinary man should be preceded by an an.  It is a well-known fact that Jack London's usage wasn't always up to par; his grammar, on the other hand, was always top-notch.  Logically, the sentence is grammatically incorrect (correct me if I'm wrong: no pun intended).  Illogically, it reads smoothly: intuitively, a native speaker understands that there's nothing wrong here.  However, those to whom English is not the first language cannot be expected to be intuitive in this case and would want an explanation. Any suggestions? I'm new to tikz-qtree, and can't find an automatic way to do something that qtree did very nicely.  With qtree, if you don't have a label for a single intermediate node, you get a straight line from the grandmother to the granddaughter.  With tikz-qtree, you get a "kink".  Compare the following:  With qtree:  \documentclass{article} \usepackage{qtree}  \begin{document}  \Tree [ Wiiiiiiiiiide [ nar- -row ] ]  \Tree [ nrw [ wiiiiiiiiiiiii- -iiiiiiiiiiiide ] ]  \end{document}   With tikz-qtree  \documentclass{article} \usepackage{tikz,tikz-qtree}  \begin{document}  \Tree [ Wiiiiiiiiiide [ nar- -row ] ]  \Tree [ nrw [ wiiiiiiiiiiiii- -iiiiiiiiiiiide ] ]  \end{document}   Is there any way to get the nice straight line in tikz-qtree, or do I have to adjust it manually?  Thanks in advance. How to use `ybar interval` without skipping the last value I have the following code  \documentclass[tikz]{standalone} \usepackage{pgfplots} \usepackage{pgfplotstable} \usepackage{filecontents}  \begin{filecontents}{testdata.dat}   -3.316988309367815e-07 0.07000000000000001   -1.948815911147526e-08 0.13   7.15864175355987e-08 0   1.799432483450127e-07 0   0.0002850606647384129 0   0.0009429559686380238 0.08999999999999997   0.00157188462064142 0   0.005259918700865051 0   0.01739392641273605 0.18   0.06208364670743995 0.55   68.69485148165926 6.970000000000001   31.217611224923 79.12   0 12.89 \end{filecontents}  \pgfplotsset{compat=newest} \pgfplotstableread{testdata.dat}\datatable  \begin{document} \begin{tikzpicture} \begin{axis}[   ybar interval,   width=10cm,   height=5cm,   enlargelimits=false,    xtick=data,   ymin=0, ymax=100,   xticklabels={0 -- 0.063,0.063 -- 0.125,0.125 -- 0.25,0.25 -- 0.5,0.5 -- 1,                1 -- 2,2 -- 4,4 -- 5.6,5.6 -- 8,                8 -- 11.2,11.2 -- 16,16 -- 22.4,22.4 -- 31.5},   x tick label style={rotate=60,anchor=east}   ]   \addplot table[x expr=\coordindex,y=0]{\datatable};   \addplot table[x expr=\coordindex,y=1]{\datatable};  \end{axis} \end{tikzpicture} \end{document}   and the output looks like this:    But the command ybar interval does skip the last value. It should look like:    I prefer to have the last value to show up in the interval 22.4 -- 31.5. How do I achieve this without inserting a row of zeros to my table? What does a "< <(...)" redirection mean? I've seen that rvm (ruby version manager) is installed using the following command:  bash &lt; &lt;(curl -s https://raw.github.com/wayneeseguin/rvm/master/binscripts/rvm-installer )   So as I understand we get the script content and pass it to the bash (I believe &lt; &lt; and &lt;&lt; is the same thing?) I am interested in the &lt; &lt; part, found following description on the net:     &lt;&lt; token    Means use the current input stream as STDIN for the program until token is seen.    This is somehow not clear for me, can someone make an example or explain it in more simple way?