I have this MWE :  \documentclass[tikz]{standalone} \usepackage[utf8]{inputenc} \usetikzlibrary{shapes.geometric,arrows,positioning,fit,calc,backgrounds} \begin{tikzpicture} \node[draw, rectangle] (r1) {R1}; \node[draw, rectangle, right= of r1] (r2) {R2}; \node[draw, rectangle, below= of r1] (r3) {R3}; \end{tikzpicture} \end{document}   I want to have this relation for the x coordinate :  r3.east.x = r2.east.x; r3.west.x = r1.west.x   How can I do that? My idea is to position r3 at south of (r1.west + r2.east)/2 and then calculate the right width.
I have the two PGF nodes foo and bar positioned in a row.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{positioning} \begin{document}   \begin{tikzpicture}[node distance=0.2cm,mynode/.style={rectangle,draw}]     \node[mynode] (foo)                {foo};     \node[mynode] (bar) [right=of foo] {bar};   \end{tikzpicture} \end{document}   How can I create a third node below those two nodes with the width of foo, bar and the node distance between them? Is there a way to do this with the fit library or is this the wrong approach?
Is there any shortcut/easy way to switch this system proxy settings? Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$ Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$.    I'm not sure how to start this proof, if it should be done algebraically or if I should try to use a proof by induction. How to make the Navigation Bar of an UINavigationController display at the bottom? How do I execute the following shell command using the Python  module?  echo "input data" | awk -f script.awk | sort &gt; outfile.txt   The input data will come from a string, so I don't actually need echo.  I've got this far, can anyone explain how I get it to pipe through sort too?  p_awk = subprocess.Popen(["awk","-f","script.awk"],                           stdin=subprocess.PIPE,                           stdout=file("outfile.txt", "w")) p_awk.communicate( "input data" )   UPDATE:  Note that while the accepted answer below doesn't actually answer the question as asked, I believe S.Lott is right and it's better to avoid having to solve that problem in the first place! Terminology of the sequences of promoters in relation to DNA strands I'm studying molecular biology and I'm trying to understand an experiment which shows the importance of promoters in the relative transcription level (RT). The image below comes from Rolf Knippers' book "Molekulare Genetik" (8th edition).    The legend says (among other things):     Die erste Zeile gibt die normale "Wildtyp"-Sequenz der 5'-flankierenden Region wieder.   Which, in english, means something like:     The first line repeats the normal wild-type sequence of the 5' flanking region.   The column on the right gives the relative transcription (RT) level, 1.0 being the highest possible level of transcription. As we can see, the lines where some parts of the "5' region" have been deleted give quite low RT levels, since some regions of the promoter are missing.  My questions are the following:  1) According to  and , I understand that RNA polymerase reads and uses both coding and non-coding strands in order to synthesize RNA. Therefore, how did they manage to use the sequences strands regions had been deleted to make a polymerase "read them" and perform transcription?  2) If the polymerase reads the template strand in the 3' -> 5' sense, shouldn't we talk of "ATAT box" or "TAAC box" instead of "TATA" or "CAAT" boxes? Does it mean that the "promoter" regions they are using on the graphic above are actually on the coding strand?  Thanks a lot for your help. I was really happy when I heard that HoTD would be continued. And even happier when the first new chapter actually came. but now after several months of waiting there still has been no other chapter. So my question is, has HoTD gone on hiatus again ? If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication) Should I use Yes/No or Ok/Cancel on my message box? Semantically, the Yes/No buttons are roughly equivalent to the Ok/Cancel buttons, but in general what would you recommend to use? Should I always use Yes/No or always use Ok/Cancel? Or does it depend on the case? Coprime cofactors of n'th powers are n'th powers, up to associates, for Gaussian integers & UFD I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1):  Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta=\varepsilon\gamma^n$, for $\alpha,\beta$ relatively prime numbers and $\varepsilon$ a unit, implies $\alpha=\varepsilon^{\prime}\xi^n$ and $\beta=\varepsilon^{\prime\prime}\eta^n$, with $\varepsilon^{\prime},\varepsilon^{\prime\prime}$ units.  My attempt: once $\alpha$ and $\beta$ are relatively prime, $\alpha\nmid\beta$ and $\beta\nmid\alpha$. Therefore $N(\alpha)$ and $N(\beta)$ are relatively prime too (where $N(a+ib)=a^2+b^2$). Therefore,  $$N(\alpha\beta)=N(\alpha)N(\beta)=N(\varepsilon)N(\gamma^n)=N(\gamma)^n=(p_1\cdot\ldots\cdot p_r)^n=p_1^n\cdot\ldots\cdot p_r^n$$  And once $N(\alpha)$ and $N(\beta)$ are relatively prime, we have (if necessary, we reorder the $p_i$'s):  $$N(\alpha)=p_1^n\cdot\ldots\cdot p_k^n=(p_1\cdot\ldots\cdot p_k)^n=N(\xi)^n=N(\xi^n)$$ and $$N(\beta)=p_{k+1}^n\cdot\ldots\cdot p_r^n=(p_{k+1}\cdot\ldots\cdot p_r)^n=N(\eta)^n=N(\eta^n)$$  And the result follows.  But I'm not sure about one step. It is $\alpha$ and $\beta$ relatively prime $\Rightarrow$ $N(\alpha)$ and $N(\beta)$ relatively prime. I know that if $\alpha|\beta$ then $N(\alpha)|N(\beta)$, but the previous assertion is not necesseraly true, and if it is, and don't know how to prove it. If it is true, the result follows, unless I made something wrong in the rest of the proof.  That's it. If you know another way of doing the exercise, please show me. Can I get a \widebar without using the mathabx package? One can use \bar to put a bar over a letter in math mode, but often the bar isn't wide enough. An alternative is \overline, but this isn't an accent, so it doesn't take into account the skew of the letter. In the following example, the bar extends too far to the left:    Now the mathabx package offers a \widebar accent, but this has several drawbacks:   mathabx changes a lot of other things, The \widebar sits slightly higher than \bar and \overline, In previewers (checked with kpdf and acroread), the \widebar appears fuzzy, which \bar and \overline don't.   So my question is: How can I put a wide bar over a letter in math mode without the above drawbacks? Night's Watch deserters are sentenced to death, as we've seen in the very first episode of Game of Thrones. A man dressed all in black is apprehended by Stark men and brought before Lord Eddard Stark. Lord Stark sees him as a deserter and sentences him to die (and then executes that death penalty).  However, there are men of the Night's Watch south of the Wall, on a legitimate mission or on some kind of allowed leave. For instance, we have Benjen Stark visiting Winterfell when the king and his party is there. There is also Yoren, who travels around Westeros in search of new recruits.  I could imagine that the Warden of the North (and other lords in the northern part of Westeros) would have a good idea of who would be allowed further south and whose appearance would raise suspicion. But I reckon the southern lords don't have as many visits from the Night's Watch, so they might not know who would be allowed to be there and who wouldn't.  How would lords who are rather unfamiliar with the specific brothers of the Night's Watch discern deserters from brothers sent out for some reason?  That ends my question, but there is a spoiler from the season 5 finale that continues the question a bit more:      Sam is sent to Oldtown to become a Maester, so he can replace Maester Aemon at Castle Black.  But he is riding together with Gilly (a woman!) and her baby (a baby!), which might raise a lot of suspicion from people who don't know what he is sent out to do.  What prevents southern lords of sentencing Sam as a deserter from the Night's Watch? I have 3 fields ( Exposed filter ) 1 Country 2 State 3 City  When a respective Country is selected in the 1st field, I want the second filter ( Dropdown or autocompete ) to display states of the country selected in the 1st field &amp; when this is done based on the state selection , I want the 3rd field to show only the cities belonging to the state selected in the second field.  How can this be done ? any help ? Is "Five dollars is all I have" correct? I want to emphasize that I only have 5 dollars, so can I use the sentence:     Five dollars is all I have.   Or must it be:      Five dollars are all I have.   Please give the explanation as well. Question is :  Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:   $X$ is not connected. $X$ is not compact $X$ is not homeomorphic to a subset of $\mathbb{R}$ None of the above.   I guess first two options are false.  We do have possibility that product of two connected spaces is connected.  So, $X\times X$ is connected if $X$ is connected. So I guess there is no problem.  We do have possibility that product of two compact spaces is compact.  So, $X\times X$ is compact if $X$ is compact. So I guess there is no problem.  I understand that this is not the proof to exclude first two options but I guess the chance is more for them to be false.  So, only thing I have problem with is third option.  I could do nothing for that third option..  I would be thankful if some one can help me out to clear this.  Thank you :) Using a comma before "but" How to make a bootable ISO image from a bootable DVD? We always say that tree levels are classical but loop diagrams are quantum.  Let's talk about a concrete example： $$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$ where $J$ is source.   The equation of motion is $$\Box \phi=-g \phi^3+J$$  Let's do perturbation, $\phi=\sum \phi_{n}$ and $\phi_n \sim \mathcal{O}(g^n) $. And define Green function $G(x)$ as $$\Box G(x) =\delta^4(x)$$  Then   Zero order:  $\Box \phi_0 = J$   $\phi_0(x)=\int d^4y G(x-y) J(y) $  This solution corresponds to the following diagram:  First order:  $\Box \phi_1 = -g \phi_0^3   $  $\phi_1(x)=-g \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 G(x-x_1)G(x_1-x_2)G(x_1-x_3)G(x_1-x_4)J(x_2)J(x_3)J(x_4) $  This solution corresponds to the following diagram:  Second order:  $\Box \phi_2 = -3g \phi_0^2\phi_1  $  $\phi_2(x)= 3g^2 \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 d^4x_5 d^4yd^4z  G(x-y)G(y-x_1)G(y-x_2)G(y-z)G(z-x_3)G(z-x_4)G(z-x_5) J(x_1)J(x_2)J(x_3)J(x_4)J(x_5) $ This solution corresponds to the following diagram:  Therefore, we've proved in brute force that up to 2nd order, only tree level diagram make contribution.  However in principle the first order can have the loop diagram, such as  but it really does not occur in above classical calculation.  My question is:   What's the crucial point in classical calculation, which forbids the loop diagram to occur? Because the classical calculation seems similiar with quantum calculation. How to prove the general claim rigorously that loop diagram will not occur in above classical perturbative calculation. How can I disable the Windows key from opening the Activities overview? How can I disable the Windows key (or Super) from showing the GNOME Shell Activities overlay? $\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$ Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost.  Let $x_1 , x_2, \dots, x_k$ be complex numbers satisfying:  $$x_1 + x_2+ \dots + x_k = 0$$  $$x_1^2 + x_2^2+ \dots + x_k^2 = 0$$  $$x_1^3 + x_2^3+ \dots + x_k^3 = 0$$  $$\dots$$  Then $x_1 = x_2 = \dots = x_k = 0$.    The statement seems obvious because we have more than $k$ constraints (constraints that are in some sense, "independent") on $k$ variables, so they should determine the variables uniquely.  But my attempts so far of formalizing this intuition have failed.  So, how do you prove this statement?  Is there a generalization of my intuition? How did Newton discover his second law? I've always assumed/been told that Newton's 2nd law is an empirical law — it must be discovered by experiment. If this is the case, what experiments did Newton do to discover this? Is it related to his studies of the motion of the moon and earth? Was he able to analyze this data to see that the masses were inversely related to the acceleration, if we assume that the force the moon on the earth is equal to the force the earth exerts on the moon?   According to , the  reads:     Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.   Translated as:      Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.   My question is how did Newton come to this conclusion? I get that he knew from  the idea of inertia, but this doesn't instantly tell us that the change in momentum must be proportional to the net force. Did Newton just assume this, or was there some experiment he performed to tell him this?