Draw two symmetrical pseudo root nodes for binary tree using tikz
Stern-Brocot tree where each node has two fathers
I am very new to LaTeX and I am writing my thesis. In a chapter, I want to show my codes, which are in Mathematica.   I use the listings package as follows:  \documentclass{article}   \usepackage{listings}   \begin{document}   \lstset{language=Mathematica}    \begin{lstlisting}   Do[i^j,{i,1,10},{j,1,10}]   \end{lstlisting}    \end{document}   However, the output is much different to what the code looks like in the Mathematica IDE (even if I use other languages in \lstset{language=&lt;name&gt;}):    I read some questions on the site but, in some of them, the font or color of the codes were not the same. Could you please help me to display something like the picture below in LaTeX?  If it is important, I am using the BiDi package. What is the right tool to process big .xel files (SQL Server Extended Events logs)? My aim is to audit all queries on an instance, who's running them and from where, how many reads, in order to consolidate/migrate on another instance.   Figured out the best way is via Extended Events to .xel files, over 2 weeks, copy those and analyse on my well-specd desktop. Set up the capture with the minimal number of details (fields), so really can't make it smaller than this - aprox 12GB in 12 files. Used "Merge Extended Event Files" in SSMS to load the .xel files and process as detailed in   Is this really the best way to do it ? Is there a better tool to load the files ? It's really slow with SSMS 17.7 - example: with only 4 filters so far (before getting into grouping and aggregates) it took over 12h to get to crunch through 50% of the logs. What are words like "Hey Man", "Hey you", "Hey Dude" called Is there a term for words that summon someone like "Man," "Dude," "Kid" in sentences such as "Hey, Dude," "Listen up, kid," etc?    I am sure there is, but I can't really remember what it is. pgfplots - draw boxplot without outliers I am creating a boxplot with pgfplots, using the builtin statistics library. I load the input data from a table file.  \usepackage{pgfplots} \usepgfplotslibrary{statistics} ... \begin{tikzpicture} \begin{axis}[boxplot/draw direction=y]    \addplot+[boxplot] table[      col sep=tab,      x expr=\coordindex,      y=aColName] {aFileName}; \end{axis} \end{tikzpicture}   Is it possible to draw the plot without outliers? Or hide them? I don't want to pre-calculate the boxplot values by hand. I have an application that reads a CSV file with piles of data rows.  I give the user a summary of the number of rows based on types of data, but I want to make sure that I don't read in too many rows of data and cause OutOfMemoryErrors.  Each row translates into an object.  Is there an easy way to find out the size of that object programmatically?  Is there a reference that defines how large primitive types and object references are for a VM?  Right now, I have code that says read up to 32,000 rows, but I'd also like to have code that says read as many rows as possible until I've used 32MB of memory.  Maybe that is a different question, but I'd still like to know. What is a word for a person with more expertise (in certain field) than an intermediate but less so than an expert? Characterising reals with terminating decimal expansions Show that a number has a terminating decimal expansion if and only if, it is rational and when in lowest terms, its denominator is coprime to all primes other than $2$ and $5$.   This is an unsolved question in my lecture notes. I can only seem to prove the converse direction for this. Would appreciate a solution for the other direction.     For converse direction:   Let the number, in lowest terms, be given by $\frac{l}{m}$ where $m = 2^ \alpha 5^\beta$, for some positive integers, $\alpha$ and $\beta$.    If $\alpha &gt; \beta$, let $k = 5^{(\alpha-\beta)}$. If $\alpha &lt; \beta$, let $k = 2^{(\beta-\alpha)}$.   Then $\frac{l}{m} = \frac{kl}{k2^\alpha5^\beta}=\frac{kl}{10^q}$ where $q = \max(\alpha,\beta)$.   Hence $\frac{l}{m}$ is a terminating decimal. Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$ Which command should I use for textual subscripts in math mode? Textual subscripts like in W_{total} should be typeset in an upright text font (so that code isn't good practice), as has been  in   . Now there are several ways to achieve this, and I'm somewhat confused which I should use:   W_{\rm total} is nice and short, and appears to work in all circumstances, but I expect LaTeX people will frown upon it :-) Is there any good reason against this? W_{\mathrm{total}} &ndash; is this the LaTeX equivalent of the \rm approach? W_{\textnormal{total}}  as total is text and not math&nbsp;... In , Ulrike Fischer suggests a more complicated approach. Even W_{\operatorname{total}} appears to work, but it seems inappropriate.   Note: I always assume here that amsmath is loaded.  I've already asked some questions above, but my main question is: which of these options is the best practice for textual subscripts?    Let me point out why I haven't mentioned \text, \textrm or \textup above. Those commands (mostly) keep the font of the context they're used in, e.g., in an italic context, \text and \textrm would give italic subscripts, and all three yield bold subscripts in a bold context.  Final note: This question is similar to, but not the same as all those  about the  between , , , ,  that have been asked before. Maybe it's still a duplicate&nbsp;... Travel to Romania with a residence permit of Germany I am a student in Germany and i have a residence permit here for 2 years, which is also a Schengen visa, I suppose. Do I need a separate visa if i wish  to travel to Romania? Remove suggested apps option in iOS 10 When visiting my profile on the mobile app, my hat scrolls with the page:    It doesn't stay linked to my icon at the top. I'm on an iPhone 6+ running iOS 9.2 and version 1.4.1 of the SE app. Given a collection of sets $\mathcal{C}$ and $E$ an element in the $\sigma$-algebra generated by $\mathcal{C}$, how do I show that $\exists$ a countable subcollection $\mathcal{C_0} \subset \mathcal{C}$ such that $E$ is an element of the $\sigma$-algebra, $\mathcal{A}$ generated by $\mathcal{C_0}$?    The hint says to let $H$ be the union of all $\sigma$-algebras generated by countable subsets of $\mathcal{C}$....although I don't know why. How do you declare an interface in C++? How do I setup a class that represents an interface?  Is this just an abstract base class? How do you handle the varying sizes of filters in your kit? I have a variety of Nikon DX lenses that take differently sized filters. Most take 52mm filters, but there are a few that take larger 67mm or 77mm filters. The advice I've read most commonly online is to buy one filter size and get filter conversion rings to step up to that size filter. In my case, the largest size appears to be 77mm.  So, I walked into a couple local photo shops and asked about 52mm-77mm filter conversion rings and they looked at me like I had two heads. One guy suggested I stack a 52mm-67m and a 67mm-77mm. I have found 52mm-77mm rings online, so I know they exist.  Is this not a common thing to do? How does the stack exchange community handle the varying sizes of filters in their kit?  Thanks, PaulH How to find solutions of linear Diophantine ax + by = c? I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan Irreducibility of reciprocal polynomials Given that  $f(x)=\sum\limits_{k=0}^n a_kx^k$ is an irreducible polynomial, show that $g(x)=\sum\limits_{k=0}^n a_kx^{n-k}$ is irreducible too. I don't see how this specific coefficient exchange would preserve the irreducibility. Is there something else I should be looking at? Find the Galois group of $x^3-5$ over $\mathbb{Q}$. In this case, the roots of $x^3-5$ are $\{\sqrt[3]{5},\omega\sqrt[3]{5},\omega^2\sqrt[3]{5}\}.$ I think $\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5})$ is the splitting field of $x^3-5.$ Then, $[\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5}):\mathbb{Q}]=3,$ I think the automorphisms are: $$id$$ $$\sigma:\sigma(\sqrt[3]{5})=-\sqrt[3]{5}$$ $$\tau:\tau(\omega\sqrt[3]{5})=-\omega\sqrt[3]{5}$$ I'm not sure about what I have done. But how to determine the Galois group based on this? And how to determine the solvability of this polynomial? If I'm not wrong, torque is perpendicular to both the radius and force i. e. It is along the axis of rotation. Questions that arise are- why do we consider the length between the axis/point of rotation while calculating torque? More importantly why is torque a cross product? What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is arbitrary. So saying $\sqrt{-1} = i$ is plainly false!  So why do people say $\sqrt{-1} = i$? Is this how it's taught in the US?