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What sequence of documents should I read to know "all of" TeX and then LaTeX? I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation  $$D_{n,0}=\left[\frac{n!}{e}\right]$$  where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation that I wrote comes from solving the following recursion, but I don't understand how exactly the author calculated this recursion.  $$\begin {align*}  D_{n+2,0} &amp; =(n+1)(D_{n+1,0}+D_{n,0}) \\ D_{0,0} &amp; = 1 \\ D_{1,0} &amp; = 0  \end {align*} $$ Is there any package or a method to force LaTeX to keep floating environments like table and figure closer to where they are declared? Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens) Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$ Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$.  Writing out several terms, I think the answer is $1/(n+1)$, but I'm struggling to prove this. I would greatly appreciate it if someone could show how to prove this. How to remove snap completely without losing the Chromium browser? I am wondering whether or not there is a way to completely remove snap from Ubuntu 19.10 without losing the ability to install important applications like Chromium.   When I just recently updated to the newest Ubuntu release I realized that the installer programmatically reinstalled snap, although I had manually removed it before. Additionally the installer removed Chromium, which was installed via the repositories, and reinstalled it via snap.   As I don't want snap to be installed on my machines for various reasons my question is if anybody knows a safe way to remove it, and to get the Chromium DEB back to the sources?  Is there a PPA? Could I use a source of an Ubuntu flavor additionally, which didn't remove the Chromium Deb from its sources? Is solving all of the exercises in a textbook a good idea? Is solving all of the exercises in a textbook a good idea? I'm particularly concerned with textbooks on mathematics. I have this obsession that I should solve all of the problems that a textbook has. It takes a lot of time and energy but usually I'm satisfied with the end result being me having a better understanding of that particular subject.   Any similar experience of this sort? How's this going to work in the long-run? I have a symlink with these permissions:  lrwxrwxrwx 1 myuser myuser       38 Aug 18 00:36 npm -&gt; ../lib/node_modules/npm/bin/npm-cli.js*   The symlink is located in a .tar.gz archive. Now when I unpack the tar.gz archive using maven the symlink is no longer valid. I'm therefore trying to reconstruct the symlink. First I create the symlink using ln but how do I set the same permissions as the original symlink? Check for missing fonts/characters in XeLaTeX? Because I began to use many Unicode characters (Japanese, some unusual diacritics, etc.) I switched from LaTeX to XeLaTeX to render my documents. Unfortunately, if a font is missing, the letters are simply left out (for example, when I had forgotten to specify the font for Japanese) without any error (and I can't seem to spot a warning either). I will read the result, but I prefer to get some early warning if some character isn't available. Is there any way to get a warning about characters that have not been rendered? An error would be even better. Which is more appropriate in the following sentence?     "I asked him to not judge her according to (or: based on) her beauty"   vs     "I asked him not to judge her according to (or:based on) her beauty"   or maybe you wouldn't choose either of the above, but something else? Is there a word for happiness made possible by a tragic situation? I frequently find myself needing a word to express happiness that acknowledges a tragic or unfortunate circumstance underpinning that happiness. Preferably the word could have both adjective and verb forms. For example, "Gregory [verb-ed | felt adjective about] the balmy February day, suspecting it was yet another consequence of catastrophic and irreversible climate change."  "Bittersweet" is close, but not precisely right. It implies ambivalence rather than genuine gladness. "Schadenfreude" is also in the neighborhood, but I don't want to imply that the tragic part of the situation is what causes the pleasure. (Also, I recognize that the word may not quite exist in English, and I don't know if this is forbidden for this forum, but I'd welcome a loan word or phrase if there's a better one.) How to remove lines included in one file from another file? Here's an example question from the :     On the last day of a one-week sale, customers numbered 149 through 201 were waited on. How many customers were waited on that day?   Possible answers: 51, 52, 53, 152, 153.   The correct answer here is 53, which is the result of (201-149) + 1 = 53.  What's the reasoning in adding the +1? In the double-slit experiment of electrons (observed by photons), is it correct to say the collapse is caused by the momentum of the photons? I'm working off the article, . I think it's well-written, but I'm not convinced about this part:     So what is causing the change in the particle’s behaviour?      In the example given above, when electrons are fired at a fluorescent screen, when we bombard the electrons with photons the interaction changes the state of the system catastrophically as, despite lacking mass, photons do carry momentum.   Here the author strongly implicates the momentum of photons as being responsible for the collapse. But is that correct to say?  I have read that the  verified that obscuring which-way information restores wave-like behavior. So it feels to me it's more about some "outer" state (perhaps well-characterized as "information") that we don't yet understand that collapses the system, not per se the photon-electron interaction.  Or am I wrong, and in fact for example, the process of obscuring in the quantum eraser experiments somehow "undoes" the photon-electron interaction or restores the quantum state through another interaction? ZF: Difference between POW and SEP? Assign multiple colors to features within a single vector layer How to show $f=g$ almost everywhere  Let two real-valued measurable functions $f$ and $g$ be such that for any measurable set $E$ the integrals $\int_E f\,d\mu$  and  $\int_E g\,d\mu$ coincide.  Could you please help.  It seems obvious. we could define the integrals as charges. can we do sth from here. but I do not know how to approach. Cross-referencing problem if counter of theorem-like environments is shared The problem is when i share numbering the \cref can't show name, even by \crefname and \autoref show all name Equation.  Here is an MWE:  %&amp;latex \documentclass{amsart}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here \usepackage{amsthm} \usepackage{hyperref} \usepackage[nameinlink]{cleveref} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \begin{document}  \numberwithin{equation}{section}  \newtheorem{The}[equation]{Theorem} \crefname{The}{Theorem}{Theorems}  \newtheorem{Dfn}[equation]{Definition} \newtheorem{Lem}[equation]{Lemma} \newtheorem{Prs}[equation]{Proposition} \newtheorem{Crl}[equation]{Corollary} \newtheorem{Rmk}[equation]{Remark}  \begin{The}\label{T1}    Assume ... \end{The}  ... \begin{Dfn}\label{D1}    Assume ... \end{Dfn}  ... \begin{Lem}\label{L1}    Assume ... \end{Lem}  ... \begin{Prs}\label{P1}    Assume ... \end{Prs}  ... \begin{Crl}\label{C1}    Assume ... \end{Crl}  ... \begin{Rmk}\label{R1}    Assume ... \end{Rmk}  \begin{equation}\label{E1}    Assume ... \bigskip \end{equation}  \noindent \cref{T1} \\ \cref{D1} \\ \cref{L1} \\ \cref{P1} \\ \cref{C1} \\ \cref{R1} \\ \cref{E1} \\  \noindent \autoref{T1} \\ \autoref{D1} \\ \autoref{L1} \\ \autoref{P1} \\ \autoref{C1} \\ \autoref{R1} \\ \autoref{E1} \\  \end{document}     How can I solve it? When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\pi^2}\,\Gamma\left(\frac{1}{4}\right)^4 $ is:     $$\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \tfrac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2\tag{A}$$   which might be regarded as a sort of Ahmed's integral under steroids.  I already have a proof of this statement (through Fourier-Legendre expansions), but I would be happy to see a more direct and elementary proof of it, also because it might have some consequences about the moments of $K(x)$ of the form $\int_{0}^{1}K(x)\,x^{m\pm 1/4}\,dx$, which are associated with peculiar hypergeometric functions. I am not very well versed on set theory or syntax, but I thought I knew the basics.  However, in a book about databases I am reading now, the author uses $2^x$ to signify "a set of $x$."  For example, $2^{\text{dogs}}$ is a set of $\text {dogs}$, etc.    The author never really explained this or why he does it, I just picked up the meaning from context.    I am not sure why the exponent operator is used, nor am I sure what the number $2$ has to do with it.  The sets being represented are NOT powers of $2$ (in size)... they come in all sizes.  Is this a valid notation?  I have not seen it anywhere before...