Does the article "a/an" apply to the parenthetical or the word that follows?
"A/An" preceding a parenthetical statement
Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? I think the answer is yes.   Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then $\{(e_{\lambda_1},e_{\lambda_2}):\lambda_1,\lambda_2\in\Lambda\}\subset\mathbb{R}^2$ is a Hamel basis of $\mathbb{R}^2$. Consider set theoretic bijection $i:\Lambda\to\Lambda\times\Lambda$ define map $\varphi:\mathbb{R}\to\mathbb{R}^2$ by its action on element of Hamel basis of $\mathbb{R}$ by equality $\varphi(e_\lambda)=(e_{i(\lambda)_1},e_{i(\lambda)_2})$. This is an isomorphism between $\mathbb{Q}$-vector spaces $\mathbb{R}$ and $\mathbb{R}^2$.  Question Could you tell me is this proof correct, and if it is not, where is the mistake?  EDIT: Thanks to GAJO we found that this proof is wrong. The correct one basis is  $$ \{(e_\lambda,0):\lambda\in \Lambda\}\cup\{(0,e_\lambda):\lambda\in\Lambda\} $$ So we had to look at bijections of the form $i:\Lambda\to\Lambda\coprod\Lambda$ and the desired linear operator is defined by equalities $\varphi(e_\lambda)=(e_{i(\lambda)},0)$ if $i(\lambda)$ lies in the first copy $\Lambda$ and $\varphi(e_\lambda)=(0,e_{i(\lambda)})$ otherwise. I want to have a page-filling and -breaking longtable, so I decided to try xltabular. Unfortunately xltabular seems to screw up the booktabs rules in the normal tabular environments, see:  \documentclass{article} \usepackage{booktabs, xltabular} \usepackage[active,tightpage]{preview} \PreviewEnvironment{tabular} \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}     Not loading xltabular resolves the issue.  Did I miss some package option or some incompatibilities? Is there a better package than xltabular which doesn't break booktabs?  (I am using TeXLive 2019 and compile with lualatex)    EDIT (by Skillmon)  I reduced the bug to the following MWE (which doesn't make much sense on its own), it looks like booktabs tries to be smart detecting something, but fails badly.  \documentclass{article} \usepackage{longtable} \let\xltabular\relax \usepackage{booktabs}  \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}   It seems to suffice that longtable is loaded and \xltabular is defined (even if it's just \relax). Is zero odd or even? \hfill \break does not work I want to plot this graph, but I don't know how to code the words in the nodes.     Here is my code:  \documentclass{article} \usepackage{amsmath} \usepackage{titletoc} \usepackage{tikz} \usetikzlibrary{trees,positioning,fit}  \begin{document}{\allowdisplaybreaks[4]}  \tikzstyle{bag} = [text width=2em, text centered] \tikzstyle{end} = []  \begin{center} \begin{tikzpicture}[sloped] \node (a) at ( 0,0) [bag] {$\ln r_0$}; \node (b1) at ( 3,1.5) [bag] {$\ln r_{10} + \sigma_1\sqrt{\Delta t_1}$}; \node (b2) at ( 3,-1.5) [bag] {$\ln r_{10}$}; \node (c1) at ( 6,3) [bag] {$\ln r_{20} + 2\sigma_2\sqrt{\Delta t_2}$}; \node (c2) at ( 6,1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c3) at ( 6,0) [bag] {$\ln r_{20} + \sigma_2\sqrt{\Delta t_2}$}; \node (c4) at ( 6,-1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c5) at ( 6,-3) [bag] {$\ln r_{20}$}; \draw [-&gt;] (a) to node [below] {} (b1); \draw [-&gt;] (a) to node [above] {} (b2); \draw [-&gt;] (b1) to node [below] {} (c1); \draw [-&gt;] (b1) to node [below] {} (c2); \draw [-&gt;] (b1) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c4); \draw [-&gt;] (b2) to node [below] {} (c5); \end{tikzpicture} \end{center}  \end{document}       Update code use  \draw [-&gt;] (a) to node [below] {} (b1.south east); \draw [-&gt;] (a) to node [above] {} (b2.north east); \draw [-&gt;] (b1.south east) to node [below] {} (c1.west); \draw [-&gt;] (b1.south east) to node [below] {} (c2.west); \draw [-&gt;] (b1.south east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c4.west); \draw [-&gt;] (b2.north east) to node [below] {} (c5.west); Showing inequality for harmonic series. I want to show that $$\log N&lt;\sum_{n=1}^{N}\frac{1}{n}&lt;1+\log N.$$ But I don't know how to show this. Difference between generalized linear models & generalized linear mixed models I am wondering what the differences are between mixed and unmixed GLMs. For instance, in SPSS the drop down menu allows users to fit either:   analyze-&gt; generalized linear models-&gt; generalized linear models &amp; analyze-&gt; mixed models-&gt; generalized linear     Do they deal with missing values differently?    My dependent variable is binary and I have several categorical and continuous independent variables. Looking out into the universe means looking back in time - how does that work? This is a question that has been gnawing on me for many years now. Back a long time ago, as I recall in reference to a scene in a popular science show on TV, I was asked the following.  The claim is that when you look out into the universe, you see the universe as it appeared at some past time. (This follows from the speed of light being finite, and looking at objects by observing their emitted light or other EM radiation.) The amount of time that one looks backwards is equal to the distance to the observed object in light-distance (so looking at something ten lightyears away means you are observing it as it was ten years ago).  Matter moves at a rate of speed less than the speed of light (it has to, since matter has mass, even if miniscule at the atomic level). So $v_{mass} \lt v_{EM}$ (probably significantly less, since $v_{EM} = c$).  So let's say you're looking at an object $10^{10}$ lightyears away by observing something for which $v=c$. To the observer, that object appears as it was $10^{10}$ years ago. But the Earth is much younger than $10^{10}$ years and we established that matter moves at a slower speed than EM radiation, so wouldn't the radiation that was emitted $10^{10}$ years ago long since have passed Earth's current position?  If, say, $v_{mass} = 0.5 \times c$ (a big assumption, but bear with me for a second), it would seem that the radiation emitted $10^{10}$ years ago would have passed Earth's current position $10^{10} - (0.5 \times 10^{10}) = 5 \times 10^{9}$ years ago, around the time when the solar system was still forming. Setting $v_{mass} = 0.25 \times c$ (which seems more realistic) means the radiation would have passed "us" around $7.5 \times 10^{9}$ years ago. So how could we be observing it now?  I'm not sure I'm posing this question in the best possible way (I'll admit I do find the concept somewhat confusing), and I'm sure that there's a simple explanation for it all. Just what am I missing?  I did find  which seems peripherally related but not quite the same thing. Qmechanic brought up  but that question seems to be about whether it is so, not why it is so. What is the difference between the "jessie" and "jessie-updates" distributions in /etc/apt/sources.list? The  gives this example file:  deb http://httpredir.debian.org/debian jessie main deb-src http://httpredir.debian.org/debian jessie main  deb http://httpredir.debian.org/debian jessie-updates main deb-src http://httpredir.debian.org/debian jessie-updates main  deb http://security.debian.org/ jessie/updates main deb-src http://security.debian.org/ jessie/updates main   What is the difference between the jessie and jessie-updates distribution entries? It used to be that the new, super wide, touch friendly context menu style that Google introduced a few versions of Chrome ago could be turned off using the start up flag "--disable-new-menu-style".  As of Chrome 28, this is no longer the case, and as a desktop user it is very irritating to see menus that previously fit in the screen not to do so anymore — I don't think I can get used to it. Is there any way around it? If the flag is gone for good, can the style of the context menus be perhaps edited with custom css like the web inspector? Why does a star beyond a certain mass limit (Chandrasekhar limit) only become a black hole? Why does a star beyond a certain mass limit () only become a black hole?   A star is first made of hydrogen, it undergoes nuclear fussion reaction combining into helium and releasing a large amount of energy. This process continues till star is made up of iron core as iron has largest value of binding energy per nucleon, after this if mass of the star is above the value of Chandrasekhar limit it becomes a black star, what is the reason for this and why is certain mass limit required? How do you do a deep copy of an object in .NET? I want a true deep copy. In Java, this was easy, but how do you do it in C#? Sometimes I have asked a question which got no answers, but is answered in the comments to the question.  How should I handle these questions? I can't accept an answer, so it is never an accepted question.  For example this one:   (ok here the answer in the comments is from me, but nevertheless I also have others of them) Acro package provides "! Missing number, treated as zero." after update to acro-v2.7b A required field that isn't really required? ...the email field in my clients data entry system is required but users of the system are allowed to enter na@na.com if the customer refuses or doesn't have an email address. Is this best practice? Have any suggestions or workarounds for their solution? The energy of an electromagnetic wave The intensity of an electromagnetic wave is only related to its amplitude $E^2$ and not its frequency. A photon has the same wavelength as the wave that's carrying it, and its energy is $h  f$.   So if a laser wave is kept at the same amplitude and the wave length is reduced, why does its intensity remain the same even though its photons now carry less energy?  Why are the intensities of electromagnetic waves so different to sound waves (and other waves travelling through a medium) which are related to $f^2  E^2$? Should Dalvik Cache be wiped in Android OS that supports ART? When switching custom ROMs (in ART), is it necessary to wipe the Dalvik Cache?  Wiping the Cache is necessary, but should wiping Dalvik (which doesn't exist in Android Lollipop and afterward) be necessary?  What impact will it cause if we wipe Dalvik (something not present)?  In ART, what does the Dalvik cache contains? (is it an empty folder?) When I used  pip install ipython   or  pip install "ipython[notebook]"   I get the following error:  Downloading/unpacking ipython   Downloading ipython-2.3.1-py27-none-any.whl (2.8MB): 2.8MB downloaded Installing collected packages: ipython Cleaning up... Exception: Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/pip/basecommand.py", line 122, in main     status = self.run(options, args)   File "/usr/lib/python2.7/dist-packages/pip/commands/install.py", line 283, in run     requirement_set.install(install_options, global_options, root=options.root_path)   File "/usr/lib/python2.7/dist-packages/pip/req.py", line 1435, in install     requirement.install(install_options, global_options, *args, **kwargs)   File "/usr/lib/python2.7/dist-packages/pip/req.py", line 671, in install     self.move_wheel_files(self.source_dir, root=root)   File "/usr/lib/python2.7/dist-packages/pip/req.py", line 901, in move_wheel_files     pycompile=self.pycompile,   File "/usr/lib/python2.7/dist-packages/pip/wheel.py", line 206, in move_wheel_files     clobber(source, lib_dir, True)   File "/usr/lib/python2.7/dist-packages/pip/wheel.py", line 193, in clobber     os.makedirs(destsubdir)   File "/usr/lib/python2.7/os.py", line 157, in makedirs     mkdir(name, mode) OSError: [Errno 13] Permission denied: '/usr/local/lib/python2.7/dist-packages/IPython'  Storing debug log for failure in /home/username/.pip/pip.log   What does this error mean? How do I fix it? INT(5) vs SMALLINT(5): numbers in parenthesis after the numeric type In MySQL table definitions is there a different between INT(5) and SMALLINT(5)?  Or do they both represent the same size? How does gravity work on Star Wars ships? Inspired by the question , how does gravity work on the Millennium Falcon and other ships its size? The passengers are not strapped in and regularly move around just easily as they would on a planet with Earth's gravity.   The answer for the Death Star refers to gravity often being "coreward" (presumably taking advantage of the natural gravity an object of such substantial mass would have), but that doesn't apply for a small ship. So how is gravity accomplished?