Bar graph with groups(inner group bars next to each other) and whitespace between groups including error bars
grouped bar chart with pgfplots
A former lecturer of mine once explained why, from a syntactic point of view, the English rule that negation and questions are formed with the auxiliary do follows from other syntactic facts about English.  More precisely, if you gave a good syntactician not familiar with English a corpus of English sentences containing everything but questions and negated phrases, then s/he could infer grammatical rules from it (such as headed-ness of phrases, what kind of arguments what word takes, etc.) From some of these rules, then, s/he could see that something specific would be wrong with the sentences "Gave you the present to Mary?" and "He went not to the cinema."  Can someone explain what exactly this something is and how it follows from other sentences?  FYI, my lecturer did this in Government &amp; Binding with X-Bar theory. This may be more suitable a question for the proposed  but as it's not online yet I'm trying it here first.   Edit: the beta is online now at . How do i create a linestring from points sort by date in PostgreSQL?  I tried:  SELECT observations.id,    ST_MakeLine(observations.geom ORDER BY observations.date) AS newgeom    FROM   observations     GROUP BY    observations.id;   This is going wrong. It returns me a linestring with same coordinates.  So, instead of a line from a to b, it gives me a to a.    What am I doing wrong? Numbering equations If I am writing a LaTeX document on TexMaker and I want to number my equation but most probably I will be adding details later on and I do not want to fall into retyping it all over again. So if for example I have an equation and currently called equation (1.2) that I want to refer later as in the following sentence "As you can see in (1.2)" but then I had to edit my document in a way that it becomes (1.3) now instead of (1.2) what should I do in order to avoid editing large number of equations and my reference to them? How to disable system beep for non-privileged user What exactly are the Python scoping rules?  If I have some code:  code1 class Foo:    code2    def spam.....       code3       for code4..:        code5        x()   Where is x found?  Some possible choices include the list below:   In the enclosing source file In the class namespace In the function definition In the for loop index variable Inside the for loop   Also there is the context during execution, when the function spam is passed somewhere else. And maybe  pass a bit differently?  There must be a simple reference or algorithm somewhere.  It's a confusing world for intermediate Python programmers. X, along with Y, "were" or "was"? I was interested in the following sentence which appeared in a   in The Washington Post, Politics, (WHO RUNS GOV).     She, along with Mark Childress, were set to be Daschle's deputies at   the White House before he stepped down.   Can someone clarify if the fragment "She, along with Mark Childress, were set to" is ungrammatical, as I think it is?  I would reword were with was because I think that the subject of the sentence - obviously, in grammatical terms - is "She", not "She" and "Mark Childress", but I'm not sure on this correction.  More precisely, I'm not so sure that "She, along with Mark Childress" is equivalent to She and Mark Childress with regard to the effects on the grammatical numbers. If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$ (a) If sup A &lt; sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.  I have argued that if sup A $\lt$ sup B, then choose an $\epsilon&gt;0$ such that sup A +$\epsilon \in B$. Since $a \le $ sup A for all $a \in A$, it follows $a \lt $ sup A + $\epsilon$ hence sup A + $\epsilon $ is an upper bound of A as well as an element of B.  (b) Give an example to show that this is not always the case if we only assume sup A $\le$ sup B.  I am having trouble extending my argument into an example which makes me think that it may not be correct.   Question: Is my argument for (a) correct? If not how would one show this fact? Can you give an example asked for in (b)? Thanks in advance.  I need to understand how I am assuming what I am supposed to prove in my approach. Can someone elaborate on that please? The other question doesn't really address this issue. Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$? I have Ubuntu 16.04 with GNOME shell. I set ctrl+shift for the "Modifiers-only switch to next source" option.   This shortcut conflicts with apps hot-keys. I can't use ctrl+shift+i to open devtools in Chrome, it just switches the layout. In VS code IDE if I press shortcuts with ctrl+shift (for example ctrl+shift+f), shortcuts work but always switch a keyboard layout. When I used Windows this problem didn't exist.  Can I set the ctrl+shift shortcut in such a way that it causes a switch action only after the keys are released and only if this 2 keys were pressed (without additional keys)? Definitions and usage of Covariant, Form-invariant & Invariant? Just wondering about the definitions and usage of these three terms.  To my understanding so far, "covariant" and "form-invariant" are used when referring to physical laws, and these words are synonyms?  "Invariant" on the other hand refers to physical quantities?  Would you ever use "invariant" when talking about a law? I ask as I'm slightly confused over a sentence in my undergrad modern physics textbook:     "In general, Newton's laws must be replaced by Einstein's relativistic laws...which hold for all speeds and are invariant, as are all physical laws, under the Lorentz transformations." [emphasis added]      ~ Serway, Moses &amp; Moyer. Modern Physics, 3rd ed.   Did they just use the wrong word? Create a page for useful links? Code examples, like in the TikZ/PGF manual I like very much the 'side-by-side' code examples in the TikZ/PGF manual...    ... and I would like to use the same thing in a manual I'm writing. I have found the source code for the codeexample environment that the TikZ/PGF manual uses for this purpose (specifically, it's located at /usr/local/texlive/2012/texmf-dist/doc/generic/pgf/macros/pgfmanual-en-macros.tex).  My (probably rather basic) question is: how can I use the same codeexample environment in my own documents? I ask because I tried compiling the example in the related question below, but the file couldn't be found. Integral around unit sphere of inner product For arbitrary $n\times n$ matrices M, I am trying to solve the integral  $$\int_{\|v\| = 1} v^T M v.$$  Solving this integral in a few low dimensions (by passing to spherical coordinates) suggests the answer in general to be  $$\frac{A\,\mathrm{tr}(M)}{n}$$ where $A$ is the surface area of the $(n-1)$-dimensional sphere. Is there a nice, coordinate-free approach to proving this formula? Objective: To separate text column from figures column in every page I don't know if this has been asked before. But so far nothing has come up with my search.  In some books, the figures are placed at the outer side of the page like in the picture below.    Is this achievable in TeX/LaTeX? That is how does one set up the page so that all text will appear in a 'text column' and the figures appear in a 'figures column' in every page? Change bend angle and swap node position on arrow Countable additivity implies countable subadditivity? Suppose $\mu$ is a measure in a ring $R$ of a set $X$. So for any mutually disjoint subsets $A_1,A_2,\ldots$ such that $\sum_{i=1}^\infty A_i\in R$, we have that $\mu(\sum_{i=1}^\infty A_i) = \mu(A_1)+\mu(A_2)+\cdots$. (i.e. countable additivity)  I'm wondering: Does that imply countable subadditivity? I.e. for any subsets $A_1,A_2,\ldots$ such that $\sum_{i=1}^\infty A_i\in R$, is it always true that $\mu(\sum_{i=1}^\infty A_i)\le \mu(A_1)+\mu(A_2)+\cdots$?  I believe it should be true, because you can take the mutually disjoint subsets $A_1,A_2-A_1,A_3-(A_1\cup A_2),A_4-(A_1\cup A_2\cup A_3),\cdots$, each of which is in $R$ (because unions and differences of sets in $R$ are again in $R$), then apply countable additivity, then use monotonicity on the fact that $A_n-(A_1\cup\cdots\cup A_{n-1})\subseteq A_n$.  Is that right? I just want to make sure I didn't make any mistake in my logic. If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $ I've tried to use the limit test: $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = C$ in the series of $a_n$ and $\frac{a_n}{a_1+a_2+...+a_n}$ but it was inconclusive.  The question does not especify the sign of $a_n$ but proving it for positives would already help.  I think the sign can change as long as the partial sums never get to be zero. Like if the series has only negative numbers the result seems to stil hold. Example of a series! Give an example of a convergent series $$\sum_{n=1}^{\infty}a_n$$ such that the series $$\sum_{n=1}^{\infty}a_{3n}$$ is divergent..  I cannot find find any kind of series... I am also be very thankful if you find a divergent series (changing the term convergent)... I'm trying to find  $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$   I tried couple of methods: Stolz, Squeeze, D'Alambert  Thanks!  Edit: I can't use Stirling. Example of a function $f$ on $X$ to $\mathbb{R}$ which is not $X$-measurable but $|f|$, and $f^2$ are measurable. Question  The following is from Bartle:  Give an example of a function $f$ on $X$ to $\mathbb{R}$  which is not $X-$measurable but is such that $|f|,$ and $f^2$ are measurable.  Relavant Information  A family $\textbf{X}$ of subsets of $X$ is a $\sigma$ algebra in case:   $\phi, \mathbb{R} \in \textbf{X}$ $X \setminus A \in \textbf{X}$ if $A \in \textbf{X}$ If $(A_n) \in \textbf{X}$, then the union $\bigcup_{n=1}^{\infty} A_n \in \textbf{X}$  Any set in $\textbf{X}$ is called $X-$ measurable.   A function is $X-$ measurable if for every real $\alpha$, the set $\{x \in X : f(x)&gt;\alpha\}$ belongs to $\textbf{X}$  Attempt  Consider the following function: $$f=\begin{cases} 1 &amp;\mbox{ if} \ x \not\in\mathbb{Q} \\                   -1 &amp;\mbox{ if} \ x \in \mathbb{Q}  \end{cases}$$ defined from $\mathbb{R}$ to $\mathbb{R}$ with $\textbf{X}$ being the Borel algebra (the $\sigma$ algebra generated by all open intervals $(a,b)$) Then if $\alpha \in [-1,1)$, the set $\{x: f(x)&gt;\alpha\}$ consists of the irrational numbers. If I can show that the irrationals are not in the Borel Algebra, then I would be done since $|f|=f^2$ is a constant and easily shown to be measurable. Unfortunately I do not know how to show that the irrationals are not in the Borel Algebra.  Edit it has been brought to my attention that the irrationals are in the Borel algebra. I think this is because the rationals, being countable, can be shown to be in the Borel algebra by constructing appropriate small neighborhoods around each rational and then taking the intersections. I now seek to construct a set that is not in the Borel algebra  Alternatively I have tried to define a set other than the irrationals that is not in the Borel Algebra and define a similar function but to no avail. I am currently self studying this book so it would be of great help to me if the answer could contain heuristics on how to tell when a set if borel measurable (although I do not require this in the answer).