So, I just wrote down the following sentence.   'Now my attempt is to understand the cause of such behaviour. Is it the childhood experiences, as we find in the movie?'  I'm not sure if I've got the punctuation correct in the last sentence with the use of question mark. If reading out, you'd expect the speaker to take a pause after 'experiences' and then provide more information. However, 'as we find in the movie' isn't really an interrogative so putting the question market after it seems erroneous. Kindly suggest the correct punctuation.
Is this the correct use of the question mark?  Should it be at the end outside of the closing bracket?  See example below     How do we deliver it? (to the learner)
Understanding the proof of infinitely many primes $\equiv 3\pmod{\!4}$ Theorem: There are infinitely many primes congruent to $3 \mod 4$. Proof: Assume that $p_1 = 3, . . . , p_n$ are primes of the form $p_j ≡ 3 \mod 4$. We will construct a new one by looking at $N = 4p_1 · · · p_n − 1$ (putting $N = 4p_1 · · · p_n+3$ would also work).  There is more to this proof but the rest of it I understand, what I don't get is where it says &quot;$N = 4p_1 · · · p_n − 1$ (putting $N = 4p_1 · · · p_n+3$ would also work)&quot;. I do not understand why we are writing $4p_1 · · · p_n − 1$. Is there a more clear way to prove this theorem? What makes scaling pixel art different than other images? Recently I was coaching a developer who's new to making pixel art games, and realized there's some "conventional wisdom" around scaling pixel art that's not obvious to a newcomer, and might not be well documented. So, here's an attempt to fix that....     I've heard I should only scale pixel art by 200%, 300% etc. not in-between sizes like 120%   Why is that? Shouldn't scaling "just work" the way it does with other images? What if I really need to make my character just a fraction larger or smaller to get the right gameplay or look? How do I fix this?  I'm told it's "bad" to use pixel art at different scales in one game - why?    For example, if I scaled up my character so each sprite texel is 4x4 screen pixels, while my background art is scaled up so each sprite texel is 6x6 screen pixels. Isn't a pixel a pixel at the end of the day? Possible Duplicate:          Here is my sync script:  ssh user@remote.host /etc/init.d/some_service stop scp user@remote.host:/var/some_service/events ./events scp ./new_data user@remote.host:/var/some_service/new_data ssh user@remote.host /etc/init.d/some_service start   As You can see, it stops service, perform some copies with scp and starts service back.  The problem: it "costs" 4 "similar" ssh connections in a row.  Is there any way to make all the work, using one physical ssh connection? Unwanted faint lines connecting dashed double line Building on the answer  by wrtlprnft, I have the following code     \documentclass[tikz,margin=1cm]{standalone}    \usetikzlibrary{decorations.markings}    \begin{document}    \begin{tikzpicture}         \tikzstyle{arrowed double line}=[             dashed, double distance between line centers=3pt,             postaction=decorate,             decoration={                 markings,                 mark=between positions 10pt and -10pt step 20pt with {                     \arrow[thin,yshift= 1.5pt,xshift=.8pt]{&gt;}                     \arrow[thin,yshift=-1.5pt,xshift=.8pt]{&lt;}                 },             },         ]         \draw[arrowed double line] (0,0) --  (1,2);     \end{tikzpicture} \end{document}   which produces this:    I am unhappy with the faint transverse lines between the dashes. I don't even know why tikz decides to draw them.  Is there a way to get rid of them?  Thanks in advance to all replies.  PS the same thing occurs at the caps of double solid lines What is the most efficient way given to raise an integer to the power of another integer in C?  // 2^3 pow(2,3) == 8  // 5^5 pow(5,5) == 3125 Over on another network site, a helpful user corrected the grammar of .  The answer now says     I believe it's valid.   where I originally wrote     I believe it valid.   Is the original sentence gramamatically (or at least idiomatically) correct or do I absolutely need a form of 'to be' here? (What's more, is 'to be' a copula or an auxiliary verb if used like this?)  I am quite certain that I have seen sentences like the original one in the past, but I am uncertain where. Given that I am frequently exposed to the somewhat made-up English of authors in the fields of sci-fi and fantasy, I might have fallen into a trap here.  Edit: I don't doubt that the correction is valid and enhances readability. What are the most downvoted questions and answers on Stack Overflow, Server Fault, Meta, and Super User history to date?  Similarly, what are the most upvoted questions and answers on SO, SF, Meta, and SU to date? PHP function substr() error So why did the price tank all of a sudden and then came back to normal? What does it mean? I was presented with the following problem:      Prove that there exist solutions to $x^2+y^2=z^n$ for all $n$, with $x,y,z, n \in \mathbb{N}$   I showed that by taking any Pythagorean triple $x^2+y^2=z^2$ and multiplying by $z^{2(n-1)}$ we get $(z^{n-1}x)^2+(z^{n-1}y)^2=(z^2)^n$, which allows me to generate solutions easily for any value of $n$. I managed to find several similar questions on this site, such as  concerning the specific case $n=3$. I notice that all of these questions take a similar approach and start with a Pythagorean triple and use it to generate general solutions.  Is there a way to prove the statement (or better yet provide solutions to the equation) without first relying on Pythagoras? onChange event doesn't trigger Do Hogwarts Houses have quotas? Not a year goes by where we don't see at least one person Sorted into each House, and it's usually a fairly even distribution.  The Sorting Hat doesn't get a chance to study the mind of each child before Sorting, so it can't Sort them relative to each other (top X in smarts go to Ravenclaw, 4 bravest to Gryffindor, etc)  What would happen if a Sorting occurred where all the witches/wizards were extremely intelligent?  Would they all get Sorted to Ravenclaw?  What if they were all extremely brave?  What if none of them were particularly smart or courageous, but they all wanted power?  We see that there's a fairly small number of beds per room in each dorm, would the Hat stop Sorting people into a House if it knew the House had no more beds?  If so, could Harry Potter have ended up in Hufflepuff if his last name had been 'Zimmers'? How important are headings in SEO I have a website that sells music artists posters. Each artists has a page containing a small bio, and a list of posters we are selling representing the artist.  Currently our HTML looks like that  &lt;h1&gt;{Artist name}&lt;/h1&gt; bio &lt;h2&gt;Posters&lt;/h2&gt; &lt;ul&gt;   &lt;li class="poster"&gt;...&lt;/li&gt;   ... &lt;/ul&gt;   We are redesigning this page, and we have a new template that we really like. Problem is that in this new version the &lt;h2&gt;Posters&lt;/h2&gt; does not appear anywhere on the page.  As you might expect, a significant part of our traffic comes from people typing something like : "{Artist name} poster" on Google.  I fear that, by removing the word "poster" from the page, we will reduce our chances of getting these people to come to the site, since Google might down rank us.  I see several options here :   Give Google a slightly altered page that contains the H2 tags. But I think this falls into the "Black Hat SEO" category. And I don't want that. Put the heading. Hide it with CSS. Everyone wins, including accessibility tools which can determine the structure of the page using the "invisible" heading. But I do not know if this is considered a good practice. Forget about the heading, and find other ways to tell Google that we really are about posters. (via keywords, but do they really have the same "weight" as an H2?) Adapt the design to our SEO need, and find a way to put that heading nicely.   Are there webmasters out there that have been facing the same issue, how did you resolve it? Favorite questions in the Android app I haven't been able to find my favorite questions in the official Stack Exchange Android app.  Looking at my accounts page gives nothing on them. Solving $\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$ I have the following problem that I don't know how to solve:  Let $\textbf{r}(t)$ be a curve and let $r(t):= |\textbf{r}(t)|$. Let the curve $\textbf{r}(t)$ be defined by: $$\textbf{a}(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ Where $\textbf{a}(t) = \textbf{r}''(t)$ is the acceleration of the particle and $G,M \in \mathbb{R}$. Suppose that $\textbf{r}(t)$ and $\textbf{r}'(t)$ are not parallel. The objective is to determine the trajectory $\textbf{r}(t)$ in polar coordinates.  So basically we have the following equation: $$\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ The thing is that I don't know how I should proceed from now on. If we'd have just normal scalar function this would be a differential equation and I think I would be able to solve it, but like this I have no idea how to approach the problem. How can I solve this?  Edit: If $$\textbf{r}(t):=(x(t),y(t))$$ then the equation turns into the following system of equations: $$x''=\frac{GM}{(\sqrt{x^2 + y^2})^3}x$$ $$y''=\frac{GM}{(\sqrt{x^2 + y^2})^3}y$$ I don't know if this helps but I think that this turns thing easier perhaps. How to prove that $\hat\sigma^2$ has $\chi^2_{n-p}$ distribution (linear regression) Consider the linear regression model:   $$Y_i=r(x_i)+\varepsilon_i\equiv\sum_{j = 1}^p x_{ij} \beta _j + \varepsilon _i,\quad i=1,\ldots,n.$$  where $x_1,\ldots,x_n\in \mathbb{R}^p$ are fixed, $E(\varepsilon_i)=0$,  $\operatorname{Var}(\varepsilon_i)=\sigma^2$. Denote $Y=(Y_1,\ldots,Y_n)^T$, $\beta=(\beta_1,\ldots,\beta_p)^T$, $X=(x_{ij})_{n\times p}$. As is known, $\hat \beta  = \arg\min\limits_{\beta  \in \mathbb{R}^p} (Y - X\beta)^T (Y - X\beta)=(X^TX)^{-1}X^TY$ if the matrix $X^TX$ is invertible, and so an estimator of $r(x)$ at $x=(x_1,\ldots,x_p)\in\mathbb{R}^p$ is given by $\hat r_n(x)=x^T\hat\beta$.  Let $$\hat{\sigma}^2 = \frac{1}{\sigma^2}\sum_{i = 1}^n (Y_i - {\hat r}_n (x_i))^2.$$ I am stucking the problem: $\hat\sigma^2$ has $\chi^2_{n-p}$ distribution. How to prove the statement? Resize (enlarge) an image without resampling (in favor of quality loss)? How to resize an image or selection without resampling the pixels? I have some pixel art that I want to enlarge without losing the hard, non-anti-aliased, pixel edges. In other words, 1px should become 4px of the same color— fake big pixels. Is there any way to achieve this procedurally?  Example (of what I don't want):  before resize:    after resize:    Open to other tools/methods beyond photoshop, thanks in advance. Removing unnecessary vertices? Substr(). Add characters for each n digits Sublime Command not working in SSH server on Ubuntu 14.04?