Question is as stated above. Can someone explain if (and how) $$\iint f(x)g(y)\,dx,dy=\int f(x)\,dx*\int g(y)\,dy$$  I found this website, but I don't get why they use $x\leq y\leq b$. It makes a bit more sense (in my mind), to have $0\leq y\leq x$, but I still don't undersand how to finish/solve this logic.
Can I argue that:  \begin{align} \left(\int_a^b f(x) \,dx\right) \left(\int_c^d g(y) \,dy\right) &amp;= \int_a^b f(x) \int_c^d g(y) \,dy \,dx \\ &amp;= \int_a^b \int_c^d f(x) g(y) \,dy \,dx \end{align}  since the same would be true for a product of finite sums?
Ubuntu 14.04 multi touch screen support Proof of greatest integer theorem: floor function is well-defined I have to prove that  $$\forall x \in \mathbb{R},\exists !\,n \in \mathbb{Z} \text{ s.t. }n \leq x &lt; n+1\;.$$  where $\exists !\,n $ means there exists a unique (exactly one) $n$. I'm done with proving that there are at least one integers for the solution. I couldn't prove the &quot;uniqueness&quot; of the solution, and so I looked up the internet, and here's what I found:  Let $\hspace{2mm}n,m \in \mathbb{Z} \text{ s.t. }n \leq x &lt; n+1$ and $m \leq x &lt; m+1$. Since $n \leq x \text{ and } -(m+1) &lt; -x$, by adding both, $n + (-m-1) &lt; (x-x) = 0$. And (some steps here) likewise, $(x-x) &lt; n+m+1$.  Now, can I add up inequalities like that, even when the book is about real analysis (and in which assumptions are supposed to be really minimal)? Or should I also prove those addition of inequalities? Thank you :D Are the known exoplanets representative of the population of exoplanets? An  states that most of the 715 newly announced exoplanets orbit very close to their host stars.   Do the theories predict that this is a representative sample and that most planets are very close to their host stars? Or is it just that our current detection techniques are predisposed to finding only those planets close to their stars (because the sensors look for dimming of the host star for example)? How to find effects of magical items? Similar to this , but specific to D&amp;D 5e, how are players meant to find out the effects of items/potions/scrolls/weapons/armor?  Say, they find a potion in some chemist's chest. It is red and smells nice. Should they perform an investigation/perception check? Or an arcana/nature check? If I tell the players specifically to do an Arcana check, then I am already hinting at the fact that this potion will have some magical effect. It could just be a natural elixir concocted by Druids to help your health regen.  Same goes for weapons. They find the legendary bow of X, which has a +3 to hit and a 10% chance of striking other enemies near the target. I describe the bow as unique, glowing, made from dragon scales and hovering on air. Would they investigate their properties? Or just I just let them fire at a wall until something happens?   Our playstyle so far has been to use active checks for a quick understanding of the item (drink a little bit of Potion as an Investigation check to find whether it heals or not, sensing magical properties of the bow with Arcana checks). When time is available, and experimentation is necessary, allow the PCs to do some creative testing and, based on their ideas, "After Y time, you realize that the bow sometimes hits several points of the wall instead of just your target".  According to the RAW, is there a specific way of checking the stats/effects of such items? \colorbox height too long \documentclass{article} \usepackage[utf8]{inputenc} \usepackage{xcolor} \usepackage{minted} \usepackage{parskip}  \definecolor{codegray}{gray}{0.9} \newcommand{\code}[1]{\colorbox{codegray}{\texttt{#1}}}  \begin{document}  \code{test code}\\ anotherline  \end{document}   This is working good on sharelatex.com and looks like I expect and want it:   but on my local project using this command gives me this:   any idea what could be causing this colorbox height going to the next line? LaTeX: optional arguments with square brackets LaTeX is trying to add a missing $ that I don't want to the following code. Minimal working example:  \documentclass[letter,12pt]{article}  \newcommand{\monkeyBeans}[2][]{   \parbox{4in}{#1} }  \begin{document}  \parbox{4in}{$[a, b]$}   \monkeyBeans[$[a, b]$]{yo monkey}  \end{document}   If I try any of the following lines in my document instead of the call to monkeyBeans, there is no error:  \parbox{4in}{$[a, b]$}  \monkeyBeans[yo monkey]{$[a, b]$} \monkeyBeans[$(a, b)$]{yo monkey} \monkeyBeans{$[a, b]$}{yo monkey}   What is the deal with square brackets in optional arguments? Is there a workaround to make CSS classes with names that start with numbers valid? Adding materials to more than one object Computing the inverse of a permutation I didn't understand the permutation and of course, I got this question wrong.     Compute the inverse of the following permutation:   $$ \begin{pmatrix}  1&amp;2&amp;3&amp;4&amp;5&amp;6\\ 2&amp;5&amp;4&amp;1&amp;6&amp;3 \end{pmatrix} $$   Could you please help me with this question and provide an explanation?  Furthermore, is it possible to compute the direct permutation, not the inverse one? If yes, how?  Thank you in advance! $\lim_{n \rightarrow\infty} ~ x_{n+1} - x_n= c , c > 0$ . Then, is $\{x_n/n\}$ convergent? Stack Overflow is a site of passers-by They says that Stack Overflow is a community-driven web-site. But, as a matter of fact, it is not.  There is no community at all. It's more like an imageboard, as a number of participants makes them pretty much like anonymous to each other despite of registration.   Stack Overflow is a site of passers-by.  Passers-by who want to solve their silly typo problem and passers-by who eager to share their vast knowledge in spotting silly typos. And - oh - sometimes to perform a civic duty and vote to close a question they have no idea what to answer to and thus calling it "not a real question".  So, as a funny consequence, whatever question can get   answered upvoted downvoted closed deleted   depends not on its quality but only on the fluctuation happened to "community" at the given moment, at a pretty random rate.  It is not only quite discouraging. It makes the very idea of collecting and improving knowledge quite impossible. 'Search on Yammer' link in search center refiners I have a heavily branded site I'm working on and can't find where the "Search on Yammer" link is being populated from. It appears within the search refiner/filter web part in the left column but none of the display templates reference this.   I'd like to be able to customize the link, placement on the page, etc. without resorting to jQuery to move elements around the page.     And the markup coming through looks like this. The refiners themselves come through a series of display templates but the Yammer link is nowhere that I can see in any of them.   &lt;div webpartid="f09e98eb-6dc3-4a9b-8dbf-65d0fbc7b7c3" haspers="false" id="WebPartWPQ4" width="100%" class="ms-WPBody " allowdelete="false" style=""&gt;     &lt;div class="ms-ref-ctrl"&gt;         &lt;div class="ms-ref-refiner"&gt;             &lt;div id="Container"&gt;                 &lt;h2 class="ms-displayInline"&gt;                     &lt;a class="ms-ref-name ms-displayInlineBlock ms-srchnav-link-selected ms-link" href="javascript:void(window.open(                     'https://www.yammer.com/search?trk_event=o365_search&amp;amp;search='+                     encodeURIComponent(Srch.ScriptApplicationManager.get_current().queryGroups.                     Default.dataProvider.get_currentQueryState().k),'_blank'));" title="Open a new tab to search for this on Yammer"&gt;Search on Yammer&lt;/a&gt;                 &lt;/h2&gt;             &lt;/div&gt;         &lt;/div&gt;     &lt;/div&gt; &lt;/div&gt; &lt;div componentid="ctl00_SPWebPartManager1_g_f09e98eb_6dc3_4a9b_8dbf_65d0fbc7b7c3_csr" id="ctl00_SPWebPartManager1_g_f09e98eb_6dc3_4a9b_8dbf_65d0fbc7b7c3_csr"&gt;             ... refiners populated through display templates ... &lt;/div&gt; Is there a max length for an array in C++?  Is it a C++ limit or does it depend on my machine? Is it tweakable? Does it depend on the type the array is made of?  Can I break that limit somehow or do I have to search for a better way of storing information? And what should be the simplest way?  What I have to do is storing long long int on an array, I'm working in a Linux environment. My question is: what do I have to do if I need to store an array of N long long integers with N > 10 digits?  I need this because I'm writing some cryptographic algorithm (as for example the p-Pollard) for school, and hit this wall of integers and length of arrays representation. Reward for Passing Review tests? Just wanted to see what the community thought of adding a reward to passing the review tests? Maybe a 1 point reward?  I know seeing the tests in the review area has helped me be extra careful when reviewing, but I thought perhaps a little boost in incentive would be cool. Should I always use a comma before a quote? How does water evaporate if it doesn't boil? When the sun is out after a rain, I can see what appears to be steam rising off a wooden bridge nearby. I'm pretty sure this is water turning into a gas.  However, I thought water had to reach 100 degrees C to be able to turn into a gas.  Is there an edge case, for small amounts of water perhaps, that allows it to evaporate? The infinite product of the sines of all positive integers is zero I tried to prove that $\prod\limits_{k = 1}^\infty  {\sin k = 0}$  Define $$P(m)=\prod\limits_{k = 1}^m  {\left|\sin k\right|}$$  then I take the logarithm of both sides $$\log P(m)=\log \prod\limits_{k = 1}^m  {\left|\sin k\right|}$$  which can be written as $$a_m=\log P(m)=\sum\limits_{k = 1}^m  {\log\,\left|\sin k\right|}$$ for any $k\in\mathbb{N}$ we have $0&lt;\left|\sin k\right|&lt;1$, so all terms of the sum above is strictly negative.  I say that the sequence diverges: $\left\{a_m\right\}_{m\in\mathbb{N}}\to\,-\infty$  This is a delicate point on which I have some doubts.  I know that  $$\int\limits_0^\pi  {\log \left|\sin x\right|{\text{ d}}x}  =  - \pi \log 2$$ and that the function $\log \left|\sin x\right|$ is periodic with period $\pi$,  so the integral from 1 to $\infty$ diverges to $- \infty$. Then the series diverges as well for the integral test for convergence.  As $\log P(m)\to\, -\infty$ we can say that $P(m)\to\,0$ as $m\to\,\infty$  Therefore $$\mathop {\lim }\limits_{m \to \infty } \prod\limits_{k = 1}^m {\left| {\sin k} \right| = \mathop {\lim }\limits_{m \to \infty } \prod\limits_{k = 1}^m {\sin k = \prod\limits_{k = 1}^\infty  {\sin k = } 0} }.\quad \blacksquare$$ What prefixes should be used on reference designators for components of various types?  I think we can all agree that "R##" refers to a resistor, "C##" refers to a capacitor, and "L##" refers to an inductor.  Beyond that, it appears to be a mishmash of conventions.    I've seen both "IC" and "U" used for chips.   "Q", "T", and "M" have been used for transistors and MOSFETs.  Ordinarily, "D##" is used for diodes, but LEDs and Zeners get special treatment on some boards.    What standards are available for reference, and who endorses them?  JEDEC, ISO, IEEE, and other standardization bodies are welcome references (though I'd prefer an inexpensive standard), but I'm also curious to see what's used at various companies other than my own.    Edit:  What I'd really like to see is a list in an answer here which complies with the standard (even if it's just a quote). How to draw circle, square, and triangle marks in TikZ picture? Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers.