I recently saw this sentence,  "the system has not been yet used in both large and small cities".  I wonder if both can be used in negative sentences like this? If not, how can "either" be used in such a sentence where two names are mentioned afterwards? we cannot say EITHER large And small cities. also both refers to positive sentences.   Thank you in advance
Hello native English speakers!  I want to ask you about either and both. I tried to find similar questions in this site but I think they were not same with these ones.  My questions are;   I was invited to two parties, but I couldn't go to ------- of them.  I took two books with me on holidays, but I didn't read ------- of     them.   I choose both for questions 1 and 2. But the right answers were either. But I think both and either are correct either. Am I right?   Thank you in advance!
How to get IOS "tossing feature" in Wolfram IOS Player to work in Mathematica Question about a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$ About a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$  If we'll take a line $f(x)=\begin{cases} ax &amp;, x\in\mathbb R\\ 0 &amp;, x=\infty  \end{cases}$ then we'll have a bijection.   But about this function: $g(x)=\begin{cases} x &amp;, x\in\mathbb R\setminus \{0\}\\ 0 &amp;, x=\infty  \end{cases}$ ? it appears to be a bijection and it's even simpler. A site (or scraper) is copying content from Stack Exchange. What should I do? of Stack Overflow, all content posted on Stack Exchange sites by their users (i.e. you wonderful people) has been provided to the whole universe under . For my fellow non-lawyers, that license basically means:   Anyone can use any Stack Exchange posts at any time without having to ask for permission Making money off of the copied content is permitted You don't even have to copy stuff from here verbatim; you can just use it as a starting point and make whatever edits you want There are just two rules you have to follow:   You have to provide . Simple links to the original post and author info are just fine. You have to  and allow other people to use your content, as long as they follow these very same rules. How meta!    (If you ever forget any of that, and want to refresh your memory, the license info is linked to in the footer of every page.)  There are, in fact, a lot of people who republish varying amounts of our content for assorted reasons. Unfortunately, there are some Stack Content Republishers Attributing Poorly and/or Excelling at Ranking (SCRAPERs, for short).  In this context, "attributing poorly" means any use that doesn't follow our attribution rules or make any other reasonable attempt at give credit. This can get pretty egregious; I've seen SCRAPERs that not only don't link back to SE originals, but also use fake author info and post dates to make it harder to find originals. By "excelling at ranking," I'm referring to copycat sites that end up higher in Google results than the original SE sites do for the same content. That's not necessarily wrong, but in some cases, it indicates inappropriate SEO hackery.  So, the question is: what can you do if you spot a SCRAPER? Question is :     Prove that the polynomial $p(x)=x^4-4x^2+8x+2$ is irreducible over the quadratic field $F=\mathbb{Q}(\sqrt{-2})$. [Hint : first use the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$ . Conclude that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$ and hence show that $p(x)$ has no linear factor over $F$. Show similarly that $p(x)$ is not the product of quadratics with coefficients in $F$.]   What I have done so far is :  Suppose $\alpha \in\mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)=x^4-4x^2+8x+2$ we would then have :  $p(\alpha)=\alpha^4-4\alpha^2+8\alpha+2=0\Rightarrow 2=\alpha(-\alpha^3+4\alpha-8)$  i.e., $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$.  so, I have used the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$  I do not understand why does he mentioned that $\mathbb{Z}[\sqrt{-2}]$ is U.F.D and all... I do not use that at all... It is unnecessarily confusing me or i am unnecessarily getting confused.. hint is actually misleading me :(  Now, I have to prove that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  i.e., suppose I have $2=ab$ in $\mathbb{Z}[\sqrt{-2}]$ then, $N(2)=N(ab)\Rightarrow 4=N(a)N(b)$   i.e.,$N(a)=1\text{ or }2\text{ or } 4$ i.e., $p^2+2q^2=1\text{ or }2\text{ or } 4$ for $a=p+\sqrt{-2}q$   $p^2+2q^2=1\Rightarrow p=\pm 1 \Rightarrow a=\pm 1$ $p^2+2q^2=2\Rightarrow q=\pm 1\Rightarrow a=\pm\sqrt{-2}$  $p^2+2q^2=4\Rightarrow p=\pm 2\Rightarrow a=\pm 2$    Once I prove that those are the only divisors then I would consider :    $p(1)=(1)^4-4(1)^2+8(1)+2\neq 0$ $p(-1)=(-1)^4-4(-1)^2+8(-1)+2\neq 0$ $p(\sqrt{-2})=(\sqrt{-2})^4-4(\sqrt{-2})^2+8(\sqrt{-2})+2\neq 0$ $p(-\sqrt{-2})=(-\sqrt{-2})^4-4(-\sqrt{-2})^2+8(-\sqrt{-2})+2\neq 0$ $p(2)=(2)^4-4(2)^2+8(2)+2\neq 0$ $p(-2)=(-2)^4-4(-2)^2+8(-2)+2\neq 0$   So, no divisor of $2$ is a root..  Thus $p(x)$ do not have a root in $\mathbb{Z}[\sqrt{-2}]$  suppose I have something like :  $$x^4-4x^2+8x+2=(x^2+ax+b)(x^2+cx+d)=x^4+(a+c)x^3+(b+d+ac)x^2+(ad+bc)x+bd$$  But then, $bd=2\Rightarrow \text { b,d  are  a divisors of 2 in $\mathbb{Z}\sqrt{-2}$}$  But then we have seen that only divisors of $2$ in $\mathbb{Z}\sqrt{-2}$ are $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  So, only possibilities are   $x^4-4x^2+8x+2=(x^2+ax\pm 1)(x^2+cx\mp 2)$ $x^4-4x^2+8x+2=(x^2+ax\pm \sqrt{-2})(x^2+cx\mp \sqrt{-2})$   But these are not possible...  This only tells that $p(x)$ is irreducible in $\mathbb{Z}[\sqrt{-2}]$ but then how do i show this is irreducible in $\mathbb{Q}(\sqrt{-2})$ I was expecting gauss lemma to help but it only works for integers and rationals...  So, please help me to clear this..  Thank you...   P.S : Proposition $11$ is Rational root theorem and Exercise $8$ is that ring of integers of $\mathbb{Q}(\sqrt{-2})$ is an Euclidean domain. A secretary with ____ good knowledge of English: "a" or no article? I'm trying to pass a test on "Articles exercise" as they call it . Reviewing my answers I got stuck about the two I posted on the picture below.  As I understand both words are uncountable (according to the ), so it looks like (or said for example ) I can't say a/an with an uncountable noun.  But this test says the opposite. Is there some trick? Or am I just dumb? Please help me to understand why I need to use "a" article in these cases?        We need a secretary with ____ good knowledge of English.          a  (marked as the correct answer)   the     a / the     no article is needed (the answer I think is correct)        We're having ____ terrible weather.         a (marked as the correct answer)     the     a / the     no article is needed (the answer I think is correct) The "Select your next badge" dialog box is not displaying correctly. It looks like this:    Freehand circles, as requested:    It seems to be an issue with box-sizing, because when it's content-box, it looks fine. I think the width and height rules for .popup-badges .all-badge-progress .badge-progress should be increased a little, for example:  .popup-badges .all-badge-progress .badge-progress {   width: 214px;   height: 96px; }   Tested on the latest Chrome and Firefox on Linux Mint. "Not being good at swimming, he didn't like it." Could we omit the word "being" here? Not being good at swimming, he didn't like it.   Could we omit the word "being" here?     Not good at swimming, he didn't like it.   Which one is grammatically correct? Nautilus does not show the left bar, how do I turn it back on? When I open nautilus does not open the left bar where it says computer and down markers such as music, documents, videos, downloads. I reinstalled nautilus from the synaptic package manager but nothing happened. What I can do to make that bar visible again? Why Does the Transaction Log Keep Growing or Run Out of Space? This one seems to be a common question in most forums and all over the web, it is asked here in many formats that typically sound like this:  In SQL Server -  What are some reasons the transaction log grows so large? Why is my log file so big? What are some ways to prevent this problem from occurring? What do I do when I get myself on track with the underlying cause and want to put my transaction log file to a healthy size? I am starting to learn using blender, and at some point I managed to do something to the 3D view camera.  Usually the camera orbits around an object, but now the rotation seems to happen (atleast very close to) around the "camera's origin". Neither the input setting in "User Preferences", the pivot point selection nor Ctrl-Alt-Shift-C (as suggested on some forum) do anything to the issue. C++: delete vs. free and performance Why does this work: returning C string literal from std::string function and calling c_str() Adobe Reader and xcolor within a table breaks the line render I'm using the xcolor package to colour the top row in a table. This is displayed properly using PDF viewers such as Preview or Skim (Mac). But using Adobe Reader 10.1.3 (either Windows or Mac) doesn't display the table line properly.   But removing the colouring for the table fixes it. Here is the example that works as expected:  \begin{table} \caption{Table showing Feature 5's Functional Tests} \hspace*{-0.5in} \begin{tabular}[h!]{|p{0.6cm}|p{4cm}|p{5cm}|p{4cm}|p{1.5cm}|} \hline {\bf Test ID } &amp; {\bf Task } &amp; {\bf Expected} &amp; {\bf Actual} &amp; {\bf Outcome} \\ \hline    5.1 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.2 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.3 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.4 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline \end{tabular}  \end{table}   And here is the example with a coloured row:  \begin{table} \caption{Table showing Feature 5's Functional Tests} \hspace*{-0.5in} \begin{tabular}[h!]{|p{0.6cm}|p{4cm}|p{5cm}|p{4cm}|p{1.5cm}|} \hline \rowcolor{gray!25} {\bf Test ID } &amp; {\bf Task } &amp; {\bf Expected} &amp; {\bf Actual} &amp; {\bf Outcome} \\ \hline    5.1 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.2 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.3 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.4 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline \end{tabular}  \end{table}   Removing the \rowcolor{} command replacing it with \cellcolor{} creates the same view. Any idea how to fix this?  I've tried printing it out of Adobe and the lines print fine, but as this will be read on as a soft copy it's pretty important. .  Any idea how to fix this? I have too many sections in my presentation such that the left-hand-side navigation bar is full and only shows the first sections in Fig. 2, initial situation in Fig. 1.  Pseudocode    clean the left-hand-hand side navigation bar when exceed the threshold and start a new one there when on sufficient page  this feature occurs with Beamer Berkeley appendix so it is possible also with the main document   Code which problem shown in Fig. 2   \documentclass{beamer} \usetheme{Berkeley}  \logo{\includegraphics{logo.png}} \usepackage{hyperref} % https://tex.stackexchange.com/a/346486/13173 \newenvironment{slide}[1] {\begin{frame}[environment=slide,allowframebreaks] \frametitle{\insertsection-#1} } {\end{frame}} % fragile only if code % allowframebreaks not by default   \begin{document}  \title{Sensory system. Primary sensory modalities. Cortical modalities. Kinds of sensory disturbances. Spinal cord disorders. Spinal cord transverse and partial lesion. Cauda equina damage.} \author{Leo}  \section{Cortical modalities} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities 2} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities 3} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities 4} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \end{document}   Fig. 1 Situation at the first page,  Fig. 2 Situation at Section 6 wheere the navbar has not changed,  Fig. 3 Samcarter's answer output + one slide addition but without \part{} - fails on sufficient page count       Output: the left-hand-side navigation bar is not refreshed when the amount of sections exceed its capacity on suffcient enough page     Testing Samcarter's proposal  I include one slide before Samcarter's body so I get the wrong output in Fig. 3.  Solution: Eventually correct output by just adding \part{} in the place where the new navbar is wanted.   TeXLive: 2016 OS: Debian 8.7 Memory layout of a .NET array What is the memory layout of a .NET array?  Take for instance this array:  Int32[] x = new Int32[10];   I understand that the bulk of the array is like this:  0000111122223333444455556666777788889999   Where each character is one byte, and the digits corresponds to indices into the array.  Additionally, I know that there is a type reference, and a syncblock-index for all objects, so the above can be adjusted to this:  ttttssss0000111122223333444455556666777788889999         ^         +- object reference points here   Additionally, the length of the array needs to be stored, so perhaps this is more correct:  ttttssssllll0000111122223333444455556666777788889999         ^         +- object reference points here   Is this complete? Are there more data in an array?  The reason I'm asking is that we're trying to estimate how much memory a couple of different in-memory representations of a rather large data corpus will take and the size of the arrays varies quite a bit, so the overhead might have a large impact in one solution, but perhaps not so much in the other.  So basically, for an array, how much overhead is there, that is basically my question.  And before the arrays are bad squad wakes up, this part of the solution is a static build-once-reference-often type of thing so using growable lists is not necessary here. The function $f(z)=z+\frac{1}{z}$ maps $\Omega$ biholomorphically on the upper plane The question states:     Let $\Omega$ be the upper half plane minus the unit disk. Show that the map $f: z \mapsto z+1/z$ maps $\Omega$ biholomorphically on the upper plane. Describe the preimages of the lines $\mathrm{Im}(z)=\text{a constant}&gt;0$.   So how exactly does one show the mapping is biholomorphic. I'm not sure where I should start, so unfortunately I haven't made any progress Fit a table into one page I am looking for a quick way to sleep without closing the lid, or using the mouse.  If it matters, I have Lion version OS and a MacBook Air. Find all solutions to $x^2\equiv 1\pmod {91},\ 91 = 7\cdot 13$ I split this into $x^2\equiv 1\pmod {7}$ and $x^2\equiv 1\pmod {13}$.  For $x^2\equiv 1\pmod {7}$, i did: $$ (\pm1 )^2\equiv 1\pmod{7}$$ $$(\pm2 )^2\equiv 4\pmod{7}$$ $$(\pm3 )^2\equiv 2\pmod{7}$$ Which shows that the solutions to $x^2\equiv 1\pmod {7}$ are $\pm1$.  For $x^2\equiv 1\pmod {13}$, i did: $$ (\pm1 )^2\equiv 1\pmod{13}$$ $$(\pm2 )^2\equiv 4\pmod{13}$$ $$(\pm3 )^2\equiv 9\pmod{13}$$ $$ (\pm4 )^2\equiv 3\pmod{13}$$ $$(\pm5 )^2\equiv {-1}\pmod{13}$$ $$(\pm6 )^2\equiv 10\pmod{13}$$Which shows that the solutions to $x^2\equiv 1\pmod {13}$ are $\pm1$.  Thus, I concluded that the solutions to $x^2\equiv 1\pmod {91}$ must be $\pm1$. I thought that $\pm1$ were the only solutions, but apparently I am incorrect! How do I go about finding the other solutions to this congruence? Which browsers are officially supported, and what else do I need? Which browsers are officially supported by Stack Exchange?  Also, what else is needed in order to use every feature of the site?