How to color a single logical page among the eight printed on a physical page?
How to change the background color only for the current page?
algebraic element over a field Let $F$ be a subfield of a field $K$, $a$ an element of $K$.  Prove that $a$ is algebraic over $F$ iff $F[a]=F(a)$.  For the first direction I've tried this: $a$ is algebraic over $F$ $\Rightarrow$ there is a polynomail $f$ over $F$ such that $f(a)=0$ $\Rightarrow$ $1,a,a^2,...,a^m$ are linearly dependent $\Rightarrow$ $\deg(F[a])&lt;m$.    but it contains $f(a)$ so they are equal..  is that correct? about the other direction.. i simply have no clue Can a character use the Misty Step spell to escape from the Iron Bands of Bilarro? Can a character escape the  (DMG p. 177, listed in the SRD as the ) with the  spell? To clarify, I am not asking if the bands prevent teleportation. I am asking whether the bands are teleported with the character as he teleports using misty step. I came to know about , it's a website that allow search for flights by country. I mean you just enter from like "Germany" and to like "Belgium" and you get best offers from all airports between the two countries. That is great. And I was able to arrange a good-price trip to Brussels.   Now I want to arrange another trip to Turkey, but Whichbudget is not helping much this time. And I can't find any other websites that allow "search by country".   Update: Not duplicate to  as his question is broad to the extent of a state. But my question is broad to the extent of a whole country + less price + "Any Time". If anyone checks , he'll understand what I'm saying. Especially that whichbudget allows "Any Time" option, not just +/-3 days.  Does anyone know more similar websites? 'pagesize' attribute ignored in hot network list Steps to reproduce:   Open  in Guest/Incognito mode or with cookies disabled. This will have 30 questions per page by default. Click the  link and as expected, you'll see questions 1-50 with page size of 50. Go the the  and see the bug: you're in the second page of 30 items per page, i.e. you'll see questions 31-60 instead 51-100 as expected.     It is very annoying. Please fix it.  My user-agent is: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.15; rv:70.0) Gecko/20100101 Firefox/70.0  There are no plugins installed affecting the site. It happens both when logged in or not.  The mechanism used to work before, but some weeks/months ago it stopped, enough to bother me to post here :)  Here are all modified "privacy" settings from about:support page:  privacy.clearOnShutdown.offlineApps true privacy.clearOnShutdown.openWindows true privacy.clearOnShutdown.siteSettings    true privacy.cpd.offlineApps true privacy.cpd.openWindows true privacy.cpd.passwords   true privacy.cpd.siteSettings    true privacy.donottrackheader.enabled    true privacy.history.custom  true privacy.item.cookies    true privacy.sanitize.migrateClearSavedPwdsOnExit    true privacy.sanitize.migrateFx3Prefs    true privacy.sanitize.pending    [{"id":"shutdown","itemsToClear":["cache","cookies","offlineApps","history","formdata","downloads","sessions","siteSetti privacy.sanitize.sanitizeInProgress ["cache","cookies","offlineApps","history","formdata","downloads","sessions","siteSettings","openWindows"] privacy.sanitize.sanitizeOnShutdown true privacy.sanitize.timeSpan   0 privacy.trackingprotection.enabled  true privacy.trackingprotection.introCount   20 Ward model implementation: specular component too weak I'm trying to implement the Ward shading model, the anisotropic, computationally efficient version:       This is how I made it:    float alphaX=0.5, alphaY=0.5; // asinotropic roughness float minFloat= 1.e-6;   vec4 Ward(vec3 L, vec3 N, vec3 V) {     vec3 H= normalize(L+V);     float NdotL= max(dot(N,L),0.0);     float NdotV= max(dot(N,V),0.0);      vec3 e= vec3(1.0,0.0,0.0);     vec3 T= normalize(cross(N,e));     vec3 B= normalize(cross(N,T));     float HdotT= max(dot(H,T),0.0);     float HdotB= max(dot(H,B),0.0);     float HdotN= max(dot(H,N),0.0);      float beta= -2.0 * ( pow(HdotN/alphaX, 2.0) + pow(HdotB/alphaY,2.0) ) / (1.0 + HdotN);     float den= max(minFloat, 4.0 * pi * alphaX * alphaY * sqrt(NdotL * NdotV) );     vec4 specular= Ps * 1.0/den * exp(beta);     return (Pd + specular) * NdotL; }   But I get a result similar to a lambertian surface or Oren-Nayar model, where the specular component is absent: Is it standard to say $-i \log(-1)$ is $\pi$? I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared  $$ \pi = -i \log(-1)$$  which really bothered me. Multiplying on both sides by $2i$:  $$ 2\pi i = 2 \log(-1) = \log(-1)^2 = \log 1= 0$$  which is clearly false.  I guess my error is $\log 1 = 0$ when $\log$ is complex-valued.  I need to use $1 = e^{2 \pi i}$ instead.  So my question: is it correct for WA to say $\pi = -i \log(-1)$?  Or should be they specifying "which" $-1$ they mean?  Clearly $ -1 = e^{i \pi}$ is the "correct" value of $-1$ here. Show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$. The problem is:  Suppose that $f(z)$ is continuous on a domain $D$ that contains the unit circle, and that $f(z)$ satisfies: $$\vert f(e^{i\theta})\vert \leq M \; \forall \theta \, \in [0, 2\pi )$$ and $$\bigg \vert \int _{\vert z \vert=1} f(z) \text{d}z \bigg \vert =2\pi M $$ show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$.  I've tried lots of different approaches such as defining a function $g(z)=\frac{f(z)}{\bar{z}}$ and trying to show that $g'(z)=0$ so that $g$ is constant on the circle, or trying to use some bounding theorems for analytic functions to show that $g$ is constant but this requires $g$ to be analytic. Some other methods were using the Cauchy-Riemann equations but I couldn't find any reason why $g$ should be analytic.  Any help would be greatly appreciated, thanks! Do you have advantage when multiple sources give advantage and only one gives disadvantage? As I understand it:   if you have Advantage from two different sources, you simply have Advantage (no stacking) if you have advantage from one source, and disadvantage from one source, they cancel out.     So what if you have advantage from two or more sources, and disadvantage from one source?  My assumption is that you simply have advantage, or put another way, you have Advantage on a roll when the # of Advantages > # of Disadvantages (and vice versa).  Is that correct? Number of ordered pairs $(a,b)$ such that $ab \le 3600$ Find number of ordered pairs $(a,b)$ such that $ab\le 3600$ and $a,b \in N$  My attempt : Well, all $a,b \leqslant 60$ are solutions. These 3600 solutions. After that I have no idea how to count the number of remaining solutions...   I have a feeling that my approach is $very$ bad. But I had to give as it would be marked as off-topic. Presentation of SL$(n,\mathbb{Z}_p)$ show that a distance function is continuous Is there any way to check whether a file is locked without using a try/catch block?    Right now, the only way I know of is to just open the file and catch any System.IO.IOException. Find $f(x)$ for $f(f(x))=\sin x$ Find all (if that's possible) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(f(x))=\sin x.$$  This is a problem a friend gave me. He said he saw it on a book a long time ago, but he couldn't remember what book. I literally have no idea how to solve this so I can't post my attempt here. Don't show pending suggested edits to me after I run out of votes Now that I've reached 2k on Stack Overflow, I get access to the "Suggested Edits" review queue.  Although I only have 20 edits per day, the "suggested edit awaiting" icon in the top bar is always there and it bothers me a lot!  My suggestion is to apply one of three solutions:   Remove the suggested edits icon when we reach the maximum (50) edits Remove the Highlight (orange) on the suggested edits Remove the suggested edits part when we click on it   What do you think? For a presentation with Beamer, I would like to have a slide where a long list of formulas (maybe mixed with text) scrolls automatically within a certain part of the slide. This should look like e.g. closing credits in movies.   I have tried the idea from  based on animateinline from the , but whenever I use environments like \begin{equation} ... \end{equation} or commands like \textcolor{red}{...} I get an error:  TeX capacity exceeded, sorry [main memory size=3000000]. \end{frame}   Ideally, the formulas/the text should be given within the document itself, but if a solution with an external PDF works fine, I'll take it!  Any ideas how that might work? "Me and my wife" or "my wife and me" When I restart httpd, I get the following error.  What am I missing?  [root@localhost ~]# service httpd restart Stopping httpd:                                            [  OK  ] Starting httpd: Syntax error on line 22 of /etc/httpd/conf.d/sites.conf: Invalid command 'SSLEngine', perhaps misspelled or defined by a module not included in the server configuration   I have installed mod_ssl using yum install mod_ssl openssh  Package 1:mod_ssl-2.2.15-15.el6.centos.x86_64 already installed and latest version Package openssh-5.3p1-70.el6_2.2.x86_64 already installed and latest version   My sites.conf looks like this  &lt;VirtualHost *:80&gt; #    ServerName shop.itmanx.com     ServerAdmin webmaster@itmanx.com      DocumentRoot /var/www/html/magento     &lt;Directory /var/www/html&gt;         Options -Indexes         AllowOverride All     &lt;/Directory&gt;      ErrorLog logs/shop-error.log     CustomLog logs/shop-access.log &lt;/VirtualHost&gt;  &lt;VirtualHost *:443&gt;     ServerName secure.itmanx.com     ServerAdmin webmaster@itmanx.com      SSLEngine on     SSLCertificateFile /etc/httpd/ssl/secure.itmanx.com/server.crt     SSLCertificateKeyFile /etc/httpd/ssl/secure.itmanx.com/server.key     SSLCertificateChainFile /etc/httpd/ssl/secure.itmanx.com/chain.crt      DocumentRoot /var/www/html/magento     &lt;Directory /var/www/html&gt;         Options -Indexes         AllowOverride All     &lt;/Directory&gt;      ErrorLog logs/shop-ssl-error.log     CustomLog logs/shop-ssl-access.log     &lt;/VirtualHost&gt; Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$ When app is moved to SD card, where is the app data stored on SD card? What is difference between: "I have been married with (was married with, married) her for three years." What is the difference between the three sentences below?   I have been married with her for three years. I was married with her for three years. I married her for three years.