Prove that $\sin A+\sin B+\sin C\leq \frac{3}{2}\sqrt3$
Maximum value of $\sin A+\sin B+\sin C$?
Affiliate Ads Are Coming To The Network Some time ago, we announced that . This has gone well and we're extremely thankful to everyone for their support and feedback. Because it has been successful so far, we're going to be extending the program to the majority of our graduated sites. Well, we've pretty much already done that, but there are quite a few sites that have ads enabled, yet still display nothing. The thing is, we don't really have an abundance of clients offering products that a  would find relevant, but Amazon has stand mixers, a selection of super fine chinois, deliciously sharp cutlery and enough cast iron to fill the Gulf Of Mexico with . Beginning as early as this week (week of 5 February 2018), our ad team is going to be hand selecting affiliate ads to be displayed on select sites. The use of inventory will of course depend on the availability of relevant ads to display which means:  Some sites might see very few, if any ads. Some topics simply don't come remotely close to orbiting anything worth advertising. Furthermore, while graduated, some sites simply don't have enough activity for ads to make sense.  Since these are hand-selected, it could take a while for ads to appear initially.  These would of course yield to inventory from clients if clients had products that were relevant to any given topic.   Nothing changes when it comes to the . If you're used to occasionally seeing a community promotion ad, you'll just notice that you'll also occasionally see an ad for something that is likely to be of interest to someone interested in the topic that your site covers. In fact, to be clear, this is simply a continuation of the announcement we made in November of 2016. We just wanted folks to know that we're going to be continuing the program where there's good potential to show relevant advertisements, even if we don't have clients lined up with a demand for it. Where can I give feedback about a certain ad on [site]? Open a meta post on your per-site meta and let us know. Please remember, the bulk of advertisements are an effort to get something small back from folks that hit the sites through search engines and are extremely unlikely to ever become contributors. Their soufflé stopped falling, and we'll probably never see them again. Maybe we can get a few bucks once they realized that copper ions help egg whites stiffen properly and buy a bowl or a proper bloom whisk. However, you're the experts on these topics and we're very likely going to listen to you. What sites, specifically? Are any sites exempt? If you're a former Stack Exchange 1.0 site and / or subsequently joined the network under a special charter, nothing changes - don't panic :) For the rest, we're just extending what we're currently doing with ads on graduated sites to also include those where there's an opportunity to make sensible use of affiliate ads. If a site has less than 15k (average) daily visits, or fewer than roughly 50k combined total posts - it might not yet be time to invest in selecting ads to be shown there. But, part of this exercise is also seeing what works best. I don't really have feedback, just some suggestions Ads aren't just an opportunity to sell stuff, they're an opportunity to really show people that we understand a given topic. That, of course, makes suggestions from people that do that stuff every day quite valuable. Post on your per-site meta and our community / ad critters will take a look. Questions? Comments? Kvetches? Let us know in an answer or comment below. If anything about the plan changes, we'll update this post to let folks know. $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$ Given the following definition of normal subgroup     A subgroup $H$ of a group $G$ is said to be normal if, for every $g\in G$: $$gH=Hg$$   I've tried to show that $H\mathrel{\unlhd} G$ if and only if we have $gHg^{-1}=H$ for every $g\in G$:  (if) Let $H\mathrel{\unlhd} G$, $g\in G$; if $l\in gHg^{-1}$, there exists $h\in H$ such that $l=ghg^{-1}$ and since $gh$ = $h'g$ for some $h'\in H$, then $l=ghg^{-1}=h'(gg^{-1}) = h'\in H$. Let $h\in H$; we can write $h$ as $hgg^{-1}$, and since $hg = gh'$ for some $h'\in H$, we obtain $h = hgg^{-1} = gh'g^{-1}\in ghg^{-1}$. So $gHg^{-1}=H$.  (only if) Let $gHg^{-1}=H$, $g\in G$; if we take an $l\in Hg$, $l = hg$ for some $h\in H$, we note immediately that $l=hg=(gh'g^{-1})g=gh'\in gH$. What should I do to show that $gH\subset Hg$?  I'm also wondering if this mess is correct, and if it is possible a more elegant proof of this fact. Combine multiple results in a subquery into a single comma-separated value I've got two tables:  TableA ------ ID, Name  TableB ------ ID, SomeColumn, TableA_ID (FK for TableA)   The relationship is one row of TableA - many of TableB.  Now, I want to see a result like this:  ID     Name      SomeColumn  1.     ABC       X, Y, Z (these are three different rows) 2.     MNO       R, S   This won't work (multiple results in a subquery):  SELECT ID,        Name,         (SELECT SomeColumn FROM TableB WHERE F_ID=TableA.ID) FROM TableA   This is a trivial problem if I do the processing on the client side. But this will mean I will have to run X queries on every page, where X is the number of results of TableA.   Note that I can't simply do a GROUP BY or something similar, as it will return multiple results for rows of TableA.   I'm not sure if a UDF, utilizing COALESCE or something similar might work? What book has criminals wear a ring? I remember reading one of my father's books when I was younger.  It was about a guy who ended up getting in trouble with the law and was required to wear a ring that could not be removed and would shock the wearer any time they committed a crime (No matter how minor.)    The book was probably published in the early to mid 80s.  I thought the name of the book was 'The Ring', but I am unable to locate anything.  The cover had a car with a single wheel driving on a metal I-beam. Nested regression model I hear a lot of people saying that neutrino is the lightest subatomic particle but according to me a photon must be the lightest as nothing can travel faster than light because it gets heavier and heavier and bends time. Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition. How can I upload files asynchronously with jQuery? What's this =! operator? How do gas molecules constantly bounce without losing energy? From a related question (), i asked myself this: How do gas molecules constantly bounce off each other without losing energy?  If you drop a ball, it bounces a bunch of times, but the height of each bounce gets shorter and shorter. Because it loses energy in each bounce.  So I don't see how a pressurized gas doesn't lose energy. As I understand it, pressure is a manifestation of lots of molecules bouncing off each other. So over time, shouldn't the pressure drop as the molecules lose speed? and eventually all the molecules will settle in a pile on the floor.  I think another way to phrase this is, how do elastic collisions not lose any energy in the exchange? My understanding of the 2nd law of thermodynamics is that some energy is always "lost" when it's converted from one form to another, or transferred from one object to another. I.e., no transfer/conversion of energy is ever 100% efficient. Question about proof that $(a + b)^p = a^p+b^p$ in a field of characteristic $p$. Definition charactersitic $p$: a field has characterstic $p$ is its prime field is isomorphic to $\mathbb{Z}/p\mathbb{Z}$.  Proposition: let $K$ be a field of characteristic $p$. Then for any $a, b \in K$ we have $(a + b)^p = a^p + b^p$.  Proof: (from my text) because $p \choose i$ is divisible by $p$ for $0 &lt; 1 &lt; p$ it is zero $\mod p$, so $p \choose i$  $a^{p - i}b^i$ is zero (this is the part I don't understand), so it follows from Newton's binomial law.  I understand that the prime field of $K$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$, so if $a, b$ are in the prime field, then $p \choose i$  $a^{p - i}b^i$ is zero, but that is not necessarily the case is it? Keyboard layout/language changes in every boot public class doublePrecision {     public static void main(String[] args) {          double total = 0;         total += 5.6;         total += 5.8;         System.out.println(total);     } }   The above code prints:  11.399999999999   How would I get this to just print (or be able to use it as) 11.4? I need to prove that there are infinitely many numbers  of the form $x = 111....1$ such that $31\mid x$  what I tried - I wrote x as   $\sum_0^{n-1} 10^i$  I know that $(10,31) = 1 $  now I'm stuck .. any help will be appreciated Proving that $\lim_{n \rightarrow\infty} \int_{0}^{\frac{\pi}{2}} \sin(t^n) dt =0$ It is clear that $\lim_{n \rightarrow\infty} \int_{0}^{1} \sin(t^n) dt =0$. (which is not what is to be proved here)  I don't know how to proceed with the remaining part of the integral ie $\lim_{n \rightarrow\infty} \int_{1}^{\frac{\pi}{2}} \sin(t^n) dt$ (I tried substitution and partial integration).  Can anybody give me a hint ? Frequent Write to Non-volatile Memory Ever since adding an iTunes gift card to my account, I'm getting verification requests like crazy. I can't even download free apps. It takes me to my payment info and when I try to select none, I continually get a message under the payment info that says "you must verify your payment info to continue"  I don't want a credit card on file and would really like the none option to work. My ID shows I still have iTunes credit. How do I force Python's print function to output to the screen? Custom right-click action in Unity Launcher keeps cursor busy for 20 seconds To have a "minimize window" option through right click on an icon in the Unity launcher (Ubuntu 14.04), I followed the detailed instructions  about changing a .desktop file and made a custom Firefox launcher in ~/.local/share/applications/ folder. The relevant part of the .desktop file is:  Actions=NewWindow;NewPrivateWindow;Minimize  [Desktop Action NewWindow] Name=Open a New Window Exec=firefox -new-window OnlyShowIn=Unity;  [Desktop Action NewPrivateWindow] Name=Open a New Private Window Exec=firefox -private-window OnlyShowIn=Unity;  [Desktop Action Minimize] Name=Minimize Windows Exec=sh /home/myusername/Documents/xdotool_sh/minimize.sh firefox OnlyShowIn=Unity;   The desktop action "Minimize" invokes a simple shell script, minimize.sh that has the following content:  #/bin/bash name=$1 for i in $(xdotool search --class "$name"); do     xdotool windowminimize $i done   The script uses xdotool, that can be installed from the official repositories, to find all the firefox windows, iterate over them and minimize them.  The script works and the launcher right menu option "Minimize Windows" works as well, but as soon as the windows are minimized, the mouse pointer gets in "busy" mode and stays like this for about 20 seconds (although the mouse actions are still responsive).   Does anyone know why starting a shell script from a right menu option in Unity could lead to this behaviour?  EDIT: Apparently the wait period is inevitable, as explained in Jacob Vlijm's answer. Since the mouse remains responsive, avoiding the transformation of the pointer in the spinning wheel is a partial esthetic workaround, as explained on .  EDIT2: giving the system a fake window is a better solution, as explained by Jacob below. Should I vary input field lengths in a single page checkout form?