Why $R/(a+bi)$ is a finite field with characteristic $p$ where $a^2+b^2=p$
Quotient ring of Gaussian integers
How to show a binomial random variable dominates another binomial random variable with a smaller success value? Let $X\sim B(n,p_h)$ and $Y\sim B(n,p_\ell)$ be two random variables following a respective binomial distribution, where $p_h&gt;p_\ell$. I want to show that  $$P(X\ge\alpha)\ge P(Y\ge\alpha),$$ for any $\alpha\in\{0,1,\dots,n\}$. In other words, I want to show  $$P(X\ge\alpha)=\sum_{i=\alpha}^{n}\binom{n}{i}p_h^i(1-p_h)^{n-i}\ge P(Y\ge\alpha)=\sum_{i=\alpha}^{n}\binom{n}{i}p_\ell^i(1-p_\ell)^{n-i}.$$  The statement to be proven is obviously true, but proving it is extremely difficult (at least for me). Can anyone give me some guidance on proving it?  It would be nice if someone can give an algebraic proof too, that is, show that  $$\sum_{i=\alpha}^{n}\binom{n}{i}p_h^i(1-p_h)^{n-i}\ge \sum_{i=\alpha}^{n}\binom{n}{i}p_\ell^i(1-p_\ell)^{n-i}.$$  Thank you very much! My Ubuntu version 18.04 brings me a GNU GRUB version 2.02 and I cannot access my entry password panel? SELECT DISTINCT on multiple columns Supposing we have a table with four columns (a,b,c,d) of the same data type.  Is it possible to select all distinct values within the data in the columns and return them as a single column or do I have to create a function to achieve this? Join tables from two different server Prove that $$\cos(\pi/11)+\cos(3\pi/11)+\cos(5\pi/11)+\cos(7\pi/11)+\cos(9\pi/11)=1/2$$ using Euler's formula.  Everything I tried has failed so far.  Here is one thing I tried, but obviously didn't work. $$\Re e \{e^{\frac{\pi}{11}i}(1+e^{\frac{2\pi}{11}i}+e^{\frac{4\pi}{11}i}+e^{\frac{6\pi}{11}i}+e^{\frac{8\pi}{11}i}) \}=\frac{1}{2}$$ $$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{e^{2\pi i}}+\sqrt[11]{e^{4\pi i}}+\sqrt[11]{e^{6\pi i}}+\sqrt[11]{e^{8\pi i}}) \}=\frac{1}{2}$$ $$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}) \}=\frac{1}{2}$$ $$\Re e \{5e^{\frac{\pi}{11}i} \}=\frac{1}{2}$$ $$5\cos(\frac{\pi}{11})=\frac{1}{2}$$ Which isn't true :D  Thanks in advance Is it OK to require users to have a third party login account? I'm trying to include a link to some code on OneDrive; the link includes an underscore, which is causing a lot of trouble.  I tried adding the entry like this:  @misc{code,     title = {Code},     url = {https://1drv.ms/link_link} }   I use \usepackage{url} and \bibliographystyle{IEEEtran}. Everything compiles fine and the bibliography shows the correct entry. However, if you click the link, it opens a page at https://1drv.ms/link - so ignoring the underscore and everything after it, which leads to a wrong page. Even copy and paste from the PDF doesn't work since the underscore is copied as space, which again is a broken link.  How can I add the link such that it can be clicked and copied properly? I tried \_, \usepackage{underscore}, {{link}}, different bibtex entries but nothing worked. Also all the related questions/answers don't answer mine. There are many documentaries, forums, blogs and more dedicated to . I have been frantically searching for an answer to my question however none of my sources have clarity to the matter of hand. I would really love a clear explanation to: What exactly is Dark Matter? Please help me to have a clear understanding. How does the Green-Flame Blade cantrip interact with the tiefling feat Flames of Phlegethos? Run a Windows Service as a console app We are beginning to go down the path of mobile browser support for an enterprise e-commerce webapp (Java/Servlet based).  Of course there are many decisions to be made, but it seems to me the cornerstone is to be able to reliably detect mobile browsers, and make decisions on the content to be returned accordingly.  Is there a standard way to make this determination (quickly) based on the http request, and ideally glean more information about the given browser and device making the request (screen size, html capabilities, etc?).  I would also appreciate any supplemental information that would be of use from someone who has gone down this path of taking an existing large scale enterprise webapp and architect-ing out mobile browser support from the development side.  [edit] I certainly understand the request header and the information about a database of standard user agents is a great help.  For those talking about 'other' request header properties, if you could include similar standardized name / resource of values that would be a big help.  [edit] Several users have proposed solutions that involve a call over the wire to some web service that will do the detection.  While I'm sure this works, it is not a good solution for an enterprise e-commerce site for two reasons: 1) speed.  A call over the wire for every page request to a third party would have huge performance implications.  2) dependency/legal.  We'd tie our website response time and key functionality to their service, which is horrible for legal and risk reasons. I am using the 2D Animation preset and so far every new scene I made had the same bright white background. Now my two latest scenes (created with 'copy settings') have a light grey background even though they are set to white. There's also no difference in surface, nodes or strength.  There are apparently exist different background shades with an off-white color but selecting the white one doesn't help either A function that is continuous at the origin but not differentiable there Show that $f(x,y)=\dfrac{xy^2}{x^2+y^2}$ (with $(x,y)\not=(0,0)$ and $f(0,0)=0$) is continuous but not differentiable at $(0,0)$.  I tried to show continuity with an $\epsilon -\delta$ argument but I don't know how to factorize the expression so that I can have something useful.   For differentiability I think I should show that the partials are not continuous at $(0,0)$. But finding the partials is also painful. I already have the "Edit Questions And Answers" privilege. Yet, if I approve someone else's edit, I'm told that more votes will be required before the edit is accepted.  This seems a bit peculiar to me, since I could just copy the edited version and paste it myself. I was really happy when I heard that HoTD would be continued. And even happier when the first new chapter actually came. but now after several months of waiting there still has been no other chapter. So my question is, has HoTD gone on hiatus again ? What and with what was Luthor doing in the ship at the end of the movie? After Doomsday was defeated, a spec-ops team sent to the Kryptionian ship in Metropolis found Luthor submerged in water, in front of a... thing, which "held" three metal cubes. The thing looked up, roars, and dissipates into thin air.    What the hell was that all about? I am really confused. What was that creature and what was Luthor doing with it? Least rank of a matrix Let $A$ be a $7\times 7$ matrix such that it has rank $3$ and $a$ be $7\times 1$ column vector.  Then least possible rank of $A+(a a^T)$ is? ($a^T$ is transpose of the vector)  Intuitively I think it's $2$ as $a a^T$ has rank $1$, and the most it could affect is one row of $A$. Is there a catchall function somewhere that works well for sanitizing user input for SQL injection and XSS attacks, while still allowing certain types of HTML tags? I'm trying to find a movie with an underground facility with people wearing white suits. The people have been affected such that they show no curiosity. There is one person that does and asks one of his co-workers where a tube leads to.      At the end there is an uprising where everyone gets to see the surface and sunlight.   What is the name of this movie? Why are some faces in my mesh darker?