Reflexive relation on set of $n$ elements
Number of reflexive relations defined on a set A with n elements
Solve Euler Project #9 only mathematically - Pythagorean triplet The "Euler Project" problem 9 () asks to solve:  $a^2$ + $b^2$ = $c^2$ a + b + c = 1000  I find answers solving it with brute-force and programmatically, but is there a way to solve the problem ONLY mathematically? Can someone help, please?  Problem as explained in Project Euler website:  A Pythagorean triplet is a set of three natural numbers, a &lt; b &lt; c, for which,  a^2 + b^2 = c^2 For example, 3^2 + 4^2 = 9 + 16 = 25 = 52.  There exists exactly one Pythagorean triplet for which a + b + c = 1000.  Find the product abc. Is it possible to install Python 3.x in 14.04 LTS, or upgrade from Python 2.7.6 to 2.7.8? On the nature of the collapse of the wave function The collapse of the wave function by measurements is one of the most mysterious properties of quantum mechanics.    At what scale does the wave function collapse? What are the conditions for a collapse?  Can it be made precise?  Does the collapse happen instantaneously, or is it spread out in time?  What is the precise equation for the collapse?  Why is the collapse time-asymmetric, and can this explain the arrow of time? Can an illusion give a player partial cover? I'm in an argument with a player at my table, he says hiding behind some bushes while shooting arrows at my monsters gives him partial cover because they can only see his head and shoulders. So they're limited in what they can even aim at. I accepted that. And now he's trying to use minor illusion to create fake bushes/walls and continuing to claim he has partial cover.  I thought cover was based on having a solid obstacle between you and the target... Like, if a magical attack, or penetrating attack can pass through your cover, wouldn't that negate the AC bonus from partial cover? Or is it about "aiming"? 'bridges' on a PCB copper layer I've come across a design where each pad was connected using 4 'bridges' to the GND copper layar. What stands behind these 'bridges'? Why not make a full copper layer with only solder-mask defining the pads? I've seen many people who say:     This is a HTML page.   Yet I've also seen many people who say:     This is an HTML page.   Are both usages equally correct?  Or, which is the grammatically correct one?       Possible Duplicates: A complete picture of the lattice of subfields for a cyclotomic extension over $\mathbb{Q}$. Is there a good general purpose algorithm (batch of theorems) allowing one to determine the intermediate fields between $\mathbb{Q}(\zeta)$ and $\mathbb{Q}$, where $\zeta$ is some primitive root of unity?  Let $p$ be a prime. Consider the case where $\zeta=\zeta_{p}$ is a primitive $p$-th root of unity. Then the Galois extension is cyclic of order $p-1$ and $1,\zeta,\dots,\zeta^{p-1}$ is a $\mathbb{Q}$-basis for the extension. In this case for any subgroup $H$ of $G=\mathbb{Z}/(p-1)$, by considering the sum $$\alpha_H=\sum_{\sigma\in H}\sigma\zeta,$$ we can observe that $\alpha_H$ lies in the fixed field for $H$, and any automorphism $\tau$ not in $H$ (note automorphisms are identified with subgroups of $\mathbb{Z}/(p-1)$ in the natural way), $\tau$ does not fix $\alpha_H$. Therefore we can conclude that $\mathbb{Q}(\alpha_H)$ is the fixed field of $H$.  In this way we can get all of the intermediate fields of $\zeta_p$ for all odd primes $p$.  We also have a theorem that says if we have $n=p^sq^t$, then $$\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})\simeq \text{Gal}(\mathbb{Q}(\zeta_{p^s})/\mathbb{Q})\times\text{Gal}(\mathbb{Q}(\zeta_{q^t})/\mathbb{Q}).$$  So what I have yet to understand is  How can one generally find the intermediate fields between $\mathbb{Q}(\zeta_{p^s})$ and $\mathbb{Q}$ for $s\ge 1$? I would like to also understand the case where $p=2, s&gt;1$ though this might turn out to be a separate case.  EDIT: Even the case $n=pq$ is a little murky to me. Even given the isomorphism given by the Chinese Remainder Theorem I don't see a priori how to get all the "product" subfields. My idea is that you can consider the separate subfields under $\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$ and $\text{Gal}(\mathbb{Q}(\zeta_{q})/\mathbb{Q})$ separately and then consider the pairwise product of the generators of various subfields to see if you get anything new, but my idea is too inchoate. How do I access the terminal without logging in to the desktop? Stop suggesting Microsoft Edge browser in start menu when typing "internet" I use Chrome as my primary browser but must use Internet Explorer to be able to remote in to my work via Remote Desktop web client (since it uses ActiveX plugins). To start an app, I would often type out the name from the start menu unless I had a pinned link or shortcut handy. But if I type &quot;internet&quot; as if I were trying to start Internet Explorer, I will always be recommended Edge and have it highlighted by default.  I will never be able to use Edge here so stop recommending it... can this behavior be disabled somehow? Not even pinning the Internet Explorer link to the start menu prevents this. If I had any other shortcuts that started with the word &quot;internet,&quot; it still offers Edge and is preselected. At least if I type in &quot;chrome,&quot; Chrome is selected first. Windows 10 Pro 1909 (18363.900) What files should i put in a logo pack to give to clients? Should I put small, medium and large version of my logo as well as the vector file. Let $S = \bigoplus_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $&gt; 0$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. Suppose $S$ is finitely generated algebra over $S_0$. Then $S_{(f)}$ is a finitely generated algebra over $S_0$. Notation for intervals I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed . I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall someone calling the notation with $[a,b]$ and $]a,b[$ as French notation.   What are the origins of the two notations?  Is the name French notation correct? Are they used frequently in France? Or were they perhaps prevalent in French mathematical community at some point? (In this  Bourbaki is mentioned in connection with the notation $]a,b[$.)      Since several answerers have already mentioned that they have never seen $\langle a,b \rangle$ to be used for closed intervals, I have tried to look for some occurrences for this. The best I can come up with is the article on , where these notations are called Czech notation and French notation. Using $(a,b)$ and $[a,b]$ is called English notation in that article. (I am from Central Europe, too, so it is perhaps not that surprising that I have seen this notation in lectures.) I also tried to google for  or . Surprisingly, this lead me to a  where this notation is used for open interval. Word: how to show hidden ribbon tabs? How can I control which tabs are available to be opened in the MS Word 2010 ribbon?  Under Word Options > Customize Ribbon > Main Tabs there are several groups checked (Blog Post, Insert (Blog Post), Outlining, Background Removal), but they don't appear on the ribbon even though there is space for them. How is one supposed to turn them on or get to them?  The one in particular I'm after today is the Blog Post tab, which I know I can get to in a round about way via File > New > Blog Post Template. However I don't want a new post, rather an old one, which means File > New > Blog Post Template > Open existing > [select from list] > [close new blog post window] > [slap No I don't to save darn it]. Extra work for nothing is irritating; how do I get Blog Post (or Outlining or ...) to show up in that nice empty space next to AddIns? Add the font changes as options in our preferences How should artwork be setup for T-shirt printing? What does it mean to colour separate the artwork for T-Shirt printing? It is normally suggested to use Pantone colours for T-Shirt printing. Is it the Pantone Coated or the Pantone Uncoated? The Uncoated one has more subtle colours and we can't choose brighter shades if the design requires that. What is more appropriate for printing? Can we use gradients in T-Shirt printing or is it suggested to stick to flat colours only?   The reason for asking if gradients were ok was to use it as an option for this design:    I was considering it an option, as it will be difficult to colour separate since the textures are clip masked inside the lion. How to install packages in ubuntu without internet connection? How to get RGB values from UIColor? "who" vs. "whom" and when to omit the relative pronoun This is the boy who was asked by the teacher to stay back after school. This is the boy whom the teacher asked to stay back after school. This is the boy the teacher asked to stay back after school.   Are the sentences above correct ? How can I multiply each item in an array easily with PHP? How to install CUDA 8.0 on Ubuntu 16.04 with Nvidia GeForce GTX 1080 I am attempting to install CUDA 8.0 within Ubuntu 16.04 for the Nvidia GeForce GTX 1080. I had CUDA 7.5 installed, but it did not appear to be compatible with the GTX 1080. I downloaded the .run installer from Nvidia's developer page, but the installation fails and outputs the following to a nvidia-installer.log     An NVIDIA kernel module 'nvidia-drm' appears to already be loaded in     your kernel.  This may be because it is in use (for example, by an X     server, a CUDA program, or the NVIDIA Persistence Daemon), but this     may also happen if your kernel was configured without support for     module unloading.  Please be sure to exit any programs that may be     using the GPU(s) before attempting to upgrade your driver.  If no     GPU-based programs are running, you know that your kernel supports     module unloading, and you still receive this message, then an error     may have occured that has corrupted an NVIDIA kernel module's usage     count, for which the simplest remedy is to reboot your computer.   I have rebooted the computer, but that did not resolve the problem. How do I unload the 'nvidia-drm' module so that I can install the 8.0 CUDA?