Simplify $\cos 1^\circ + \cos 3^\circ + \cdots+ \cos 43^\circ$?
$\sum \cos$ when angles are in arithmetic progression
In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\rightarrow A$ such that for any subset $S$ of $A$, $f(S)\in S.$ Is this correct?  My question then is about proving this fact, so that we do not need to put it as an axiom. Now as per the research done on this single object- Axiom of Choice, I believe here that there should be some falsity in my argument. I do not find the mistake.  For any $S\in \mathcal{P}^*(A)$, since $S\neq \emptyset$, $\exists s\in S$. Define $f(S)=s$. Then $f$ is a choice function.  This was showing me that the axiom of choice is proved, but then why it had been put as an axiom? For example, in  book, the author asserts that      It is a metatheorem of mathematical logic that it is impossible to specify the function that assigns to each non-empty subset of $\mathbb{R}$, an element of itself.     There are several notes and books on axiom of choice, but here I am trying to understand through doing some argument for some problem, where problem actually arises. Valid references on origins of Machine Learning, Statistical Learning and Data Mining How to monitor who is pinging me? Is there a way to know who is pinging my PC on my network? Is it possible to monitor all ICMP packets? How can I do that? I have strange problem. My Wallch icon is huge. See the screenshot:    I removed XFCE settings and reordered items a little. How can I restore normal Wallch icon size?  Update:  My /usr/share/applications/wallch-nautilus.desktop file:  [Desktop Entry] Version=4 Name=Wallch Comment=Change desktop wallpapers automatically Exec=/usr/bin/wallch %U Icon=wallch Terminal=false Type=Application Categories=Utility;Application; MimeType=inode/directory;  Actions=Change_Wallpaper;Start;  [Desktop Action Change_Wallpaper] Name=Change Wallpaper Once Exec=/usr/bin/wallch --change TargetEnvironmet=Unity  [Desktop Action Start] Name=Start/Stop Wallpapers Exec=/usr/bin/wallch --start TargetEnvironment=Unity   And the /usr/share/pixmaps/wallch.png: Order of a group and subsequent subgroups I want to prove that if the order of a group G is greater than 1 and non-prime, then G has a subgroup other than the trivial subgroups {e} and G itself.   I am confused because I am only being able to prove the case for prime order by using Lagrange’s theorem i.e. if I have a group of order p, where p is a prime number, then the order of subgroups must divide p, hence the order can be either 1 or p.   How do I go about proving the case for non prime order let’s say x > 1? When I set up a site in multiple languages, how should I set up my URLs for search engines and usability?  Let's say my site is www.example.com, and I'm translating into French and Spanish. What is best for usability and SEO?  Directory option:  http://www.example.com/sample.html http://www.example.com/fr/sample.html http://www.example.com/es/sample.html   Subdomain option:  http://www.example.com/sample.html http://fr.example.com/sample.html http://es.example.com/sample.html   Filename option:  http://www.example.com/sample.html http://www.example.com/sample.fr.html http://www.example.com/sample.es.html   Accept-Language header:  Or should I simply parse the  header and generate content server-side to suit that header?  Is there another way to do this? If the different language versions don't have different urls, what do I do about the search engines?    UPDATE 2011-12-06  Google has new recommendations for meta tags for explicitly pointing to other language content: .  UPDATE 2012-05-25  Related but not precisely:   UPDATE 2013-06-12  includes discussion of several URL schemes directly relevant to the question. Is there a way to open a raster that is stored in ESRI V10.0 Geodatabase in QGIS V2.4?    I can open the vector layer, but there does not seems to be a way to open a raster. When training a neural network using the back-propagation algorithm, the gradient descent method is used to determine the weight updates. My question is: Rather than using gradient descent method to slowly locate the minimum point with respect to a certain weight, why don't we just set the derivative $\frac{d(\text{Error})}{dw}=0$, and find the value of weight $w$ which minimizes the error? Also, why are we sure that the error function in back-propagation will be a minimum? Can't it turn out the error function is a maximum instead? Is there a specific property of the squashing functions that guarantees that a network with any number of hidden nodes with arbitrary weights and input vectors will always give an error function that has some minima? (Re?)Install Ubuntu on a Dell laptop that had Ubuntu preinstalled I have Dell Laptop that is 15 5567 7th Generation. So it was maybe pre-installed as Dell says and I can see in the product information that this laptop is Ubuntu built in and based not Windows.  So I want to get my Ubuntu OS again. How can I do that? What is the association bonus, and how does it work? First let me explain what I mean by "transitive" Suppose that the price of product A and the price of product B has a correlation of .5 Suppose also that the price of product B and product C has a correlation of .5  One might thing that if correlation(A,B) = .5 and correlation(B,C) = .5 then correlation(A,C) > 0, but with an intuitive example we can see why this is not the case:  Suppose that A is a fruit salad made of papaya and banana, suppose also that B is a fruit salad made of banana and strawberry and, finally, suppose that C is strawberries and cream. Clearly, when the price of banana goes up, both the price of A and B will go up; when the price of strawberry goes up the price of B and C will go up; but since A and C have no ingredients in common, their price isn't correlated.  My question is which statistical concept captures this intuitive idea. Is it the concept of dimensions?  What do I mean by transitive? Let me define the binary operator X corr? Y as 1 if correlation(X,Y) != 0 and 0 otherwise, so:   A corr? B = 1, B corr? C = 1, but  A corr? C could be = 0   If corr? was transitive, then A corr? C = 1 Retrieve globalpicklist/globalvalueset without SOQL Is it possible to retrieve the values of a Global Picklist/GlobalValueSet using Schema class without a SOQL ? Is the word "management" singular or plural? Bash 'unbound variable' error when variables are referenced in multiline comment (set -u) I have an unexpected behaviour- following snippet fail with 'unbound variable' error:  #!/bin/bash set -u &lt;&lt;EOF a=a b=$a EOF   Tested on:  GNU bash, version 4.4.12(1)-release  GNU bash, version 4.2.46(2)-release Multiple cases in switch statement Is there a way to fall through multiple case statements without stating case value: repeatedly?  I know this works:  switch (value) {    case 1:    case 2:    case 3:       // Do some stuff       break;    case 4:    case 5:    case 6:       // Do some different stuff       break;    default:        // Default stuff       break; }   but I'd like to do something like this:  switch (value) {    case 1,2,3:       // Do something       break;    case 4,5,6:       // Do something       break;    default:       // Do the Default       break; }   Is this syntax I'm thinking of from a different language, or am I missing something? I am running a web site design service for local small business owner. I want avoid legal issues on the images I use. Usually, the images found on Flickr are forbidden for commercial use. Where would be a normal channel to get pictures for commercial use with minimal cost? How can I validate that the picture I used has no legal risk? Where are executables for programs stored in Ubuntu?  An application (Komodo Edit) is asking me to identify an application to be used as a web browser. I've become used to just entering the application name as a command for situations such as these, but this scenario got me thinking.   I know in Windows it would just be the relevant application folder in the 'program files' folder, but I'm assuming things are a bit different on Linux?  I thought somewhere like bin would be logical but this appears to standard Linux/Unix applications. Where would I find the binary executable for applications stored on my system? Why or how did Agen Kolar and Saesee Tiin seem to die so easily without any resistance? php sort associative arrays by keys that those keys exist in other array How can I find a saved Wi-Fi password? I have an Android phone which is connected to a secured Wi-Fi hotspot.  How can I find out the password of the hotspot from the phone?