a question about compact set, how to prove there exists f(y)=y
Contraction Map on Compact Normed Space has a Fixed Point
Composition of two Riemann integrable functions Suppose $f,g$ are two Riemann  Integrable functions .Is it true that $f\circ g$ is also Riemann  Integrable?  Trying this for a long time but not getting the answer How does the Eco-Cooler air conditioner really work? describes a device for developing countries which apparently cools the air by 5 °C without electricity. According to , it works by lowering the pressure to drag in the ambient air. Wouldn't the ambient air be at the same temperature?  Does the constriction compress the air and temporarily increase the pressure and so temperature, which then dissipates heat because it is hotter than surroundings; then after passing the constriction, the pressure returns to almost ambient but the temperature drops to below ambient?  How does the Eco-Cooler air conditioner actually work, if at all? Commenting out multiple lines of code, specified by line numbers, using vi or vim I have learned from  that it is possible to use vi/vim to comment out a specified range of line numbers.  For example, suppose I have the following bash script:  #!/bin/bash  This is my very very great script   Now suppose that I want to comment out line numbers 6 through 8 (which contain the words very, very, and great) using the # comment character.  In vi/vim, I can simply type :6,8s/^/# to obtain the following:  #!/bin/bash  This is my #very #very #great script   which comments out lines 6 through 8.  My question is, is it possible to type a similar one liner that will remove the # comment character from lines 6 through 8 (but not any other commented lines in the file)?  Having said this, I realize that there is some debate about whether I am actually using vi or vim.  In practice, I open a file script.sh with the command vi script.sh.  Also, when I type the command which vi, I obtain /usr/bin/vi.  Nevertheless, when I simply type vi and press Enter, I obtain this:  ~                              VIM - Vi IMproved ~ ~                               version 7.2.330 ~                           by Bram Moolenaar et al. ~                 Vim is open source and freely distributable ~ ~                           Sponsor Vim development! ~                type  :help sponsor&lt;Enter&gt;    for information ~ ~                type  :q&lt;Enter&gt;               to exit ~                type  :help&lt;Enter&gt;  or  &lt;F1&gt;  for on-line help ~                type  :help version7&lt;Enter&gt;   for version info   which seems to suggest that I'm actually using vim.  I am accessing a remote Ubuntu Linux cluster using SSH from my PC.  I am not using a Ubuntu Linux GUI. Why Smart's attack doesn't work on this ECDLP? Why don't self-closing script elements work? What is the reason browsers do not correctly recognize:  &lt;script src="foobar.js" /&gt; &lt;!-- self-closing script element --&gt;   Only this is recognized:  &lt;script src="foobar.js"&gt;&lt;/script&gt;   Does this break the concept of XHTML support?  Note: This statement is correct at least for all IE (6-8 beta 2). What is the best algorithm for overriding GetHashCode? In .NET, the  is used in a lot of places throughout the .NET base class libraries. Implementing it properly is especially important to find items quickly in a collection or when determining equality.  Is there a standard algorithm or best practice on how to implement GetHashCode for my custom classes so I don't degrade performance? Random piece of plastic Prove that $\omega + \omega_1 = \omega \cdot \omega_1 = \omega^{\omega_1} = \omega_1$ I am assuming already that a) the union of countably many countable sets is countable and  b) $\omega_1$ is the least uncountable ordinal, so $x &lt; \omega_1$ if and only if $x$ is a countable ordinal.   I'm not sure if it is relevant but this question also allows The Axiom of Choice.   Thanks Only actions taken are displayed on question while checking the history of Low Quality Posts review How can I disable landscape mode in Android? How can I disable landscape mode for some of the views in my Android app? Can't delete folder in Windows 7 I'm trying to delete a folder in Windows 7 and get a perplexing error message: "Could not find this item:  This is no longer located in G:\Graphics.  Verify the item's location and try again.  I can see the folder -- I can find it.  I just can't delete it.    I also get a second error message (sometimes) when I click on the folder: G:\Graphics 2009-11-17 refers to a location that is unavailable...this information might have been moved to a different location.    I'm using Windows 7; this folder is on an external hard drive.  I've emptied the folder (there were items in it); I've scanned that external hard drive for errors.  Trying to rename the folder yields the same enigmatic error message.    Is there a way to delete this folder? Windows of an application have minimum size and I can't find the window anymore I had an application (in case it matters: LibreOffice Calc] in full-size mode. I then clicked the middle of the three buttons of the top right to adjust the size. When I did that, the window became minimum size, literally a vertical line. I panicked and alt-tabbed, and now I can't even find the vertical line anymore. Opening new windows in that application opens all of them hidden to me, and closing/reopening or even restarting the OS doesn't help. I don't know how to resize my invisible windows anymore. Here's the preview of the windows in the application launcher: a vertical line (that I don't find on my screen anywhere if I click this). Recommended shutter speed for action sports? What shutter speed do professional sports photographers typically use for action sports like football and basketball? How can I install MonoDevelop v4 on 12.04? I am using Ubuntu 12.04. I can install v2.8.6.3 of Monodevelop from software center but I need at least v3 to use Monogame v3 or higher. Can anyone point me in the right direction please?   Thanks $\mathbb{Q}(\sqrt p_{1}+\sqrt p_{2})=\mathbb{Q}(\sqrt p_{1},\sqrt p_{2})$ for $p_{1},p_{2}$ primes. Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems? Since I started learning  and  last year, I've read a huge number of articles which insist that design patterns (especially in Java) are workarounds for the missing features in imperative languages. One article I found :     Most people I've met have read     by the Gang of   Four (GoF). Any self respecting programmer   will tell you that the book is   language agnostic and the patterns   apply to software engineering in   general, regardless of which language   you use. This is a noble claim.   Unfortunately it is far removed from   the truth.      Functional languages are extremely   expressive. In a functional language   one does not need design patterns   because the language is likely so high   level, you end up programming in   concepts that eliminate design   patterns all together.   The main features of functional programming (FP) include functions as first-class values, currying, immutable values, etc. It doesn't seem obvious to me that OO design patterns are approximating any of those features.  Additionally, in functional languages which support OOP (such as F# and OCaml), it seems obvious to me that programmers using these languages would use the same design patterns found available to every other OOP language. In fact, right now I use F# and OCaml every day, and there are no striking differences between the patterns I use in these languages vs. the patterns I use when I write in Java.  Is there any truth to the claim that functional programming eliminates the need for OOP design patterns? If so, could you post or link to an example of a typical OOP design pattern and its functional equivalent? Can a git submodule be created retroactively? "a" or "an" in front of a word beginning with a vowel I was writing a text when I came to say 'a union', then it struck me that I need to change the 'a' to 'an'. But for some reason, 'an union' didn't sound right.   Why doesn't it work? Is it because its a mass noun, instead of a regular noun? Unable to log in to Stack Exchange