I came across following statement:     The set of all bijective functions on a finite set forms a group under function composition.   The proof was given as follows:     We can have identity function as $\{(1,1),(2,2),(3,3),(4,4)\}$.   Since function is bijective and mapping to same set, we can have an inverse for any function by inverting the relation (changing the mapping $a\rightarrow b$ to $b\rightarrow a$).   Since the function maps to the same set, it must be closed and associative also.   I was thinking is associativity really holds in above set. For example, consider the finite set: $\{1,2,3,4\}$ and two bijective functions $f_1$ and $f_2$ as follows:   Now, $f_1(f_2(1))=f_1(3)=3$ and $f_2(f_1(1))=f_2(2)=1$. So, I feel its not bijective. Am I incorrectly evaluating associativity?
If I have $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f$ is onto and $f\circ f\circ f = f$, how can I prove that $f$ is bijective? I know that I only have to prove that it is 1-to-1 because I'm given the fact that it's onto, but how can I use the fact that $f\circ f\circ f=f$ to prove that it is 1-to-1?  Thanks!
My SoulSeek client Nicotine+ checks my open ports during initial configuration and promts that the one port number it needs is closed, cutting me off frmo about 50% of the users. Is there anyway I can open that particular port for the application to use?  RUnning ubuntu desktop 13.04. I have not manually installed any firewalls, but not sure if its included in the default installation. In the following sentence, I want to know if I've understand the correct meaning or not:     This version holds that the wave function does not so much describe the state of the physical system as the state of our knowledge of the system.    Can we say that it means the wave function describes the state of our knowledge  of the system more than it describes the state of the physical system  ?  It is worth noting that in the Cambridge Dictionary, we read:  not so much sth as sth ​ If you say that something is not so much one thing as something else, you mean it is more the second thing: They're not so much lovers as friends. I don't feel angry so much as sad. There are several distinct usages:   kernel density estimation   kernel trick   kernel smoothing   Please explain what the "kernel" in them means, in plain English, in your own words. Travelling from India to canada via Dubai and London. Do I need a transit visa What's the best way to version CSS and JS URLs? Can't clean a full /boot because of unmet dependencies I got an error message explaining my /boot is full. trying to clean up old image files always seems to fail because of disk full errors or dependency errors. Could someone explain where I’m going wrong.   $ sudo apt-get autoremove Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run ‘apt-get -f install’ to correct these. The following packages have unmet dependencies.  linux-image-extra-3.13.0-44-generic : Depends: linux-image-3.13.0-44-generic but it is not installed  linux-image-extra-3.13.0-45-generic : Depends: linux-image-3.13.0-45-generic but it is not installed  linux-image-generic : Depends: linux-image-3.13.0-45-generic but it is not installed E: Unmet dependencies. Try using -f.  $ ls /boot abi-3.13.0-32-generic         initrd.img-3.13.0-43-generic abi-3.13.0-36-generic         lost+found abi-3.13.0-37-generic         memtest86+.bin abi-3.13.0-39-generic         memtest86+.elf abi-3.13.0-40-generic         memtest86+_multiboot.bin abi-3.13.0-43-generic         System.map-3.13.0-32-generic config-3.13.0-32-generic      System.map-3.13.0-36-generic config-3.13.0-36-generic      System.map-3.13.0-37-generic config-3.13.0-37-generic      System.map-3.13.0-39-generic config-3.13.0-39-generic      System.map-3.13.0-40-generic config-3.13.0-40-generic      System.map-3.13.0-43-generic config-3.13.0-43-generic      vmlinuz-3.13.0-32-generic grub                          vmlinuz-3.13.0-36-generic initrd.img-3.13.0-32-generic  vmlinuz-3.13.0-37-generic initrd.img-3.13.0-36-generic  vmlinuz-3.13.0-39-generic initrd.img-3.13.0-37-generic  vmlinuz-3.13.0-40-generic initrd.img-3.13.0-39-generic  vmlinuz-3.13.0-43-generic initrd.img-3.13.0-40-generic   I used   kernelver=$(uname -r | sed -r 's/-[a-z]+//') dpkg -l linux-{image,headers}-"[0-9]*" | awk '/ii/{print $2}' | grep -ve $kernelver   to get   linux-headers-3.13.0-32 linux-headers-3.13.0-32-generic linux-headers-3.13.0-36 linux-headers-3.13.0-36-generic linux-headers-3.13.0-37 linux-headers-3.13.0-37-generic linux-headers-3.13.0-39 linux-headers-3.13.0-39-generic linux-headers-3.13.0-40 linux-headers-3.13.0-40-generic linux-headers-3.13.0-44 linux-headers-3.13.0-44-generic linux-headers-3.13.0-45 linux-headers-3.13.0-45-generic linux-image-3.13.0-32-generic linux-image-3.13.0-36-generic linux-image-3.13.0-37-generic linux-image-3.13.0-39-generic linux-image-3.13.0-40-generic   Then picking the first image to remove   $ sudo apt-get purge linux-image-3.13.0-32-generic Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies.  linux-image-extra-3.13.0-32-generic : Depends: linux-image-3.13.0-32-generic but it is not going to be installed  linux-image-extra-3.13.0-44-generic : Depends: linux-image-3.13.0-44-generic but it is not going to be installed  linux-image-extra-3.13.0-45-generic : Depends: linux-image-3.13.0-45-generic but it is not going to be installed  linux-image-generic : Depends: linux-image-3.13.0-45-generic but it is not going to be installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution). Is there a VB.NET equivalent for C#'s '??' operator? Is there a VB.NET equivalent for C#'s ?? operator? Why do people have darker skin in sunnier climates? I don't understand why darker skin is advantageous in hotter climates. Wouldn't it absorb more of the heat? I have heard that it reduces the incidence of cancer, but I would think absorbing more radiation would increase the risk? Why not use tables for layout in HTML? It seems to be the  that tables should not be used for layout in HTML.   Why?  I have never (or rarely to be honest) seen good arguments for this. The usual answers are:   It's good to But this is a fallacious argument; . I guess it's true that using the table element for layout has little to do with tabular data. So what? Does my boss care? Do my users care?Perhaps me or my fellow developers who have to maintain a web page care... Is a table less maintainable? I think using a table is  than using divs and CSS.By the way... why is using a div or a span good separation of content from layout and a table not? Getting a good layout with only divs often requires a lot of nested divs. Readability of the codeI think it's the other way around. Most people understand HTML, few understand CSS. It's better for SEO not to use tablesWhy? Can anybody show some evidence that it is? Or a statement from Google that tables are discouraged from an SEO perspective? Tables are slower.An extra tbody element has to be inserted. This is peanuts for modern web browsers. Show me some benchmarks where the use of a table significantly slows down a page. A layout overhaul is easier without tables, see .Most web sites that need an upgrade need new content (HTML) as well. Scenarios where a new version of a web site only needs a new CSS file are not very likely. Zen Garden is a nice web site, but a bit theoretical. Not to mention its  of CSS.   I am really interested in good arguments to use divs + CSS instead of tables. resizable serial console window? Novel about a dragon lord who has to wear gloves at all times When writing custom classes it is often important to allow equivalence by means of the == and != operators. In Python, this is made possible by implementing the __eq__ and __ne__ special methods, respectively. The easiest way I've found to do this is the following method:  class Foo:     def __init__(self, item):         self.item = item      def __eq__(self, other):         if isinstance(other, self.__class__):             return self.__dict__ == other.__dict__         else:             return False      def __ne__(self, other):         return not self.__eq__(other)   Do you know of more elegant means of doing this? Do you know of any particular disadvantages to using the above method of comparing __dict__s?  Note: A bit of clarification--when __eq__ and __ne__ are undefined, you'll find this behavior:  &gt;&gt;&gt; a = Foo(1) &gt;&gt;&gt; b = Foo(1) &gt;&gt;&gt; a is b False &gt;&gt;&gt; a == b False   That is, a == b evaluates to False because it really runs a is b, a test of identity (i.e., "Is a the same object as b?").  When __eq__ and __ne__ are defined, you'll find this behavior (which is the one we're after):  &gt;&gt;&gt; a = Foo(1) &gt;&gt;&gt; b = Foo(1) &gt;&gt;&gt; a is b False &gt;&gt;&gt; a == b True Simplifying and evaluating $\cot 70^\circ+4\cos 70^\circ$ I'm trying to update my system using apt-get, but it's giving me this error  Err http://bd.archive.ubuntu.com raring Release.gpg          Could not connect to bd.archive.ubuntu.com:80 (116.193.170.18), connection timed out Err http://bd.archive.ubuntu.com raring-updates Release.gpg   Unable to connect to bd.archive.ubuntu.com:http: Err http://bd.archive.ubuntu.com raring-backports Release.gpg   Unable to connect to bd.archive.ubuntu.com:http: Reading package lists... Done W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring/Release.gpg  Could not connect to bd.archive.ubuntu.com:80 (116.193.170.18), connection timed out W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring-updates/Release.gpg  Unable to connect to bd.archive.ubuntu.com:http: W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring-backports/Release.gpg  Unable to connect to bd.archive.ubuntu.com:http: W: Some index files failed to download. They have been ignored, or old ones used instead.   I am not a Ubuntu master but I can understand this is not a computer error. Any clues on how to fix it? Is it possible to create image data and save to a file from a script I'm currently trying to extract textures from a binary file, and ideally I'd like to save each texture to a file or embed in the .blend file. However, before saving an image, I need to be able to create an image in memory from an array of pixel data.  The PIL library contains the kinds of functions I need to use, but it is not included with Blender. Is there anything similar to PIL in Blender? bpy.ops.image seems to be close to what I need, but lacks necessary pixel set/manipulation functions. Why is "letter" not plural in "two letter words"? Why is "letter" not plural in "two letter words"?  For me it's very strange as the equivalent in French would be plural but my English friend finds it totally normal. Cross-reference packages: which to use, which conflict? How to get things off your head in Minecraft Mo' Creatures mod I installed the Mo'Creatures mod into Minecraft, then made a fish bowl. After catching a fish I went to put it on top of a bookshelf in my room, but instead of placing it, the bowl just dropped at my feet. When I tried to pick it up my character put it on his head. Now I'm stuck with a fish bowl on my head and I can't find any mention anywhere of how to get it off. There are a number of other animals from the mod that you can put on your head for some effect, but no mention of how to remove them pretty much anywhere. In case of a syntax error, pdflatex usually (in \errorstopmode) gets interactive and asks you what to do. I know I can press:   X to terminate S to enter \scrollmode (continue processing without stopping if further syntax errors arise, but stop for other errors) R to enter \nonstopmode (continue processing without stopping if further errors arise) Q to enter \batchmode (continue processing without stopping if further errors arise and suppress further output to the terminal)   Apparently, only the latter three (SRQ) can be provided implicitly by the respecive \...mode commands or the -interaction=...mode command line option. However, what I need is a mode that mimics the implicit pressing of X, that is, to make the pdflatex process terminate after the first error. Short story or novel from the 1970s or before about two warring alien tribes