Is this partition axiom equivalent to axiom of choice over ZF? Suppose we add the axiom to ZF that for any set $S$, and any partition $P$ of $S$, the cardinality of $P$ is less than or equal to the cardinality of $S$. It is known that the axiom of choice implies that axiom. My question is, is the converse true? Or are there models of ZF where that axiom holds but the axiom of choice doesn't?
Equivalent statement of AC or not? Here's a corollary of AC stating that: if there exists an onto mapping $f:A\to B$, then there exists a 1-1 mapping $g:B\to A$. What I'm confused about is that whether or not  AC can be derived from this proposition, in return? I have tried but haven't found a proof. Any help would be appreciated.
Do web browsers use different outgoing ports for different tabs? I am a girl in my early twenties traveling to USA from India and in my visa application I have provided the contact address for a relative in San Francisco, and their invitation letter. I would like to visit LA first. Will I face problems during immigration if I land in LA (with an immediate on wards ticket to SF)? I dont plan to get on the flight from LA to SFO.  Initially I was planning to fly to SFO from India and then fly to LA immediately after completing immigration at SFO airport. Finding the limit of roots products $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$ Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$ I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, so I proceed:  \begin{align*} \lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)&amp;=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{n}{n^2+k^2}\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{n^2}{n^2+k^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{1}{1+(k/n)^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}f(\frac{k}{n})(\frac{k-(k-1)}{n})\\ &amp;=\int_{0}^{1}\frac{1}{1+x^2}dx=\frac{\pi}{4} \end{align*}  where $f(x)=\frac{1}{1+x^2}$. Is this correct, are there any steps where I am not clear? Uncaught SyntaxError: Unexpected token with JSON.parse How do I install the latest  on Ubuntu? I've been looking around, and I can't find anything. Is there a Ubuntu package for node.js, or do I have to compile it myself? Progress bar with dynamic text & text color update What IDE to use for Python? What IDEs ("GUIs/editors") do others use for Python coding? When I boot in GRUB2 and press c, I have access to GRUB shell.  How to change the keyboard layout (language) of this GRUB shell?  Note that I'm booting an Ubuntu VM inside Virtualbox. Bash variable scope Please explain to me why the very last echo statement is blank? I expect that XCODE is incremented in the while loop to a value of 1:  #!/bin/bash OUTPUT="name1 ip ip status" # normally output of another command with multi line output  if [ -z "$OUTPUT" ] then         echo "Status WARN: No messages from SMcli"         exit $STATE_WARNING else         echo "$OUTPUT"|while read NAME IP1 IP2 STATUS         do                 if [ "$STATUS" != "Optimal" ]                 then                         echo "CRIT: $NAME - $STATUS"                         echo $((++XCODE))                 else                         echo "OK: $NAME - $STATUS"                 fi         done fi  echo $XCODE   I've tried using the following statement instead of the ++XCODE method  XCODE=`expr $XCODE + 1`   and it too won't print outside of the while statement. I think I'm missing something about variable scope here, but the ol' man page isn't showing it to me. Lightsabers are not made of light, nor have they anything to do with lasers (even if there are people who mistakenly refer to them as "laser swords").   Blaster bolts are also not lasers. They travel significantly slower than light, they travel even slower than ordinary bullets fired by gunpowder. (called slug throwers on the Star Wars universe)  Was there any use of lasers ever presented in the Star Wars universe, either as we use them today, or as weapons?  By laser I mean something which behaves like laser does, even if it's not called by that name in a movie or book. If something is called a laser (in the script or by fans) but clearly doesn't work like a laser (for example, it travels slowly, and in visible blocks) it doesn't count. Algorithm over 2 pages I am writing a few rather large pseudocode functions using algorithm and algorithmic, i.e. my code looks something like:  \begin{algorithm}                      \caption{Coolest Algorithm ever}           \label{findme}                           \begin{algorithmic} [1]                   % enter the algorithmic environment \REQUIRE    Here are a few variables \\             variable 2 \\             variable 3  \ENSURE     this is the output \\             output 2  \STATE $some cool code here$ \STATE $some cool code here$ \STATE $some cool code here$ \STATE $some cool code here$ \end{algorithmic} \end{algorithm}   Unfortunately this code does not fit on one page and I am definitely not able to split it into some subfunctions or so (I already did that as far as possible).  So now I need to split up the algorithm into two pages, but I have no idea how.  The easiest way I can imagine is just to start a new algorithm on the new page, but this time letting the line counter start at line 35 or so instead of 1. But how can I do that?  Please not that there is already a similar question here on this site, but it deals with the package algorithmicx. Why is Ubuntu Software Center dropping previously provided applications? Ubuntu Software used to provide Pinta, KolourPaint, Audacity, and others, but they are no longer available there. I have to use apt to install them.   Why has Ubuntu Software 3.20.5 stopped providing applications that it used to?  Ubuntu 16.04 Ubuntu Software Center 3.20.5  Thanks in advance for your help! How can I recover my external hard drive's free space? I had these partitions on an external hard drive:  /dev/disk2 (external, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *2.0 TB     disk2    1:                        EFI EFI                     209.7 MB   disk2s1    2:                  Apple_HFS Android                 63.9 GB    disk2s2    3:                  Apple_HFS RiptideHUP              968.0 GB   disk2s3    4:       Microsoft Basic Data RiptideF                967.8 GB   disk2s4   My aim was to delete disk2s2 and then resize disk2s3 to occupy the leftover space by deleting disk2s2.  I deleted disk2s2 using this command:  diskutil  erasevolume  "Free Space"  ""  /dev/disk2s2   Now, to resize the volume, I used:  diskutil  resizevolume  /dev/disk2s3  R   However, I got this error:  Started partitioning on disk2s3 RiptideHUP Error: -69743: The new size must be different than the existing size   I did a bit of research and realised the free space was created before disk2s3 RiptideHUP so it cannot resize itself. Is there still a way I can reclaim the Free Space?  NOTE: Using diskutil list now shows:  /dev/disk2 (external, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *2.0 TB     disk2    1:                        EFI EFI                     209.7 MB   disk2s1    2:                  Apple_HFS RiptideHUP              968.0 GB   disk2s3    3:       Microsoft Basic Data RiptideF                967.8 GB   disk2s4 OpenLayers Plugin Error: latitude or longitude exceeded limits I'm aware the 'traceback' OpenLayers plugin error has been dealt with before, so apologies for raising the issue once again. Unfortunately for me, the solutions posted were way over my head, e.g. using workarounds and code posted on github, etc. I find it all mind-boggling.  If anyone could post a step by step answer for an absolute beginner like me I would be eternally grateful! Thank you. I hope that one day I can contribute something.  The error message reads:   An error has occured while executing Python code:  Traceback (most recent call last):   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 48, in addLayer     self.__plugin.addLayer(self)   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 190, in addLayer     self.__setMapSrsGoogle()   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 249, in __setMapSrsGoogle     extMap = coodTrans.transform(extMap, QgsCoordinateTransform.ForwardTransform) QgsCsException: forward transform of (0.884883, 2.549197) PROJ.4: +proj=longlat +datum=WGS84 +no_defs +to +proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +wktext  +no_defs Error: latitude or longitude exceeded limits Exploring ocean depths: is pressure an issue? Should it be? An upcoming adventure will have the party exploring the depths of the ocean. Breathing underwater shouldn't be an issue for them, as there are both items and rituals to take care of that, however, I was wondering what to do about about pressure, since none of these options mention it. I'll have to split my question into two, since they are closely related.   Is pressure an issue in 4e adventures? Is it mentioned in any of the published material? Has it been ruled anywhere that options that enable water breathing also grant immunity to pressure? Should it be an issue? This is somewhat subjective, but I really am wondering what other people have done. Would it be unresonable to assume that these options grant immunity to pressure? Should the players work to find another solution? Is Autopilot Openstack 16.04 available? Difference between standard error and standard deviation I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error? How to assign symbols to functions Importing users that will be installed by programs