Strange applications of the Axiom of Choice
Is there a *simple* example of how the axiom of choice can lead to a counterintuitive result?
about how to open a URL in your default browser using the Terminal. Basically, you use the command   xdg-open http://google.com   My default browser is Firefox. But lets say that I wanted to open a specific website in Chrome, instead of Firefox. Is there a command to open it in Chrome instead of Firefox? I asked a  recently, looking for examples of an online tool. Per the , I answered my own question with a few examples that I had already found. I did this so that these examples would be included in the answers, along with the rest.  The question has received only down votes and insults saying that this shouldn't be done.   Should I not answer my own questions? What does 1x1 convolution mean in a neural network? I am currently doing the Udacity Deep Learning Tutorial. In Lesson 3, they talk about a 1x1 convolution. This 1x1 convolution is used in Google Inception Module. I'm having trouble understanding what is a 1x1 convolution.  I have also seen  by Yann Lecun.  Could someone kindly explain this to me? Repeating something with (1/n)th chance of success n times Is there anything that can be said about how many attempts it will take to correctly guess a random number out of 1000 numbers? If the number wouldn't change the probability would just increase every guess, giving in the end a probability of 1.  If the number changes every guess tough, you just have a 1/1000th chance everytime. Can it be calculated how many attempts it would probably take, on average?  (I'm sorry, i'm not very skilled in probability maths) Currently, the only way to vote to delete (and undelete, since you can't see deleted posts) in the Android app is to open the post in a browser, and then vote to delete from there.  Can we get delete vote functionality within the app, as well as the ability to delete our own posts?  Worth to mention: users , so it's possible in the API.  Related: "Access denied" when trying to end a process on Windows 8 More section headings? I am a novice in LaTeX and I have one question.  Can someone tell me what if you want more heading's in LaTeX. Like 5 or 10 or even 15. Is there a package for that? How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals? How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals? queries with db_select don't respect integers When I check "SHOW PROCESSLIST" in phpMyAdmin I can see lots of queries where argument-integers are being inserted as strings (with single quotation marks).  I'm wondering if this is due to the way phpmyadmin reports the queries (if it puts every argument between quotation marks) of if it is db_select that puts every argument between quotation marks.  An example of my code where integers are being used as arguments:  $select-&gt;join('taxonomy_index', 't_'.$key, 't_'.$key.'.nid = n.nid AND t_'.$key.'.tid = :tid_' . $key, array(':tid_'.$key =&gt; (int)$tid)); $select-&gt;condition('t_'.$key.'.tid', (int)$tid);   As you can see I explicitly write (int)$tid but in my query log I can see the following:  ... WHERE (t_0.tid = '10') ...   The big question is: How can I be sure that integers are being treated as integers in my queries? I don't want MySQL having to transform strings to integers. Where can I find legendary crafting materials? "Closed" (square) root symbol I found a cool trick in the  which allows the square root sign to be altered such that it &quot;closes&quot; over its contents. Here's an example:  (&quot;closed&quot; root on the left, normal one on the right) The way they do this is as follows: % New definition of square root: % it renames \sqrt as \oldsqrt \let\oldsqrt\sqrt % it defines the new \sqrt in terms of the old one \def\sqrt{\mathpalette\DHLhksqrt} \def\DHLhksqrt#1#2{% \setbox0=\hbox{$#1\oldsqrt{#2\,}$}\dimen0=\ht0 \advance\dimen0-0.2\ht0 \setbox2=\hbox{\vrule height\ht0 depth -\dimen0}% {\box0\lower0.4pt\box2}} The cool thing is that (as far as I've tested it) this works regardless of the math font and with pdftex as well as xetex. However what is not possible is mulitple roots, such as \sqrt[3]{a}. Those produce incorrect output when this &quot;trick&quot; is being used. So my question is, is somebody able to tweak this code so that it works for multiple roots as well? If I'm floating in space and I turn on a flashlight, will I accelerate? Division ring if and only if it has no proper left ideals. Collapsing a stream of multiple lines into one line I have a file containing a a large number of lines, each of which contains a bunch of numbers which are separated by spaces. I process this data in a pipe in some way, and then I want to collapse the multiple lines into a single line of all the numbers separated by spaces.  Is there a standard command-line utility I can use to do this? It seems like most line-by-line utilities won't mess with the newlines... I hope to receive my PhD in Mathematics in May of 2015, so I am currently applying for jobs.  Due to life circumstances, I'm fairly certain that I will only be staying at my next position for about 2 years.  Does this mean I shouldn't even try to apply for tenure-track positions?  Will a hiring committee not even consider me for the position if I'm not willing to commit to staying with them for a certain amount of time?  I would love to get a teaching/lecturing position at any institute of higher education (4-year research university, liberal arts school, community college), but as I look for openings, a lot of them are tenure-track assistant professor positions.  Should I go through the effort of sending in an application, or will that be a waste of my, and the committee's, time?  Thanks for any advice you have!  Edit (to address Nate's comment):  My wife is currently applying for 2 year graduate degree programs in various cities in the US, so I'm applying for jobs in the same cities.  When she finishes her program, we plan to move outside of the US (Africa or Southeast Asia), where I do hope to remain in academia, teaching mathematics to college-aged students. When functions commute under composition Cardinality of equivalence relations in $\mathbb{N}$ I came across this long proof on this site:    But I would like to know whether my direction can work.  Say we want to find the cardinality of all equivalence relations in $\mathbb{N}$. Since it is a subset of all relations in $\mathbb{N}$, I conclude it has a cardinality smaller or equal to $\aleph$. Now, define an injective function from $P(\mathbb{N})$ to the set of equivalence relations by matching each subset of $\mathbb{N}$ with the identity relation (which is an equivalence relation in $\mathbb{N}$.  Therefore the cardinality of all equivalence relations in $\mathbb{N}$ is greater or equal to $\aleph$ and using CSB we get the desired result.  Seems legit? Proof that a Combination is an integer Get width of a given text as length Why is $c+q$ still a primitive root modulo $q$? Question: Let $p$ and $q$ be distinct odd prime numbers.  By considering primitive roots, we need to show $\exists c\in\mathbb{Z}$ with the property that:  $\bullet$ $c^n\equiv 1\pmod{p}$ whenever $n$ is a multiple of $p-1$, and   $\bullet$ whenever $n$ is not divisible by $q-1$, then $c^n\not\equiv 1\pmod{q}$  I am only interested with the beginning of this answer.  Answer: We let $c$ be a primitive root modulo $q$.  We are going to use Fermat's little theorem so we need $(c,p)=1$  If $(c,p)\neq 1$, then we have $p\mid c$, but then we have $(c+q, p)=1$  and since $c+q$ is a primitive root modulo $q$, we can replace $c$ with $c+q$ for $(c,p)=1$ for using Fermat's little theorem.  I understand the general method and argument, but...  Contention: Why is $c+q$ still a primitive root modulo $q$?  When is this generally true? Is there an identity for when things remain a primitive root?  Why is $(c+q)^{\phi(q)}\equiv 1\pmod{q}$?