We know that set of rational numbers is countable. We have bijection betwee $\mathbb{N} $ and $\mathbb{N} \times \mathbb{N }$, whose graphical representation is beow:       Also the bijection between $\mathbb{N} $ and $\mathbb{Z }$ exists. I am unable to find the bijection between $\mathbb{N} $ and $\mathbb{Q} $. Can anyanybody tell me the bijection between $N$ and $Q$ with its graphical representation.
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such a bijection?
Server login attempts NOT blocked after configuring fail2ban So, I installed (and configured) fail2ban on three Ubuntu VPS servers. I configured (imo) a maximum of 3 failed login attempts (via Putty as root) within 2 minutes. Then I restarted the fail2ban service (according to ).    But when I look at my auth.log files in /var/log/ I still see 4, 5 or more failed attempts from certain IP's. So obviously I did something wrong as the IP is not blocked after 3 failed attempts.   It might have to do something with iptables, but that's where it gets a bit "foggy" for me..   Not sure how to setup a max-three-failed-login-attempts rule..  Additional: I see that the login attempts mostly have different ports for every attempt. Does that influence the count to max-three?  Because I see a port being "poked" twice and then no more attempts. Then the IP tries a different port (again 2 times). (all bots of course) Why is it called a quadratic if it only contains $x^2$, not $x^4$? Should it not be bidratic or didratic etc? Can you cast a 9th level spell if your multiclass spell slots have 9th level spell slots? This is a lake of approximately 13 ha with 81 sampled depth points in about 10 transect lines:    Previously in ARCgis with a 50 ha lake and about a 100 depth sampling points, I got decent output with the TopotoRaster tool. However, no such exact equivalent seems to exist with QGIS or open source software.  In QGIS, I got the interpolated raster below, using the TIN method in the Raster Interpolation plugin. However, this seemed to be a lucky outcome, when I tried to repeat, I got different results (unsatisfactory) in spite of not really varying any parameter.    These are more urban (constructed) tanks than lakes, therefore, they are fairly regular in their bed profile, hence the low point sampling density. Some articles seemed to suggest Inverse Distance Weighting (IDW) as the most suitable technique, yet I seemed to get the worst results with that (possibly owing to low point density).  Are there any heuristics which can be employed here considering the size of the lake, its regular profile and the point sampling density to arrive at which interpolation method is apt? (between Kriging, IDW, Bilinear, Cubic convolution, TIN or spline)  Or is it always a bit of trial and error? Determining the number of bit strings of length n that contains no adjacent 0s • $C_n =$ this number of bit strings  • A binary string with no adjacent 0s is constructed by   Adding “1” to any string w of length $n-1$ satisfying the condition, or Adding “10” to any string v of length $n-2$ satisfying the condition   I still don't find that intuitive at all. Why adding up $C_{n-1} + C_{n-2}$ results in $C_n$? I have tried my best and asked my friends for help, but they find the formula unintuitive as well. Could anyone here come up with a clear explanation? Thanks! Proving that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$ Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$ How does one install PlayOnLinux? How do I install PlayOnLinux on Ubuntu? I want to describe my business and want to refer to it in a sentence.  I am trying to say:      "In 2015 we established a business (that is) named X".   Intuitively, the "that is" part is redundant, and makes the sentence more cumbersome. But I cannot think of a reason why we should be able to drop the verb.  Is the version without the "that is" part grammatically correct? How can it be explained? Scanner issue when using nextLine after nextXXX How to select even or odd elements based on class name if you create html layout like so  &lt;ul&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt; &lt;/ul&gt;   and try to select odd elements with 'a' class like so $$('.a:odd') you will get empty array, and if you do $$('.a:even') you will get all four li elements with 'a' class.. It really strange.. But im new to mootools, maybe im doing something wrong..    So my question is how to select first and third li elements with a class. I know that i can do it with function like this  $$('.a').filter(function(item, index) { return index%2; }  but its too complicated for such small task as selecting odd or even elements.. When I try to play multiplayer I type in the only server I know, which the game says was right, then click Play and it loads. Then it says Can't connect to server: bad log in. How can I fix this? Sheafs and closed immersion Force "low power mode" on mac Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be $$ d(A,B) = \inf \{ \|x-y\| : x \in A, \; \; y \in B \} $$ For any $A,B$, I want to prove that $d(A,B) = d( \overline{A}, \overline{B} )$. My Attempt Put $\alpha = d(A,B)$. Therefore, $\alpha = \|x_0 - y_0 \|$ for some $x_0 \in A$ and some $y_0 \in B$. But, notice $x_0 \in \overline{A}$ and $y_0 \in \overline{B}$ by definition. hence $$ \| x_0 - y_0\| \geq \inf\{ \|x' - y'\| : x' \in \overline{A}, \; \; y' \in \overline{B} \} =  d( \overline{A}, \overline{B} )$$ $$\therefore d(A,B) \geq d( \overline{A}, \overline{B} )$$ I am stuck trying to show the other direction: $d(A,B) \leq d( \overline{A}, \overline{B} )$ Can someone help me? thanks a lot Disabling module with config sync and removing with composer I have an issue disabling modules, I've searched but cannot find something useful:  - Developer A disable module and via composer remove it  - I pull his changes so i run composer install that remove the code, then drush cim to update configuration, but it complains that the module is missing while it's trying to delete its configurations.  What's the right way to do this? I can't import config before running composer install because along the uninstalled module there could be also new modules needed for the configuration import. Do particles in a sound wave ever move transversally? I'm trying to visualize how sound waves work and I was curious about something.   So sound moves in longitudinal waves, which I think I understand.  There is a really good  explaining this.  In the video the air particles are depicted as moving from left to right (longitudinally). But do they ever move up and down even a little bit? Are they on some kind of fixed axis where they can only move longitudinally and never move even the slightest bit in a vertical direction? For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$ Here are my thoughts so far:    $C_p = \langle x \mid x^p =e\rangle$ $C_q = \langle y \mid y^q =e\rangle$ $C_p \times C_q$ has elements of the form $(x^a,y^b)$ There are $p$ possible values for $x^a$ and $q$ possible values for $y^b$. So there are $pq$ possible elements in $C_p \times C_q$. $C_{pq} = \langle z\mid z^{pq} = e\rangle$ and there are $pq$ elements in this group. For an isomorphism from $C_p \times C_q$ to $C_{pq}$ we send $(e,e)$ to $e$ to ensure that the identities are mapped to each other. However, I'm not sure how to define an isomorphism for the other elements. Units and Nilpotents 2016: a year in closing Consequence of the Hahn-Banach Theorem From wikipedia's page on the Hahn-Banach Theorem:  If $V$ is a normed vector space with linear subspace $U$ (not necessarily closed) and if $z$ is an element of $V$ not in the closure of $U$, then there exists a continuous linear map $ψ : V → K$ with $ψ(x) = 0$ for all $x$ in $U$, $ψ(z) = 1$, and $||ψ|| = dist(z, U)^{−1}.$  How is this a consequence of the Hahn-Banach Theorem?