What are the number of possible partitions of a set containing n elements? This question rises immediately if we try to enumerate the number of possible equivalence relations on a set with n elements.
Number of equivalence relations on a set If a set has $n$ elements then what are maximum number of equivalence classes and equivalence relations possible on it?
Are there any versions of t-SNE for streaming data? How do I earn SC2's "Hurry up, it's raid night" achievement? The "Hurry up, it's raid night" achievement in Starcraft 2 involves "beating Wings of Liberty, on normal, in 8 hours or less."  Is there a recommended mission path to take? Which mission should I take for the branching paths? Does it count time spent on the Hyperion in between missions, or is it exclusively time elapsed on missions. Does replaying a mission affect the time? General tips / hints from those who've unlocked the achievement?  Edit: Just got this achievement tonight, finally! The answer was perfect. I recently upgraded to 12.04 from 11.10 and since upgrading have been unable to log in to my user profile. The upgrade went okay and I can login to a guest session fine but whenever I try to login to my profile, after entering my password, I just return to the login screen.  I've changed my password in Root (passwd &lt;username&gt;) and can log in to tty1 with no issues, it's just in GUI I'm having problems.  I'm using a HP dv7 laptop, 32-bit Ubuntu install, Intel® Core™2 Duo CPU P7350 @ 2.00GHz × 2, Nvidia graphics.  Any help would be appreciated. Do you "gain" your 1st level? I know that for a Banach space $X$ and a linear functional $T:X\rightarrow\mathbb{R}$ in its dual $X'$ the following holds: \begin{align}T \text{ is continuous } \iff \text{Ker }T \text{ is closed}\end{align} which probably holds for general operators $T:X\rightarrow Y$ with finite-dimensional Banach space $Y$. I think the argument doesn't work for infinite-dimensional Banach spaces $Y$. Is the statement still correct? I.e. continuity of course still implies the closedness of the kernel for general Banach spaces $X,Y$ but is the converse still true? How can I find out which process is locking a file or folder in Windows?  For instance, when trying to delete a folder, Windows reports this:     The action can't be completed because the folder is open in another program   Happens the same with a file, but how do I find out what program or application is currently using it and preventing me from deleting the file or folder? probability question ("birthday paradox") $n$ people attend the same meeting, what is the chance that two people share the same birthday? Given the first $b$ birthdays, the probability the next person doesn't share a birthday with any that went before is $(365-b)/365$. The probability that none share the same birthday is the following: $\Pi_{0}^{n-1}\frac{365-b}{365}$. How many people would have to attend a meeting so that there is at least a $50$% chance that two people share a birthday?   So I set $\Pi_{0}^{n-1}\frac{365-b}{365}=.5$ and from there I manipulated some algebra to get $\frac{364!}{(364-n)!365^{n}}=.5\iff (364-n)!365^{n}=364!/.5=.....$   There has to be an easier way of simplifying this. Table with 100 % width and vertical / horizontal alignment I want to draw the following table in LaTeX. In the first column the text should be left aligned. The text in all other cells should be centered. The first problem occurs with D in the first row. If I write a \centering in front of D, I get a LaTeX error. The second problem is, that I want to vertically align the text in all cells but I don't know how. I have found a solution but it doesn't work if the width of the first column is specified.  \documentclass{article} \usepackage{blindtext} \usepackage{tabularx}  \begin{document}   \begin{tabularx}{\textwidth}{|p{4cm}|X|X|X|X|}     \hline                &amp; \centering A &amp; \centering B &amp; \centering C &amp; D\\     \hline     \blindtext &amp; 123 &amp; 123 &amp; 123 &amp; 123\\     \hline   \end{tabularx} \end{document} How to disable captive portal detection? How to remove exclamation mark on Wi-Fi or signal icon in Lollipop? I hope this isn't a duplicate.  Similar questions that I found are over a year old.  I have seen cases where shortly after an upvote (perhaps on an old answer?), the summary in my user activity dropdown shows inconsistent counts:    Particularly, the Slid a div offscreen using jQuery answer shows +40, when it was only updated once yesterday.  Now, if this is a monthly total for this question, then it is correct, but when I refreshed later I got:    Now it shows +10, and other entries completely disappear.  So, what is this supposed to show me?  At the very least I am confused about what I am supposed to see here.  Here is another example:    This shows +20 for the same question as before, after I received a single upvote last night.  What gives?  You can see from my reputation that there was only a sinle upvote recently: Assume I wield a Whip in one hand and a dagger in the other.  If a creature moves from 5ft to 10ft it has technically moved out of my reach (dagger), but at the same time it is still inside my reach (whip). When does it provoke opportunity attacks from me?  I am fully aware that this would not enable me to make off-hand attacks since the whip is not a light weapon, just to cut that issue out of the question. How do I get the local machine name? Is it a bad practice to catch Throwable? Terminology of the sequences of promoters in relation to DNA strands I'm studying molecular biology and I'm trying to understand an experiment which shows the importance of promoters in the relative transcription level (RT). The image below comes from Rolf Knippers' book "Molekulare Genetik" (8th edition).    The legend says (among other things):     Die erste Zeile gibt die normale "Wildtyp"-Sequenz der 5'-flankierenden Region wieder.   Which, in english, means something like:     The first line repeats the normal wild-type sequence of the 5' flanking region.   The column on the right gives the relative transcription (RT) level, 1.0 being the highest possible level of transcription. As we can see, the lines where some parts of the "5' region" have been deleted give quite low RT levels, since some regions of the promoter are missing.  My questions are the following:  1) According to  and , I understand that RNA polymerase reads and uses both coding and non-coding strands in order to synthesize RNA. Therefore, how did they manage to use the sequences strands regions had been deleted to make a polymerase "read them" and perform transcription?  2) If the polymerase reads the template strand in the 3' -> 5' sense, shouldn't we talk of "ATAT box" or "TAAC box" instead of "TATA" or "CAAT" boxes? Does it mean that the "promoter" regions they are using on the graphic above are actually on the coding strand?  Thanks a lot for your help. $p=4n+3$ never has a Decomposition into $2$ Squares, right? Primes of the form $p=4k+1\;$  have a unique decomposition as sum of squares $p=a^2+b^2$ with $0&lt;a&lt;b\;$, due to .  Is it correct to say that, primes of the form $p=4n+3$, never have a decomposition into $2$ squares, because sum of the quadratic residues $a^2+b^2$ with $a,b\in \Bbb{N}$ $$ a^2 \bmod 4 +b^2 \bmod 4 \le 2? $$  If so, are there alternate ways to prove it? Why does the (relativistic) mass of an object increase when its speed approaches that of light? I'm reading Nano: The Essentials by T. Pradeep and I came upon this statement in the section explaining the basics of scanning electron microscopy.     However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. At such velocities, one needs to do relativistic correction to the mass so that it becomes[...]   We all know about the famous theory of relativity, but I couldn't quite grasp the "why" of its concepts yet. This might shed new light on what I already know about time slowing down for me if I move faster.  Why does the (relativistic) mass of an object increase when its speed approaches that of light? Texture mapping with multiple UV maps So, as a newbie relatively new to blender, I started doing some texture mapping (blender internal Render). I created a texture for the main object and a couple of transparent decals to decorate it. (basic effects made in GIMP). Simple stuff, just to know how it works.  After unwrapping, having 2 UV maps made, creating both new material and textures, enabling UV mapping coordination I always end up with:  - all - 1 faces of the cube being covered nicely in the main texture  - 1 face covered in blank material with a decal in the middle of it  - the whole thing partly coated by a huge "stain" of a decal  So, the problem is that I can't layer textures correctly. Is there a way to have one texture covering everything and more transparent ones on top of it? I am working on a bigger project right now and having this ability would be useful Which are the word orders that can be found in English? What do they do with all the plates and cups in Star Trek? After an AJAX request, sometimes my application may return an empty object, like:  var a = {};   How can I check whether that's the case?