We have the set $A=\{1,2,3,4,5\}$. what is the number of functions from $A$ to $A$? I think the domain of the function should contain all the elements of $A$ but range of the function may have any elements of $A$ or not having them. so for each of the numbers from $1$ to $5$ we have two possibilities $(1)$ existing in the range of function. $(2)$ not existing in the range. so the number of functions from $A$ to $A$ is $2^5=32$. is my justification right?
Let $X$ and $Y$ be finite sets. Then the set $Y^X$ is finite and $\#(Y^X) = (\#Y)^{\#X}$.  I can see how to do it combinatorially. Let $y \in Y$. There are $\#(X)$ choices for such y to come from x, therefore we have $\#Y^{\#X}$ choices. I am not sure how to do it bijectively.
Story about main character with regeneration powers I read a book 10+ years ago where the world was ruled by various factions, each one had a special ability of some sort. The main characters faction had regenerative ability. Another faction had the ability to speed up or slow down time. I am not sure of the plot, but I know at one point he is injured so badly he starts growing a clone which he attempts to kill, but comes back later in the story.    Any ideas? My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. Unusual integral I got a clock as a gift recently. It has a very novel face in that the hour positions are given by a complex formula. For the most part, I have been able to verify the calculations presented as accurate, but the two o'clock identity has me stumped.  $$ \frac{\gamma}{\displaystyle{\int_0^\infty\frac{e^{-x^2}-e^{-x}}{x}\,dx}} $$  I know that $\gamma$ is the . And WolframAlpha tells me that $\int_0^\infty\frac{e^{-x^2}-e^{-x}}{x}\,dx=\frac{\gamma}{2}$ which  makes sense because  $$ \frac{\gamma}{\displaystyle{\int_0^\infty\frac{e^{-x^2}-e^{-x}}{x}\,dx}}=\frac{\gamma}{\displaystyle{\frac{\gamma}{2}}}=2 $$  It is not clear how I would show $\int_0^\infty\frac{e^{-x^2}-e^{-x}}{x}\,dx=\frac{\gamma}{2}$. Could anyone shed some light on this or point me to a source (book, article, etc...) where I can read up on this. The usual internet (listed above) resources have not been helpful to me. If-else working, switch not Show that $E(X)=\int_{0}^{\infty}P(X\ge x)\,dx$ for non-negative random variable $X$ Centre of Lie Algebra $sl_2(\mathbb{F})$ For $L=sl_2(\mathbb{F})$ i.e. matrices with trace zero,  what is the centre i.e. $Z(L)$= {$x\in L : [x,y]=[y,x] \ \forall\ y \in L$}.  I will have to find matrices $A \in L$ such that $AB=BA$ for all $B \in L$. But how to appoach this considering trace is zero here. I have this weird object (vascular tree) which is empty inside, but closed from the top and the bottom:  However, it is empty inside, which makes me unable to use boolean on a bigger object (I want to make a hole in an object through which this object will go through). So I thought I should fill it and then try to boolean. But, how do I fill it?  Thank you sincerely. Why is Ubuntu 14.04 so slow on my laptop? Data recovery from an accidental format on ext4 partition When I was installing Mint Debian edition unlike the classic edition, the installation automatically formated my home partition when I did not specify to format.  So the formatting previously was ext4 as is now. I believe the data is still there as it was a quick format.   I have now booted the computer up on a live USB to prevent writing on it.  Ran testDisk. Is there anyway to recover to a previous superblock so i can recover my data? I would like to connect a USB Mouse, Keyboard or Storage Device to my Nexus 4. So far I know, for this USB OTG must be supported. Unfortunately I can not find any actual information about this.  Does anybody know whether USB OTG is supported by Ubuntu-Touch? If yes, does anybody know if it would work with a Nexus 4 device? fitting a smaller range when we bought our house, it had a very used 40" gas range. now that range has died. replacing with a new 40" range is very pricey ($2k+), and our cooking needs don't justify the size anyhow.  i'm looking for ideas on how to fit a smaller range. i would imagine that extending the counter top and cabinets would be a pretty extensive change. we've found we can get a 36" range for around 1/2 the cost. would it be feasible to add some sort of storage along side the smaller range to fill in the space?  thanks. i know this is an open-ended question with no specific answer, i'm just looking for ideas. How do I convert a string to a byte[] in .NET (C#) without manually specifying a specific encoding?  I'm going to encrypt the string. I can encrypt it without converting, but I'd still like to know why encoding comes to play here.  Also, why should encoding even be taken into consideration? Can't I simply get what bytes the string has been stored in? Why is there a dependency on character encodings? Can someone suggest a way to compare the values of two dates greater than, less than, and not in the past using JavaScript? The values will be coming from text boxes. Maximum Likelihood Estimation (MLE) in layman terms Could anyone explain to me in detail about maximum likelihood estimation (MLE) in layman's terms? I would like to know the underlying concept before going into mathematical derivation or equation. Why can't a Federation Starship's transporter history be used to bring someone back to life? apt-get update fails to fetch files, “Temporary failure resolving …” error SVG donut slice as path element (annular sector) vertical alignment of a turned cell I know using turn command and a sample code like this:  \usepackage{rotating}  \begin{turn}{90} Supervised Learning \end{turn}}   we can rotate a text inside a cell, however, I couldn't find a way to vertically center this rotated text. Any suggestions is appreciated.  Here is the sample MWE:  \begin{table}[]     \centering     \begin{tabular}{|&gt;{\centering\arraybackslash}m{0.5cm}|c|&gt;{\centering\arraybackslash}m{8cm}|}         \hline         \multicolumn{2}{|c|}{Classification} &amp; References \\ \hline         %\multirow{2}{*}{{\begin{turn}{90}Supervised Learning\end{turn}}}            \multirow{2}{*}{\rotatebox[origin=c]{90}{Supervised Learning}}  \end{tabular}     \caption{A Classification}     \label{tab:Machine_Learning_models} \end{table} require_once () or die() not working I need to have a list of all available view modes for an entity type so I can use it as the #options in a select field.  Any ideas? Thanks