Transform right skewed distribution to normal
Appropriate data transformation
"Above"/"below" before/after a noun I have seen sentences similar to the following:  (1) See the reference above. (2) See the reference below.  And,  (3) See the above reference.  But not,  (4) See the below reference.  Are all these forms acceptable? Which is/are preferred in formal writing? How do I start a process from C#? How do I start a process, such as launching a URL when the user clicks a button? Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous Suppose that $X$ is infinite and that $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous.     My attempt:  Let $|A|=n$. We will prove by induction on n. It's clear that the the theorem is trivially true for $n=0$. Assume the theorem is true for all $n=k$. For $n=k+1$, then $|A \setminus \{a\}|=k$ for some $a \in A$. Thus $X \setminus (A \setminus \{a\}) \sim X$ by inductive hypothesis, or $(X \cap \{a\}) \cup (X \setminus A) \sim X$, or $\{a\} \cup (X \setminus A) \sim X$. We have $\{a\} \cup (X \setminus A) \sim X \setminus A$ since the theorem is true for $n=1$. Hence $X \setminus A \sim \{a\} \cup (X \setminus A) \sim X$. Thus $X \setminus A \sim X$. This completes the proof.       Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help!     Update: Here I prove that the theorem is true for $n=1$.  Assume that $A = \{a\}$ and consequently $X \setminus A= X \setminus\{a\}$. It's clear that $|X \setminus A| \le |X|$. Next we prove that $|X| \le |X \setminus A|$. Since $X$ is infinite, there exists $B \subsetneq X$ such that $B \sim X$ (Here we assume Axiom of Countable Choice). Thus $|X|=|B|$. There are only two possible cases.   $a \in X \setminus B$   Then $B \subseteq X \setminus \{a\}=X \setminus A$ and consequently $|X|=|B| \le |X \setminus A|$. Thus $|X| \le |X \setminus A|$ and $|X \setminus A| \le |X|$. By Schröder–Bernstein theorem, we have $|X \setminus A| = |X|$. It follows that $X \setminus A \sim X$.   $a \in B$.   Let $b \in X \setminus B$. We define a bijection $f:X \setminus \{a\} \to X \setminus \{b\}$ by $f(x)= x$ for all $x \in X \setminus \{a,b\}$ and $f(b)=a$. Thus $X \setminus \{a\} \sim X \setminus \{b\}$. Since $b \in X \setminus B$, it follows from Case 1 that $X \setminus \{b\} \sim X$. Hence $X \setminus \{a\} \sim X \setminus \{b\} \sim X$. Thus $X \setminus \{a\} = X \setminus A \sim X$.  To sum up, $X \setminus A \sim X$ for all $|A|=1$. Do professors get paid for supervising students? Ubuntu 16.10 Dual Boot error - 'grub-efi-amd64-signed' package failed to install View approximate number of users online Is there any way to view the number of users online at any given time (or even an approximation)?  This could be really useful to provide an indication of whether your question is likely to get answered any time soon.  It could also provide a bit of an indication for why your question is getting very low views - if there are only a few users online, then that could be the reason for little interest in a question, whereas if there are lots of users online and the question still doesn't garner many views, then maybe it needs to be rephrased or something.  I know that this can't be used definitively, but it could be used as a rough guide, and could also just be an item of interest.  Perhaps it could show the number of users that have been seen in the past 5 / 10 minutes? (Shouldn't be too hard, since each user page already shows the last time that they were seen). Vector. Properties It is given in a question that $\vec x+\vec a=(\vec a\cdot \vec x)\vec b$ and it is stated that $\vec a$, $\vec b$ and $\vec x$ are vectors.  It is also stated that $\vec a\cdot \vec b\ne 1$.  The question asks to express $\vec x$ in term $\vec a$ and $\vec b$.  How do you this? Is there a way to buy SO reputation? Viewer-independent copyable spaces at the beginning of a line? I need to include a couple of Python code listings, where the indentation of the lines (using some number of spaces) is significant. I would like for the code listings to be copyable, so the spaces at the beginning of the line need to be be copied along with the text.  This question has been asked in various ways before (e.g. , or ). Those questions focus on making line numbers uncopyable, though.  Making the spaces at the beginning of a line copyable seems to be harder: "I am not sure it is possible to specify in the PDF (at least in a viewer-independent way) that the indentation should be copied too" (). At , Philippe Goutet suggests a solution (turning the spaces into visible spaces, and coloring them in the background color so that they appear invisible) that works using Acrobat Reader, but not all readers. He says "It works under Acrobat Reader and it's extremely pleasant to be able to quickly copy/paste code without problem (perhaps the problem can be circumvented by writing direct PDF code to tell that it's a space, I've never had the time to try)".  Is it possible to produce a PDF with a code listing with copyable real spaces at the beginning of a line?  Minimal example: The line return x should start with four spaces.  \documentclass{article}  \begin{document} \begin{verbatim} def myfunction(x):     return x \end{verbatim} \end{document}   I know that I could attach the code to the PDF as a file, but that's not what I want. Do eyeballs exhibit chromatic aberration? Fairly straightforward question. If not, why not?  I suspect that if they do, it is not perceived due to the regions of highest dispersion being in one's region of lowest visual acuity. I'm trying to build a datapack that lets you craft xp bottles from glass bottles at the expense of 7 xp per bottle (because they drop 3 to 11 xp so the average is 7). I have almost everything working except for a check for if they have too little xp. If they have 7 or less xp, I would like to remove one bottle of enchanting from their inventory and give them one glass bottle, because the function for removing xp is called after they've crafted one and a scoreboard has counted it. My remove function currently looks like this:  #create dummy scoreboard to multiply bottles crafted by seven to get xp to be removed scoreboard objectives add 7xp dummy scoreboard players set @s 7xp -7 scoreboard players operation @s xpCrafted *= @s 7xp  #how do I get the value in xpCrafted into the command to remove xp? xp add @s -7 points  #cleanup scoreboard players reset xpCrafted scoreboard players reset 7xp scoreboard players set @e xpCrafted 0 scoreboard objectives remove 7xp   And my tick function like this:  execute as @e[scores={xpCrafted=1..}] at @s run function craft_xp:removexp   As the inline comment says, how do I get that value in xpCrafted into my xp removing command, and also how should I check to make sure the player has enough xp for the operation in the tick command (or elsewhere)? Presumably I would run a different command to remove the extra bottle if they didn't have enough xp, but I don't know how because there doesn't seem to be any type of if/else structure in commands. Finally, should a question like this be in two different questions or is one ok on this forum? Algorithm to detect overlapping periods How to integrate $\int x^2 \sin^2(x)dx$ I don't know how to integrate $\int x^2\sin^2(x)\,\mathrm dx$. I know that I should do it by parts, where I have  $$ u=x^2\quad  v'=\sin^2x \\  u'=2x \quad v={-\sin x\cos x+x\over 2}$$  and now I have  $$ \int x^2\sin^2(x)\,\mathrm dx = {-\sin x\cos x+x\over 2} x^2 - \int 2x{-\sin x\cos x+x\over 2}\,\mathrm dx\\ ={-\sin x\cos x+x\over 2} x^2 - \int x(-\sin x\cos x+x)\,\mathrm dx\\  $$  so I have to calculate $$ \int x(-\sin x\cos x+x)\,\mathrm dx=-\int x\sin x\cos x\,\mathrm dx+\int x^2\,\mathrm dx$$  I know that $\int x^2\,\mathrm dx = {1 \over 3}x^3+C$ but I don't know what should I do with $$\int x\sin x\cos x \,\mathrm dx$$ Should i use parts again or what? Please help. I need to analyse data and am unsure whether or not I should use a multinomial logistic regression analysis or a binary logistic regression analysis. I have just conducted a latent class analysis and am interested in seeing if my variables can predict class membership. There are only two classes and so my response variable only has two categories; however the variables range from having 2 to 3 to 4 and are mostly nominal categorical data. How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? How to prove that $\mathbb{Q}$ ( the rationals) is a countable set I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it equivalent to find a bijection from $\mathbb{Q}$ to $\mathbb{Z}$? Solve $$\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$$ without complex integration.  This integral can be very easily solved with contour integration, but how can you solve it without taking a tour in the complex plane? File upload to Heroku not working How to find the id of the last added feature using pyQGIS? Is it possible to programmatically (in python) access the attribute info of the last added feature in QGIS 2.0 before edits are saved? I'm failing to find a way to get the ID and subsequently other attributes of my feature of interest (the one most recently added).  idx = layer.fieldNameIndex('myField') myList = [] lastEdited = ID THAT I'D LIKE TO RETRIEVE for feature in layer.getFeatures():     if feature.id() == lastEdited:         print feature.attributes()[idx]   From this I'd like to get 'myField' of my most recent addition.   UPDATE: Following lead of this , I was able to achieve my goals via the Python console, but am failing to get it to work for my plugin.  def updateTableWidget(self, fid):     idx = layer.fieldNameIndex('myField')     myList = []     for feature in layer.getFeatures():         if feature.id() == fid:             myList.append(feature.attributes()[idx])   Via the python console, I was able to connect to my layer like so:  layer.featureAdded.connect(updateTableWidget)   After adding a QgsMessageLog to my function, the attributes were correctly identified. I now have a beginner python issue, however. Calling the function in my plugin as shown below gives me TypeError: updateTableWidget() takes exactly 2 arguments (1 given):  self.updateTableWidget()   The function was called from feature form python file. "Baggage" versus "luggage"