Why do p-values have uniform distribution for normally distributed data
Why are p-values uniformly distributed under the null hypothesis?
Is product of two closed sets closed? If $A$ is closed in $X$ and $B$ is closed in $Y$,   does it follow that $A \times B$ is closed in $X \times Y$ ? Check if macro is fully expandable Question  Is there a way to check whether a macro is fully expandable (or rather "safe in an expansion-only context" )?    Consider this code:  \def\a{Just a string} \def\b{\a} \def\c{\def\unsafe} \def\d{\c}   How could I check which of the macros (a-d) are safe in an expansion-only context? By looking at them I know that a and b are whereas c and d are not but if I wanted to know the same for a macro I haven't written myself this could get quite useful.      Background  I am working on a way to detect whether some input is a valid number in PGF. For this I developed  which makes use of passing the input into \pgfmathfloatparsenumber. The problem I have run into is that said macro appears to somehow manages to expand the input until there is an error (if the input is in fact not safe in an expansion-only-context). I tried using protected, noexpand and similar but somehow PGF manages to circumvent those.    So the idea is to check whether the input is safe before actually passing it to PGF. The problem is: I don't know how I'd go about that... Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$.  Show that $p(x)+p&#39;(x)+p&#39;&#39;(x)+\cdots+p^{(n)}(x)\geq 0$ for all $x$ where $p^{(i)}(x)$ is the $i^\text{th}$  derivative.  My interest: I know that, we can rewrite the LDE as follows  $$ p(x)+p&#39;(x)+p&#39;&#39;(x)+\cdots+p^{(n)}(x) = Lp(x) $$  where $L := I + D + D^2  + \cdots + D^{(n)}$. Can we say anything about a linear operator of this kind so that it does not change the sign of the input it takes? I can try to solve the question by writing for all the derivatives and factoring them using $p(x)$, but I think there should be a clever way of showing this by the properties of $L$. Can I figure out a solution by just looking at $L$ and the sign of $p(x)$ as in the question? Where should I look for that? How do you handle uneven tiles while rendering a tile map? Here is what I want to do with my tile map that I am unsure of. As you can see the top walls are way larger then the bottom and side ones (this is also an issue with my corners which are odd shapes [more like an L then a square] as well as larger then the 40x40px that my current tiles are).    I have also attached the tile sheet I am using for the tiles to show you what the corners and tops look like compared to the rest of the tiles.  What I’m thinking is I might be able to just draw them using the same array I have if I use the tiles width and height instead of a set WxH? I dont know how well this will work, but were the blocks that will have different values will be blocks the player/NPCs can’t step on then it may not be an issue, again I’m not even sure if that is a viable solution.  My code in case you want to see how I'm currently handling it:  (pastebin) (edit:im not handling it, meant how I'm currently doing the map)  tl;dr - how can I take take my 2d looking tile map and change it to have different size tiles? or is there a better way to do what I am trying to do? Is there a name of a method that I should be looking for while googling? How do I boot into recovery mode? I guess I can live with the fact that the  changes size drastically when he transforms, but why does his shirt rip off and not his pants?  I think that we need a little better answer than "it keeps us tied to the fact that inside he's still human". Why is King Tommen Baratheon the "First of his Name" rather than at least third? How can I upload files asynchronously with jQuery? How I can recover files when the folder shows empty but the files are not deleted? Reading order for Dragonlance? What would be the material or shaders node set up for a smooth gem in blender cycles engine?  I want a result similar to this image: The conditions $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$ make the group $G$ abelian? Unable to locate package Examples of provably${}^n$ unprovable statements Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement is provable, then it is provably provable). Let $T_1=T$ and $T_{n+1}=T_n+{\sf Con}(T_n)$. (We need stronger theories if we want to prove unprovability.) Given this, simply by considering the truth values of the relevant statements, we get the following classification of statements by unprovability:   $T\vdash A$ ($A$ is provable) $T_2\vdash T\not\vdash A$ ($A$ is provably unprovable) $T_3\vdash T_2\not\vdash T\not\vdash A$ ($A$ is provably unprovably unprovable)   &nbsp;&nbsp;&nbsp;&nbsp;...  &nbsp;&nbsp;&nbsp;&nbsp;$\omega$. For all $n$, $T_n\not\vdash\dots \not\vdash T_2\not\vdash T\not\vdash A$ (we can't say anything about $A$'s provability)  We can refine this hierarchy by also considering the status of $\lnot A$. Let's call a class $m,n$ statement one such that $A$ is at level $m$ above and $\lnot A$ is level $n$ in the list. Since $T\vdash\lnot A$ implies $T_2\vdash T\not\vdash A$, the only nonempty classes at level $1$ are the class $1,2$ statements (provable statements) and the class $2,1$ statements (refutable statements). Independent statements are in class $2,2$. The class $1,1$ is nonempty iff the theory is inconsistent.  My question is: Are there any (preferably "natural") statements that fall further into this hierarchy? I know many examples of independent statements, like the continuum hypothesis or the axiom of choice relative to $\sf ZF$, or the Paris–Harrington theorem relative to $\sf PA$. But I know no examples of even any class $2,3$ statements. I expect there may also be natural examples of $\omega,\omega$ statements. My guess is that all the classes $m,n$ with $m,n\ne1$ are nonempty, but examples elude me. I read this at least 20 years ago, and I believe it was originally published in the '60s.   The main character is a human, but somehow "inherits godhood" from an alien, whose species is very long lived. IIRC this is one of the old-style god thingies, i.e. "god of thunder" not "omniscient omnipotent omnipresent transcendent being".  The plot somehow revolved around identifying and fighting some enemy which, I believe, was in turn a member of the alien species who acted as the avatar of the opposite god in the alien pantheon.  The only thing I remember for sure is that the enemy was identified as "green green" or "gringrin" (sadly, I read this in translation so I am not sure what the original meant) Is it possible to set the GPU fan speed using the default free ati driver?  My card (X1900) was deprecated in the fglrx drivers a while back.  The free drivers seem to work with 3D, Compiz and such, but running graphics intensive programs makes my GPU heat up and produce artifacts, while the GPU fan never seems to increase. (When set to 100% in Windows it makes a massively louder noise.) The PS3 version of Portal 2 comes with a code to link it to a Steam account. Apparently this allows you to play on both the PC and PS3 with one purchase.  Can this be used to play co-op between the two platforms? I'm using the beamer miniframes theme (with subsection=false) and the compress option. Currently the navigation shows the dots for each subsection side-by-side. I would prefer if there would be a horizontal space (e.g. with the width of a dot) between the dots of each subsection. Is this possible? Right now it looks like:  Section 1 oooooo  And I would prefer it to look like:  Section 1 ooo ooo  MWE: \documentclass[compress]{beamer}   \useoutertheme[subsection=false]{miniframes}   \begin{document}   \section{Sec 1}   \subsection{}   \begin{frame}[t]     hi   \end{frame} \subsection{}   \begin{frame}[t]     hi   \end{frame} \end{document}` "I use to", or "I used to" Which is the correct sentence, if there is a correct one?     I use to be a hitman.        I used to be a hitman.   I've read the second sentence recently in a book, but I was sure it should be I use to be a hitman. Biblatex with Biber: Configuring my editor to avoid undefined citations Running the minimal example  \documentclass{article} \usepackage{filecontents} \begin{filecontents*}{\jobname.bib} @ARTICLE{example,   author  = {Other, Anthony Norman},   title   = {Some things I did},   year    = {2014},   journal = {J.~Irrep. Res.},   volume  = {1},   number  = {1},   pages   = {1-10} } \end{filecontents*} \usepackage[backend=biber]{biblatex} \addbibresource{\jobname.bib} \begin{document} Hello\cite{example}. \printbibliography \end{document}   I get the warning There were undefined references.  I have read  and know that I need to run:   LaTeX Biber LaTeX   However, my editor is only set up to run BibTeX. How do I go about setting up my editor/IDE to be able to run Biber, and how do I run the LaTeX/Biber/LaTeX cycle?  Answers (sorted alphabetically by editor name)                    Answer guidelines   Each answer should be for one editor. If the editor is cross-platform, if possible give a single answer with notes covering the minor platform variations. Please edit the question to include new answers in the 'link list' Each answer should be 'stand alone', i.e. don't say 'It's almost the same as editor Y but ...' for the editor part Instructions for 'build tools' such as arara or latexmk are welcome but should explain how to set up the editor in question as not all editors allow simple addition of arbitrary tools