I've seen this notation pop up in some Bayesian statistics notes but despite my best efforts with Google I've not had any luck discovering the meaning. The only helpful context that I have is that $y$ is the observed data and $\theta$ is some sort of parameter. As for giving an example of this notation in use, I've seen De Finetti's theorem summarized as "$Y_1 , ... , Y_n | \theta$ are i.i.d. and $\theta \sim p(\theta)$ if and only if $Y_1 , ... , Y_n$ are exchangeable for all $n$".  EDIT: Note, I'm not asking how something like $Y \sim N(\mu, \sigma^2)$ is read, I know exactly what that means. It's the $p$ parts that are confusing me.
What does $\dot\sim$ notation (dot over tilde) mean, in the context like $x \mathrel{\dot\sim} \mathcal N(0,1)$?  Turns out it is easier to find how to typeset it correctly:  that one should type \mathrel{\dot\sim} instead of simply \dot\sim to fix the spacing issue -- than to find what it actually means. It has only been used  on CV until now; is it standard?
Reverse Cauchy Schwarz for integrals Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that $$\sqrt{\int_a^bf^2 \int_a^b g^2}\leq \frac{1}{2}\left(\sqrt{\frac{M_1M_2}{m_1m_2}}+\sqrt{\frac{m_1m_2}{M_1M_2}}\right)\int_a^bfg$$  I can't make any significant progress on this one... Thanks for any hint. Why null cast a parameter? Experimental observation of matter/antimatter in the universe Ordinary matter and antimatter have the same physical properties when it comes to, for example, spectroscopy. Hydrogen and antihydrogen atoms produce the same spectroscopy when excited, and adsorb the same frequencies. The charge does not make a difference in the potential (regardless if it's generated by a proton or an antiproton) nor in how the positron behaves in this potential (being its mass equal to the mass of an electron)  How can astronomy evaluate if a far galaxy is made of matter or antimatter, given that from the spectroscopy point of view, they behave in the same way? In other words, how do we know that an asymmetry exists between matter and antimatter in the universe? Biblatex-apa style is missing a comma in the references. why? I have been trying to solve this problems for hours, but there is no solution in sight.  Until a few weeks ago, my biblatex-apa style was working like a charm. Although, after installing the updates and Biber, the reference section became a problem. It should display a journal article by two authors with a comma after the first author, but it doesn't:     Weber, M. &amp; Ruch, W. (2012). The role of character strengths in adolescent romantic relationships: An initial study on partner selection and mates’ life satisfaction. Journal of adolescence, 35(6), 1537–1546. doi:10.1016/j.adolescence.2012.06.002   it should look like this:     Weber, M., &amp; Ruch, W. (2012). The role of character strengths in adolescent romantic relationships: An initial study on partner selection and mates’ life satisfaction. Journal of adolescence, 35(6), 1537–1546. doi:10.1016/j.adolescence.2012.06.002   This is how my document looks like:   \documentclass[12pt, onecolumn]{apa6}   \usepackage[utf8]{inputenc} \usepackage[american]{babel} \usepackage{csquotes} \usepackage[style= apa, backend=biber]{biblatex} \bibliography{testbib} \DeclareLanguageMapping{american}{american-apa}   \begin{document}  Some text \parencite{Weber2012}.  \printbibliography \end{document}   And the BibTeX file:  @article{Weber2012, abstract = {}, author = {Weber, M. and Ruch, W.}, doi = {10.1016/j.adolescence.2012.06.002}, file = {}, issn = {1095-9254}, journal = {Journal of adolescence}, number = {6}, pages = {1537--1546}, pmid = {22749517}, publisher = {Elsevier Ltd}, title = {{The role of character strengths in adolescent romantic relationships: An initial        study on partner selection and mates' life satisfaction.}}, url = {http://www.ncbi.nlm.nih.gov/pubmed/22749517}, volume = {35}, year = {2012} }    I have tried various things so far, but none of it helped: e.g,   \renewcommand*{\finalnamedelim}{ \finalandcomma}   or  \renewcommand\finalandcomma{\addcomma}   Even in the biblatex manual, the reference examples of two authors is without the comma (it works for three or more authors). There should be a comma, according to . Why is that comma not there? And how could I insert that into my references? $\text{Tor}_0^R(M,N)\simeq M\otimes_R N$ I'm trying to understand why $\text{Tor}_0^R(M,N)\simeq M\otimes_R N$.  Let $\dots\to F_1\to F_0\to M\to 0$ be a free resolution of $M$. Then $$ F_1\to F_0\to M\to 0$$ is exact. Hence $$ F_1\otimes_R N\to F_0\otimes_R N\to M\otimes_R N\to 0$$ is exact. Call the left map $f_1$ and the right map $f_0$. Then $M\otimes N\simeq F_0\otimes N/\ker (f_0)\simeq F_0\otimes N/\text{im}(f_1)$.  But if I understand the definition correctly, to show the result we need that $M\otimes N\simeq H_0(\mathcal F)=\ker f_0/\text{im}(f_1)$. But why does $\ker f_0=F_0\otimes N$? If it does at all... Import Large Excel File Can PostGIS and QGIS provide dependable multi-user editing? After considering a few options, I'm trying to give my 15-20 QGIS 2.2 users the ability to concurrently view/edit one or two basic point PostGIS layers, largest one has just 16,000 records. I've installed PostgreSQL 9.1 / PostGIS 2.0 and basic testing seems to work ok but haven't unleashed it to the masses yet.  We'll typically only have 2-3 editors at a time, but possibly 15 viewers. Edits will all be in QGIS, viewers will be QGIS and ArcGIS (but we don't have SDE).  As I said, these are just basic point files, and  edits will just be adding/deleting points and some table edits. I am not worried about editing the same features at the same time.  Can the combination provide dependable multi-user editing for two basic point files? Probability of two uniform random numbers being more than $\frac{1}{2}$ apart Two numbers are chosen randomly and independently from the unit interval $(0,1)$.  What is the probability that they differ by more than $\frac{1}{2}$? Hint: Think of the $2$ numbers chosen as coordinates in the Cartesian plane. Draw a square with vertices at $(0,0)$,$(1,0)$,$(1,1)$, and $(0,1)$. Allowable choices for the $2$ points lie below the line $y=x-\frac{1}{2}$ or the above $y=x+\frac{1}{2}$. Find the combined area of these regions.  I need help on this. I know what $y=x-\frac{1}{2}$ and $y=x+\frac{1}{2}$ looks like but I am not understanding the problem as a whole In Star Wars: "The Force Awakens," how could this lightsaber have turned up where it did? In The Force Awakens:      Maz Kanata tells Rey that the lightsaber she just picked up and had visions from belonged to Anakin and Luke and now it's hers.   The lightsaber that Luke got from Obi-Wan Kenobi in A New Hope was lost/destroyed when Luke fought Darth Vader in Star Wars: The Empire Strikes Back.  In Return of the Jedi Darth Vader compliments Luke on building a new lightsaber and tells Luke that his skills are now complete.   How could that possibly be the lightsaber that Anakin used originally, and that was then passed on to Luke, when that lightsaber no longer exists?  Writers notes, director's notes, interviews, cast interviews, etc. are welcome. Thanks. I'm working on a map in Bedrock Edition, and I need to have a command block at the end that says "Congratulations [player], you win!" Because it's a map, I cant just have it say a specific player's name, and I've seen maps that do exactly what I want mine to do. How do I format the command to allow this? If $A\simeq\prod A_i$ then $\sum e_i=1$ Given $A$ commutative ring with unity. If we have rings $A_1,\dots, A_n$such that $A\simeq A_1\times \dots \times A_n$ there are $e_1,\dots,e_n$ idempotent elements of $A$ such that $e_1+\dots+e_n = 1$ and $e_i e_j =0$ if $i\neq j$.  I'm not sure how general the proof should be written...  The isomorphic relation suggest that if $f$ defines the isomorphism then there is a $(a_1,\dots, a_n)\in \prod A_i$ such that $f(a_1,\dots,a_n)=1\in A$. Could it work thinking of the idemmpotent terms as $e_i=(0,\dots,0, a_i,0, \dots,0)?$ This should satisfy $e_i e_j =0$ if $i\neq j$. But I have two problems with it (1) I don't see why $e_i$ should idempotent as no assumptions in regard to the existence of idempotent elements in $A_i$ has been made. (2) The sum $\sum e_i=1$ has no reason to be true. In the Law of Reflection, the angle of incidence is equal to angle of reflection. Why is this true? This is clearly true experimentally, but how does one prove this true mathematically? Why there isn't a feature that lets any user on SO to request other user to answer their question? When I browse through other questions to find my answer, sometimes I'll find answers similar to what I'm looking for, and though they don't exactly answer my question, the author might be able to do so. Could we have a feature on SO that will let me request that another user answer my question, in case I find someone who could potentially answer my question? Can you re-equip equipment to the same creature? Why is the folder size (in properties) different from the total file/folder sizes in the folder? Here's my problem  I'm trying to do a cleanup of my hard drive, and I'm constantly seeing this scenario:  (1) Aggregate folder size: I check the folder size viewing properties right clicking the folder  (2) Individual folder/file sizes: I check the individual file/folder sizes belonging to that folder viewing properties on the individual file/folder sizes, and then adding them up (with hidden and system files shown).   These two figures are always different, sometimes by a large margin.   For instance, my C drive now, by method (1), shows a consumed space of 42.4 GB.   However, by method (2), by inspecting the respective files/folders, the total consumed space adds up only to 33.1 GB. This is a 28% discrepancy (over the smaller file size) and I have no idea where the rest of the space is going.   My hard drive is constantly in danger of being full (it's bootcamp on a mac with 50 GB of partitioned space), and there's not much I've got installed in terms of software, nor do I have large media files like movies, downloads, temp files, application temp files, torrents, etc.   Most of it's just consumed by the files in C:/Windows, but last I checked, I don't know enough to know which files to delete, and I've heard that you really can't do anything about the space boggling winsxs and system32.   Can you guys recommend a starting point to free up disk space?  Thanks!!  Baggio. $N$-body simulation in General Relativity How would one perform an $N$-body simulation in General Relativity (GR) for something like galaxy formation or galactic dynamics?  Suppose one wants to simulate the rotation curve $v(r)$ for galaxies with an $N$-body simulation, and wants to do it in the framework of GR / the Einstein equations. I would say $\infty - \infty=0$ because even though $\infty$ is an undetermined number, $\infty = \infty$. So $\infty-\infty=0$. When people are asked to match monchromatic violet light with an additive mix of basic colours, they (paradoxically) mix in red.  In fact, the CIE 1931 color space chromaticity diagram shows this effect begins at about 510nm (greenish-cyan), where people mix in no red.  From that point on, the higher the frequency of the light source, the more red they mix in.    This effect is reflected by the red curve of the , which has an additional bump in the area of blue light.  However, that curve does not match the actual spectral sensitivity of red cones.  So where does this additional perception of red at higher frequencies come from? Suppose $f$ and $g$ are entire functions with $|f(z)|\leq|g(z)|$ for all $z$.  How can we show that $f=cg$ for some complex constant $c$?  Thanks for any help :) What is the procedure for taking down Titans?