Python character alias to function
Unicode identifiers in Python?
n-dimensional vector space isomorphic to the n-th product of a field as abelian groups Let $V$ be a finite dimensional vector space (of dimension n) over a field $F$. I need to show that $V$ is isomorphic to $F^n$ as abelian groups. However, I don't really understand what does "isomorphic as abelian groups" mean, my (poor) attempt to solve this was:  Let $f:V\to F^n$, $f((v_1,v_2,\ldots,v_n)):=v_1+\ldots+v_n$, clearly $f$ is a group homomorphism, but it is not biyective since, for example, in the case where $dim(V)=2$ and $F=\mathbb{R}$ we have $f((2,-2))=0=f((3,-3))$, I've also tried it by defining $f$ to be the product of a vector's entries but obviously it didn't work so I'm stuck!   I think this should be an easy problem but since I don't quite understand it, it's giving me problems. Thank you for your help. Was Tracy Mills a part of John Doe's plan from the beginning? In , John Doe is shown as being extremely meticulous and patient while carrying out his plan of murdering and/or torturing people in accordance with the seven deadly sins.  At the end of the film, it is revealed that John Doe had killed and decapitated Tracy Mills, the wife of Detective David Mills.  Was Tracy Mills his intended target from the beginning?  Did John Doe decide to kill Tracy only after Detective Mills and Somerset discovered his residence? Change footer in beginning of chapters I want the footer of all pages to be "Page N°X" on the right. I used this:  \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} % clear all header and footer fields \fancyhead[R]{\leftmark} \fancyfoot[R]{\footnotesize Página N$^\circ$ \thepage\ } \renewcommand{\headrulewidth}{0pt} \renewcommand{\footrulewidth}{0pt}   It works on every page but the chapter beginning ones. Can not unlock my Android device as have no internet connection? VMware Linux Server -- how can you tell if you are a vm or real hardware? Occasionally (very occasionally) when returning to the main page of a site by clicking on the logo I get the following message:     It was not possible to perform this tag search at this time due to an unexpected error.   This is odd because I'm not performing a tag search.  So far this has happened on both Movies &amp; TV and Cooking.  I have heard that others are getting this same message. I need a CD or USB stick, that will install Ubuntu on a completely headless machine. Stick CD in and restart, no keyboard or screen should be involved.  There are a few obstacles:   The language selection menu when you first boot the CD. The fact that the CD menu waits. The installer's asking questions during installation. Why do I have to solve for $x$ in this proof, if I choose not to find a counterexample? How to make left-right transition of overlay image ffmpeg Does the appositive in this sentence need to be set off with commas? On the freelancing website Upwork, there is a US Punctuation and Mechanics test one can take to boost credibility on one's profile for potential clients. These tests are administered by third-party companies that often hire non-native English speakers, so they're known to have some inaccuracies. Unfortunately, because you're allowed to retake the exams after a certain period passes, the correct answers are never revealed. One of the questions on this exam is the following:  Richard’s wife(A) Abigail(B) is an incredibly talented cook(C) and gardener. Comma at (A) Comma at (B) Comma at (C) Comma at (A) and (B) No commas needed  I answered &quot;No commas needed&quot;, but I suspected afterwards that the answer key was looking for the answer choice above it, &quot;Comma at (A) and (B)&quot;. After searching this question online, I found a website with answers for the various questions on the test, and the site claims that &quot;Comma at (A) and (B)&quot; is in fact the correct answer for this particular question. Is the answer key wrong? I feel like there's really no need for any commas, but on the other hand, the appositive Abigail isn't restrictive because it doesn't offer terribly &quot;essential&quot; information, suggesting it could be set off by commas. HTML/Javascript: how to access JSON data loaded in a script tag with src set latex gives me the error: Ouch---my internal constants have been clobbered As the title states, compiling with latex produces the error, right away:  Ouch---my internal constants have been clobbered!---case 14   I have no idea what this means... I use TexLive 2012. I updated my texmf.cnf and ran fmtutil-sys and texhash. I guess texhash may have messed it up?  Any ideas?  As requested a minimal working example:  \documentclass{article} \begin{document}     Any tex file. \end{document} The existence of the empty set is an axiom of ZFC or not? I found in the Wolfram MathWorld page of the  that this is one of the Zermelo-Fraenkel Axioms, however on the page about these  I read that it is an axiom that can be deduced from the Axiom of Subsets and the Axiom of Foundation (or Axiom of Regularity), so, the existence of the Empty Set is an axiom of ZFC or not? SQL IN Clause 1000 item limit Strange arrow mark with TikZ edge and anchors. When I draw an edge between two nodes with anchors and globally set option [-&gt;], I get an additional arrow tip symbol at the beginning of the last edge in the path. For example in  \documentclass{article} \usepackage{tikz}  \begin{document} \begin{tikzpicture}     \node (A) at (0,0) {$A$};     \node (B) at (1,0) {$B$};      \draw[-&gt;] (A.north east) edge (B.north west);      \draw (A.south east) edge[-&gt;] (B.south west); \end{tikzpicture} \end{document}   the first \draw command produces incorrect output, while the second produces correct output:    Am I doing something wrong, or is this a bug in TikZ? I see many posts on both this forum and elsewhere the discuss the use of a wide angle on a crop sensor.  For example, they discuss how at 14mm lens is 14mm on a 35mm film or full frame, but with a 1.6x crop sensor it is an effective 28mm.  My question is, is a 14mm on a crop sensor the same image as a 28mm on a full frame.  Or does "effective" include some other conotation.  I read that a 14mm can create distortion on a full frame and some vignette, so people seem to say using it on a crop takes out some of those edge problems. My confusion is how the lens is actually taking in the extra image.  Reading kenrockwells blog I learned that wide angle keeps lines straight, while fish eye actually skews the lines.    So with this as my understanding, I am still a bit confused how a lens that is curved in order to get a greater angle of view, will be the same as a cropped version of a less curved lens.  Or does "effective" only refer to the objects to the sides but not to the distortion of the image.  Or, if it is different, is it so small that for photography it does not matter?  A technically description would be welcome as long as a basic understanding of the situation.  Thanks!  Edit: To be more clear after the comments,  I am specifically talking about the end resulting 2d projection from 3d space.  I have read both those previous answers.  the first one is closer to what I am talking about.  However, it is still confusing for me.  I have experience with 3d modeling and projection matricies, which might be confusing me with this.  But for example, i dont understand the images having a 50 mm lens being drawn Starting from the sensor, so it shows a different fov for crop and full frame . If the lens is the same, it takes the same amount of information from the world, which makes a ray from a certain degree off the last lens projected into a smaller space, there is no projection from the sensor, so drawing the line from the sensor doesnt make sense to me.  The distance of the sensor to the rear of the lens must have some impact on the light projected but is not represented in the images.  Further, again from that first answer it says cropping is the same as zoom, but from what i understand from how perspective works, it is different, since a wide angle will project lines differently than a small fov to the same size, so cropping the center of a wide angle and zooming into something is very different.  This is easily recreated in an opengl application with varying fov projection matrix, i imagine in painting class people learn this as well.  A fisheye would be a good example, since if cropping was the same as zooming, a fish eye lens would keep the center exactly like a normal lens and then a gradient weighted towards the outside would rapidly create a warped perspective, but from what I see, it is even.  To me those images just look like its comparing cropped with full frame in regards to orthogonal projections. My understanding is that when using a Bayesian approach to estimate parameter values:   The posterior distribution is the combination of the prior distribution and the likelihood distribution. We simulate this by generating a sample from the posterior distribution (e.g., using a Metropolis-Hasting algorithm to generate values, and accept them if they are above a certain threshold of probability to belong to the posterior distribution). Once we have generated this sample, we use it to approximate the posterior distribution, and things like its mean.   But, I feel like I must be misunderstanding something. It sounds like we have a posterior distribution and then sample from it, and then use that sample as an approximation of the posterior distribution. But if we have the posterior distribution to begin with why do we need to sample from it to approximate it? I came across the following question:    This can be easily flagged for too broad or opinion based, but I wonder on which stack-exchange site can one ask answers for such a question? What is the best collation to use for MySQL with PHP? I wanted to try out different desktop environments so I installed XFCE, KDE, GNOME, etc on top of Unity in Ubuntu 12.04  After a while I decided that I didn't like those other DEs and would stick to Unity. So I changed my default DE to Unity by configuring gdm.  sudo dpkg-reconfigure gdm   Now I am able to choose my DE at login time and all of DEs are working properly.   But the strange thing is that my boot up screen says 'lubuntu', my login screen is KDE, and my desktop is Unity.   How and why is this happening? Why didn't my gdm configuration have any effect? Does login and boot up screen need to be configured separately from the DE?  There are other similar questions here but they are not the same as this one. I do not want to remove the other enviroments I'm quite happy with having a list of DEs to select from. I want to know how to set proper defaults.