Examples of Non-Hausdorff Topology
"Naturally occurring" non-Hausdorff spaces?
Can't get unicode symbols in math mode I am trying for about a week to launch unicode-math package with XeLaTeX and got nothing. PDF is created well but all Cyrillic symbols in math mode are skipped. I tried using different math fonts - no progress at all. In log file, I found low-level error:  Missing character: There is no [cyrillic letter from input] in font cmmi12!   But all fonts used in document are Unicode ones.  Here is a file I want to be processed well (of course it is UTF-8).  \documentclass[12pt]{book} \usepackage{polyglossia} \setdefaultlanguage[spelling=modern]{russian} \setotherlanguage{english} \defaultfontfeatures{Ligatures={TeX}} \setmainfont{CMU Serif} \setsansfont{CMU Sans Serif} \setmonofont{CMU Typewriter Text}    \usepackage{amsmath, amssymb} \usepackage[russian]{hyperref}  \usepackage{unicode-math} \setmathfont{Latin Modern Math}  \frenchspacing  \begin{document} Просто буквы % Plain letters $$Память: M_{доп}(n) = \Theta(N)$$ % Memory: M_add(n) = \Theta(n) \end{document}   Looking for your help. I am doing , and am having a few problems with function notation. I understand the form f(x).  What does the form f: x ↦ y mean? Could you also give one or two examples?  And, if possible, state your source. Thank you. My local hackerspace has a CNC machine.  I can give them my Gerbers and they can print out the PCB.  The machine does not plate VIAs.  I need to do this after the CNC machine is finished.  What is the best process and materials to use? I think every single one of us has experienced a dark dot in his/her sight when looking at a white surface such as a white ceiling, there's something in our eye that with a little focus we can see it, and when we move our eyes around, that thing floats around quickly in your sight. It's like a substance on eye's surface, I want to know what is it called. Hope I'm clear enough. Companion affinity change notification Is there a detailed user's documentation for KDE Plasma 5? I'm getting frequent inaccurate location as the same location, can I make this stop? Number of publications needed to get into a computer science PhD program? I've looked at the resumes of the type of candidates that get into computer science PhD programs at the top few universities (e.g. Stanford, MIT etc.) and a lot of them seem to have multiple publications (2-4) done during their undergraduate years. Is this representative of the entire pool? Or is this just a self-selected group that puts their resumes online.  I currently have one first-author workshop paper (5 pages), and I wonder how this might affect my chances at entry into these top universities. Casting two cantrips, both modified by metamagic? Can you cast one cantrip, modifying it with "Twinned Spell" (Fire Bolt) to target 2 creatures, and then cast another cantrip as a Bonus Action (Fire Bolt), using "Quickened Spell" to target one of the previous creatures or another creature? How do I know if a laptop will work with Ubuntu? I want to buy a laptop at some point in the near future, but not knowing beforehand whether it works with Ubuntu makes the whole operation risky and tiresome. I would like to know if there is some kind of official hardware database where I can enter the name of the laptop I'm interested in and see if it's 100% compatible out of the box, or in case it isn't what is it exactly that doesn't work (ideally also if there is a way to fix it).  Does Canonical have a project along these lines? Why do some actors in credits appear as - Character (As Actor) How does "Reputation" work? On Stack Exchange, users may gain a certain level of reputation.  What does reputation do? How can a user gain or lose reputation?   See also   in the Help Center  - Maximum amount of votes a user can use in a day and reputation requirements to vote.  - How to get an exact overview of the actions that got you your reputation.  - A full list of all privileges granted at each reputation level on all sites. Trajectory of a photon around a Schwarzschild black hole? Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius $r_{s}$) (see  for illustration). Its impact parameter is $b$ and its distance of closest approach is $r_{0}$ with $$b^2=\frac{r_{0}^{3}}{r_{0}-r_{s}}$$.  Then its trajectory in polar coordinates is defined by :  $$\frac{d\varphi}{dr} = \frac{1}{r^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{r}\right)\frac{1}{r^2}}}$$  Consequently : $$\varphi\left(r\right) = \int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}$$  and one can compute the total deviation using :  $\Delta\varphi = 2\times\left(\lim_{r\to+\infty}\varphi\left(r\right)-\frac{\pi}{2}\right)$  But my question is : how can I plot/draw the trajectory using the integral expression of $\varphi\left(r\right)$ ?    Because if I compute : $$f\left(r\right) = 2\times\left(\int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}-\frac{\pi}{2}\right)$$ I obtain $f\left(r_{0}\right) = -\pi$, and then $f$ increases up to zero, crosses zero, and tends to its positive value at infinity $\Delta\varphi$. It does not make sense for me and I do not understand how to compute the trajectory from that.    If I compute : $$g\left(r\right) = \int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}$$ it starts from $0$, and increase up to $\frac{\pi}{2}+\frac{\Delta\varphi}{2}$.    I would like to compute the trajectory in the $\left(x, y\right)$ plane, so how to use the values of $f\left(r\right)$ or $g\left(r\right)$ to compute the function $y\left(x\right)$ ? Last night I was grilling chicken thighs on a barbecue and I was told that if there was any pink inside them, they are not safe to eat. My question is twofold:  (a) Is this true?  (b) Is there a way to tell that doesn't involve cutting the thighs open to have a look at the colour? In naive set theory, I believe ∅ = {∅} = {{∅}} is correct, but just wanted to make sure that I understood this correctly.   ∅ is an empty set, so having an empty set as an element of a set that contains nothing else is pretty much the same thing as e.g. marking the number 2 as 2.000000, right? Doesn't the symbol ∅ intrinsically imply that this is an empty set which contains it self? Move all files within subfolders to parent folder Firstly, I'm aware there are tons of similar questions, and I know because I must have tried adapting at least 5 of them, but I'm still here asking because my crappy grasp of CMD (which I actually thought I was getting good at up until this point) is frustrating me with generic, unhelpful errors, and I can't understand what part of the syntax is wrong. This is likely due to the fact that other questions deal with the folder/filename structures of their respective OPs, and when I try to adapt the code to my own folder/file structure it refuses to work. Therefore, I'm here asking for a CMD for loop solution that'll actually work for my own file and folder structure.  I have a parent folder which contains many other folders, in which there are files:  Parent Folder      Subfolder            File1            File2      Subfolder            File1            File2      Subfolder            File1            File2   ...and so on.  I'm wanting to use a CMD for loop in combination with the move command to extract all the files in the subfolders to the parent folder, leaving each subfolder empty.  When giving the CMD for loop please break down what each part of the command is doing, as this'll help make sure I understand the code and hopefully minimise the trouble I have getting it to work for myself. Install Numpy for Python3.4.0 on Ubuntu 14.04.01 LTS How to get the current date/time in Java OS: UBUNTU 18.04  I had installed Oracle Java 8 earlier. But decided to switch to open JDK 11. it's successfully installed. java -version openjdk version "11.0.2" 2019-01-15 OpenJDK Runtime Environment (build 11.0.2+9-Ubuntu-3ubuntu118.04.3) OpenJDK 64-Bit Server VM (build 11.0.2+9-Ubuntu-3ubuntu118.04.3, mixed mode, sharing)  Then I set the JAVA_HOME sudo nano /etc/environment  File changes: PATH="/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/$ JAVA_HOME="/usr/lib/jvm/java-11-openjdk-amd64/"  Then executed: source /etc/environment  echo $JAVA_HOME /usr/lib/jvm/java-11-openjdk-amd64/  But the moment I exit the terminal and then launch the terminal again: I get this output:  ▶ echo $JAVA_HOME /usr/lib/jvm/java-8-oracle  which you can see is not what is expected. Please help me out Because of  I'd like to prevent apt-get upgrade and Update Manager from updating the "libgtk2.0-0" package.   How can this be achieved?