Open sets having open images in $R^2$
$\|h(x)-h(y)\|\geq 3\|x-y\|$ prove that image of open is open
Awaken monit daemon every few hours for all monitored processes If I create just a basic hello.tex file an compile it using   pdflatex hello.tex   The pdflatex program is creating a plethora of file which is a little annoying. Currently it writes all those files about which I really don't care:  hello.aux, hello.log, hello.nav, hello.out, hello.snm, hello.toc   Is there a way to tell pdflatex to only write hello.pdf if the build is successful ? If not, can I specify the output directory of those files ? Maybe I could just write them in /tmp. I am wondering why matrix factorization techniques in the machine learning domain almost always expect the provided matrix to be non-negative. What is the advantage of this constraint?  Background: I want to use matrix factorization algorithms for a sparse user-item matrix containing positive and negative implicit feedback. Is there any another possibility to set interactions with negative indications apart from fields that denote that no interaction happened between the user and the item? "This is a favorite question" tooltip when not favorite Evaluate $$\int \frac{2-3x}{2+3x}\sqrt\frac{1+x}{1-x}d x$$ What substitution should I use ? $\sqrt\frac{1+x}{1-x}$ suggests $x=cos2\theta $ but its not useful in $\frac{2-3x}{2+3x}$ What is the correlation if the standard deviation of one variable is 0? Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex How do I delete Windows after an Ubuntu install? I downloaded Ubuntu 12.04 LTS to my Dell Dimension 4600 with Windows XP installed.  I now have a neat dual-boot system, but I want to get rid of the XP OS and keep Ubuntu as my only OS.  How do I do this? What is the difference between a function decorated with  and one decorated with ? From some comments in the answers for  (now deleted, 10k only), there's confusion around the usage of I vs. me:  While the sentence, "the other attendees are myself and Steve," is agreed to be incorrect, there's confusion about whether the correct form is "the other attendees are me and Steve" or "the other attendees are Steve and I."  (I've always used the heuristic of removing the other people from the sentence, so I always thought "the other attendee is me" would be correct, instead of "the other attendee is I." Is this true, or am I using a flawed heuristic?) Disable OS X startup sound How can I permanently disable the startup sound (or chime) in Yosemite?  (I am using a late-2014 13" macbook pro)  I already tried several options found here and there :    installed StartNinja.app => did not work installed Auto-Mute.app  => did not work tried to install the old StartupSound prefPane => did not work tried the command "sudo nvram SystemAudioVolume=%80" => volume does not change (I also tried other values than 80, such as 00, 01, or nothing)   NOTE: I know that it is possible to mute the next startup sound by turning the volume down, but this is not a permanent solution. Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear? This is an exercise in complex analysis:     Let $\Omega\subset{\Bbb C}$ be open and bounded, and $\varphi:\Omega\to\Omega$ a holomorphic function. Prove that if there exists a point $z_0\in\Omega$ such that   $$ \varphi(z_0)=z_0\qquad\text{and   }\qquad \varphi'(z_0)=1 $$   then $\varphi$ is linear.   I'm trying work out the case $z_0=0$ first, in which $$ \varphi(z)=z+\sum_{n=2}^{\infty}a_nz^2. $$ It suffices to show that $a_n=0$ for all $n\geq 2$. If let $$ \varphi(z)=z+a_2z^2+O(z^3) $$ then  $$ \varphi^k(0)=z+ka_2z^2+O(z^3), $$ and $$ \varphi^k(0)=0,\quad (\varphi^k)'(0)=1. $$ If one can show that $\{ka_2\}_{k=1}^{\infty}$ is uniformly bounded, then one at least has $a_2=0$. But I don't know how to go on. Any idea? Why do Google search results include pages disallowed in robots.txt? I have some pages on my site that I want to keep search engines away from, so I disallowed them in my robots.txt file like this: User-Agent: * Disallow: /email  Yet I recently noticed that Google still sometimes returns links to those pages in their search results.  Why does this happen, and how can I stop it? Background: Several years ago, I made a simple web site for a club a relative of mine was involved in.  They wanted to have e-mail links on their pages, so, to try and keep those e-mail addresses from ending up on too many spam lists, instead of using direct mailto: links I made those links point to a simple redirector /  script running on my own site.  This script would return either a 301 redirect to the actual mailto: URL, or, if it detected a suspicious access pattern, a page containing lots of random fake e-mail addresses and links to more such pages.  To keep legitimate search bots away from the trap, I set up the robots.txt rule shown above, disallowing the entire space of both legit redirector links and trap pages. Just recently, however, one of the people in the club searched Google for their own name and was quite surprised when one of the results on the first page was a link to the redirector script, with a title consisting of their e-mail address followed by my name.  Of course, they immediately e-mailed me and wanted to know how to get their address out of Google's index.  I was quite surprised too, since I had no idea that Google would index such URLs at all, seemingly in violation of my robots.txt rule. I did manage to submit a removal request to Google, and it seems to have worked, but I'd like to know why and how Google is circumventing my robots.txt like that and how to make sure that none of the disallowed pages will show up in their search results. Ps. I actually found out a possible explanation and solution, which I'll post below, while preparing this question, but  in case someone else might have the same problem.  Please do feel free to post your own answers.  I'd also be interested in knowing if other search engines do this too, and whether the same solutions work for them also. Are there rules about using "that" to join two clauses? He will understand that I was not joking.   He will understand I was not joking.  Which of the sentences is correct? Are there any specific rules about the use of &quot;that&quot; in the sentences I reported as example? Positrons versus holes as positive charge carriers : [The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.]  and:  [Dirac's idea is completely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge — the charge of the ionic lattice — that cancels out the electric charge of the sea.]  It always confused me to think of holes as positive charge carriers in semi-conductors as not being real: real electrons move from one lattice-position to another lattice-position, which effectively looks like a positive hole in the lattice that is moving in the other direction, but in reality a real electron moves, the hole is kind of an "illusion".  On the other hand the positrons are always introduced as real hard-core particles.   The quotes from the Wikipedia article make me unsure: how should I look upon these phenomena?   Edit: holes in a Dirac sea give rise to real pos. entities in one case and to unreal pos. entities in another - how can we distinguish, is it a matter of formalism? Disable mouse speed check between monitors in windows 8.1 update 1 In windows 8.1 and below, the mouse would seamlessly move through all my monitors.  However, in windows 8.1 update 1 the mouse needs to be going at a certain speed before it goes to a different monitor. How do I disable this?  A google search has turned up nothing, so I'm not sure what exactly is going on Can creatures without hands (and as a consequence fingers) use rings? Pretty much what it says on a label. I have a monster for my next adventure that has a pretty interesting ring in his treasure. Since it is intelligent, and would be able to use the ring, I don't see why the ring should be lying there in a chest waiting to be looted.  Monster would use the ring, if it only had hands or even fingers. Hold a second, are fingers even necessary in 5e?  For reference, assume it only has grasping / manipulating tentacles like a  (although it is not one); Said tentacles are very flexible, capable of fine manipulation, and have varying thickness. Calculate the last digit of $3^{347}$ I think i know how to solve it but is that the best way? Is there a better way (using number theory). What i do is: knowing that  1st power last digit: 3 2nd power last digit: 9 3rd power last digit: 7 4rh power last digit: 1 5th power last digit: 3    $3^{347} = 3^{5\cdot69+2} = (3^5)^{69} \cdot3^2 = 3\cdot3^2=3^3=27 $ so the result is $7$. Small snap loop devices visible in gnome-disk-utility OR what is the function of snap ubuntu-core I just realized using gnome-disk-utility (3.18.3.1) that my 16.04 installation is featuring several small loop devices.  In detail:   78 MB Loop Device (/var/lib/snapd/sn_ntu-core_352.snap -> /dev/loop0) 76 MB Loop Device (/var/lib/snapd/sn_ntu-core_216.snap -> /dev/loop1) 79 MB Loop Device (/var/lib/snapd/sn_ntu-core_423.snap -> /dev/loop2) 705 KB Loop Device (/var/lib/snapd/snaps/tpad_12.snap -> /dev/loop3) 684 KB Loop Device (/var/lib/snapd/snaps/tpad_11.snap -> /dev/loop5) 705 KB Loop Device (/var/lib/snapd/snaps/tpad_13.snap -> /dev/loop6)   Image showing gnome-disk-utility    My questions are as follows:   What are those loop devices used for? Is that an expected behaviour in 16.04? Can  I remove them? And if so - how (just by unmounting)? Is there any risk in removing the 2 installed snaps mentioned below (tpad and ubuntu-core)   UPDATE  Installed snaps  According to snap list I do have   tpad (1.8.1) ubuntu-core (16.04.1)   installed. I remember I did tinker with snaps for a short time when  I realized this option - but am pretty unsure if  I  installed those 2 snaps - or if they are pre-/auto-installed.  tpad is a Terminal text editor with GUI-like user interface  I  did install manually, as  I don't use it  I removed it now via sudo snap remove tpad. As a result the 2 loop devices pointing to tpad are gone in gnome-disk-utility.  So - at the current point  I am still having 1 snap installed called ubuntu-core and I don't know what it is used for. The summary of this particular snap is   The ubuntu-core OS snap Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma\left(A^T\right)$ where $A^T$ is the transpose matrix of $A$.