Prove/Disprove that $\left|\phi^{-1}(\{a\})\right| = \left|\phi^{-1}(\{b\})\right|$ for every $a, b \in f(G)$
Number of preimages of a group element under a homomorphism.
I have installed Linux Bash on Windows 10 (available from Microsoft), but I've forgotten my password.  I've searched for a solution but I only find suggestions that involve doing something before the system boots, like accessing recovery mode, but that will not work in this case as this Ubuntu is just an application within a Windows system and doesn't have its own boot process.  How can I reset my password? Checking from shell script if a directory contains files From a shell script, how do I check if a directory contains files?  Something similar to this  if [ -e /some/dir/* ]; then echo "huzzah"; fi;   but which works if the directory contains one or several files (the above one only works with exactly 0 or 1 files). How to customize Windows 10 "6 Folders" in My Computer I would like to customize the folders in the Default Folders group of My computer. It is not necessary to have the same folders appear on the left sidebar and in the middle view. How can I do that ? Terminal appears with much delay when shortcut is used When I press Ctrl+Alt+T the Terminal opens after around 30 seconds. That's pretty annoying. If I open it with Alt+F2 and enter gnome-terminal the window appears instantly as usual.  I'm using Ubuntu 16.04 with gnome3.  Thank you for your help in advance! Which is more appropriate in the following sentence?     "I asked him to not judge her according to (or: based on) her beauty"   vs     "I asked him not to judge her according to (or:based on) her beauty"   or maybe you wouldn't choose either of the above, but something else? In English, I sometimes hear a phrase like green is the new black.  Literally, this makes absolutely no sense whatsoever.  What does it mean when someone says X is the new Y, when this makes no sense when taken literally?   gives plenty of examples:          I've found definitions online for specific examples, such as , but overall I don't understand what X is the new Y means, and I can't find it explained anywhere.  What does X is the new Y mean?  As far as I'm aware, this phrase exists only in English. I needed to edit a .config file (which I know for a fact is not creating these problems, just to clear that up) and it wouldn't let me save it, so I took ownership of /etc and all contents with the command chown -hR username /etc and that let me edit the .config file but now whenever I try to install any packages or use any sudo commands, it doesn't work  (I've checked for errors in the sudo file in sudoers.d and there was nothing wrong with that).   So how do I return it to the previous owner? In a question related to introductory Galois theory, I was asked to given an example of a tower of fields $F \subset K \subset E$ such that $E$ is a splitting field for some polynomial $f(x) \in F[x]$, but that $K$ need not be a splitting field. The example that comes to mind for me is $\mathbb Q \subset \mathbb Q(\sqrt[3]{2}) \subset \mathbb Q(\omega_3, \sqrt[3]{2})$.  So I know that the minimal polynomial for $\mathbb Q(\omega_3, \sqrt[3]{2})$ over $\mathbb Q$ is $x^3-2$. This polynomial does not split in the intermediate field $\mathbb Q(\sqrt[3]{2})$.  I wanted to make sure that I'm justifying that this intermediate field is not a splitting field correctly. Now I may be confused about the books terminology. When we say something is not a splitting field, we always have to refer to a specific polynomial correct? I've read a term called a normal extension which I think refers to one in which every polynomial splits, but my book doesn't mention these.   If splitting field means with respect to the polynomial $x^3-2$, then I've already demonstrated that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field. Also, it couldn't be a normal extension either because that polynomial doesn't split.  Does it sound like I'm understanding this correctly or are there some subtleties I might be missing? Can humans eat grass? Can a human eat grass and digest it? Could it be possible to use it as food just like other plants such as wheat or beans? When a polished piece of metal (or steel in particular) is heated to incandescence, how do its reflective properties change?  Given a mirror-like surface, would the object temporarily cease to act like a mirror at a certain temperature?  Would the specular reflection of light turn into diffuse reflection; in other words, would the surface temporarily become matte/dull?  I am not asking about permanent changes to the object such as melting, but rather can there be a temporary change in the reflectivity only while the the object is hot, that disappears when the object is cooled down?  Background of the question: I have a laser scanner that usually has trouble scanning shiny surfaces with high specular/ low diffuse reflectivity; and it seems to work a lot better for hot steel pieces than for cold ones. Glowing steel also looks less shiny to me than cold steel, but of course that is hard to tell with all the glowing 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. Strange floating-point behaviour in a Java program In my program I have one array with 25 double values 0.04 When I try to sum these values in a loop I get following results:  0.0 + 0.04 = 0.04 0.04 + 0.04 = 0.08 0.08 + 0.04 = 0.12 0.12 + 0.04 = 0.16 0.16 + 0.04 = 0.2 0.2 + 0.04 = 0.24000000000000002 0.24000000000000002 + 0.04 = 0.28 0.28 + 0.04 = 0.32 0.32 + 0.04 = 0.36 0.36 + 0.04 = 0.39999999999999997 0.39999999999999997 + 0.04 = 0.43999999999999995 0.43999999999999995 + 0.04 = 0.4799999999999999 0.4799999999999999 + 0.04 = 0.5199999999999999 0.5199999999999999 + 0.04 = 0.5599999999999999 0.5599999999999999 + 0.04 = 0.6 0.6 + 0.04 = 0.64 0.64 + 0.04 = 0.68 0.68 + 0.04 = 0.7200000000000001 0.7200000000000001 + 0.04 = 0.7600000000000001 0.7600000000000001 + 0.04 = 0.8000000000000002 0.8000000000000002 + 0.04 = 0.8400000000000002 0.8400000000000002 + 0.04 = 0.8800000000000002 0.8800000000000002 + 0.04 = 0.9200000000000003 0.9200000000000003 + 0.04 = 0.9600000000000003   Why on earth could that happen?! MacBook Pro Trackpad Suddenly Stopped Working Completely The trackpad on my Macbook Pro of 2 years has suddenly stopped working out of blue.  The trackpad doesn't move, respond to any gestures nor do any clicks work. This is really frustrating as I used gestures a lot to navigate around my laptop, and even though I use a bluetooth mouse it makes using my laptop unnecessarily laborious.  Does anyone know what may be causing the problem, and any solutions? Application of Kolmogorov's Zero-One Law Most of the books I read, they only state some examples of tail events. One of them is  $[\sum_n X_n \ converges]$. My main problem is to show that this is indeed a tail event. Why do RAW images look worse than JPEGs in editing programs? I've found that when you load a RAW image into an editing program such as Lightroom/Aperture, the image is usually worse than if you just took the image as JPEG. Now I understand that the camera does some magic during the JPEG conversion. But I'm trying to understand what that "magic" is.   If I'm looking to implement that "magic" myself on the desktop, what kind of settings should I be trying? I find that the RAW files have more extreme contrast. The dark areas for example are way darker than JPEG. Why is this? And what is the best way to fix this? Starting a new online business I have just started to work on an online business mode website. My dream is to make it live and bring revenue from the website. However, the project is in Incubator phase.   In this situation, can i register it as a Business, Open a Business Bank Account and start making the Purchases such as Servers, Software's etc.  Whether i have to pay Tax for my start up business ?  Location : New Jersey, US Converting spoken word audio files into text I've been working on transcribing a lecture series from audio to text files.  There are just over 20 lectures, and each one is about an hour long.  Using VLC Player, I'm able to slow down the speaker's words to a rate that I can keep up with.  However, it's extremely slow going, and it takes so much concentration that I can barely get through a third of a lecture before I have to take an extended break.  The thought had occurred to me that I would be better served by using some kind of speech recognition software, and then proofreading the computer-generated result.  This is likely to be a one-time thing, so while I don't mind using commercial software to get the job done, it would be preferable if it wasn't too expensive (or free; free is always good).  Command-line solutions are fine, too (:  Any suggestions? Prove that the range of the entire function $z^2+\cos(z)$ is all of $\mathbb{C}$. Prove that the range of the entire function $z^2+\cos(z)$ is all of $\mathbb{C}$.   I'm aware this question has been  already, but the explanations were a little shakey and referenced a google sample preview of a text of whose terminology I wasn't very familiar with, so I would like some more input if possible.  Here's my attempt:  Let $f(z)=z^2+\cos(z)$. Through the power series of the cosine function $$ f(z)=z^2+1-\frac{z^2}{2!}+\frac{z^4}{4!}-\cdots=1+\frac{z^2}{2!}+\frac{z^4}{4!}-\cdots, $$ so $$ g(z):=\frac{f(z)-1}{z} $$ has a removable singularity at $z=0$ and is effectively an entire odd function. Picard's little theorem asserts that the range of $g$ is all of $\mathbb{C} $: if there exists some $z_0$ such that $g(z)\neq z_0$ for all $z\in\mathbb{C}$, then $-z_0$ is also missed by oddness of $g$, which is a contradiction (since $g(0)=0$ we rule the possibility that $z_0=0$). (I'm not sure about the following step) It follows that $zg(z)$ and so $zg(z)+1=f(z)$ is surjective.  Is this method valid? Would I need further justification to show that $z\mapsto z$ and $z\mapsto g(z)$ surjective imply its product -- $z\mapsto zg(z)$ -- is also surjective (if this is even true). Any input is greatly appreciated.  Edit: After thinking about this for a moment longer, I realize this logic would also show that the range of any entire even function is all of $\mathbb{C}$, which seems like quite a strong conclusion. This makes feel like there are holes in my logic. Auto update specifick plugin I've created a plugin and I want to auto-update this plugin.  I've been looking in the WordPress Codex and found this following piece of code:  add_filter( 'auto_update_plugin', '__return_true' );   I added this code into a new Must-have Plugin but it doesn't work. It does not auto-update all my plug-ins..  Anyone that have experience with auto updating? Is there a Raster Equivalent of the Split tool?