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Aligning equations within multline Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated Let $A$ be a ring that is not a field and $M$, $N$ modules over $A$. What conditions on $M,N$ (and maybe also on $A$) guarantee that $\operatorname{Hom}_A(M,N)$ is finitely generated? Is it enough to assume $M,N$ finitely generated? Endofunctors of $\textsf{FinVec}$ which are not $C^n$ Let $a$ be an element of a group G. Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.   My Try:  I proved the backward direction. For forward one, given $g\in G, a(gag^{-1})=(gag^{-1})a$. I was trying to prove that $a\in Z(G)$. In order to prove that, I must show that $ag=ga$. But failed. Am I on the right track? Can anybody please give me a hint? I want to try it myself. Fastest converging method for calculating e How do I assign the output of a command to a variable? Avoid footnotes from appearing before mark I have these \footnotetext and \footnotemark as illustrated in the figure below. I used these tags to enforce my own numbering (and because using only \footnote was not working the way I expected).    The code I use for generating the mark is:   \caption{Lista de notas presentes em um piano de 88 teclas,  com notação em Inglês, frequência em Hertz e comprimento de  onda em centímetros.\protect\footnotemark[1]}   Is there any way I can enforce my \footnotetext to appear in the same page as my \footnotemark? Let $X$ be a set and consider the collection $\mathcal{A}(X)$ of countable or cocountable subsets of $X$, that is, $E \in \mathcal{A}(X)$ if $E$ is countable or $X-E$ is countable. If $X$ is countable, then $\mathcal{A}(X)$ coincides with the power set $\mathcal{P}(X)$ of $X$. Now suppose that $X$ is uncountable. Assuming the axiom of choice, we can conclude that $\mathcal{A}(X) \ne \mathcal{P}(X)$, since $|X| = |X| + |X|$. So the question is:     Can we prove in ZF that $\mathcal{A}(X) \ne \mathcal{P}(X)$ for every uncountable set $X$?   I'm assuming that a set $X$ is uncountable if there is no injective function $f : X \rightarrow \mathbb{N}$. Biblatex: Don't show n.d. when no year is given I am using biblatex/biber with several bibliographies for my book, and the misc-type, which is also a single bibliography (as a chapter), for legal documents. These documents often do not have a proper date or rather the date is part of the title. However, Latex gives me a "n.d." output, and I am wondering what I can do for it to not give a "n.d." when there is no year given?    To make clear, the reason for why I don't want to use the "year" field is that I deal with very different kinds of documents, which may either have no date or the date is in the title, because of which I prefer to simply "hard-copy" the whole thing in the "howpublished"-section and need to suppress the year, if none is provided.  MWE:  \usepackage[autocite=footnote, language=german, style=authoryear-ibid, sorting=nyt, backend=biber]{biblatex} \usepackage[bottom]{footmisc} \bibliography{lit.bib} \makeindex \begin{document}  Test\autocite[2]{draftnotice}.  \printbibliography[title={EU Legal Documents}, type=misc1]  \end{document}   My lit.bib is as follows:  @misc1{draftnotice,     Author = {{COMMUNICATION FROM THE COMMISSION}},     Howpublished = {Draft Commission Notice of 2014 on the notion of State aid pursuant to Article 107 (1) TFEU}}   Check out my MWE on overleaf In Harry Potter and the Prisoner of Azkaban, Harry sneaks into The Three Broomsticks and overhears Madam Rosmerta, the Minister for Magic, and several Hogwarts teachers discuss his parents' death and the fact that the Potters were under the protection of the Fidelius Charm. Professor Flitwick explains the charm:     ‘An immensely complex spell,’ he said squeakily, ‘involving the magical concealment of a secret inside a single, living soul. The information is hidden inside the chosen person, or Secret-Keeper, and is henceforth impossible to find – unless, of course, the Secret-Keeper chooses to divulge it. As long as the Secret-Keeper refused to speak, You-Know-Who could search the village where Lily and James were staying for years and never find them, not even if he had his nose pressed against their sitting-room window!’   Prisoner of Azkaban - pages 152-153 - Bloomsbury - chapter 10, The Marauders Map   In Harry Potter and the Deathly Hallows, once Ron returns to Harry and Hermione after his and Harry's fight weeks earlier, he tells Harry and Hermione about the Taboo:     ‘NO!’ roared Ron, causing Harry to jump into the hedge and Hermione (nose buried in a book at the tent entrance) to scowl over at them. ‘Sorry,’ said Ron, wrenching Harry back out of the brambles, ‘but the name’s been jinxed, Harry, that’s how they track people! Using his name breaks protective enchantments, it causes some kind of magical disturbance – it’s how they found us in Tottenham Court Road!’   ‘Because we used his name?’   ‘Exactly! You’ve got to give them credit, it makes sense. It was only people who were serious about standing up to him, like Dumbledore, who ever dared use it. Now they’ve put a Taboo on it, anyone who says it is trackable – quick and easy way to find Order members! They nearly got Kingsley –’   Deathly Hallows - page 316 - Bloomsbury - chapter 20, Xenophilius Lovegood   So in hindsight the trio understands that it was the Taboo that caused the Death Eaters to find them at Tottenham Court Road, and then later, of course, they're caught by Snatchers because Harry says "Voldemort" out loud. However, the trio says "Voldemort" numerous times while they're staying at Grimmauld Place and Death Eaters appear and stake the place out (granted, it could be that the Death Eaters simply knew that Harry had inherited Grimmauld Place from Sirius after the Ministry examined Sirius's will). They cannot get inside the house, though. I'm completely unclear as to the status of Grimmauld Place and the Fidelius Charm -- the house was under the charm, but the Secret Keeper (Dumbledore) had died. Nevertheless, like eleventy-one people were privy to the secret by the time of Dumbledore's death, and I have never found a satisfactory answer (to me) to exactly how the Fidelius Charm works.   Would Voldemort's Taboo spell be able to trump the Fidelius Charm?  &#9733; I'm looking for a canon-based answer (Harry Potter novels, the three supplemental books, JKR interviews, and Pottermore) and do not prefer an answer from the HP Wikia or the Wikipedia. Show that $n$ lines separate the plane into  $\frac{n^2+n+2}{2}$ regions if no two of these lines are parallel and no three pass through a common point.   I know we start with the base case, where, if we call the above equation P(n), P(0), for 0 lines would be 0.  But I really have no idea how to begin the inductive step.  How do we know what k+1 we're supposed to arrive at?  Thanks! When will my device get the Android 4.4 update (KitKat)? Android 4.4 (KitKat) was officially announced on October 31, 2013.  (You can review  at the official Android Developer's site or the .)  Of course, not all devices will get this update right away and others won't get it at all. Each manufacturer and carrier often choose to add their custom modifications which tend to delay the official release.  When will each device get Android 4.4, either officially or unofficially, if ever?  See also: How to ensure that all traffic goes through VPN connection once it has been started? I connect to the internet through a VPN provider along with other precautions in order to ensure privacy (against traffic analysis by agencies of a repressive government). I currently use  to configure and establish the VPN connection. Tunnelblick does not allow automatically connecting to the provider at or before login so I need to connect manually each time, and sometimes the connection is lost.  I need any and all traffic to go through the VPN connection at all times; whenever the VPN isn't connectable for whatever reason, I want the fallback to be no connection. I need to ensure   that OS X does not connect to the internet on my regular internet connection, or at all, during boot (until the login screen, for time synchronization and other possible "call home" stuff etc.). What services, if any, try to do this, and how can I practically analyze and change what's going on with networking during boot? Is there a way to connect to VPN earlier than right after login? that if my VPN connection is temporarily lost, applications do not continue to communicate over my regular unencrypted connection.    I basically want the system to act as if the network connection is lost altogether if the VPN connection fails, but currently they just continue to work as if nothing happened. What can I do to my system so that once a VPN is established, all traffic can only flow through that connection and no traffic can flow if the connection drops? Comparing and contrasting, p-values, significance levels and type I error GNOME: supend on laptop lid close no longer works since 19.10 upgrade I'm quite positive that this used to work flawlessly in 19.04 (and past releases) but no longer does after a release-upgrade to 19.10 (kernel 5.3.0-24, ThinkPad X1 Carbon gen 5 in case this matters).  I have the corresponding switch set in GNOME-Tweaks and have also attempted to manually edit /etc/systemd/logind.conf as suggested , namely uncomment and adjust the following lines, followed by a reboot:  $ grep -v '^#' /etc/systemd/logind.conf [Login] HandleLidSwitch=suspend HandleLidSwitchExternalPower=suspend HandleLidSwitchDocked=suspend LidSwitchIgnoreInhibited=yes   (strictly speaking, only the first one ought to be required but, just in case, I also added lines 2 &amp; 3, as well as #4 for good measure...  Looking at the output of systemctl status systemd-logind, I see the following:  Dec 17 17:56:50 x1c5 systemd-logind[8959]: Lid closed. Dec 17 17:57:03 x1c5 systemd-logind[8959]: Lid opened.   So the lid events are correctly detected.  To rule out any issue with the suspend process itself, I have also set my power button to suspend in Settings->Power->Suspend &amp; Power Button, for testing purposes.  Pressing the power button now results in this in the output of systemctl status systemd-logind (after a successful suspend-resume cycle):  Dec 17 18:13:58 x1c5 systemd-logind[8959]: Power key pressed. Dec 17 18:14:28 x1c5 systemd-logind[8959]: Delay lock is active (UID 121/gdm, PID 9215/gsd-power) but inhibitor timeout is reached. Dec 17 18:14:39 x1c5 systemd-logind[8959]: Operation 'sleep' finished.   Bottom line: lid events are detected correctly and suspend works flawlessly - but not on lid close...  Any idea how I can get the suspend on lid-close feature back to life?  I can of course use the workaround with the power button but it's a little too easy to forget and result in laptop overheating in bag etc...  --- edited to add ---  Got a little further courtesy of , although the issue isn't the exactly as described.  Here's a smoking gun:  $ systemd-inhibit --list --mode=block WHO       UID  USER      PID  COMM           WHAT                                                     WHY                                                         MODE  gdm       121  gdm       2231 gsd-power      handle-lid-switch                                        External monitor attached or configuration changed recently block gdm       121  gdm       2252 gsd-media-keys handle-power-key:handle-suspend-key:handle-hibernate-key GNOME handling keypresses                                   block user      1000 user      3017 gsd-media-keys handle-power-key:handle-suspend-key:handle-hibernate-key GNOME handling keypresses                                   block   This suggests that the lid close action is being blocked because "External monitor attached or configuration changed recently".  I do connect my laptop to a monitor using a USB-C cable but as I write this, it's been undocked and running on batteries for nearly 3 hours (GNOME settings only shows my laptop display as available).  However it looks like systemd or whichever component is responsible for this still incorrectly believes the external display is attached and - per the ticket - suspend isn't supported by GNOME in such scenarios...  Interestingly the gsd-power process (PID 2231) is owned by gdm rather than my UID.  Killing it restores the suspend-on-lid-close functionality.  Need to figure out why this process exists... When actually connected to external display, systemd-inhibit --list --mode=block reports a 2nd gsd-power process owned by the interactive user; which appears correct and sufficient. Weird future tense usage I am now reading  book and have noticed a couple of cases of weird (for non-native speaker) future tense usages.   The point of the kata is to train your fingers and your brain. I'll do a kata or two every day, often as part of settling to work. I might do it in Java, or in Ruby, or in Clojure, or in some other language for which I want to maintain my skills.`   The phrase about doing a kata or two every day looks perfect for Present Simple. Why does the authoer use Future Simple instead?   However, a professional also knows that there will be times when he will fail, his risk calculations will be wrong, his abilities will fall short; he'll look in the mirror and see an arrogant fool smiling back at him.   Shouldn't we use Present Simple after when/if in this kind of cases? I am especially curious about this case as sometimes I forget to switch to Present Simple after if/when and I thought it is a critical error (which I didn't expect to see in a book). How can I provide anonymous user to access my REST GET API without providing any auth information? The Drupal 8 REST UI configuration page asks me to select Authentication providers for GET method; otherwise, it doesn't allow me to export the REST request.  How can I provide anonymous user to access my REST GET API without providing any auth information? How to prove wave nature of large object? How can I explain how to play D&D to new players without them falling asleep from boredom? Some of my friends just started playing Dungeons &amp; Dragons [5e] for the first time and don't know anything about how to play. I have tried explaining to them the things that are completely necessary and teach them the rest while playing but I could not get them to understand. None of them have a players handbook and they can't afford one and they really want to play. I really want a simple way to explain the basics and since I have a habit of explaining things in ways that are much too complicated. I have already told them the basic rules so all I need to know is a simple way to explain how to play.