Theorem 17.5 Gallian: A polynomial generates a maximal ideal if and only if it is irreducible.
Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
I am looking for a JavaScript array insert method, in the style of:  arr.insert(index, item)   Preferably in jQuery, but any JavaScript implementation will do at this point. What's this I hear about First Edition Unix being restored? that the source code of First Edition Unix had been restored from tapes.  The response was amazement and a request for more information.  What have people done?  Who did it?  When?  And how? How are dual-role films shot? How to disable the physical sim and only use the eSIM on iPhone XS/iPhone XS Max? I'm wondering if there is a way to turn off the Physical sim and use only the eSim on my iPhone XS Max. I travel a lot and I'm loving the use of eSIM.   I understand that I can turn off the eSIM if its required, but I wanted to know if there was any way to turn the Physical Sim off and leave only the eSim ON.  Note - I have an unlocked iPhone XS Max. Difference between decimal, float and double in .NET? What is the difference between decimal, float and double in .NET?  When would someone use one of these? Lasso and statistical signficance of selected variables When I create a clan in the multiplayer menu, it says clan created but I cannot add or invite people to my clan. I go to create a new one assuming the request was canceled or not entered correctly, it says clan name already exists as if someone else created it.   Is this a bug? Did I do it wrong? Or am I missing something?   I'm on the PS3. Will this also happen on my Xbox 360 Version? RMSE vs Standard deviation in population multisite: allow only specific users to update sites Confused about the possibility of different differentiable structures. In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.  He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.   My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept? Some of code I read in trigger is like this    for(account a: [select id  from account where id in: mapAccount.keySet()]) {     //code      }    some are like   List&lt;Account&gt; listAccount = [select id  from account where id in: mapAccount.keySet()];   for(Account a: listAccount ){   //coode   }   Are those same from performance view?which one is better? Show all inbox items (notifications) in app The SE app currently only shows some inbox items:   Answers Comments Chats   When I see those I assume I'm all caught up, but it turns out there are more on the web which aren't displayed in the app:   Bounty expiration Elections Edits on your own posts Maybe others?   I was about to tag this  but realized it's probably intentional.  I imagine the reason for not showing them is because they're not something you can act on from within the app. Bounties can't be offered or awarded1 and elections can't be voted on.  But then again chats can't be replied to either. Badge descriptions can't be viewed (on the Achievements side). Selecting those launches the browser. Why then shouldn't bounties or elections launch the browser too? Hiding them just means I'll miss it if I haven't visited in browser recently.  1 Though accepting answers does award the bounty I have a realy strange problem in Ubuntu 11.10. I have by accident my password set to empty. If I wan't to install software now the system asks for a password, but I haven't one anymore. If I enter my old password it sais it's wrong and if I leave it empty it also does. Now I wan't to set my password back, but the system asks for the old password before I can change it. That password doesn't exist, so I can't do anything anymore! Can someone help me please? Syntax error due to using a reserved word as a table or column name in MySQL Let $f(x)$ and $g(x)$ be two continuous functions on $[0,1]$ and  $$\int_{0}^{1}f(x) dx= \int_{0}^{1}g(x)dx = 1$$   Show that there exist  $[a,b]\subset [0,1]$, such that  $$\int_{a}^{b}f(x) dx= \int_{a}^{b}g(x)dx = \frac{1}{2} $$  The question can be solved by considering the fundamental group of $S^1$, now I am wondering if we can solve it by real-analysis.  Here is the topological solution:  Assume that for any $[a,b]\subset[0,1]$, we always have  $$(\int_{a}^{b}f(x) dx\neq \frac{1}{2}) \quad \vee \quad(\int_{a}^{b}g(x)dx \neq \frac{1}{2}) $$ Consider mapping $$\phi:D\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\},\,\,(x,y)\mapsto(\int_{y}^{x}f(t) dt,\int_{y}^{x}g(t) dt)$$ where $D=\{(x,y)|0\leq x\leq y\leq 1\}$.  Let $a$ be path from $(0,0)$ to $(0,1) $in $D$ and $b$ be path from $(0,1)$ to $(1,1) $in $D$, then $ab$ is a path from $(0,0)$ to $(1,1) $in $D$ and $$\phi\circ (ab):[0,1]\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\} $$ Notice that $$(\phi\circ (ab))(0)=(\phi\circ (ab))(1)=(0,0)$$  so $\phi\circ (ab)$ is a loop based on $(0,0)$ in $\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}$.  For any $t\in [0,\frac{1}{2}]$, it's not hard to get that $$(\phi\circ (ab))(t+\frac{1}{2})=(1,1)-(\phi\circ (ab))(t)$$ is equivalent to $$(\phi\circ (ab))(t+\frac{1}{2})-(\frac{1}{2},\frac{1}{2})=-((\phi\circ (ab))(t)-(\frac{1}{2},\frac{1}{2}))$$  Define retraction  $$r:\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}\rightarrow S^1,\quad (x,y)\rightarrow\ \frac{(x,y)-(\frac{1}{2},\frac{1}{2})}{||(x,y)-(\frac{1}{2},\frac{1}{2})||}$$  Then $r\circ \phi\circ (ab)$ is a loop on $S^1$, such that $$(r\circ \phi\circ (ab))(t+\frac{1}{2})=-(r\circ \phi\circ (ab))(t),\quad\forall t\in [0,\frac{1}{2}]$$  So $&lt;r\circ \phi\circ (ab)&gt;$ is not trivial in $\pi_1(S^1)$.   However, $ab$ is path homotopic to $c$ in $D$, where $c$ is the path between $(0,0)$ and $(1,1)$ in $D$. In this case, $r\circ \phi\circ (ab)$ is a point-path in $S^1$ by $$(\phi\circ c)(t)=\phi(t,t)\equiv (0,0)$$  which leads to contradiction. Why don't brakes come with 'Safety Levers' any more? It used to be very common to see bikes with dual pull brake levers (if that's even the right name) as shown below. However now it seems almost impossible to find brakes like this.  Even finding a picture to post was hard.  It seems like it would be quite useful to be able to brake from various hand positions.  There was an answer I read a while back (can't find it now) that stated that there was a reason they got rid of those, but no explanation as to why. So any technical reason, or is it just for looks? Count days between two dates, excluding weekends (MySQL only) I'm using a BooleanToVisibilityConverter in WPF to bind the Visibility property of a control to a Boolean. This works fine, but I'd like one of the controls to hide if the boolean is true, and show if it's false. I'm running macOS Catalina 10.15.4 on a 2018 MacBook Pro 15", which has the T2 chip (if anything else is relevant, let me know and I'll update the question.)  I'm a developer and I'm in the process of writing and debugging a C command-line utility that's supposed to take a few miliseconds to run and exit.  My internet connection went down while I was coding (WiFi was still connected, but my ISP was down), and afterwards I noticed that, after compiling the app, the first time I ran it, it took a few seconds to run rather than the expected fraction of a second.  I turned off WiFi (so there was no active network connection), compiled my app again, and ran it -- this time it ran as quick as I would expect.  Turn on WiFi, compile and run again, first run of the app takes longer than expected.  Maybe I'm paranoid, but it sure sounds like macOS is trying to ask Apple's servers for blessing to run my app. Is this the case, or could there be another explanation? If it's indeed the case, can anyone link to an article explaining what's going on under the hood?  UPDATE: as requested, here are timings for an average of three runs of the following command, under different conditions:  touch file.c &amp;&amp; make &amp;&amp; time ./file &amp;&amp; time ./file  The first measurement is the average time for running file for the first time after compilation (the third command in the chain). The second measurement is the average time for running file for the second time after compilation (the fourth command in the chain).  WiFi on, with internet: 347 ms/3 ms.  WiFi on, without internet: 5024 ms/3 ms.  All network connections offline: 24 ms/2 ms. robots meta not blocking indexing We have a staging version of our website to test changes on at trailheadpaddleshack.ca/staging1. This never appeared in search before. Recently the staging site has appeared on Google and is affecting our search results.  I'm trying to figure out how the pages got there and how to remove them. The pages have always had &lt;meta name="robots" content="noindex, nofollow"&gt; in the head.  I am kinda new at this but was under the impression this should prevent Google from showing my site in results. I am pretty sure the results appeared in google after I accidently copy pasted some codes from the staging site to the live site that contained links to pages on the staging site. If anyone can point me in the right direction to figure out what happened and prevent it from happening again would be much appreciated.  robots.txt looks like this:  User-agent: * Disallow: /calendar-2/action~posterboard/ Disallow: /calendar-2/action~agenda/ Disallow: /calendar-2/action~oneday/ Disallow: /calendar-2/action~month/ Disallow: /calendar-2/action~week/ Disallow: /calendar-2/action~stream/ #Begin Attracta SEO Tools Sitemap. Do not remove sitemap: http://cdn.attracta.com/sitemap/4035112.xml.gz #End Attracta SEO Tools Sitemap. Do not remove   I also tried adding an X-Robots-Tag header and submited the site to be re-crawled. Did that a few days ago and I still see no changes. Here are the HTTP headers according to "Fetch as Google":  HTTP/1.1 200 OK X-Robots-Tag: noindex,nofollow Vary: Accept-Encoding Transfer-Encoding: chunked Date: Sun, 24 May 2015 16:26:49 GMT Server: LiteSpeed Connection: close X-Pingback: http://trailheadpaddleshack.ca/staging1/xmlrpc.php Content-Type: text/html; charset=UTF-8 Link: &lt;http://trailheadpaddleshack.ca/staging1/?p=170&gt;; rel=shortlink   I am now faced with a bunch of results I need to remove from Google asap as they contain out of date information and are affecting our search results. Webmaster Tools has something for removal of a single URL but I am looking to remove the entire /staging1/ subfolder. Any tips?