What do we mean by $CP^2$ with reverse orientation?
Orientation on $\mathbb{CP}^2$
"To delete" is to "deletion" as "to discard" is to what? "To delete" is to "deletion" as "to discard" is to what? A surjective endomorphism (of a Noetherian ring) is injective. The problem is stated as follows:  "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective."  Regardless of the right solution, I don't understand why is the following wrong:     We have $\theta: R\to R$. By the isomorphism theorem $R/\ker\theta\cong\operatorname{Im}\theta$. Since $\operatorname{Im}\theta = R$, it follows $\ker\theta =0$, so it's injective.   Harsh criticism will be appreciated. Thanks. Is there a CSS parent selector? Display number with leading zeros Given: a = 1 b = 10 c = 100  How do I display a leading zero for all numbers with less than two digits? This is the output I'm expecting: 01 10 100 Legend inconsistent in QGIS map print layout After I did styling my vector data set to be classified by graduated circle and categorized by type, it appears beautifully consistent in the map canvas as shown below:   But when I create a print map layout and after adding the map, I add the legend and it come out ugly inconsistent like this:    So, I would like to ask for help to make the legend in the map print layout consistent as the legend shown in layer panel. I use the latest version QGIS 3.10.2 64bit. What is the relationship between MIME types and File .extensions? I may have the wrong idea here, but I thought that:    MIME types are identifying codes, embedded inside a file. File .extensions are idenifying codes, suffixed to the file's name.   I thought, from something I heard in the dim dark ages, that Linux was .extension agnostic... ( maybe it was in the early days, and things have changed since then...? )  I've recently come from the Windows world, where, at the Operating-System level, a file .extension is the only way (as far as I know) to associate it with its relevant Application program.    Because I don't know why, I find it a bit disconcerting that a file named  "fred" and a file named "fred.txt" both open up in a Text Editor.    Is there a clear-cut hierarchy at work here? How to make beveled 90 degree corners on a curve better? I know the title is a mouthful, but this is the problem: I made a curve with 90 degree corners, like so:  I then beveled it, and this was the result:  As you can imagine, this isn't very pretty. I'm looking for something like this:  How can I achieve this look? Change the selected value of a drop-down list with jQuery I have a drop-down list with known values. What I'm trying to do is set the drop down list to a particular value that I know exists using jQuery.  Using regular JavaScript, I would do something like:  ddl = document.getElementById("ID of element goes here"); ddl.value = 2; // 2 being the value I want to set it too.   However, I need to do this with jQuery, because I'm using a CSS class for my selector (stupid  client ids...).  Here are a few things I've tried:  $("._statusDDL").val(2); // Doesn't find 2 as a value. $("._statusDDL").children("option").val(2) // Also failed.      How can I do it with jQuery?     Update  So as it turns out, I had it right the first time with:  $("._statusDDL").val(2);   When I put an alert just above it works fine, but when I remove the alert and let it run at full speed, I get the error     Could not set the selected property. Invalid Index   I'm not sure if it's a bug with jQuery or Internet Explorer 6 (I'm guessing Internet Explorer 6), but it's terribly annoying. Is there a proof from the first principle that the Lagrangian L = T - V? ASP.NET MVC Relative Paths In my applications, I often have to use relative paths. For example, when I reference JQuery, I usually do so like this:  &lt;script type="text/javascript" src="../Scripts/jquery-1.2.6.js"&gt;&lt;/script&gt;   Now that I'm making the transition to MVC, I need to account for the different paths a page might have, relative to the root. This was of course an issue with URL rewriting in the past, but I managed to work around it by using consistent paths.  I'm aware that the standard solution is to use absolute paths such as:  &lt;script type="text/javascript" src="/Scripts/jquery-1.2.6.js"&gt;&lt;/script&gt;   but this will not work for me as during the development cycle, I have to deploy to a test machine on which the app will run in a virtual directory. Root relative paths don't work when the root changes. Also, for maintenance reasons, I cannot simply change out all the paths for the duration of deploying the test - that would be a nightmare in itself.  So what's the best solution?  Edit:  Since this question is still receiving views and answers, I thought it might be prudent to update it to note that as of Razor V2, support for root-relative urls is baked in, so you can use   &lt;img src="~/Content/MyImage.jpg"&gt;   without any server-side syntax, and the view engine automatically replaces ~/ with whatever the current site root is. How to assign a SSL Certificate to IIS7 Site from Command Prompt Can you advise me whether it is possible or not to assign a SSL Certificate to a website in IIS7 using the APPCMD application?  I am familiar with the command to set the HTTPS Binding  appcmd set site /site.name:"A Site" /+bindings.[protocol='https',bindingInformation='*:443:www.mysite.com']   and how to obtain current mappings   %windir%\system32\inetsrv\Appcmd   but can not seem to find any way to map a site to a certificate (say the certificates hash for example) How do I remove an incomplete or broken snap installation of nextcloud on a Raspberry Pi? Show that a holomorphic function satisfying $|f(z)|\leq c|z|^n$ must be $f(z)=a\cdot z^n$ for some a. Let $f:\mathbb{C}\to\mathbb{C}$ be a holomorphic function such there exists a $c\in\mathbb C$ with $|f(z)|\leq c\cdot |z|^n$ for all $z\in\mathbb C$. Show that there exists an $a\in\mathbb C$ with $f(z)=a\cdot z^n$.   Proof.  My general idea is to show that the taylor series only has one non-zero coefficient:  $\displaystyle |a_k|\leq\frac{k!}{2\pi}\oint_{|w|=r}\frac{|f(w)|}{|w|^{k+1}}dw\leq \frac{k!}{2\pi}\oint_{|w|=r}\frac{c\cdot|w|^n}{|w|^{k+1}}dw\leq \\\displaystyle\leq \frac{c\cdot k!}{2\pi}\oint_{|w|=r}|w|^{n-k-1}dw$  Now using the standard estimate since $\oint=2\pi r$:  $\displaystyle \leq c\cdot k!\cdot r^{n-k}\longrightarrow\begin{cases}0,\ k&gt;n\\c\cdot n!,\ k=n\\ \infty,\ k&lt;n\end{cases}$ as $r\to\infty$  In conlusion the taylor series for now looks like $f(z)=\sum_{k=0}^{n-1}a_k z^k+ a_n z^n$ with $|a_n|\leq n!\cdot c$. What do I do with the remaining summands?  Edit: I would say that for $k&lt;n$ we have $|z|^k&gt;|z|^n$ for $z\in D_1(0)$. So since the estimate has to hold for all $z\in \mathbb C$ we can't have that either. Stacks for Unity Launcher I'm looking forward to emulate the behavior of OS X Dock Stacks so that I can have, for instance, one stack for browsers and another one for IDES. People at ask ubuntu are doing a lot of cleaver stuff with custom Launchers and Quicklists (see ). So has anyone figured out how to have stacks in Unity? Anything fancier than a static Quicklist over a dummy folder icon? How can the pixel values of a raster image be extracted under a polygon fishnet? I have a single band raster image of 30m by 30m resolution. I have already created a polygon fishnet (grid) of 300m by 300m with point labels over that. How can I extract the mean pixel value of the raster image overlaying each feature in the fishnet grid? Is it possible with Arcgis?  Please inform me. Organize desktop icons on Windows into categories or sections Reactjs convert html string to jsx Divisibility Proof with Induction - Stuck on Induction Step I'm working on a problem that's given me the run around for about a weekend. The statement:  For all $m$ greater than or equal to $2$ and for all $n$ greater than or equal to $0$, $m - 1$ divides $m^n - 1$.  Using induction over $n$, the base step comes easy since $m^n - 1$ is $0$ when $n = 0$.  My induction hypothesis is letting $k \geq 0$ and assuming that $m - 1$ divides $m^k - 1$. In order to show that $m - 1$ divides $m^{k+1} - 1$, I obviously need to use the induction hypothesis. However, no matter where I try to use the fact that $m^k - 1 = (m-1)a$ for some $a$ in the integers, the expression $m^{k+1} - 1$ always becomes more difficult to get to $m^{k+1} - 1$ being equal to $(m-1)b$ for some $b$ in the integers.   In other words, I can't figure out any actually helpful way to apply the induction hypothesis with the goal of proving the next step.  Any help would be appreciated! How to create a SQL Server function to "join" multiple rows from a subquery into a single delimited field? Occasionally (very occasionally) when returning to the main page of a site by clicking on the logo I get the following message:     It was not possible to perform this tag search at this time due to an unexpected error.   This is odd because I'm not performing a tag search.  So far this has happened on both Movies &amp; TV and Cooking.  I have heard that others are getting this same message.