Vector bundle on a quadric
Vector bundle on a quadric $Q$
How can I repair grub? (How to get Ubuntu back after installing Windows?) Unparsed HTML shows on search result page When you search on any of the SE network sites, with a tag (so that a description is shown); it shows the tag info content box, with the highlight-search-term span elements.  For eg. I searched ; ; and the results are as shown:                  Notice the &lt;span class='search-highlight'&gt; boxes around searched tag terms. Suppose we have a natural number $n \ge 0$.  Given natural numbers $\alpha_1,\ldots,\alpha_k$ such that   $k\le n$ $\sum_i \alpha_i = n$   what is the maximum value that $\Pi_i \alpha_i$ can take?  I'm quite sure that there is a theorem telling me the result, but I cannot find it. For sure an upper bound is $n^k$ but I'm searching for a real upper bound. I'm pretty sure that upper the bound should be $n^2$, but I don't know I could prove it. How to reverse a user input without using array or any library function(any function for reversing)? How to speed up 4e combat I DM a group of four PC's and even "easy" battles take at least an hour to resolve. Normal encounters take even longer! I want to speed up combat so my players can do more in a session.  I've been trying a lot to speed up combat, we are already using:   Initiative cards that the players keep up 30 second bonus rule: if someone take his turn within 30 seconds, he receives a token. This token can be used to give a +1 on attack or damage, after rolling. Completely removed healing surges Checked all character sheets, they are roughly equal Roll damage and attack at the same time   I even introduced an experienced player, which helped my party to play more strategic, but speed hasn't improved.  Also the party is Lv 8, nearing Lv 9.  This is not an issue of the DM's turn taking too long. My turns take about 60 seconds tops. The 30 second rule helps a lot, most of the the players turns are within or near 30 seconds.   The major issue appears to be the sheer number of rounds, averaging on 25-30 rounds needed to resolve even a "normal" encounter.  What can I change to resolve combat faster?  I'm looking for both 4e solutions, system agnostic and home brew solutions. Redirect to SSL only if browser supports SNI How to model effectively using exact measurements? I always struggle when trying to create a floor plan or furniture. How do I model using exact measurements. Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this? Run a batch file in a completely hidden way I'm looking for some way to run a batch file (.bat) without anything visible to the user (no window, no taskbar name, .etc..).  I don't want to use some program to do that, I'm looking for something cleaner. I've found  that uses VBScript, but I don't really like using VBS, either. What do you call a group of people that move a lot? I can't think of the word to describe it. Something similar to "wanderer" or "roamer".   It's often used to describe people that don't stay in one place... not "migratory"... Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$ Which is the preferred way to convert an Enum to a String in .NET 3.5?   Enum.GetName Enum.Format ToString   Why should I prefer one of these over the others? Does one perform better? As a(n) noun vs as only Why does $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ instead of $1$? On Wikipedia, it says that $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ :     It [e] is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, ... ()   When I evaluate $(1+\frac{1}{n})^n$ for $n = 10^8$, I get approximately $2.718281798347$ which indeed is pretty close to $e$.  But when I try to "solve" the limit using the laws of limits, I get  $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = \left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^n$$  because of the power law  $$ = \left(\lim_{n\to\infty}(1)+\lim_{n\to\infty}\left(\frac{1}{n}\right)\right)^n$$  because of the addition law  $$=\left(1+0\right)^n = 1$$  but that would mean that $e=1$, which is obviously not true.  What am I missing / doing wrong?  Thanks in advance. When to spell out numbers Evaluating $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using polar coordinates. I want to evaluate the integral $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using the substitution $x=r\cos\theta$ and $y=r\sin\theta$ where the Jacobian is given as $|J(r, \theta)|=r$ and thus $dxdy=rdrd\theta$.  I need to set the range for $r$ and $\theta$. For $\theta$, it seems obvious that it ranges from $0$ to $\frac{\pi}{2}$ to cover all parts of the rectangle. But I am confused with the range for $r$. Would it be OK to set minimum $r$ as $0$ and maximum $r$ as $\sqrt{2}$, which is the diagonal of the square domain? I think I am confused with the fundamentals of integration. How do I install and manage RPMs? I have a burning desire to install RPMs onto my Ubuntu installation - is this possible? Can I use Yum - or another RPM Package Management tool? How to take a screenshot with an Android device? Is there a way to take a screenshot of an Android device and save it as an image file? From 2012 onward the SQL Server docs show they support  which I'm trying to use instead of a LIMIT.  The following works fine in PostgreSQL to sample a result set,  SELECT * FROM ( VALUES (1),(2),(3) ) AS t(x) OFFSET 0 ROWS FETCH NEXT 1 ROWS ONLY;   However, with SQL Server, I get  Msg 153, Level 15, State 2, Line 4 Invalid usage of the option FIRST in the FETCH statement.   What's going on here? Does SQL Server support the standardized OFFSET .. FETCH? When you watch a video in YouTube, you can seek it with the thumbnail preview. Many Windows' and MacOS' video players have this feature, but what about Linux's? I've already read , but the information in it seems out-of-date.  is still alive but the project is neglected (see ), and actually the player doesn't work on the latest Ubuntu or Linux Mint.  Is there any video player which has the feature and works on the latest linux distributions? Or must I use Windows' one with VirtualBox? If only VLC would implement this feature, but it seems that won't happen as far as I read .  YouTube's seek preview: