I heard a person on television say the following...     "Pittsburgh need to win this game."   To me, this sounds incorrect.  I think it should be "needs to."  If however, the announcer had said...     "The Steelers need to win this game."   it sounds right to me.  So, I have always gone by the following  plural = need; singular = needs  However, this again does not make sense for some words (e.g. I = need).  Is there some hard and fast rule on when need should be used vs. needs?
I'd say Microsoft have a way of bending the rules and I know that McLaren have won the championship. While this sounds strange, I believe it is correct English (sorry, I'm not native).   But when it's a small company, would you still use it this way? Is a company always plural, or are small companies singular? I.e., would you say Bakery Johnson makes fine bread or Bakery Johnson make fine bread? Is it My book seller, Woody's, have moved or is it has moved?
List user's folder access permissions Let $G$ be a finite group and suppose there exists $f\in\text{Aut}(G)$ such that $f^2=\text{id}_G$, i.e., $f$ is its own inverse, and such that $f$ has no fixed points other than the identity $e$ of $G$, i.e., $f(x)=x\Rightarrow x=e$. Show that $G$ is necessarily abelian.   While trying to do this exercise I noticed two facts.  First, $g$ and $f(g)$ have the same order because $o(f(g))|o(g)$ and, applying $f$ again and using $f^2=\text{id}_G$, $o(f(f(g)))=o(g)|o(f(g))$ and, once the order of any element is $\geq 1$, it follows that $o(g)=o(f(g))$. Also, it's easy to see that $g$ and $f(g)$ commute.  Second, there cannot exist such an automorphism if the order of $G$ is even, because $f(e)=e$ and we can form pairs like $\{g,f(g)\}$ with $f(g)\neq g$ that are invariant under $f$, i.e., $f(\{g,f(g)\})=\{g,f(g)\}$. But once the order of $G$ is even, proceeding with the construction of the pairs, we'll end up with just one element $\neq e$, so we must have some $\gamma\in G\setminus{e}$ with $f(\gamma)=\gamma$, contradicting the hypothesis.  Let $n$ be the order of $G$ and let's fix an enumeration of the elements of $G$, say $G=\{g_1,\ldots,g_n\}$. My approach was the following. For each $\sigma\in S_n$, let $x_{\sigma}=\prod_{i=1}^ng_{\sigma(i)}$. Then we have that $f(x_{\sigma})=x_{\sigma^{\prime}}$. If it's shown that $f(x_{\sigma})=e\;\forall\;\sigma\in S_n$, then we'd have $x_{\sigma}=e\;\forall\;\sigma\in S_n$, which implies that $G$ is abelian. The problem is that $S_n$ has $n!$ elements while $G$ has $n$ elements, so there repetitions among the $x_{\sigma}$, and we cannot apply directly the reasoning of the last paragraph.  Is this the right way, or is there an easier manner to solve this? Libraries module. Where to extract? What are the left and right ideals of matrix ring? How about the two sided ideals? What are the left and right ideals of matrix ring? How about the two sided ideals? Why is carbon dating limit only 40,000 years? How can I stop mobs from despawning? I am using Spigot, some plugins, and I want to make like a zoo on a map, with mobs behind glass cubes. The problem is that when I load my map, every time some of the mobs just vanish, empty cages. I put eggs again, they spawn, I go to my other map in the multiverse (/mv tp my_other_map) and when I come back, empty cages again, always the same ones...   I've  read it's a question of distance of me (distance in cubes unit), if I'm too far away, the mobs will despawn... Is there a way for them to stay? Are +1/+1 counters different from +1/+1 effects? Merge two data frames while keeping the original row order How can I prevent false score reports to global highscore tables? Browser and mobile games commonly have global highscore tables. It's also common for those tables to contain scores of 2,147,483,647 &mdash; where people have figured out the webservice call that reports scores and used it to record a fictitious score.  For simple puzzle games, we can defend against this by including a record of every move the player made (and any random seeds used to generate the level) with the score-reporting call. The entire game can then be reproduced and verified on the server.  However, this quickly becomes infeasible for anything larger than Pac-man.  How else can cheating of this kind be prevented? Why do wheels appear to revolve opposite to the direction they are rotating? When viewing cars that are driving along side of us, sometimes their wheels appear to be turning backwards even though they are traveling in the same direction as our car. Why do they look that way? Prove upper bound for recurrence I am working on  problem 3 from MIT's Fall 2010 OCW class 6.042J. This is covered in  which is about recurrences.  Here is the problem:  $$A_0 = 2$$ $$A_{n+1} = A_n/2 + 1/A_n, \forall n \ge 1$$  Prove   $$A_n \le \sqrt2 + 1/2^n,  \forall n \ge 0$$  I have graphed the recurrence and the upper bound and they seem to both converge on $\sqrt2$.    Also, if you ignore the boundary condition $A_0 = 2$ then you find that $\sqrt2$ is a solution to the main part of the recurrence. i.e. $\sqrt2 = \sqrt2/2 + 1/\sqrt2$.  The chapter and videos on recurrences have a lot to say about a kind of cookbook solution to divide and conquer recurrences which they call the Akra-Bazzi Theorem. But this recurrence does not seem to be in the right form for that theorem. If it were in the form $A_{n+1} = A_n/2 + g(n)$ then the theorem would give you an asymptotic bound. But $1/A_n$ is not a simple function of $n$ like a polynomial. Instead it is part of the recurrence.  Also, the chapter has a variety of things to say about how to guess the right solution and plug it into an inductive proof, but I haven't had much success. I have tried possible solutions of various forms like $a_n = \sqrt2+a/b^n$ and tried solving for the constants $a$ and $b$ but to no avail.  So, if someone can point me in the right direction that would be great. I always assume that the problem sets are based on something taught in the videos and in the text of the book but I am having trouble tracking this one down.  Bobby Origin/reason for the expression "on the bus" instead of "in the bus" What does the size of the font translate to exactly? I have a 40 pixel font that I'm trying to translate into custom images. I'm trying to match the size of the image exactly, but I'm having some problems getting exactly the right size. Basically, what it comes down to is this. When a font says it is X pixels, what does this physically mean? TOC entries and labels for included PDF pages I am embedding full pages from another PDF document (using the pdfpages package), but I would like to be able to declare my own section headings before each embedded document (for e.g. hyperlinking / TOC purposes). pdfpages has an experimental feature which lets me do this, but the problem is that I want to put all of these embedded documents inside a section, so I need to insert a section heading and a subsection heading on the first document I embed. Unfortunately pdfpages doesn't seem to be able to do this. Is there a way for me to do this?  Edit: Thank you for the suggestions. I am also in need of something similar for general labels — at the moment I am abusing the addtolist option of \includePDF but the objects I'm labelling aren't really figures or tables. GUI/Unity crashing in 16.04 LTS after updates 2018-01-04, compiz segfaults I created one event receiver and deployed it, but only users who are site collection admins are able to view this web part. I have included the SPSecurity.RunWithElevatedPrivileges but the same thing. What am I missing?  Below is my code   public override void ItemAdding(SPItemEventProperties properties)     base.ItemAdding(properties);     SPSecurity.RunWithElevatedPrivileges(delegate()        {SPWeb web = properties.OpenWeb();         base.ItemAdding(properties);         string name= web.Lists[properties.ListId].Fields["first name"].InternalName;          var list = web.Lists["location names"];    });} Did Daenerys mess up regarding caravans? Ubuntu 19.10 change desktop environment I've been trying to install 12.04 for a while now, I'm running on Windows 7 and I already made the partition (it's dynamic). I've tried booting from a disc and also the Windows installer. I haven't tried used any repair-boot programs or anything. Every time I tried to load from either, I get an error:  command=C:\windows\sysnative\bcdedit.exe /set {03f9077f-661f-11e1-845a-d08b36406c3d} device partition=U;  retval=1  stderr=An error has occured setting the element data. The request isn't supported  stdout=   Then after that, it makes a log file with information that I don't understand.  If it is simply due to the fact of it being a dynamic partition, someone please help. If it is more extensive, anyone with details on possible solutions or definite ones please express them. How are Python's Built In Dictionaries Implemented? Does anyone know how the built in dictionary type for python is implemented?  My understanding is that it is some sort of hash table, but I haven't been able to find any sort of definitive answer.