"Prove that there exist 4 people in the waiting room who all know each other..."  This is a part of a math problem. Should the "exist" be "exists" or is it correct the way it currently is?
I know the usage of     This is a new car.   This is singular.     These are some books for you.   These is plural.  Shall we use   There exists some people who agree with me. There exist some people who agree with me.   Which one is the correct usage and why?
Why girth of Petersen Graph is five? (Proof) I want to know, what is the mathematical proof that shows the shortest cycle in Petersen Graph has length five? Fields - Proof that every multiple of zero equals zero This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details).  Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies that any parentheses are meaningless. Therefore, we will not use parentheses. Therefore, we will not use $\textit{associativity}$ explicitly.  By $\textit{identity element}$, $F \ne \emptyset$. Now, let $a \in F$. It remains to prove that $0a = 0$. \begin{equation*} \begin{split} 0a &amp;=  0a +  0      &amp;&amp; \quad \text{by }\textit{identity element }(+    ) \\    &amp;=  0a +  a + -a &amp;&amp; \quad \text{by }\textit{inverse element  }(+    ) \\    &amp;=  0a + 1a + -a &amp;&amp; \quad \text{by }\textit{identity element }(\cdot) \\    &amp;= (0 + 1)a + -a &amp;&amp; \quad \text{by }\textit{distributivity   }        \\    &amp;= (1 + 0)a + -a &amp;&amp; \quad \text{by }\textit{commutativity    }(+    ) \\    &amp;=      1 a + -a &amp;&amp; \quad \text{by }\textit{identity element }(+    ) \\    &amp;=        a + -a &amp;&amp; \quad \text{by }\textit{identity element }(\cdot) \\    &amp;=  0            &amp;&amp; \quad \text{by }\textit{inverse element  }(+    ) \end{split} \end{equation*} QED  PS: Is "Let $(F, +, \cdot)$ be a field." ok? Besides, I would not want to call $F$ a field, because $F$ is just a set. Also, what do you think about using adverbs like "Now"? How would you have said the associativity-thing? Safari Extension to Manage Tabs? Induction and electromagnetic fields I've got a few questions on induction and electromagnetic fields. My current understanding of induction and electromagnetic fields is that, when electricity/current flows through a wire, it creates an electromagnetic field. If you twist that wire into a coil then you can concentrate that field. You can then induce current into a different coil by placing it inside of the electromagnetic field created by the first coil.  So here are my questions:  Q1. Does the electromagnetic field contain any energy?  EDITED QUESTION  Q2. Electromagnetic fields are the same as waves, so does that mean that when current is induced, they get the energy from electromagnetic waves being exchanged, or do they induce current directly from the electromagnetic field?   Q3. If electromagnetic fields and waves are the same, then doesn't that mean that you can induce current from waves? Therefore transferring electricity over long distances?  Q4. What spectrum of light are the electromagnetic waves that are in a field created by electricity?  Q5. Finally, why is it not possible to induce current from the Earth's magnetic field? I thought Maxwell said that electric fields and magnetic fields were the same thing.   Thank you to anyone who answers.  Edited Q2: So my original question (although asked poorly) still stands. Where does the current that gets induced in the secondary coil come from? How does the electricity transfer from the primary coil to the secondary? For example, is it taken from the stored energy in the e.m field? Does the field just act as a bridge between the 2 coils? Or do the 2 coils exchange e.m waves which carry the electricity that is being transferred from coil 1 to coil 2? Or is it none of these things and something completely different? Last time I asked a similar question but that was about svn related versioning info. Now I am wondering how to query windows "File version" attribute about eg. a dll. I payed attention to wmi and win32file modules as well without success. How do I install drivers for an Intel HD Graphics? In System Info in 11.10, the graphics driver wasn't detected. I use ACER Aspire 4750 with Intel Core i3-2340M and Intel HD Graphics 3000 graphic card  How to install the driver for my Ubuntu? unable to reset root password of mysql Is it possible to level grind in Normal Mode? So in Awakening, we can use Reeking Boxes... but what of Three Houses? I'm not far off from the beginning of the game- I have just recently finished the mock battle, actually, but I noticed this calendar mechanic in the game (and in extension, the activity points stuffs). Apparently, the days just... jump.  With this info, I assume there's only a limited number of days before reaching the so-called War Phase... so this brings me back to the question: are there maps or battles that I can replay again and again so that I can level grind the characters, or are there only a limited amount of maps/battles that I can participate in? Prove the inequality and limitation Suppose $\{a_k\}$ ,$\{b_k\}$ and $\{\xi_k\}$ are non negative, and for all $k\ge 0$ , we have $$a_{k+1}^2\le(a_k+b_k)^2-\xi_k^2$$   Prove   1.$$\sum_{i=1}^{k}\xi_i^2\le(a_1+\sum_{i=0}^kb_i)^2$$   2. If $\{b_k\}$ also satisfy $\sum_{k=0}^{\infty}b_k^2\lt+\infty $, then $$\lim_{k\to\infty}\frac{1}{k}\sum_{i=1}^k\xi_i^2=0$$   After doing this transformation:$$\xi_k^2\le(a_k+b_k)^2-a_{k+1}^2$$ $\Rightarrow$ $$\sum_{i=1}^{k}\xi_k^2\le(a_1+b_1-a_2)(a_1+b_1+a_2)+(a_2+b_2-a_3)(a_2+b_2+a_3)+\cdots(a_k+b_k-a_{k+1})(a_k+b_k+a_{k+1})$$ then I can't move forward any more. What are the uses of "using" in C#? User  answered the wonderful  question by mentioning the using keyword. Can you elaborate on that? What are the uses of using? Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?  There are a number of equivalent definitions for the measurability of a function and the most obvious one would be to show that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{R}$.  Thus my strategy has been to consider an arbitrary irrational $y$ and use the density of the rationals in the reals to show that there exists some open set $O$ such that $m^*(O-\{x\in E : f(x)&gt;y\})&lt; \varepsilon$.  I would do this by choosing some rational $q&lt;y$, close enough to $y$, such that the set of values in $E$ which get sent under $f$ to values strictly greater than $q$ and less than or equal to $y$ is small enough.  But that doesn't in general seem possible since I could keep sending intervals of some fixed positive length to smaller and smaller regions just beneath $y$.  However conversely, such a function would not be enough for a counter example, since finding such a function only rules out this strategy, but doesn't necessarily entail non-measurability.  Is there a better definition of the measurability of a set I should use? or does the statement actually not imply measurability?  Thanks. I have a long sequence of hex digits in a string, such as      000000000000484240FA063DE5D0B744ADBED63A81FAEA390000C8428640A43D5005BD44   only much longer, several kilobytes.  Is there a builtin way to convert this to a bytes object in python 2.6/3? I am considering buying a Dell XPS 15 laptop. The laptop has a Nvidia 420m card which should work with Linux but I keep reading that the Nvidia Optimus technology isn't supported on Linux. I am not really interested in switching from Nvidia to Intel to save power but need to know that the Nvidia card will in fact work if I installed Ubuntu.  If anyone has experience using a Nvidia card with Optimus technology or even better the exact laptop in question (Dell XPS 15 with 1GB NVIDIA® GeForce®  GT 420M) it would be great.  A major problem holding people back from adopting Linux is this sort of hardware issue. I am a long term Ubuntu user and supporter who can't afford to make a mistake with a purchase like this. I don't want to spend £500+ to find I have no graphics acceleration because Windows 7 is not an option. Is it possible to edit a layer in QGIS while there is an active query? I am new to Quantum and I am digitizing polygon layers using the editing features. I would like to apply simple queries ('date > 2012/08/01'), however while the query is active I cannot edit the layer. Is this possible? How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$? How to prove $\gcd(a^2, b^2) = (\gcd(a, b))^2$?  My attempt:  Let $\gcd(a, b) = d$.  Then $d|a$ and $d|b$ then $d^2|a^2$ and $d^2|b^2$. i.e $d^2$ divides $a^2 ~~\&amp;~~ b^2$. Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer. So I'm making this summary of a course where I've only been getting information from one book. Sometimes I want to show what page from the book a statement is taken from but I can't figure out a clean way of doing it.  Basically I would like to have something (in function) similar to:  "Statement[Book p123]"  Do you have any suggestions? Why is the main page showing only the user login form? Allow deleting a tag wiki or excerpt Sometimes a tag wiki excerpt or body ends up with unacceptable or useless content: plagiarized text that must be removed, or a totally unhelpful dictionary definition. (For example:  — and I've seen this situation many times before.)  The only way to get rid of that content is to submit replacement text that meets the minimum length requirement. This is not always easy as writing a decent wiki, and especially a decent excerpt, requires subject knowledge.  Currently the users or even moderators who are alerted to unacceptable content in a tag wiki cannot remove that content without providing their own, which they are not always in a position to do. Yet these wikis would be better off with no content than with either unacceptable or unhelpful content, or some content written by a non-expert in a hurry.  Please provide a way for users to blank a tag wiki excerpt or body. I'm not sure what the reputation requirement should be: same mechanism as for tag wiki edits (suggested edits with approval), or only done directly by a user with the trusted user (20k) privilege, or moderator-only. Contraction of quotient ideal is quotient of contractions?