"We propose the session focuses on ..." or "We propose the session focus on ..." Which is correct? I understand a verb in English needs to match the sentence subject. Is the subject here "we" or "session", or am I seeing this incorrectly?
When should I use the subjunctive mood? In which cases should I use the subjunctive mood?     I suggest that every applicant fill out the form carefully.        If she were rich, she would live on Long Island.
Hi I make this class A amplifier but how to replace R5 (10 ohm/10 W) with transistor because its getting really hot and how to achieve up to 10 Watt with +/- supply. I have +/-25v with 4A please help How to center an element horizontally and vertically Euclid Lemma proof reasoning This question is more concerned with understanding the reasoning behind mathematical proving rather than explaining this specific proof. I understand why this proof works.   This is the proof of Euclid Lemma from Wikipedia:      This   states that if $x$ and $y$ are relatively prime integers (i.e. they   share no common divisors other than 1) there exist integers $r$ and   $s$ such that      $rx+sy=1$. Let $a$ and $n$ be relatively prime, and assume that   $n|ab$. By Bézout's identity, there are $r$ and $s$ making      $rn+sa=1$. Multiply both sides by $b$:      $rnb+sab=b$. The first term on the left is divisible by $n$, and the   second term is divisible by $ab$, which by hypothesis is divisible by   $n$. Therefore their sum, $b$, is also divisible by $n$.   I am mostly concerned with this statement:      Multiply both sides by $b$:   My question is why this multiplication by $b$ happens. Is it because the person proving this observed that multiplying the equation by $b$ he would get a sum of two multiples of $n$? Or is there a mathematical rule by which we know that the next step is multiplying by $b$?  I am aware that this question might seem trivial to people here, but I just want to make sure. I am trying the fallowing exercise : Solve $P(X^2 -2)=P(X)^2 -2$ with P a monic polynomial (non-constant)  My attempt :  Let P satisfying $P(X^2-2) = (P(X))^2-2$  Then $Q(X)=P(X^2-2) = (P(X))^2-2$   Therefore,  $$Q(X^2-2) = (P(X^2-2))^2-2 = (P(X)^2-2)^2-2 = Q^2-2$$  As X is a solution, by defining the sequence:  $(P_n)_{n \geq 1}$ with $P_1 = X$ and for all $n \geq 1, P_{n+1} = P_n^2-2$  We obtain a sequence of polynomials which are solutions.   But I don't know how to prove it's the only one. If someone have an idea to prove it or an another method to solve the problem ?   Thank you in advance for your time. Finite abelian group contains an element with order equal to the lcm of the orders of its elements I will quote a question from my textbook, to prevent misinterpretation:     Let $G$ be a finite abelian group and let $m$ be the least common multiple of the orders of its elements. Prove that $G$ contains an element of order $m$.   I figured that, if $|G|=n$, then I should interpret the part with the least common multiple as $lcm(|x_1|,\dots,|x_n|)=m$, where $x_i\in G$ for $0\leq i\leq n$, thus, for all such $x_i$, $\exists a_i\in\mathbb{N}$ such that $m=|x_i|a_i$. I guess I should use the fact that $|x_i|$ divides $|G|$, so $\exists k\in \mathbb{N}$ such that $|G|=k|x_i|$ for all $x_i\in G$. I'm not really sure how to go from here, in particular how I should use the fact that $G$ is abelian. Determining confidence interval with one observation (for Poisson distribution) Rejecting a PhD offer after having accepted it. Advice? I have accepted a PhD offer from a very good university to do a project very close to my interests. I have told the project supervisor I accept it. In the meanwhile I have managed to secure a PhD offer from another 2 universities much better in world rankings and with better research groups and better faculty. Additionally these 2 universities do research exactly on what I want to devote my PhD research.  My only commitment to the first university is a mail confirming I accept. Nothing else. I understand declining the offer after already having accepted it is quite dishonest but it is a huge decision that will affect my future in a very significant level.   I would like your opinion and I would also like to ask for possible consequences of such an action. As far a postdocs concerned, in the first university there is no chance I ever get a postdoc since they do not actually do research on my area (except for the prospective supervisor and even he not as much).  UPDATE I have decided to do as I was afraid to. I have decided to choose a different university than the first one that I have already accepted the offer. My offer acceptance included:    An online "I accept" statement. An e-mail I sent stating "I accept the scholarship and this University is my first option"   I do not have any legal commitments since I have not signed anything. The supervisor of the first university though is really nice to me and continuously was sending me mails. How do I very nicely tell her I will not be joining in the end? Additionally the place I will go in the end will be for research in Mathematics and not Physics. Can I use this as a part of a sensical excuse?  WHAT I HAVE CHOSEN AND WHAT THE CONSEQUENCES WERE  I think that I should let people, especially other beginning grad students know what happened with my case. I have chosen to nicely tell the first university I got accepted that I will not be attending and will not be pursuing a PhD there since I decided I have different research interests (which is true). I had applied there in the first place since I was not sure what I wanted to do and because it was a nice program. Despite that, I got an offer from a World top university to work on 100% my research interests. I explained this to them, the first uni, and they happily let me go and wished me good luck, especially the supervisor who faught a lot secure me funding. I was lucky. Why does C++ have header files and .cpp files? Why is the installer not showing any partitions? Why $f(x) = \sqrt{x}$ is a function? Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a rule of corre- spondence that assigns to each element in set A exactly one element in B.", so $f(x) = \sqrt{x}$ is not a function? What should be in the sources.list for Kali Rolling Linux? This is exact content from my sources.list:  #  # deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free #deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free   I couldn't install the Synaptic package manager and was told to modify sources.list.  I googled a lot and added some repos into it and ran update and upgrade command only to see Kali crash... I had to recover it.  Please mention the exact content that should be there to ensure that Kali Linux works. I am a Pakistani national and need to transit through Dubai International Airport (DXB).  I am not planning to leave the airport and both of my flights (arrival and departure) are from Terminal 3.  I am pretty sure that I do not need transit visa at the DXB, however it is no where mentioned on any of their official website.   Can anyone please provide me such a link stating that transiting without leaving the airport does not require visa at DXB, as I need to use this link as a proof for a visa application?  Thanks If dark matter only interacts with gravity, why doesn't it all clump together in a single point? I'm a complete layperson. As I understand, dark matter theoretically only interacts with the gravitational force, and doesn't interact with the other three fundamental forces: weak nuclear force, strong nuclear force, and electromagnetism.  Those are my understandings going in. If I'm wrong, please correct me. I've done some googling, and I haven't found anything confirming or denying that dark matter is affected by either of the fundamental nuclear forces.  So since dark matter only interacts with gravity, what causes any dark matter particle to be repelled from another? If they can pass freely through each other, and they are gravitationaly attracted to each other, why don't such particles clump together in a single 'point' in space?   It seems to me that particles occupying a single 'space' are philosophically not distinct particles, but I don't know how actual physics would play into this.  Edit , author's credentials unknown, but implicitly claims to be a physicist or astronomer, says "...[P]hysicists generally take all dark matter to be composed of a single type of particle that essentially interacts only through gravity."  Edit 2 The author is this , "Professor of Science on the physics faculty of Harvard University." command not running in cron (systemctl suspend) <lightning:recordViewForm will not display Canonical has recently announced , which is Ubuntu with a specific interface for mobile devices.  Where can I get phones with this version of Ubuntu pre-installed? If no devices are available yet, where can I pre-order one? When punctuating the dimensions of something, if both dimensions share the same units, does the punctuation occur after each measurement or just after the last one. For example, if I'm looking at a sheet of paper, which is correct: {8 1/2 x 11"} or is {8 1/2" x 11"}? When I play Black Ops 2, it seems that my xbox controller is inverted. Example: if I press up, my character looks down and vice versa. It was not like this before. Does anyone know how to reset the controller back to its original settings? How to create an Atlas based on point attributes? Essentially I am wanting to produce an atlas based on a categorical field in a point layer.   i.e. I have a point layer of childcare providers with the categorical field "Provision". I've categorized each feature in this field with "After School Club" , "Breakfast Club" etc, and I now want to produce a set of maps that iterate through each category and show only the points for each. One map of after school clubs, one map of breakfast clubs etc. The extents might be subtly different.  I could do it one by one but it seems like there should be a way to produce an atlas based on the extents of each category? (I do feel i'm missing something obvious :) )   Or alternatively is there a way of automating the creation of a polygon layer and using that as a hidden coverage for the atlas?  EDIT: I've made a little progress with this  - you can use rule based styling to switch on and off features relevant to the current atlas coverage feature.  it actually works fine if all you want to do is show a different set of points. I'm now looking at tying that back to a color scheme and reactive legend. Stopping xcode from indexing