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Dimension of cone of projective variety Should initial e's after h-dropping be capitalized? A new Victorian-set British novel begins sentences containing h-dropping with lower-case e's, as in " 'e took my money, 'e did." This seems incorrect but I'm at a loss to find the rule. Plain integer partitions of $n$ using $r$ parts Division of number $n$ on parts $a_1,...,a_r$ where $a_1 \le ... \le a_r$ we call a plain if $a_1 = 1$ and $a_i - a_{i-1} \le 1$ for $2 \le i \le r$. Find enumerator (generating function) for plain divisions.  my try  The hint was to use bijection between plain divisions and some commonly know enumerator. I tried to use enumerator of divisions on different parts: $$ (1+x)(1+x^2)...(1+x^r)$$ where number of plain divisions is  $$[x^n](1+x)(1+x^2)...(1+x^r) $$ let function  $$f(n,r) = [x^n](1+x)(1+x^2)...(1+x^r) $$  For some first divisions it works. For example: $$f(4,3) = 1 $$ $$f(6,3) = 1 $$ $$f(11,5) = 2$$ But when I tried to find bijection, I failed. I found that this function isn't correct because $f(15,6) = 4$ but should be equal to $3$ because: $$15 = 1,1, 2, 3, 4, 4 \\ 15 = 1, 2, 2, 3, 3, 4\\ 15 = 1, 2, 3, 3, 3, 3 $$. There I stucked. When an account is deleted, don't auto delete its negatively scored posts on the meta site Meta is different. Negatively scored questions are not always bad questions, and sometimes they have merit and draw good answers, especially discussions.  When an account (in the main site) is deleted, either by self deletion or by a moderator, all questions with score of -1 or less are being deleted regardless of their answers, both in the main and meta site.  Can you please change this behavior so that the questions on the per-site meta will survive and won't be auto-deleted? Could an argument with false Premises and a true Conclusion be logically valid? Creating a large amount of convex polygons using existing coordinates I am new to ubuntu and i would like some help. I have a laptop with windows 7 installed but i would love to have ubuntu too because it's a great OS. So i would like to install ubuntu on an external usb hard drive (320 - 500 GB) and i have some questions about that :   Do i need a usb external hard drive which will be compatible with ubuntu - linux ? or i can do it with any hard drive  If i make a partition on it for ubuntu staff will i be able to use it and save other staff in the usb hard drive (movies,songs,documents etc) ? If i have ubuntu installed on the usb hard drive will i be able to boot ubuntu from any pc ? Are my changes going to be saved normally or my settings will be lost after shut down ?   Waiting for your answers, if something is not explain well tell me to try explain it better . Thank you! I would like to remove selected commit log entries from a linear commit tree, so that the entries do not show in the commit log.  My commit tree looks something like:  R--A--B--C--D--E--HEAD   I would like to remove the B and C entries so that they do not show in the commit log, but changes from A to D should be preserved. Maybe by introducing a single commit, so that B and C become BC and the tree looks like.  R--A--BC--D--E--HEAD   Or, ideally, after A comes D directly. D' representing changes from A to B, B to C and C to D.  R--A--D'--E--HEAD   Is this possible? if yes, how?  This is a fairly new project so has no branches as of now, hence no merges as well. Why and how is the speed of light in vacuum constant, i.e., independent of reference frame? Explain what is a transistor in plain English! That's a lot of ___ vs. those are a lot of ___ That's a lot of cars!   Those are a lot of cars!      The context is when a someone sees many objects (in this example, cars) and makes the exclamation.   According to , lot can handle either singular or plural verbs.  I agree with the author who states that that's sounds better, but the post doesn't seem definitive either way.     Are both grammatically correct and do they have the same meaning? Is LED properly implemented in the constant current circuit? Why is the camera tilting around the z axis when I only specified x and y? My goal is to program a camera to point towards the mouse cursor. I attached the following script to the Main Camera.   using UnityEngine; using System.Collections;  public class CameraController : MonoBehaviour {      public float sensitivity;      // Update is called once per frame     void Update () {         transform.Rotate (Input.GetAxis ("Mouse Y") * -1 * sensitivity, Input.GetAxis ("Mouse X") * sensitivity, 0);     } }   I "told" the camera to rotate around the x and y axes. Whenever it does this, the camera tilts sideways. When I checked the rotation of Main Camera, I saw that the z rotation had changed. Why is it rotating around the z axis when I left it at 0? How does Natural Selection shape Genetic Variation? Background  Importance of the additive genetic variance  As stated , the fundamental theorem of Natural Selection (NS) by Fisher says:     The rate of increase in the mean fitness of any organism at any time ascribable to NS acting through changes in gene frequencies is exactly equal to its genetic variance in fitness at that time.   NS reduces the additive genetic variance  On the other hand, NS reduces additive genetic variance (discussion about the origin of this knowledge can be found ). The genetic variance of a population for multiple traits is best described by the G-matrix ( is a post on the subject).  What is a G-matrix  A G-matrix is a matrix where additive genetic variance of trait $i$ can be found at position $m_{ii}$. In other words, the diagonal contains the additive genetic variance for all traits. The other positions $m_{ij}$, where $i≠j$ contains the additive genetic covariance between the traits $i$ and $j$.  Question  How can one model how the G-matrix changes over time because of selection (assuming no mutation)? I understand the idea of a focal length for a single lens, i e the distance from the lens to the point at which parallel rays of light converge. However, in the case of a photographic lens with multiple lens elements, where exactly is the focal length of the lens as a whole measured from? Font size setting is missed in iPhone application Can you add a setting for choosing font size in iPhone mobile app?   It's hard to read questions. Does the Rogue's Assassinate feature grant criticals on multiple attacks? Does the Rogue's Assassinate feature (auto-crit on surprised targets) allow for multiple crits if you make multiple attacks that round, either through two-weapon fighting or extra attacks gained through multiclassing?  As an example: my 3rd level Human (Variant, TWF feat) Rogue-Assassin rolls high on his stealth roll, and gets up behind his target.  With two daggers, he gets 2 attacks per round if he uses his bonus action as well.  Do both of these attacks qualify for Assasinate's auto-crit feature?  That is: would I do 2 criticals for       4d4+4d6+2*DEX damage   or a single critical and a non-critical for       3d4+4d6+2*DEX damage?   Additionally, if I gained the Extra Attack feature through multiclassing, would I get crits on all 3 of my attacks, or just the first one? Find three Poisson-distributed random variables, pairwise independent but not mutually independent Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$. Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.   My try :  I consider $h(z)=\frac{f(z)}{g(z)}$. If I prove that $h(z)$ is entire, using the fact that $|h(z)|\leq 1$, the result follows immediately from Liouville's theorem.  To prove that $h(z)$ is entire, I have to prove that $h(z)$ has removable singularities at possible zeros of $g(z)$.  Suppose $g(z_0)=0 \Rightarrow |g(z_0)|=0 \Rightarrow |f(z_0)|=0$, BUT what if $z_0$ be a root of of order $k$ of $g(z)$ (i.e. $f^{(n)}(z_0) = 0 $ for all $ 0 \leq n \leq k-1$) and be a zero of order $m$ of $f(z)$ and $k&gt;m$ ?  And if that's not the case, how to rigorously show that $z_0$ is a removable singularity ?  Thank you all in advance. Exercise 3.5.12. Let $ f: \mathbb N \times \mathbb N \to \mathbb N$ be a function, and let c be a natural number. Show that there exists a function $a:\mathbb N\to \mathbb N$ such that $$a(0)=c$$ and $$a(n++)=f(n,a(n))\;for\;all\;n\in \mathbb N,$$ and furthermore that this function is unique. Tao gives the hint: first show inductively, by a modification of the proof of Lemma 3.5.12, that for every natural number $N\in \mathbb N$, there exists a unique function $a_N:\{ n\in \mathbb N:n\lt N\}\to \mathbb N $ such that $a_N(0)=c$ and $ a_N(n++)=f(n,a(n))$ for all $n\in \mathbb N$ such that $n\lt N$.  In my opinion, we could solve it by proving that $\forall n\in \mathbb N,a(n)\in \mathbb N$ according to mathematcial induction.  The property P(n) is:$$\forall n\in \mathbb N, a(n)\;is\;unique$$ P(0) is true. And if P(n) holds, P(n++) is true because different natural numbers must have different successors. $$a(n++)=f(n,a(n))$$ $$a(m++)=f(m,a(m))$$ $$n++=m++\Rightarrow n=m\Rightarrow a(n++)=a(m++)$$ It's similar to the proof of  Proposition 2.1.16 (Recursive definitions): Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to the natural numbers. Let $c$ be a natural number. Then we can assign a unique natural number $a_{n}$ to each natural number $n$, such that $a_{0} = c$ and $a_{n{++}} = f_{n} (a_{n})$ for each natural number $n$.  This proof indicates both the existence and uniqueness of the function $a$. I don't know why Tao says it's informal.  But regarding the hint, how do we get the function f even if we've got the conclusion from the hint. Also it seems like the function a(n) can be diffentent considering different N. I don't know if the hint is correct.