"...and all would have to be accounted for." Improper sentence ending at 'for'. Please suggest alternative
When is it appropriate to end a sentence in a preposition?
A malevolent computer with AI takes over a woman's house with the intent of hybrid computer/human procreation. The evil computer seals off the house with the woman in it and keeps her prisoner while it prepares. I read the tale sometime in the late 80's or the early 90's. It was one of many of my missing books that got yardsaled by my in-laws when I wasn't looking. It may also have been made into a movie, but I didn't see the film. Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability $p$ and loses one dollar with probability $1 - p$. Let $f_{n}$ be the probability that he or she first becomes broke at time $n$ for $n = 0, 1, 2\ldots$. Find a generating function for these probabilities.  Is it binomial distribution $?$ and then use the $\operatorname{mx}\left(s\right) = rx\left(\mathrm{e}^{s}\right)$ of the binomial distribution $?$. Help$\ldots!$. I am correcting my quiz. Prove that the subsequence of a convergent series with nonnegative terms converges If I add a new case class, does that mean I need to search through all of the pattern matching code and find out where the new class needs to be handled?  I've been learning the language recently, and as I read about some of the arguments for and against pattern matching, I've been confused about where it should be used.  See the following:  Pro:  and    Con:   The comments are pretty good in each case, too.  So is pattern matching something to be excited about or something I should avoid using?  Actually, I imagine the answer is "it depends on when you use it," but what are some positive use cases for it and what are some negative ones? Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. Convert NSDate to NSString How do I prove $F(a)=F(a^2)?$ Let $E$ be an extension field of $F$. If $a \in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$.  let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and $k$ be the degree of the minimal polynomial $q(x)$ of $a^2$ over $F$.   Since $a^2 \in F(a)$, We have $F(a^2) \subset F(a)$, then $k\le n$  In order to prove the converse:  $q(a^2)=b_0+b_1a^2+b_2(a^2)^2\ldots+b_k(a^2)^k=0$   implies   $q(a)=b_0+b_1a^2+b_2a^4\ldots+b_ka^{2k}=0$ Then   $p(x)|q(x)$, because $p(x)$ is the minimal polynomial of $a$ over $F$.  If I prove that $n|2k$ we done, since $k$ is odd, we have $n|k$ and $n\le k$ and finally $n=k$.  So I almost finished the question I only need to know how to prove that $n|2k$ It should be only a detail, but I can't see, someone can help me please?  Thanks I have a site which is running under SharePoint 2013. Via REST I am trying to retrieve the file and folder directory for that site. There are several similar questions on this site related to recursively retrieving all of the files and folders for a given a folder:       The answers provided therein led me to the following URI:     /_api/web/Lists/GetByTitle('Documents')/Items?$expand=Folder&amp;$select=Title,FileLeafRef,Folder/ServerRelativeUrl   Which from what I can tell does retrieve all of the files and folders from the given starting path.  However, I am missing something. When reading through the returned XML I cannot figure out where a given file goes. Meaning that I have info on how to retrieve file bros.pdf, but cannot determine that it is in the documents/super/mario/ folder.  For the file items I've tried to retrieve the ServerRelativeUrl property, but have not succeeded though the folders seem to. Is there some identifier in the XML that I am missing? When doing a recursive RESTful query how do I determine where a given file belongs? What is this operator: &= Does it matter what I choose for serialVersionUID when extending Serializable classes in Java? Passing Javascript variable to Working with large images and transform in the VSE Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $\overline{x}\in X$.      The n-th power is defined inductively: $$ T^{n+1}(x):=T(T^{n}(x)), n\in\mathbb{N}. $$ And for a function $T\colon X\to X$, $q$-contractive means  $$ \exists 0\leq q&lt;1~\forall~x,y\in X: d(T(x),T(y))\leq q d(x,y). $$    Now to the proof.  I think I have to show, that $T$ is q-contractive, because then it follows with Banach, that $T$ does have exactly one fixed point. So I have to show, that there exists a $0\leq q&lt;1$ so that for all $x,y\in X$ it is $$ d(T(x),T(y))\leq q\cdot d(x,y). $$ And most likely I have to use the q-contractivity of $T^n$, i.e. that there exists a $0\leq q &lt;1$ so that for all $x,y\in X$ it is $$ d(T^n(x),T^n(y))\leq q\cdot d(x,y). $$  Can you help me? Find what packages are installed from a repository How does gravity escape a black hole? Shapefile projections not matching up in QGIS? To start off with, I'm very new to QGIS. I'm attempting to place a shapefile with locations for 3 substations on a map of Peniche, Portugal. The locations for the substations are given in latitude and longitude coordinates.   When I start off by importing the substation shapefile, I make sure that both the layer CRS and project CRS are set to EPSG:32629, which gives each substation point the correct coordinates, however the scale is at 1:1, which is not correct. As soon as I then add a raster map of the region or a shapefile of Portugal using the same CRS the position of the substations also becomes incorrect.   I'm sure there's a simple solution I'm not getting, but I've been trying for a while and have had no success. Here's a dropbox link to the files lcm versus phi in RSA In textbook RSA, the Euler $\varphi$ function $$\varphi(pq) = (p-1)(q-1)$$ is used to define the private exponent $d$. On the other hand, real-world cryptographic specifications require the Carmichael lcm function $$\lambda(pq) = \operatorname{lcm}(p-1,q-1)$$ to define $d´$. It is clear that the $d´$ divides $d$, and therefore using $d´$ may be more efficient than using $d$. My question is: Are there any further reasons, e.g. regarding security, why one should not use $d$? A story about a whole crew of aliens who visit Earth, befriend a child It was a book I read when I was a kid, so I am assuming it was a book aimed at kids/young adults. A whole crew of aliens of different species meet an Earth kid. He befriends them. One of the aliens comes from a planet with dozens/hundreds of different sexes/genders. At one point, the Earth kid is talking about a swamp:  Alien: &quot;I live in/near a swamp on my home planet&quot; Kid: &quot;Oh, do you come from a swamp planet?&quot;  The alien points out that just because there is a swamp on his planet doesn't make it a swamp planet, there are dozens of biomes on his planet, just like Earth. Earth isn't just a swamp planet. The Pearson correlation coefficient of x and y is the same, whether you compute pearson(x, y) or pearson(y, x). This suggests that doing a linear regression of y given x or x given y should be the same, but I don't think that's the case.   Can someone shed light on when the relationship is not symmetric, and how that relates to the Pearson correlation coefficient (which I always think of as summarizing the best fit line)? How do I assign a value to a whole array of integers?