Eight dollars is the price of a movie these days.  Why dont use ''are''?
I am having trouble with the following sentence:      "The complexity and diversity of the new platforms is worrying us."   If "complexity" and "diversity" constitute two subjects, then the verb should presumably be "are"; however, the sentence feels more natural with "is". Is this correct? Is there a linguistic term for this occurrence?    The following questions address the general topic of compound subjects, but not specific cases like this:             And the following question seems to have somewhat inconclusive answers:         So this question is not really a duplicate of these three.
Incompatibility between TikZ and babel What part of speech is handsome in the sentence:     The bridegroom looked so handsome. What's a word for lack of confidence? I'm trying to find a word that describes a lack of confidence. Something that gives a sense of worthlessness maybe. I've dried looking up in Thesarus.com for similarities to lack of confidence and antonyms to self-worth, but with no luck. Anyone have some ideas? Isomorphism between $\Bbb R$ and $\Bbb R(X)$? My questions are:     $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ?      $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields?   $ $    $ $  For $1.$, here are some comments:   This is true for $\Bbb C$, see . The key point here is the fact that $\Bbb C$ is algebraically closed, of cardinality $2^{\aleph_0}$, and of characteristic $0$. The same argument doesn't work for $\Bbb R$. We can actually find a subfield $F$ of $\Bbb R$ such that $F \cong F(X) \cong F(X,Y) \cong \cdots$, see .  that $\Bbb Q(X)$ does not embed in $\Bbb Q$ (as a field).  This is equivalent to: Is there an injective ring morphism $\Bbb R[X] \hookrightarrow \Bbb R$ ? (By taking the fraction fields).   As for 2. :   They are isomorphic as abelian groups, and actually as $\Bbb Q$-vector spaces. They are not isomorphic as $\Bbb R$-algebras, because they haven't the same transcendence degree over $\Bbb R$. Let's try to see how to construct such a field isomorphism. We could try$^{[1]}$ to find a subset $B$ of $\Bbb R$ such that $\Bbb R = \Bbb Q(B)$ and $C \subsetneq B \implies \Bbb R \neq \Bbb Q(C)$.  The set $B$ must contain transcendental elements, pick such an $x_0 \in B$.  Then $\Bbb R = \Bbb Q(B) = \Bbb Q(B \setminus \{x_0\})(x_0)$ and we could try$^{[2]}$ to find a field isomorphism $\Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$, so that $\Bbb Q(B \setminus \{x_0\})(x_0) \cong \Bbb Q(B)(x_0) \cong \Bbb R(X)$.  For ${[1]}$, I tried to use Zorn's lemma on $\mathscr B = \{B \subset \Bbb R \mid \Bbb R = \Bbb Q(B)\}$ and the partial order $B_1 ≤ B_2 \iff B_1 \supset B_2$. But this is not clear to me why this should be an inductive set.  For ${[2]}$, my idea was to define $f : \Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$ by choosing a bijection $b : B \setminus \{x_0\} \to B$ (these sets should be uncountable), and then trying to extend $f\vert_{B \setminus \{x_0\}} = b$ to a field morphism.     Edit for $^{[1]}$: in  article (MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS, by LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RUZICKA) the theorem 2.4. proves that $\Bbb R$ has no minimal generating set as field (or $\Bbb Q$-algebra, i.e. as ring). The argument uses Artin-Schreier theorem. Minimal generating sets of modules are   .  I think that $B$ is a minimal generating set of a subfield $K \subset \Bbb R$ over $\Bbb Q$ if and only if any element of $B$ cannot be written as a rational fraction of other elements of $B$. Under these conditions and using (transfinite) induction, we can probably prove (and this was my first idea) that any function $f: B \to \Bbb C$ such that $b$ and $f(b)$ are conjugate over $\Bbb Q$ (i.e. have the same minimal polynomial) extends to a (unique) field morphism $\sigma : K = \Bbb Q(B) \to \Bbb C$.  Actually, the collection $\mathscr B$ defined above doesn't have to be an inductive set, at least if we replace $\Bbb R$ by $K =\Bbb Q(\sqrt 2, \sqrt[4]{2},\sqrt[8]{2},\dots)$, because the descending chain $\left(E_n = \{\sqrt[2^m]{2} \mid m \geq n\}\right)_{n \geq 1}$ satisfy $K = \Bbb Q(E_n)$, but $K \neq \Bbb Q\left(\bigcap_{n≥1} E_n\right)=\Bbb Q$, so the chain has no upper bound w.r.t. $≤$ in $\mathscr B$. How to find original community description Before joining a community you can easily see the community description. For example, for example before joining the Programming Puzzles &amp; Code Golf Stack Exchange you see the description:     Programming Puzzles &amp; Code Golf Stack Exchange is a question and   answer site for programming puzzle enthusiasts and code golfers.   However, after joining I can no longer see this description. I can go to the on-topic page and see a different description:     Programming Puzzles &amp; Code Golf is for programming contests and   challenges. We welcome questions from beginners and experts alike.   Yet, I cannot seem to find the original description anywhere. Is there a way to find the original description while being logged into a Stack Exchange site?  Based on Rene's answer, I can click the Tour link on each site, but StackOverflow (and maybe other sites) do not have a tour link. Instead, I'd have to go there by manually typing it in the address bar. Is there a reason that this has been removed from the Stack Overflow page? How is Leibniz's rule for the derivative of a product related to the binomial formula? Possible Duplicate:          The  describes the expansion of the $n$th power of the sum $(a+b)$:  $$(a+b)^n = \sum_{k = 0}^n {n\choose k}a^kb^{n-k}$$  In calculus, there is a generalization of the product rule called , which describes the expansion of the $n$th derivative of the product of two functions $f$ and $g$:  $$(fg)^{(n)} = \sum_{k = 0}^n {n\choose k}f^{(k)}g^{(n-k)}$$  It's not difficult to prove using the product rule and induction, what I'm really trying understand here is why the formula is true. Is there a way to prove it using the binomial formula? How are these two formulas related? Is there a combinatorial proof for Leibniz's rule like there is for the ?   I understand this question is a bit vague, but I hope you'll agree that these two formulas look so similar it just feels like there should be some relation between the two. Of course this is not necessarily true, but I'm hoping it's not just a crazy coincidence. “...than I am” vs. “...than me” I read an English grammar article in which the author talked about the correct use of pronouns.  He writes:     If the extended sentence is “She’s more likely to ask him than I am,” in which the comparison is between the subject and the writer, I is correct. However, if the intent is to convey that the man referred to as him is more likely to be asked something by the subject than the writer is, the correct pronoun form is me, but that distinction should be clarified with a revision like “She’s more likely to ask him than ask me.”   Before reading this piece, I thought that the two sentences had the same meaning. While the second example is quite clear for me (“She’s more likely to ask him than ask me”), I don't think that I have correctly understood the first one. Could anyone explain it to me, please? Why copy constructor not invoked here Unzip File: Permission Denied I am trying to unzip a file (Facebook download-your-info zip file), but here's what I got:   Is there a way I can force-unzip it?  Update: I tried sudo unzip file.zip -d destination_folder in terminal, and it unzipped the file, but what came out, all the files have size 0 bytes:   Update 2: I do have permission to the directory and the file.    (I'm really sorry if my question is asked badly. Let me know down in the comments :) ) Coprime cofactors of n'th powers are n'th powers, up to associates, for Gaussian integers & UFD I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1):  Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta=\varepsilon\gamma^n$, for $\alpha,\beta$ relatively prime numbers and $\varepsilon$ a unit, implies $\alpha=\varepsilon^{\prime}\xi^n$ and $\beta=\varepsilon^{\prime\prime}\eta^n$, with $\varepsilon^{\prime},\varepsilon^{\prime\prime}$ units.  My attempt: once $\alpha$ and $\beta$ are relatively prime, $\alpha\nmid\beta$ and $\beta\nmid\alpha$. Therefore $N(\alpha)$ and $N(\beta)$ are relatively prime too (where $N(a+ib)=a^2+b^2$). Therefore,  $$N(\alpha\beta)=N(\alpha)N(\beta)=N(\varepsilon)N(\gamma^n)=N(\gamma)^n=(p_1\cdot\ldots\cdot p_r)^n=p_1^n\cdot\ldots\cdot p_r^n$$  And once $N(\alpha)$ and $N(\beta)$ are relatively prime, we have (if necessary, we reorder the $p_i$'s):  $$N(\alpha)=p_1^n\cdot\ldots\cdot p_k^n=(p_1\cdot\ldots\cdot p_k)^n=N(\xi)^n=N(\xi^n)$$ and $$N(\beta)=p_{k+1}^n\cdot\ldots\cdot p_r^n=(p_{k+1}\cdot\ldots\cdot p_r)^n=N(\eta)^n=N(\eta^n)$$  And the result follows.  But I'm not sure about one step. It is $\alpha$ and $\beta$ relatively prime $\Rightarrow$ $N(\alpha)$ and $N(\beta)$ relatively prime. I know that if $\alpha|\beta$ then $N(\alpha)|N(\beta)$, but the previous assertion is not necesseraly true, and if it is, and don't know how to prove it. If it is true, the result follows, unless I made something wrong in the rest of the proof.  That's it. If you know another way of doing the exercise, please show me. How did Harry's blood being inside Voldemort save him? There are, of course, an incredible amount of theories on how Harry Potter survived. Problem is that the most important one, the one that is canon, is the one I don't get: how Harry's blood inside Voldemort saves him?  Because all that I have read so far, leaves me with questions. I understand why Voldemort wanted Harry's blood, but why does it save Harry? Why is that necessary to keep the sacrifice alive, while it's still inside Harry? And if Harry lost it at 17 (I have seen some different ideas on this as well), why does Voldemort still have it at 69 (give or take a few)?  To clafiry, the parts I'm talking about:     "For a fleeting instant, Harry thought he saw a gleam of something like triumph in Dumbledore's eyes"    (GoF, right after Harry tells Dumbledore that his blood now goes through Voldemort.)  And:     "He took your blood believing it would strengthen him. He took into his body a tiny part of the enchantment your mother laid upon you when she died for you. His body keeps her sacrifice alive, and while that enchantment survives, so do you"   (Albus Dumbledore in DH)  I just don't get why the blood transfusion thing suddenly gave Harry the chance to survive the whole thing, and it's never really been satisfyingly answered, not to my knowledge at least, as every answer I've read raises more questions. What is the difference between center of mass and center of gravity? What is the difference between  and ?  These terms seem to be used interchangeably.  Is there a difference between them for non-moving object on Earth, or moving objects for that matter? interaction between mathematical structures From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one... Is string in array? What would be the best way to look in a string[] to see if it contains a element. This was my first shot at it. But perhaps there is something that I am overlooking. The array size will be no larger than 200 elements.  bool isStringInArray(string[] strArray, string key) {     for (int i = 0; i &lt;= strArray.Length - 1; i++)         if (strArray[i] == key)             return true;     return false; } I needed to change the SID of an Oracle XE database (not the Service Name) to match a production database.  When I tried searching online, most of the pages were describing changing or adding a service name through tnsnames.ora; that's not what I needed to do. Finder: keep sort-by per-directory Some directories make sense to be sorted by date modified, while others make more sense to be kept sorted alphabetically.   How can I make it so that finder will apply a specified sort by selection on a per-directory basis?   If no such feature exists (In which case I hope somebody at APPL is paying attention), I suppose using/remapping the keyboard shortcuts (Defaults being Cmd+Ctrl+5 for Modified and Cmd+Ctrl+1 for Alphabetical by Name) is the next best thing. How can I adjust Cycles material displacement amount? Why are there $736$ matrices $M\in \mathcal M_2(\mathbb{Z}_{26})$ for which it holds that $M=M^{-1}$? I'm currently trying to introduce myself to cryptography.  I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for encryption and the inverse matrix $M^{-1}$ for decryption. The book recommends to use a matrix $M$ for which it holds that $M=M^{-1}$, since we then have the same key for encryption and decryption.  In the book, they use a $2 \times 2$ matrix $M$ over $\mathbb{Z}_{26}$ as example, and state that for there are 736 $2 \times 2$ matrices for which it hold that $M=M^{-1}$.  I'm trying to pick up on as much as possible when reading things, since I find it counter-productive for learning to skip something, when you don't get the theory behind it. Can someone enlighten to me, as to why it is that there are 736 possible $2 \times 2$ $M=M^{-1}$ matrices and how to find them? Consider the integral equation $\phi(x)-\frac{e}{2} \int_{-1}^{1} x e^{t} \phi(t) d t=f(x) .$ Then  there exists a continuous function $f:[-1,1] \rightarrow(0, \infty)$ for which solution exists there exists a continuous function $ f:[-1,1] \rightarrow(-\infty, 0)$ for which solution exists for $f(x)=e^{-x}\left(1-3 x^{2}\right)$, a solution exists for $f(x)=e^{-x}\left(x+x^{3}+x^{5}\right),$ a solution exists  what I tried $f(x)=f(x)+\frac{e}{2} \int_{-1}^{1} x e^{t} \phi(t) d t$ $\phi(x)=f(x)+\frac{e x}{2} \int_{-1}^{1} e^{t} \phi(t) d t$ $ \begin{array}{l}L e t \quad c=\int_{-1}^{1} e^{t} \phi(t) d t \\ \Rightarrow \phi(x)=f(x)+\frac{x e}{2} c\end{array} $ $c=\int_{-1}^{1} e^{t}\left(f(t)+\frac{c e}{2} t\right) d t$ $c=\int_{-1}^{1} e^{t} f(t)+\frac{\operatorname{ce}}{2} \int_{-1}^{1} t e^{t} d t$ $c=\int_{-1}^{1} e^{t}f( t)+\frac{\operatorname{ce}}{2}\left[t e^{t}-e^{t}\right]_{-1}^1$ $c=\int_{-1}^{1} e^{t} f(t)+\frac{c e}{2}\left(2 e^{-1}\right)$ $c=\int_{-1}^{1} e^{t} f(t) d t+c$ $\int_{-1}^{1} e^{t} f(t) d t=0$ Option (3) and (4) satisfies this Relation so, (3) and (4) seems true  to me.is i am right? I am stuck  for option (1) and (2)  is there any result or theorem ? I need to execute a directory copy upon a user action, but the directories are quite large, so I would like to be able to perform such an action without the user being aware of the time it takes for the copy to complete.  Any suggestions would be much appreciated.