Can the adjectivized nouns have a plural form?
How do you use plural or singular form of an adjective?
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. Before 1994 the last Triwizard Tournament had been held in 1792, where the three headmaster judges were injured by a rampaging cockatrice.  The tournament is held every 5 years, which would put the next one in , the year . Did they carry on with the tradition, or was it deemed too risky again? I have windows 7 in Arabic. Now I dowloaded QGIS 2.14.3-2 in French.  But when I installed it shows Arabic menu/tools...  How can I change the menu language to French? Norms Induced by Inner Products and the Parallelogram Law Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$.  It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ for some (real) inner product $\langle \cdot, \cdot \rangle$, then the parallelogram equality $$ 2\lVert u\rVert^2 + 2\lVert v\rVert^2 = \lVert u + v\rVert^2 + \lVert u - v\rVert^2 $$ holds for all pairs $u, v \in V$.  I'm having difficulty with the converse.  Assuming the parallelogram identity, I'm able to convince myself that the inner product should be $$ \langle u, v \rangle = \frac{\lVert u\rVert^2 + \lVert v\rVert^2 - \lVert u - v\rVert^2}{2} = \frac{\lVert u + v\rVert^2 - \lVert u\rVert^2 - \lVert v\rVert^2}{2} = \frac{\lVert u + v\rVert^2 - \lVert u - v\rVert^2}{4} $$  I cannot seem to get that $\langle \lambda u,v \rangle = \lambda \langle u,v \rangle$ for $\lambda \in \mathbb{R}$.  How would one go about proving this? Symmetric powers of modules Log4j Logging to a Shared Log File Is there a way to write log4j logging events to a log file that is also being written to by other applications. The other applications could be non-java applications. What are the drawbacks? Locking issues? Formatting? Do the Imperials know they are evil? What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters? What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?  I believe the answer for this is 6. As we can write the group elements as below   (a)(b)(C) (ab)(c) (ac)(b) (bc)(a) (abc) (bac)   Can we generalize that for any $S_n$ onto $n$ the number of automorphisms will be $n!$ Also I cannot find this anywhere in my text, can we have a permutation $S_n$ onto $m$ where $n\neq m$? (for eg: is it possible to have a permutation group say $S_5$ on say 2 elements?) One of the new features in SQL Server 2012 codenamed Denali is the   I know a good bit about regular old row-store indexes, like the b-tree structure, differences in storage between the leaf level and b-tree pages, affects of included fields, optimizing to use them, order of keys, etc.  I'm having difficulty getting any good info on the internals of a columnstore index.     How is it structured? Is there a b-tree?  Some other structure in place? How is data organized? What sorts of specific operators are best suited to use it? Any other anti-patterns to be avoided when using them?   A lot of what I can find out about them are basically the exact opposite of a "normal" index, i.e. no ordering of keys, no included fields, nonclustered ONLY.  Any insights are appreciated. Inspired by a StackOverflow question ().  Is there a DBMS that actually supports the = NULL syntax? How to edit command line in full screen editor in ZSH? I am having an error compiling the following code with xelatex and lualatex:  ** WARNING ** "seac" character deprecated in Type 2 charstring. ** ERROR ** Type2 Charstring Parser: Parsing charstring failed: (status=-1, stack=5)  Output file removed.  \documentclass{article}  \usepackage{xunicode} \usepackage{fontspec}  \setmainfont[Mapping=tex-text,Ligatures={Common}, Numbers={OldStyle}]{IM FELL English PRO} \begin{document}  Buongiorno a tutti. Trallallà.  \end{document}   The error disappears when I omit the accented à. It also doesn't happen when I use other fonts.  Could you suggest any workaround?  Thank you, A What happened to the Grid at the end of Tron Legacy? invalid EFI file path on windows 8 How should I validate an e-mail address? What is $\arctan(x) + \arctan(y)$ I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$  which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by which I mean, has different definitions for different domains:  $$g(x) = \begin{cases}\arctan\left(\dfrac{x+y}{1-xy}\right), &amp;xy &lt; 1 \\[1.5ex]  \pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &amp;x&gt;0,\; y&gt;0,\; xy&gt;1 \\[1.5ex]  -\pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &amp;x&lt;0,\; y&lt;0,\; xy &gt; 1\end{cases}$$  Furthermore, When I plot the function $2\arctan(x)$, it turns out that the book definition is correct. I don't understand how such peculier definition emerges. Thank you. How to open port via SSH tunnel? There is a system B with the following open servers: I have a web server listening on port 80 and a ssh server listening on 22. However, only port 22 is publicly available. Now, I would like to create some kind of tunnel, so that I can access B:80 from A. However, my client computer A that would like to connect to the system B is not publicly open, either.  So, all I have is a client computer A from where I would like to access the server B and, there, an open port 22. On A, no port is open or can be opened.  What (I think) I would need is to open locally (on A) some port that connects in some way through port 22 of B to port 80 on B.  Is this possibly without using any man-in-the-middle open servers with multiple ports? At the start of A New Hope are Yoda and Obi-Wan really the last remaining Jedi? I have never read any Star Wars books and have no knowledge outside of the Star Wars films, but after seeing Episode III I am left wondering if there are any other Jedi that escape death and remain in hiding like Yoda did. I mean with regard to the "official" events in the Star Wars universe. When do I use the following:   home.php frontpage.php Under Settings > Reading Settings  Front Page Posts Page    I wish to clarify this as I find myself using "Placeholder Pages", empty pages just for the sake of a menu item and to display blog listings.   If I use a Static Front Page (Set under Settings > Reading Settings > Front Page) how do I access my posts page? Do I need to have a "Placeholder Page" titled blog and use page-blog.php? Why are C character literals ints instead of chars? In C++, sizeof('a') == sizeof(char) == 1. This makes intuitive sense, since 'a' is a character literal, and sizeof(char) == 1 as defined by the standard.  In C however, sizeof('a') == sizeof(int). That is, it appears that C character literals are actually integers. Does anyone know why? I can find plenty of mentions of this C quirk but no explanation for why it exists.