Showing an inequality in the complex plane
Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$
What are all the common undefined behaviours that a C++ programmer should know about?  Say, like:  a[i] = i++; Why does list(my_list) modify the object? Should questions having lots of downvotes be downvoted further? Is down-voting an already heavily down-voted/closed question kind of like kicking a dead horse? Is this bad practice? What is the best way to modify a list in a 'foreach' loop? $f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors Let $f(x,y)=4x^3y^2$  How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most rapidly from the point $(2,1)$? And, what is the rate of change of $f$ in the direction given in the second question?  My Work (What I have done so far):  For the first question: $\nabla f(x,y)=12x^2y^2$i$+8x^3y$j$\nabla f(2,1)=48$i$+64$j$u=(3/5)$i$-(4/5)$j $\therefore D_u f=\nabla f\bullet u=0$  I'm not entirely sure if this is right? Can someone please verify?  For the second question: $u=(3/5)$i$-(4/5)$j $\therefore -||\nabla f(2,1)||=$? How do I find this? And am I doing this right?  I have no idea what to do for the third question. Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$. Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.   I do not understand how to go about completing this problem or even where to start. How to perform a true GIS clip of polygons layer using a polygon layer in R? Elementary central binomial coefficient estimates How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}&lt;\binom{2n}{n}&lt;\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ &gt; 1 ?     Does anyone know any better elementary estimates?   Attempt.  We have $$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}=\prod_{k=0}^{n-1}\left(1+\frac{k}{2(n-k)}\right).$$ Then we have $$\left(1+\frac{k}{2(n-k)}\right)&gt;\sqrt{1+\frac{k}{n-k}}=\frac{\sqrt{n}}{\sqrt{n-k}}.$$ So maybe, for the lower bound, we have $$\frac{n^{\frac{n}{2}}}{\sqrt{n!}}=\prod_{k=0}^{n-1}\frac{\sqrt{n}}{\sqrt{n-k}}&gt;\frac{2^n}{\sqrt{4n}}.$$ By Stirling, $n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$, so the lhs becomes $$\frac{e^{\frac{n}{2}}}{(2\pi n)^{\frac14}},$$ but this isn't $&gt;\frac{2^n}{\sqrt{4n}}$. How to redirect apache /var/www to the home directory and access all the directories and files in my home directory? I have installed Apache(LAMP) web server on my ubuntu machine and I want to do the above mentioned thing.I have tried creating symbolic links in the /var/www directory but it only shows files but not directories inside the directory linked.I want to redirect /var/www to my home folder so that I can access all the files and directories contained therein.Please help me. Let $(X,d)$ be a metric space. How to prove that for any closed $A$ a function $d(x,A)$ is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof:  $$ |d(x,a) - d(y,a)| \leq d(x,y) $$ for any $a\in A$ - but we cannot just replace it by $|d(x,A) - d(y,A)|\leq d(x,y)$ since the minimum (or infimum in general) can be attained in different points $a\in A$ for $x$ and $y$, so we only have that $$ |d(x,A)-d(y,A)|\leq d(x,y)+\sup\limits_{a,b\in A}d(a,b) $$ which does not mean continuity. How can I calculate how long it will take to transfer a file over LAN? I want to know about the method with which we can calculate data transfer speed.   If I am sending a 1 GB file through a 1 Gb/s LAN connection, from one computer to another, how much time will it take to transfer that file? Is `sudo pip install` still a broken practice? While watching  play for the gold medal in London 2012, I noticed Williams was going about her business while Sharapova was grunting each time she hit the ball.  After doing some , grunting in tennis dates back to 1962. Although men grunt, women draw more attention due to their high-pitched grunts. In addition, there is an initiative to eliminate excess grunts in the next generation of tennis players.  Another quote from said research says:     "I have not found in my 32 years anybody who says grunting or not   grunting has an empirically positive or negative effect on performance   -- it's all about belief system."   What is this "belief system?"  If grunting is not related to performance, why do tennis players do it excessively? Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie.  length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal to $|b-a|$.  It seems intuitive, but I don't have any idea how can I show this in the formal way. How to remove the “AMD Testing use only” watermark? I've installed the  (beta) following the step in  for Ubuntu Quantal Quetzal. My system is 64 bit and my graphic card is an ATI RadeonHD 6670, this g.c. is Officially Supported (Catalyst &amp; Open Source), you can confirm that from this .  I don't have any problem, except the AMD testing use only watermark. I see the following frame in any stage into the OS (logged, unlloged, etc.) except in the terminals.     I found different versions of how to remove this image, but this change according to the system, so I want an answer from this popular (trusted) site.   How to solve this issue in Ubuntu 12.10 32b? This procedure is different in a 64b system? using gedit through ssh connection - should I have gedit installed on the server? I wanted to try out using graphical tools over ssh. Just to learn a new thing :)  so my understanding is this, I connect like this: ssh -port -x user@server  But I'm running ubuntu server so it has no gui and thus no gui tools. My understanding is I've to install gedit on server. But also xserver?  Can someone clear up this concepts to me? Exact sequence of sheaves with non exact sequence of global sections Let $X$ be some topological space. By $\mathcal{F}_i$ we denote some sheaves of abelian groups on $X$. The sequence of sheaves and morphisms  $$\mathcal{F}_1\longrightarrow \mathcal{F}_2\longrightarrow \mathcal{F}_3\longrightarrow... $$ is said to be exact if for each $x\in X$ the corresponding sequence of stalks $$(\mathcal{F}_1)_x\longrightarrow (\mathcal{F}_2)_x\longrightarrow (\mathcal{F}_3)_x\longrightarrow... $$ is exact. However if the sequence of sheaves is exact than the sequence of global sections is not necessarily exact! (The most famous example is the sequence of sheaves $$0\longrightarrow\mathbb{Z}\hookrightarrow \mathcal{O}\stackrel{\exp}{\longrightarrow} \mathcal{O}^*\longrightarrow0$$ considered as sheaves on $\mathbb{C}-\{0\}$, where $\mathcal{O}$ is a sheaf of holomorphic functions, $\mathcal{O}^*$ is a sheaf of holomorphic functions with no zeros).  So, could you give me some easy examples of such phenomenon ? It seems that given a statement $a = b$, that $a + c = b + c$ is assumed also to be true.  Why isn't this an axiom of arithmetic, like the commutative law or associative law?  Or is it a consequence of some other axiom of arithmetic?  Thanks!  Edit: I understand the intuitive meaning of equality. Answers that stated that $a = b$ means they are the same number or object make sense but what I'm asking is if there is an explicit law of replacement that allows us to make this intuitive truth a valid mathematical deduction. For example is there an axiom of Peano's Axioms or some other axiomatic system that allows for adding or multiplying both sides of an equation by the same number?   In all the texts I've come across I've never seen an axiom that states if $a = b$ then $a + c = b + c$. I have however seen if $a &lt; b$ then $a + c &lt; b + c$. In my view $&lt;$ and $=$ are similar so the absence of a definition for equality is strange. What is the difference between "citizen" and "denizen" Citizen: 1. A legally recognized subject ornational of a state or commonwealth, either native or naturalized. 2. An inhabitant of a particular town or city.  Denizen: 1. An inhabitant or occupant of a particular place.  Same thing? I'm too spoiled by Windows utilities that take a digital snapshot of your entire drive, which you can then restore from in the event of a drive crash.  (e.g. like Time Machine for Mac OS X).  Is there a similar way of doing this in Linux?