How to show that $\displaystyle{\cot z-\frac{1}{z}}$ is bounded on the given circle
Is $\cot(z)-\frac1z$ bounded on a given circle?
Crash course on algebraic geometry with view to applications in physics Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?   What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references. Soving a Complex Integral along a circle proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem What does Minecraft's scary music mean? Apparently it means there is a dungeon around but I can never find any?  If it does mean dungeons how far away will it trigger? When constructing the matrices for the two CNOT based on the target and control qubit, I can use reasoning:  "If $q_0$==$|0\rangle$, everything simply passes through", resulting in an Identity matrix style $\begin{bmatrix}1&amp;0\\0&amp;1\end{bmatrix}$ in the top left. "If $q_0==|1\rangle$, we need to let $q_0$ pass and swap $q_1$, resulting in a Pauli X $\begin{bmatrix}0&amp;1\\1&amp;0\end{bmatrix}$ in the bottom right.  $CNOT \equiv \begin{bmatrix}1&amp;0&amp;0&amp;0\\0&amp;1&amp;0&amp;0\\0&amp;0&amp;0&amp;1\\0&amp;0&amp;1&amp;0\end{bmatrix}$  "If $q_1$==$|0\rangle$, everything simply passes through", results in leaving $|00\rangle$ and $|10\rangle$ unaffected. "If $q_1==|1\rangle$, we need to let $q_1$ pass and swap $q_0$, mapping $|01\rangle$ to $|11\rangle$ and  $|11\rangle$ to $|01\rangle$  $CNOT \equiv \begin{bmatrix}1&amp;0&amp;0&amp;0\\0&amp;0&amp;0&amp;1\\0&amp;0&amp;1&amp;0\\0&amp;1&amp;0&amp;0\end{bmatrix}$  This all seems to check out, but here comes my question: I would like to know if there is a more mathematical way to express this, just as there would be when combining for instance two hadamard gates: $H \otimes H \equiv \frac{1}{\sqrt{2}}\begin{bmatrix}1&amp;1\\1&amp;-1\\ \end{bmatrix} \otimes \frac{1}{\sqrt{2}}\begin{bmatrix}1&amp;1\\1&amp;-1\\ \end{bmatrix} = \frac{1}{2}\begin{bmatrix}1&amp;1&amp;1&amp;1\\1&amp;-1&amp;1&amp;-1\\1&amp;1&amp;-1&amp;-1\\1&amp;-1&amp;-1&amp;1 \end{bmatrix}$  And a bonus question:  How I can, using notation like "CNOT", show which qubit is the control bit and which qubit the target bit? My interest in combinatorics was recently sparked when I read about the many things that the Catalan numbers count, as found by Richard Stanley. I picked up a copy of Brualdi's Combinatorics, and while browsing the section on counting sequences I found a nice little puzzle that has definitely puzzled me.     Let $m$ and $n$ be nonnegative integers with $n\geq m$. There are $m+n$ people in line to get into a theater for which admission if $50$ cents. Of the $m+n$ people, $n$ have a $50$-cent piece and $m$ have a $\$ 1$ dollar bill. The box offices opens with an empty cash register. Show that the number of ways the people can line up so that change is available when needed is   $$ \frac{n-m+1}{n+1}\binom{m+n}{m}. $$     I first noted that the first person to enter must be one of the $n$ with a half-dollar. Now the register has a half-dollar change. The second person can be either a person with a half-dollar or a dollar. In the first case, the register will now have two half-dollars, in the second case, the register will now have one dollar bill. So it seems to me that when one of the $n$ people with a half-dollar enters, the number of half-dollars in the register increases by $1$, and when one of the $m$ people with a bill enters, the number of half-dollars decreases by $1$ but the number of bills increases by $1$.   I tried to model this by looking at paths in $\mathbb{Z}^2$. The $x$-axis is like the number of half-dollars, and the $y$-axis is the number of bills. You start at $(0,0)$, and you can take steps forward $(1,0)$ or backwards diagonally $(-1,1)$ corresponding to who enters, but you must always stay in the first quadrant of the plane without crossing over the axes. The goal is to make $m+n$ moves, and I figured maybe the number of such paths is counted by $\frac{n-m+1}{n+1}\binom{m+n}{m}$, but I'm not sure how to show this. I don't know if this observation simplifies the problem at all, as I don't know how to finish up. I'd be happy to see how this problem is done, thank you. How do amortization schedules work and when are they used? Specifically, I notice that on a mortgage statement, my payment is "amortized" so that at the beginning of the loan, I am paying almost entirely interest and very little principal.  However, on a credit card balance, it seems interest is only charged on the balance.  Why do amortization schedules work the way they do, and are they typically used for most loans, and why? Has it been practically proven that quantum superposition exists ? If yes, how does it even work? I am having the same trouble as others getting my Atheros AR8131 Ethernet card recognised by Ubuntu 12.10 on my new HP Pavillion P7-1449 PC. I found the same question where the answer suggested the following:   Download Tarball compact-wireless-3.6.8-1-snpc Run  ./scripts/driver-select alx make sudo make install    The problem is when I run the make command I get the following:  make -C /lib/modules/3.5.0-21-generic/build M=/home/chris/Downloads/compat-wireless-3.6.8-1-snpc modules make: *** /lib/modules/3.5.0-21-generic/build: No such file or directory.  Stop. make: *** [modules] Error 2   Found a link to an alternative patch file (compat-wireless-2012-09-25-pc) but then I get the similar error:  make -C /lib/modules/3.5.0-21-generic/build M=/home/chris/Desktop/compat-wireless-2012-09-25-pc modules make: *** /lib/modules/3.5.0-21-generic/build: No such file or directory.  Stop. make: *** [modules] Error 2   Any help you could provide would be greatly appreciated. Are repository lists secure? Is there an HTTPS version? Are repository updates secure?  As a bear of little brain from the developer side, I cannot understand why the repository list is http://security.ubuntu.com and the other http(unsecured) sites listed in /etc/apt/sources.list.  Without a certificate chain match this appears as "ask any responder for a list of packages to update" instead of "ask the ubuntu.com site..."  Can any network choose to spoof the update sites, and is this a common practice to provide a locally cached and vetted copy? What is the problem that gets worse after you try to solve it? Find objects between two dates MongoDB Error when using TikZ-Feynman package Today I downloaded and installed the new  package to draw Feynman diagrams. I included the packages   \usepackage{tikz}  \usepackage{tikz-feynman} \tikzfeynmanset{compat=1.0.0} \usetikzlibrary{shapes,arrows,positioning,automata,backgrounds,calc,er,patterns}   into my preamble (changed them already many times).  However, already a simple sample code like  \begin{tikzpicture} \begin{feynman} \vertex (a) {\(\mu^{-}\)}; \vertex [right=of a] (b); \vertex [above right=of b] (f1) {\(\nu_{\mu}\)}; \vertex [below right=of b] (c); \vertex [above right=of c] (f2) {\(\overline \nu_{e}\)}; \vertex [below right=of c] (f3) {\(e^{-}\)}; \diagram* { (a) -- [fermion] (b) -- [fermion] (f1), (b) -- [boson, edge label'=\(W^{-}\)] (c), (c) -- [anti fermion] (f2), (c) -- [fermion] (f3), }; \end{feynman} \end{tikzpicture}   directly gives the error:     Package pgfkeys Error: I do not know the key '/graph drawing/node distance'.   Can you please help me? Thanks in advance! Proving Eigenvalue squared is Eigenvalue of $A^2$ The question is: Prove that if $\lambda$ is an eigenvalue of a matrix A with corresponding eigenvector x, then $\lambda^2$ is an eigenvalue of $A^2$ with corresponding eigenvector x.  I assume I need to start with the equation $Ax=\lambda x$ and end up with $A^2 x=\lambda^2 x$ but between those I am kind of lost. I have manipulated the equations several different ways and just can't seem to end up where I need to be. Help would be greatly appreciated as I believe this will be on a test tomorrow. How to find out the variant of my Galaxy Nexus? Sometimes I create Material Groups to be used as appendable reusable assets. Unfortunately sometimes parameters I want to control using a Group Node's slider are not available for routing to a Group Node's input socket. Some examples of these are the Color Ramp and the Mapping Node.  So here's what I want to do:    Can this be done? If so, how?  Thanks! Motivation  I want to include a list of related equations, say, for a proof, in my LaTeX document. As far as I know, I have two good options, eqnarray and align.   Question  What is the difference between eqnarray and align, and how do I know which I should be using? Or does it matter at all? I mean 100+ MB big; such text files can push the envelope of editors.  I need to look through a large XML file, but cannot if the editor is buggy.  Any suggestions? Today in a match the setter set the ball too low for me to spike it, so I just dinked it over.  The bottom of the ball was lower than the top of the net at the point where I made contact.  When the defender made contact, again, the bottom of the ball was lower than the top of the net (the top of the ball was over the top of the net though).  I felt that this wasn't technically a block and that he shouldn't have been allowed to hit it a second time, but I wasn't sure so I just let it go.  Was this a legal block or should it have been our point when he hit it a second time in a row?  Edit:  I don't think I was clear about the part where, after I hit the ball, the ball travelled over the net onto the other team's side of the court.  The other player did not reach over the net so there is no issue there.  It's just an issue of whether or not their first contact was a block or a hit.  I believe it was not a block but I can't find a specific ruling to address it. Copy and set-mark in Bash as in Emacs? I would like to be able to copy and paste text in the command line in Bash using the same keyboard bindings that Emacs uses by default (i.e. using C-space for set-mark, M-w to copy text, C-y, M-y to paste it, etc.).  The  says that Bash comes with some of these key bindings set up by default.   For example, yanking (C-y) works by default on my terminal. However, I can't get the set-mark and copy commands to work, and they don't seem to be bound to any keys by default.  Usually, the way a user can define her own key bindings is to add them to .inputrc. So I looked and found the following bash functions in the documentation that I presume can help me define the Emacs-like behavior that I want (i.e. set-mark with C-space and copy with M-w).  copy-region-as-kill ()      Copy the text in the region to the   kill buffer, so it can be yanked right   away. By default, this command is   unbound.   and    set-mark (C-@)      Set the mark to the point. If a   numeric argument is supplied, the mark   is set to that position.   If I understand correctly, the above means that copy-region-as-kill is not bound to any keyboard sequence by default, while set-mark is bound to C-@ by default.   I tried C-@ on my terminal, but I don't think it runs set-mark because I don't see any text highlighted when I move my cursor. In any case, I tried adding  keyboard bindings (M-w and C-) to the functions  copy-region-as-kill and set-mark above in my .inputrc and then reloading it with C-x C-r, but this didn't work. I know that my other entries in .inputrc work because I have other user-defined keybindings defined in it.  Is there anything I am doing wrong? Am I missing anything?