Reflection of Light from a surface
Explain reflection laws at the atomic level
How should we deal with duplicate answers? Texmaker GUI is messy with high resolution screen How to execute file I'm editing in Vi(m) Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'? Show that when $BA = I$, the solution of $Ax=b$ is unique I'm just getting back into having to do linear algebra and I am having some trouble with some elementary questions, any help is much appreciated.  Suppose that $A = [a_{ij}]$ is an $m\times n$ matrix and $B = [b_{ij}]$ and is an $n\times m$ matrix and $BA = I$ the identity matrix that is $n \times n$.  Show that if for some $b \in \Bbb{R}^m$ the equation $Ax = b$ has a solution, then the solution is unique. Properly discontinuous action: equivalent definitions Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is equivalent to, having given $G$ the discrete topology and in the $X$ locally compact Hausdorff case, the map $G \times X \rightarrow X \times X$ given by $(g, x) \mapsto (x, gx)$ being proper (i.e. closed and preimage of compact sets is compact) plus the action being free.  I have managed to prove one direction, that is, if the action is proper and free with $G$ having the discrete topology then it is properly discontinuous. I'm having trouble though with the other direction. Here is an attempt: let's denote by $\rho : G \times X \rightarrow X \times X$ the map $\rho(g, x) = (x, gx)$. Suppose $K \subset X \times X$ is compact. We wish to show $\rho^{-1}(K)$ is compact. Let $(g_i, x_i)$ be a net in $\rho^{-1}(K)$. Then $\rho(g_i, x_i) = (x_i, g_i x_i)$ admits a convergent subnet, so passing to it we may assume $x_i \rightarrow x$ and $g_i x_i \rightarrow y$. Essentially we must now find a way to prove $g_i$ converges, but I can't seem to do this. Any hints? Expected log value of noncentral exponential distribution hello everyone I have a little problem, how do I bring up the menu in the circle? when I select another object the menu becomes non-existent My disk already has 4 primary partitions, how can I install Ubuntu? Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that: $$ 2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant (2^\mathfrak{m})^\mathfrak{m} = 2^{\mathfrak{m} \cdot \mathfrak{m}} = 2^\mathfrak{m} $$ The last step relies much on the axiom of choice, because the fact $\mathfrak{m}\cdot\mathfrak{m} = \mathfrak{m}$ does. Moreover,  states that in $\sf{ZF}$ this fact is equivalent to $\sf{AC}$.  Note, by a cardinal I mean an equipollence class of an arbitrary set, not necessarily admitting a well-ordering.  That has motivated my question: what can we say about relationship of the initial equality and the axiom of choice in pure $\sf{ZF}$?    To avoid question about definition of cardinals without $\sf{AC}$, we may consider the following statement instead:     For any infinite set $X$ exists a bijection $\; F \, \colon X^X \rightarrow 2^X$   where $X^X$ is the set of all functions from $X$ to $X$, $\; 2^X = \mathcal P(X)$ is the power set, and an infinite set is such that cannot be bijectively mapped onto a finite ordinal. What is the best technique to calculate a confidence interval of a binomial experiment, if your estimate is that $p=0$ (or similarly $p=1$) and sample size is relatively small, for example $n=25$? How should an educational institute woo good faculty? What steps must an educational institution wanting to recruit good faculty take? There is a cycle in the academia:    Good doctorates from top universities invariably join other top universities as faculty members.  Good students aspire to study in top universities under excellent faculty members.   Suppose a new educational institution or department springs up and desires to set up a top-quality group, how does it break into this cycle? I suppose this could take some years, but is money-power the only thing that could be wielded by the institute to achieve its end? How can I reorder my divs using only CSS? How to create an iso from current installation I installed Ubuntu 14.04.1 on one of my laptops and now I want distribute that exact installation to a few of my friends. I tried making an iso with Gparted on a live session but it makes a 500 GB file (I've only used around 8 GB) .   Thanks in Advance ! Listings - Only highlight between delimiters I want to implement a function name color highlighting and I use the listings package. My 'regex' for this is def %s( where def (including space) is the first delimiter %s is the function name and ( is the second delimtier. Now my aim is to color only %s in purple. So basically I want to only touch the part between the delimiters and the delimiters should stay untouched.  Here is what I tried so far:  \documentclass{article} \usepackage{xcolor,listings} \usepackage[ngerman]{babel} \usepackage{fontspec} \setmonofont{Courier New}  \begin{document}  \lstset{language=Python,     escapebegin=\color{orange},     keywordstyle=\color{blue}\bfseries,     commentstyle=\color{orange!80!red},     stringstyle=\ttfamily\color{gray},     %moredelim = [is][\color{blue}\bfseries def \mdseries\color{purple}]{def\ }{(},     moredelim = [s][\color{purple}]{def\ }{(},     showstringspaces=false}‎ \lstset{literate=%    *{0}{{{\color{red}0}}}1     {1}{{{\color{red}1}}}1     {2}{{{\color{red}2}}}1     {3}{{{\color{red}3}}}1     {4}{{{\color{red}4}}}1     {5}{{{\color{red}5}}}1     {6}{{{\color{red}6}}}1     {7}{{{\color{red}7}}}1     {8}{{{\color{red}8}}}1     {9}{{{\color{red}9}}}1 }  ‎‎\begin{lstlisting}[basicstyle=\ttfamily,mathescape] def fancy_function_name(page_url = 'tex.stackexchange.com'):     return page_url.split('.')[0] \end{lstlisting}  \end{document}   This code will generate:   You can see that it matches the right place but is too greedy and colors the delimiters too. So I thought about the uncommented line: %moredelim = [is][\color{blue}\bfseries def \mdseries\color{purple}]{def\ }{(}, If I use this one and uncomment the line below I get this result:    It's nearly what I wan't but I cannot add the last, now missing ( in black color.  Does anyone has a solution for this without pretouching the code and add all the @@ symbols? Question: Let $a, b \in \mathbb{R}$ with $a &lt; b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.  I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $\displaystyle \lim_{x \to a} f(x)$ exists and if $\displaystyle \lim_{x \to a} f(x) = f(a)$. I must not be understanding it, since I am not sure how to begin showing this... Should I be trying to show that $x$ is both greater than or equal to and less than or equal to $\displaystyle \lim_{x \to a} f(x)$ ? Use Unity API from another Thread or call a function in the main Thread Connecting two Ethernet PHY without magnetics? In a (tentative) new design I want to connect two 100 Mbit/s Ethernet PHY similar to , separated by few inches on the same PCB, with same ground plane (but different power supplies). I have the choice of one of my PHY, but the other is embedded in an as-yet-unspecified PCIe to Ethernet IC (perhaps Gigabit but used in 100 Mbit/s mode), and it is critical that this PHY thinks there is a bona fide 100 Mbit/s Ethernet connection.  I could use the following         &ndash; Schematic created using   but I would like to achieve the same without the magnetics, in order to save on cost, purchasing headaches, and if at all possible power consumption.  I can imagine an RC network, perhaps as simple as in this , found thanks to a ; or maybe engineered with attenuation and lowpass, but that will not emulate the fact that with real magnetics, when there is a negative pulse on TX1P, there is a positive pulse on TX1M and RX2M. I am uncertain about if that would prevent some PHYs from working normally.  Anything to suggest that one is confident will work for practically any PHY around? Motivation for Ramanujan's mysterious $\pi$ formula The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$   Does anyone know how it works, or what the motivation for it is? I was looking for a tree or graph data structure in C# but I guess there isn't one provided.  explains a bit about why.  Is there a convenient library which is commonly used to provide this functionality?  Perhaps through a strategy pattern to solve the issues presented in the article.  I feel a bit silly implementing my own tree, just as I would implementing my own ArrayList.     I just want a generic tree which can be unbalanced.  Think of a directory tree.  C5 looks nifty, but their tree structures seem to be implemented as balanced red-black trees better suited to search than representing a hierarchy of nodes.