How I get value "&" QueryString?
Escaping ampersand in URL
Can you raise $\pi$ to a real power to make it rational? Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$. Given any set $X$, let $P(X)$ be the set of all subsets of $X$, and let $\{0,1\}^X$ be the set of all functions $X \rightarrow \{0,1\}$. Construct a bijection (and its inverse) between P(X) and $\{0,1\}^X$.    Let us define $f: P(X) \rightarrow \{0,1\}^X$ by $f(\alpha) = g_{\alpha}$ where $\alpha \in P(X)$ and $g_{\alpha} : X \rightarrow \{0,1\}$ is defined by  $$g_{\alpha}(x) =\begin{cases}        1 &amp; x \in \alpha \\       0 &amp; x \in X-\alpha     \end{cases}$$  We need to check if this function is well-defined. Let $\alpha, \beta \in P(X)$. If $\alpha=\beta$, then $\alpha$ and $\beta$ correspond to the same set, and so correspond to the same function since both $g_{\alpha}$ and $g_{\beta}$ send that set to 1 and the ($X-$that set) to 0.   If we have a function $g_{\alpha} \in \{0,1\}$, then $\alpha$ is a set in $X$ that is sent to 1. But since $P(X)$ is the set of all sets of $X$, we must have $\alpha \in P(X)$. So the function is surjective.  Finally, if $g_{\alpha}=g_{\beta}$ then both $g_{\alpha}$ and $g_{\beta}$ send the set $\alpha$ or $\beta$ to 1. So $\alpha$ and $\beta$ must be same set. In other words, the function is injective; and hence bijective.   The inverse function is $f^{-1}: \{0,1\}^X \rightarrow P(X)$ where $f^{-1}(g_{\alpha}) = \alpha$.     Another way to prove this is to first construct the inverse function, show that it is well defined, and then show that $ff^{-1}$ and $f^{-1}f$ are the identity functions, right?  So if we let $g_{\alpha}=g_{\beta}$, then $\alpha=\beta$, because otherwise $g_{\alpha} (\beta)=0$ and $g_{\beta} (\beta)=1$; a contradiction.   Now we have $f \circ f^{-1} (g_{\alpha}) = f(\alpha) = g_{\alpha}$ and $f^{-1} \circ f(\alpha) = f^{-1}(g_{\alpha}) = \alpha$ , and so $f$ and $f^{-1}$ must be bijective. How to save user-entered line breaks from a TextArea to a database? How do I save user-entered line breaks from a &lt;textarea&gt; HTML element to a database?  It always removes the line breaks. Best Algebraic Topology book/Alternative to Allen Hatcher free book? Allen Hatcher seems impossible and this is set as the course text?  So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online.   Any good intro to Algebraic topology books?  I can find a big lists of Algebraic geometry books on here. On a very old thread on Maths overflow someone recommended that a person should read James Munkres Topology first, then you should read Allen Hatcher book.   It just seems like Rudin's book but crammed with ten times more material. No sound ALC294 Asus ROG Strix 512 Ubuntu 20.04.01 Can you use Dispel Magic on the source of a spell (ie, the caster)? Example sentence:     I love when your dog just lets you sit there to pet them. You don’t necessarily know if they are enjoying it, but they love you enough to just sit there with you for a bit.   Is this correct? We assume the words "you" and "your" refer to the speaker of the sentence, and not to the listener, as second-person usually does. But it also refers to dog owners in general. I have always been curious about this. Clickable Table Footnote I'm trying to have some clickable footnotes within tables. I tried using \footnote{}, it does create a link but there's no footnote text associated with it. \footnotemark[number] along with \footnotetext[]{} does create a fotenote but then there's no hyperlink. (I would like to make it clear that I'm trying to have footnotes at the bottom of the page and not footnotes within the table.)  Here's a small code snippet:  \documentclass{report}  \usepackage{hyperref} \begin{document}  \begin{table}[h!]     \centering     \begin{tabular}{|r|r|r|}         \hline         A\footnote{Test!} &amp; B\footnotemark[4] &amp; C\footnotemark[5]\\         \hline         123 &amp; Tim &amp; Jane\\         \hline         456 &amp; Tom &amp; Jung\\         \hline     \end{tabular}     \caption{A table} \end{table}  This is a clickable footnote\footnote{You clicked me!}.  \footnotetext[4]{This is B} \footnotetext[5]{This is C}  \end{document} What is this "answer your own question" jazz? In many C/C++ macros I'm seeing the code of the macro wrapped in what seems like a meaningless do while loop.  Here are examples.  #define FOO(X) do { f(X); g(X); } while (0) #define FOO(X) if (1) { f(X); g(X); } else   I can't see what the do while is doing.  Why not just write this without it?  #define FOO(X) f(X); g(X) Looking for a command to create the geometry line symbol (\overline with arrows on both ends) I need a way of encoding the geometry line symbol. $\overline{AB}$ gives me the line segment symbol and $\overrightarrow{AB}$ gives me the ray symbol, I need a way to get a bar over the letters that has arrows on both ends. $\overrightleftarrow{AB}$ does not work.   Thanks! How do I start skype in the system tray? Using Xpath With Default Namespace in C# I've got an XML document with a default namespace. I'm using a XPathNavigator to select a set of nodes using Xpath as follows:     XmlElement myXML = ...;   XPathNavigator navigator = myXML.CreateNavigator(); XPathNodeIterator result = navigator.Select("/outerelement/innerelement");   I am not getting any results back: I'm assuming this is because I am not specifying the namespace. How can I include the namespace in my select? If every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal operator Let $V$ be a finite-dimensional inner product space over $\mathbb{C}$ and $T: V \to V$ a linear transformation.   Show that if every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal.   We need to show that $\forall v \in V,\ TT^*v=T^*Tv$.  I started by picking $v$ to be an eigenvector of $T$, hence $Tv=\lambda_1 v$, and $T^*v=\lambda_2 v$. Therefore: $$TT^*v=T(\lambda_2 v)=\lambda_2 Tv=\lambda_1 \lambda_2 v=...=T^*Tv$$  But I'm stuck when $v$ is not an eigenvector.  (Also, this question is taken from a chapter about Jordan forms, but I can't see how it relates) How to find a function in $ L_{2}(0,\infty)$ but not $L_{q}(0,\infty)$, $q\neq 2 $ How do I join two lists in Java? I just ordered the Nvidia GTX card. I have a dilemma, though. Should I keep using the driver which is available in "additional drivers" in Ubuntu, or should I install the driver from the Nvidia site?  So which driver is the best for me? Number of possible marriage arrangements if Marriage Condition is satisfied Suppose I have a set of $n$ boys and each subset of $1\leqslant k \leqslant n$ boys knows at least $k$ girls then this means that the set of boys $B$ satisfies the marriage condition.  I am doing an exercise from a Graph Theory book and it says we can show by induction on $n$ that the number of possible marriage arrangements is $\geqslant t!$ if every individual boy knows at least $t \leqslant n$ girls or there are $\geqslant t!/(t-n)!$ possible marriages if each individual boy knows at least $t&gt;n$ girls.   I am struggling to see how the induction hypothesis can be implemented to prove the $n+1$ case. When you swap from the $n$ to $n+1$ case the conditions change and I can't see the link between them.  Would anyone be able to point me in the right direction? Downgrading from OS X 10.9.2 to 10.9 On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $