Counting in other bases
What is the best base to use?
Let $K\subseteq \mathbb{R}^{n} \times \{ 0 \}\subseteq \mathbb{R}^{n+1} $ and $v=(0,0,\cdots,1) \in \mathbb{R}^{n+1}$.      For every $x \in K$,  let $$Lx=\left \{ tv+(1-t)x \; | \ t \in [0,1] \right \}$$      Define    $$ \text{Cone}(K)=\bigcup_{x \in K}Lx. $$   namely the union of all segments from $v$ to every point of $K$.      According to all definitions that I see on books, Cone is defined:       However the exercise I'm working on  asks to find under what circumstances  $$\text{Cone}(K)=\bigcup_{x \in K}Lx$$ is isomorphic to $(K \times [0,1]) /(K \times \{ 1 \})$ (quotient space).   For example if $$K=\{(1/n,0):n\in\mathbb{N}\} \cup (0,0)$$ I can see both of sets form the same graph but I'm still unable to derive a homeomorphism between these two objets. I was curious about how other people use the this keyword. I tend to use it in constructors, but I may also use it throughout the class in other methods. Some examples:  In a constructor:  public Light(Vector v) {     this.dir = new Vector(v); }   Elsewhere  public void SomeMethod() {     Vector vec = new Vector();     double d = (vec * vec) - (this.radius * this.radius); } Self-references in object literals / initializers If you copy the contents of   httpd.conf   and then paste it into a cat command.. such as this one..  #!/bin/bash cat &gt; /test &lt;&lt; EOF pasted here.. EOF   you run into this error:  -bash: command substitution: line 1: unexpected EOF while looking for matching `'' -bash: command substitution: line 4: syntax error: unexpected end of file   perhaps the solution is to escape the dollar signs and maybe even quotes and so on..  but given that this is such a large file.. can dollar signs be automatically be escaped ?  is my only option to escape them via another program and then give it to the cat command ? Why didn't Harry himself drink Polyjuice Potion instead of all others in Deathly Hallows Part 1? In Harry Potter and the Deathly Hallows Part 1, it is shown that members of the Order of Phoenix come to rescue Harry from his house at Privet Drive. There, they all drink Polyjuice Potion and assume the form of Harry to confuse death eaters in identifying who is the real Harry.  An obvious, more safe, better choice would be to change Harry into someone else. Maybe a muggle, like his cousin Dudley. And then, safely take him to the Weasleys, maybe apparating. That way, death eaters won't ever find out. Isn't that correct?  Why didn't they take this path, and follow a rather complicated and risky one? Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$ Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction. I am trying to do a new environment or command (i tried both) where i can put in the name, description and code of a Method in c++ and it gives me a new section with the MethodName, the code listed and the description below. But i am not able to use the listings environment in a new command or new environment.. Consider my MWE   \documentclass[a4paper,11pt]{article} \usepackage{listings}  \newcommand{\method}[3] { \section{#1}\label{#1} \begin{lstlisting} #2 \end{lstlisting} #3 }   \begin{document}  \method{MethodName}{Method Void MethodName()}{Description of the Method blablabla}  \end{document}   How can I fix this?   I would prefer to do it using a macro since I don't need an ending. Image looks embossed when converted to SVG Fluent and Query Expression — Is there any benefit(s) of one over other? Batch update/delete EF5 Why I can't read a read only file? Why does  get a pass, but I can't ask my own book recommendation question? Can I ask my own book recommendation question in a way that is acceptable to the community, like the C++ book list question? Proof: $\|X\|_2\leq \sqrt{n}\|X\|_\infty$ (Euclidean Norm)  Proof that $\|X\|_2\leq \sqrt{n}\|X\|_\infty$, if $x \in \mathbb{R^n}$.   With that, proof $\|X\|_\infty \to 0 \implies \|X\|_2\to 0$  I'm stuck:  $\|X\|_2= \sqrt{x^2_1+x^2_2+...+x^2_n}\leq\sqrt{n}\sqrt{x^2_1+x^2_2+...+x^2_n}$  I don't know how to use the fact: $\|X\|_\infty=\max(|x_1|,|x_2|,...,|x_n|)$ If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ path connected? Possible Duplicate:          As the topic,if $A\subset\mathbb{R^2}$ is countable, does $\mathbb{R^2}\setminus A$ path connected??? I know the answer is it is path connected but not sure how to prove it. Can Mjolnir "stick" to anything? Is it necessary to use commas in lists before the conjunction? I have seen many sentences like this:     "Capitalism, Socialism, and Anarchism"   and this:     "She was happy, and cheerful"   Do we have to use a comma before "and", or is it incorrect?  If it is not, then why? Successfully connected to a bluetooth speaker, but how do i direct sound there? SQL query to convert columns into rows What is it called when you use many words to describe the word you want to use? What is it called when you use many words to describe the word you want to use?  I know there is a technical word that's used in foreign language acquisition that means using many words to describe a word you don't know.  I'm doing it now, just talking and beating around the bush until someone can tell me the word I'm looking for. Monotonicity in argmin function I have the following equation $$f(a)=\text{argmin}_{t\in\mathbb{R}} \sum_{i=1}^n\frac{1}{2}(1-a)(b_i-t)^2+a| b_i-t |, a\in[0,1]$$  How does one prove that the function is monotone in a?