Trouble proving a function homeomorphism in S. G Krantz's Topology I am learning Topology from SG Krantz ( Dolciaini expository text) and I am unable to prove the function in following example to be homeomorphism.   I am unable to prove it either 1-1 or onto. I have proved it continuous.  Inverse of f is  ($ \frac{y} {\sqrt{ \frac{t^2} {x }-1}}$,$ \frac{x} {\sqrt{ \frac{t^2} {y }-1}}$). It is continuous everywhere when x$\neq$ t or -t or y$\neq$ t or -t. But I have no reason to believe that it must happen. S It's my humble request to help proving me 1-1 , onto and continuity of inverse. Thank you.
Homeomorphism between the unit disc and the unit square I know that the function that describes the homeomorphism is: $f(x,y) =       \begin{cases}     \frac{x^{2}+y^{2}}{\max(|x|,|y|)}(x,y) &amp;\quad\text{if } (x,y)\ne  (0,0) \\       \text{} (0,0) &amp;\quad\text{if } (x,y) =  (0,0) \\       \end{cases}$  But it's impossible to me to show that $f(x,y)$ is an homeomorphism. Mostly i can't see the injectivity.
What's that ship in the opening credits of 'Star Trek: Enterprise'? Applying an ellipsis to multiline text How to change boot priority? $\mathbb{R}P^4$ and $\mathbb{R}P^6$ do not admit fields of tangent $2$-planes Filling triangle doesn't seem to work with PGF/TikZ I was wondering if anyone had any clue on why my triangles are not filled in red within the following piece of code:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc,3d}  \usepackage[active, pdftex, tightpage]{preview} \PreviewEnvironment[{[]}]{tikzpicture} \newcounter{number}  \begin{document} \begin{tikzpicture}[x  = {(-0.5cm,-0.5cm)},                     y  = {(0.9659cm,-0.25882cm)},                     z  = {(0cm,1cm)}] \begin{scope}[canvas is yx plane at z=0] \foreach \x / \y / \label in {1 / 0.4 / 1, 0.9 / 0.4 / 0, 1 / 0.5 / 1, 0.7 / 0.3 / 0, 0.8 / 0.4 / 0, 0.9 / 0.5 / 0, 1 / 0.6 / 1, 0.7 / 0.4 / 0, 0.4 / 0.4 / 0, 0.6 / 0.4 / 0, 0.5 / 0.4 / 0, 0.8 / 0.5 / 0, 0.9 / 0.6 / 0, 1 / 0.7 / 1, 0.7 / 0.5 / 0, 0.5 / 0.5 / 0, 0.4 / 0.5 / 0, 0.6 / 0.5 / 0, 0.8 / 0.6 / 0, 0.9 / 0.7 / 0, 1 / 0.8 / 1, 0.7 / 0.6 / 0, 0.6 / 0.6 / 0, 0.5 / 0.6 / 0} { \addtocounter{number}{1} \pgfmathparse{\value{number}}\let\k\pgfmathresult \draw (8 * \x, 8 * \y) node[inner sep=0, outer sep = -0.1pt,color=black,fill,circle] (node-\k) {}; };  \foreach \x / \y / \z / \trash in { 4/5/8/0, 9/11/16/0, 9/16/17/0, 11/10/18/0, 11/18/16/0, 10/8/15/0, 10/15/18/0, 8/5/12/0, 8/12/15/0, 5/2/6/0, 5/6/12/0, 2/1/3/0, 2/3/6/0, 17/16/24/0, 16/18/23/0, 16/23/24/0, 18/15/22/0, 18/22/23/0, 15/12/19/0, 15/19/22/0, 12/6/13/0, 12/13/19/0, 6/3/7/0} { \draw[fill=red]  (node-\x) --  (node-\y) --  (node-\z) -- cycle; } \end{scope} \end{tikzpicture} \end{document}) Floating Objects and Weight The Situation:  A ball is placed in a beaker filled with water and floats. It is also attached to the bottom of the beaker via a string.   The Question:  The ball is attached to the beaker, thus making the ball and beaker (and string included) a system. However, it is floating. Does the ball add to the weight of the beaker? Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows (I'm on a mobile device, so I can't type absolute values. Bear with me.) $$\left|x_{n+1} - x_n\right| \leq \frac{1}{n^2}.$$ So, we have $$ \left|x_m -x_n\right| \leq \sum_{k=n}^{m} \frac{1}{k^2}$$ which is itself Cauchy, etc., etc. But, of course, I can't just use absolute values like that. One thing I have shown is that $x_n$ is bounded. Inductively, one may show $$ \limsup_{n \to \infty} x_n \leq x_k + \sum_{k=n}^{\infty} \frac{1}{k^2},$$ although I'm not sure this helps or matters at all. Thanks in advance.  Disclaimer I've noticed that asking a large number of questions in quick succession on this site is often frowned upon, especially when little or no effort has been given by the asker. However, I am preparing for a large test in a few days and will be sifting through dozens of problems. Therefore, I may post a couple a day. I will only do so when I have made some initial, meaningful progress. Thanks. How to calculate relatedness in haplodiploid organisms (mainly full sisters and full brothers)? I have tried to calculate the relatedness for haplodiploid organisms, but cannot understand the calculations behind full sister and full brother.  (taken from  I have managed to solve the other cases. Since female will have ZZ:  female will share 1/2 of her genes with daughter female will share 1/2 of her genes with son female will share 1/2 of her genes with mother female will share 1/2 of her genes with father  Since male will have only Z:  male will share 1 (100%) of his genes with his daughter male won't share anything with his son (unfertilized eggs leads to male in haplodiploidy system) male will share 1(100%) of his genes with his mother male won't share anything with his father (unfertilized eggs leads to male in haplodiploidy system)  Even though I figured out how others work, I wasn't able to calculate r for fullsister fullbrother? How can it be 3/4,1/4,1/2,1/2? It seems that the question has eventually become a series of questions....    An example goes as:     The 1st and (the) 2nd paragraphs of the article are extremely long.   Another example:     What are the situation, (the) task and (the) result of your story?   Is it necessary to use the in ()? Is it "re-offend" or "reoffend"? I want to know whether there is a hyphen in the word re-offend, or if it is spelt reoffend. I looked in Oxford English dictionary and the word "reoffend" appears, but then I checked Merriam-Webster and the word is not in there.  So is reoffend a word or is it re-offend?  I'm in Australia where we observe UK English if that matters. Characterization convex function. TikZ rectangular node with different rounded corners help with some Hilbert style proofs in a propositional logic axiom system. I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. How to instruct Windows 8 NOT to perform a fast shutdown? I have enabled fast boot in my copy of Windows 8 but I would like to be able to specifically instruct Windows not to perform a "fast shutdown" whenever I want (without disabling fast boot altogether).  I need to be able to do this because I dual-boot Ubuntu and therefore can't mount an NTFS partition as read/write if Windows has performed a "fast shutdown". (Well okay, I can force the partition to be mounted read/write, but then I could run into data loss/corruption problems.) How to construct a relative path in Java from two absolute paths (or URLs)? Given two absolute paths, e.g.  /var/data/stuff/xyz.dat /var/data   How can one create a relative path that uses the second path as its base? In the example above, the result should be: ./stuff/xyz.dat I need to enter the command line alone with no GUI to install the official NVIDIA drivers. I'm trying to get my computer to work over HDMI and someone on Youtube shows how but you have to be in the Nvidia settings program. This is something I just realised I don't entirely know the answer to, so I'm going to ask it here as I think it's interesting.  In most non-scientific writing on depth of field, diagrams generally show camera and subjects as being perfectly parallel, e.g.    However, is this a more accurate representation of the focal plane?    Are there ways to optically alter the shape of the focal plane?  Note: Obviously these diagrams are two dimensional, but I'm assuming in the second diagram the shape would be spherical with the sensor at the center. I have a filename in a format like:     system-source-yyyymmdd.dat   I'd like to be able to parse out the different bits of the filename using the "-" as a delimiter. Most riders seem to say it's their groin, neck or lower back that starts hurting first after a decent distance has been ridden. For me it is my wrists.  I have no history of wrist pain (strains, breaks etc) so it can't be explained by that and I try to adjust my hand positions across my handlebars as often as I can.  My question has 2 parts:   1. Is there anything I can do to minimise pain in terms of riding position, gloves, exercises etc which will reduce soreness?  2. Why might this pain be occurring?  *I do all of my riding on paved roads on a road (Specialized Allez) bike. Difference between >> and >>> in verilog?