Save list as template using client object model in SharePoint Online
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$S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective? $S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is  a continuous function.  I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(x) : x \in S^2\}  \: \: \mathrm{ and }\: \: T_{\mathrm{min}}:= \mathrm{inf}\{ T(x) : x \in S^2\}$$ are reached by $T$ with the argument that $S^2$ is closed and bounded and therefore after the Heine-Borel-Theorem compact, and therefore since T is continuous it assumes its max and min in $S^2$.   Now i have answer the following:     1) Is there a point $x_0 \in S^2$ with $T(x_0)=T(-x_0)$ ?      2) Is there a value $T \in(T_{\mathrm{min}}, T_{\mathrm{max}})$ which is only assumed at exactly one point of $S^2$ ?   I wanted to show that $T$ is injective to get that 1) is false and 2) is true but i didn't get anywhere. Any ideas? Thanks in advance! How to draw a crisp, opaque hairline in JavaFX 2.2? I've read some great on-topic example questions on Area 51.  My questions:   Should a proposed site's "Great On-Topic Example" questions be automatically included in its beta? Will a proposed site's "Great On-Topic Example" questions be automatically included in its beta? I have a 32-bit machine, can I use programs that are meant for 64-bit processors? I am using a 32-bit machine, but I am having to use a relatively obscure semantic database which only comes in 64-bit versions.  What are my options for running it in my local dev environment?  Currently when I try to run it, I get an error message that looks like this:  cannot execute binary file   and when I check my machine info, I get this:  $ uname -a Linux slabrams-desktop 2.6.32-29-generic #58-Ubuntu SMP Fri Feb 11 19:00:09 UTC 2011 i686 GNU/Linux The computed moments of log normal distribution can be found . How to compute them? Why doesn't Quicksilver create sonic booms? Permission denied, are you root? I get this error whenever I try to install programs using the terminal:  home@ubuntu:~$ apt-get install myunity E: Could not open lock file /var/lib/dpkg/lock - open (13: Permission denied) E: Unable to lock the administration directory (/var/lib/dpkg/), are you root?   Also I'm unable to install updates using the terminal. Forest tree not symmetric Twin Paradox, calculating spacetime intervals from both perspectives I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me.  The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path.    According to Alice: $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$  Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$.     The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations.    According to Bob: $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$  Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated! Why cannot this question with an open bounty be closed? I'm struggling a bit with functions(limits, squeeze theorem, etc).  I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would appreciate if some experienced people here could advise me which books would be better to master the basics so I can move on with derivatives, etc. Limits is my main problem, so I need a book which explain this in depth so I can master it. Then I'll keep going on with the same book for other chapters.  Below is the list which I found :   Apostol T.M. Calculus Vol. 1 Burn R.P. Numbers and Functions: Steps into Analysis  Courant R. and John F. Introduction to Calculus and Analysis I - One of the better calculus text in print.  Crowell B. Calculus  Dawkins, P Calculus I and Calculus II  Garrett P. Calculus  Ghorpade S.R. and Limaye B.V. A Course in Calculus and Real Analysis  Gill G.S. Calculus Bible  Guichard D. Whitman Calculu.  Hardy G.H. A Course of Pure Mathematics  Hwang A.D. Calculus for Mathematicians, Computer Scientists, and Physicists: An Introduction to Abstract Mathematics  Santos D. Differential Calculus  Shapiro B.E. Calculus and Analysis  Spivak M. Calculus  Strang G. Calculus  Thomas G.B. and Finney R.L. Calculus and Analytic Geometry  Tranter C.J. Advanced Level Pure Mathematics  Tranter C.J. Techniques of Mathematical Analysis  Veeh J.A. Lectures Notes on Calculus  So far I narrowed my list down to Hardy, Spivak and Apostol based on further research, but I haven't got my hands on these books yet to compare them and I'm not sure if these books will be good for me.  Thanks. How to get the current date without the time? Is a string literal in c++ created in static memory and destroyed only when the program exits? Good textbooks on homological algebra If $$\tan(\frac{\pi}{4}+\frac{\theta}{2})=\tan^3(\frac{\pi}{4}+\frac{\alpha}{2})$$, then prove that $$\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$$  I tried using the fact that  $\frac{\cos A}{1-\sin A}=\tan(\frac{\pi}{4}+\frac{A}{2})$ but now not able to eliminate $\cos \alpha$ and $\cos \theta$. How should I proceed? Joining two rigged models I currently trying to join two rigged (and animated) models, which both has separate meshes, separate bones and separate animations. The first mesh is the horse while the second mesh is human:   So I am interested to join human model onto horse model to make a rider:   I think the best way would be to make horse as a parent and human as a child during the join process. In that case I would like that all horse animations (there are 3 of them - idle, walk and run) would remain as they are, while I can lose human animations, as I will need different ones for it. I was also thinking to make human root bone as a child of one horse back bones, that when horse moves up and down it would move entire human mesh together. It would be also good to keep the same weights that human bones would not start deforming horse mesh and vice versa.  I tried to use Ctrl+J by selecting various combinations between bones and meshes in object mode, however, I noticed that it messes up scaling and rotation of the child object. So I was wondering how it would be possible to properly join these models to create a rider? How can I prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain?  Also, I need to prove that its field of fractions is isomorphic to the field of rational functions $\mathbb{Q}(t)$?    (The question is taken from ) Some days ago I realized that  (as well as ) never throw an IOException when writing, flushing or closing.  Instead it sets an internal flag (trouble=true) when an error occurs. It's not possible to get the exact exception, but only if there was some exception ().  My question is: why would one want to have such behavior? Isn't that bad API design? How to fix "Duplicate sources.list entry" warning? Every time I try to do some update (sudo apt-get update) it says:   Reading package lists ... ready W: Duplicate sources.list entry http://archive.canonical.com/ubuntu/ need /partner amd64 Packages (/var/lib/apt/lists/archive.canonical.com_ubuntu_dists_precise_partner_binary-amd64_Packages) W: Duplicate sources.list entry http://archive.canonical.com/ubuntu/ need /partner i386 Packages (/var/lib/apt/lists/archive.canonical.com_ubuntu_dists_precise_partner_binary-i386_Packages) W: You have to run apt-get update to correct these problems   sources.list:   ls /etc/apt/sources.list.d/ as requested:  milton@milton-HuronRiver-Platform:~$ ls /etc/apt/sources.list.d/ emesene-team-emesene-stable-precise.list precise-partner.list precise-partner.list.save upubuntu-com-chat-precise.list upubuntu-com-chat-precise.list.save What library do I have to use such that the document can render \lt and \gt as less than and greater than signs, respectively? \documentclass{article} %\usepackage{amsmath} It does Not work ! \begin{document} \[ x\lt y \gt z \] \end{document}  What library do I have to use such that the document can render \lt and \gt as less than and greater than signs, respectively? Edit I am using MathJax library in my site under Asp.net Mvc 3. Web application is vulnerable to XSS and XSRF attacks so &lt; and &gt; should be avoided as the input from the users. My database will save LaTeX commands that can be rendered by both LaTeX compiler and MathJax. Therefore, I need \gt and \lt that already work in MathJax. Source:  Rendered: