I changed my name in minecraft and cant login
I changed my username and now I cannot play online
Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. I define a map $$\phi:D_{2n} \rightarrow \mathbb{Z}_2 \times D_n$$ by $R \mapsto (0,r^{\frac{n+1}{2}})$ and $M \mapsto(1,m)$. Then I am stuck at showing the map is bijective. By the way, do we need to show that $(0,r^{\frac{n+1}{2}})(1,m)(0,r^{\frac{n+1}{2}})^{-1}=(1,m)$ ? If we can prove this, then what can we conclude? How can I delete an email address from Outlook's email history? Can't download any app from AppStore! I'm using iPhone 6s 16 GB. My storage isn't full but still I can't download anything. Every time I wanted to start a download it keeps telling:  app name* could not be download at this time ARIMA equation interpretation I'm trying to replicate ARIMA (1,0,1)(1,0,1) equation in excel as a formula but I am not able to understand the interpretation of white noise residual e(t) or u(t).If could help me understand the interpretation and how to derive it from raw data it'll be really great.  Thanks Cancel misclicked flags How to use \def to redefine \end{split}? When I use \def to redefine \end{split}, there is an error.  \documentclass{article} \usepackage{amsmath}  \begin{document} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bs{\begin{split}} \def\es{\end{split}}  \be     \bs         a\\         b     \es \ee  \end{document}   The error is:      Runaway argument?      a\ b \es \ee       ! Paragraph ended before \split was complete.   However, if I don't use \def\es{\end{split}} and just modify the code to be  \documentclass{article} \usepackage{amsmath}  \begin{document} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bs{\begin{split}} \def\es{\end{split}}  \be     \bs         a\\         b     \end{split} \ee  \end{document}   it's OK. What's the problem? Thank you! How to simulate a rope? I'm trying to simulate a rope (that later might get pulled on the ends*) that interacts with some obstacles. So far I created a plane as the floor and a cube as an obstacle, and activated "collision" in the physics menu for both of them.   *(In the end the goal is simulating a .)  For modelling the rope, I tried using a long cylinder with enough subdivisions, and activate "cloth" in the physics menu. This resulted in the cylinder collapsing like a sleeve of a sweater, but I would like the cross section to remain roughly circular. Is there a way to directly avoid this?   Another approach I tried now was just using a a line made up of vertices and edges (with the idea of later adding modifiers to give it a thickness), but this just fell through all the collision objects as it has no faces. Is there good way to do this? Is the uncertainty principle just saying something about what an observer can know or is it a fundamental property of nature? I ask this question because I have read two different quotes on the  that don't seem to match very well. There are similar questions around here but I would like an explanation that reconciles these two interpretations specifically:   Feynman talks about the uncertainty principle in one of his lectures and mentions it as the reason why electrons don't crash into the atom's nucleus: If they did they would have an exact location and momentum which is not allowed by the uncertainty principle. In saying this it is clear that the uncertainty principle is a fundamental property of nature because it has an effect on where an electron can reside. Recently I read - somewhere else but I forgot where exactly - an account of the uncertainty principle where there was explained how we can measure position of a particle by firing another particle into it, the collision disturbs the velocity of the observed particle therefore we can not know its momentum anymore.   Now, 2) very much seems like a limitation of what the observer can know, while 1) attributes a fundamental property of nature to it (electrons don't crash into the nucleus). What is the correct way to think about this? Creating a Movie from a Series of Plots in R Area of intersection of two Annulus Given two separate annulus with centers $[C_1,C_2]$ and their corresponding radii being $[R_1,r_1]$ and $[R_2,r_2]$ respectively, larger radius being $R$. There are methods to look at whether they are overlapping or not. If they overlap how to find the overlapping area. Is there any close form expression for it?  equation of first annulus   $r_1^2&lt;(x-x_{c1})^2 + (y-y_{c1})^2 &lt;R_1^2$   equation of second annulus  $r_2^2&lt;(x-x_{c2})^2 + (y-y_{c2})^2 &lt;R_2^2$  If there are three circles overlapping, we get overlapping area as solution to set of 4 in-equations by least squares approach(Though we don't end up getting total area but, area within tangents of circles ). Contraction of quotient ideal is quotient of contractions? Can't boot into Ubuntu in Windows 10 / Ubuntu dual boot How can I scale all images in a folder to the same width? I want to scale all images in a given folder to the same width (but different appropriately scaled heights). How can I do this using  a GUI-based or command-line tool?  For bonus points, is it possible to restrict which images are scaled based on their initial width (that is, only scale images that have a width > x and/or a width &lt; y)? I'm trying to have a surface with soft body hold obects that are falling on it with rigid body physics, but for some reason, no what matter what mass I give it, the objects just fall trough the softbody enabled surface.  Is there a toggle or a value I need to change to make them stay there? I'm trying to boot Canonical's Azure 16.04 .vhd file in a VirtualBox VM and it hangs at "non blocking pool is initialized".  It works in Hyper-V, so it may only be a bug in virtual box (5.0.20). Numbering side-by-side equations or inline equations What is the correct way to pluralize an initialism in which the final word is not pluralized by adding the letter "s"? Desoldering/soldering 1/4" chip with eight pins What is the need of complex functions in wave analysis? It is commonly known that waves can be express in terms of sine or cosine function. But when I study further, I seen that for analyising the waves, it is common to use complex functions in the form $$y=y_{_0}e^{i(kx-\omega{t})}$$ where $y_{_0}$ is the amplitude, $k$ is the wave number, $\omega$ is the angular velocity and $x$ &amp; $t$ are position an time respectively. Ofcourse, I know that the function $e^{ix}$ can be written in the form $\cos{x}+i\sin{x}$ and so it is a periodic function with period $2\pi$ but my question is for what purpose we define it in terms of complex numbers? It seem to be more convenient to use real functions for real variables such as amplitude, electric and magnetic field of an electro magnetic wave, and also in quantum mechanics. What actually this interpretation means or what is the advantage of such functions? Are employments in universities affiliated (or named) with a specific religion restricted to people with the same religion? I just saw an ad in mathjobs.org about an opening in a Catholic university, see . I am not christian. So I was wondering to be eligible for employment in a university or college affiliated with a specific religion, should the applicant  have the same religion, or other people from other religions (or even non-religious people) are eligible to apply as well?