Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$?
$\mathbb{Q}^{\times}$ is the group of rational number without $0$ under multiplication, and $\mathbb{Z}^{n}$ is the free abelian group of rank $n$. Show that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$. I tried to assume that there is an isomorphism and to get a contradiction.. didn't succeed
How are first digits of $\pi$ found? Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am talking about accurate digits by either multiplication or division or any other operation on numbers.  Then how are the first digits of $\pi$ found -     3.1415926535897932384626433832795028841971693993...   In fact, more than 100,000 digits of $\pi$ are found (sources - )  How is that possible? If these digits of $\pi$ are found, then it must be possible to compute $\pi$ with some operations. (I am aware of breaking of circle into infinite pieces method but that doesn't give accurate results.)  How are these digits of $\pi$ found accurately? Can it be possible for a square root of some number to be equal to $\pi$? Images are escaping from the help center privilege page Images included in the fully expanded privilege pages are escaping from the body.    This is most noticeable on the  (which I know is going away soon anyway):    But is also apparent on:       In the edit page preview, the images sit correctly within the borders.  Chrome 27, Windows 7 When should we consider a dataset as imbalanced? I'm facing a situation where the numbers of positive and negative examples in a dataset are imbalanced.  My question is, are there any rules of thumb that tell us when we should subsample the large category in order to force some kind of balancing in the dataset.  Examples:   If the number of positive examples is 1,000 and the number of negative examples is 10,000, should I go for training my classifier on the full dataset or I should subsample the negative examples? The same question for 1,000 positive example and 100,000 negative. The same question for 10,000 positive and 1,000 negative. etc... It's been some weeks now, whenever I try to used my Youtube app, it tells me it's impossible to connect to Youtube. Since it's an iPad 1, I have had trouble finding information online.  Did Google or Apple stop supporting the app? Is it finally obsolete? Is there another app I could use that works on iOS 5?  Just to be clear, I did not tinker with my device. I did not change its parameters, I did not jailbreak it, it connects to the Internet just fine and I've tried the app on different Internet connections. Also, this is not a login problem, the app just refuses to let me browse videos, whether I'm logged in or not.  Thanks for your help. Baking cloth simulation use only 20% of cpu ressources, how to get 100%? Baking a cloth use only 20% of CPU, inactive process of cpu is 78%, is there a way to use those extra 78% ? Blender 2.75 Windows 10 Some doubts about photons How to determine the amount of experience needed for leveling up Related questions under this topic might include:   How should I base how much experience points it takes to get to the next level? What factors are involved in this? How do I keep the player not entertained long enough till he levels up? How can I vary the way the player gets experience points? The most common way to refer to non-immediate family? Is it true that The Doctor can only regenerate 12 times? "Repository doesn't support architecture" when trying to update on Landscape client Making the background of an image transparent in Gimp I have some jpg images and I need for each of them to have a transparent background.  I have GIMP as a graphic tool.   I need to put them in a web page and I need to see just the main element and have a transparent background. I'm trying to fill empty space in my mesh, and I looked at the knife tool page on the Blender Wiki, and I'm not sure if there's a way to make it so that a new edge is made using the knife tool. I took a screenshot, and hopefully it'll explain better than I can of what I'm trying to do. Oh and I'm using Blender version 2.71, by the way. I love the subset of collective nouns known as the terms of venery. These are collective nouns specific to a particular group of animals. Some of the more inventive examples are: a murder of crows, a crash of rhinos, a mischief of mice, and a puddling of ducks (specifically swimming ducks).  Is there a standard for these group names and is there a central reference for them as with other zoological taxonomy?  I can see in the Wikipedia articles that the practice has its origins in Medieval Hunting tradition, but the Kangaroo (Mob), for instance would have been unknown at that time, so how do new collections come into being? Remove numbering from Section and Subsection Why upload to academic preprint sites like arXiv? The recent question  made me realize that I don't have any understanding of why one would upload to a preprint site.  (I don't work in a field where they seem to be common.)  Why would one be concerned about getting a manuscript on arXiv, if it's already under review with a journal?   Is it because of the long publishing timeline of journals?  (That could explain why computer science &mdash; which publishes much more in conferences &mdash; doesn't seem to do much with them.) Can we have an option to disable notifications on community wiki posts? Summation of Central Binomial Coefficients divided by even powers of $2$ Whilst working out  the following summation emerged: $$\sum_{m=0}^n\frac 1{2^{2m}}\binom {2m}m$$ The is equivalent to  $$\begin{align} \sum_{m=0}^n \frac {(2m-1)!!}{2m!!}&amp;=\frac 12+\frac {1\cdot3}{2\cdot 4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}+\cdots +\frac{1\cdot 3\cdot 5\cdot \cdots \cdot(2n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot 2n}\\ &amp;=\frac 12\left(1+\frac 34\left(1+\frac 56\left(1+\cdots \left(1+\frac {2n-1}{2n}\right)\right)\right)\right) \end{align}$$  and terms are the same as coefficients in the expansion of $(1-x)^{-1/2}$.  Once the solution  $$ \frac {n+1}{2^{2n+1}}\binom {2n+2}{n+1}$$ is known, the telescoping sum can be easily derived, i.e.    $$\frac 1{2^{2m}}\binom {2m}m=\frac {m+1}{2^{2(m+1)-1}}\binom {2(m+1)}{m+1}-\frac m{2^{2m-1}}\binom {2m}m$$  However, without knowing this a priori, how would we have approached this problem? Modernist Cuisine recommends that for longer sous vide cooking (times greater than 36 hours) that bags are vacuum sealed rather than just using the water displacement method to remove the air from a ziploc bag.  What is the reason for this? Will I be losing something essential if I just use a regular ziploc bag without vacuum sealing? The question is:      Prove that if $p \in \mathbb{Z}$ is irreducible, then $p$ is also prime.    Irreducible is defined as follows: $n \in \mathbb{Z} $ is irreducible iff the only way to write $n=ab$ for some integers $a$ and $b$ is if $a = \pm1$ or $b = \pm1$. Prime is defined as follows: an integer $n$ is prime iff whenever $a,b \in \mathbb{Z} $ and $p\mid ab$ then it must be true that $p\mid a$ or $p\mid b$ (or both).    Hint: Prove that if $n\mid ab$ but $n$ does not divide $a$, then $\gcd(a,n) = \pm1$. Then use Bézout's theorem and prove $n\mid b$. Windows 8 removes Grub as default boot manager I have a Sony VAIO S series with Windows 8 preinstalled, and I installed Ubuntu 12.10 on a new partition. When the PC boots Grub is display, and I can choose between Windows 8 (didn't worked at the beginning, had to change Grub settings) and Ubuntu 12.10. When I select Ubuntu 12.10 everything is OK, but when I use Windows 8, and I restart Grub isn't displayed anymore (Windows 8 boots immediately).  I booted a Ubuntu 12.10 LiveUSB, used efibootmgr, and found out that Windows 8 (each time it boots) changes EFI default bootloader from Grub to Windows Boot Manager.  How do I change this in Windows 8? I read something about bcdedit, but it does not work or I don't know how to use it...