im trying to show that $D_6  \cong D_3\times \mathbb Z_2$ as groups.  $D_n$ is the is the dihedral of order 2n  I know that it is possible to form all of the points in $D_6$ and in $D_3$ as compositions of $R$ the rotation operation (such that every vertex move one to the left) and T the reflection on one of the symmetry axis, so i tried to write it done and joining point that act the same, but for no use.
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?
barchart with pgfplot - missing bar I am using PGFplot to make a barchart. There are 4 entries in my csv file and unfortunately I have only 3 bars in my figure.   my LaTeX code:  \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}[             x tick label style={/pgf/number format/1000 sep=},                 xlabel=Sequence,             ylabel=Heigth,                 ymin=0.7,             enlargelimits=0.05,             legend style={at={(0.5,-0.15)},                 anchor=north,legend columns=-1},             ybar interval=0.7,             width=\textwidth,             ]             \addplot table [col sep=comma,trim cells=true,y=height] {numericaltests/data/test.csv};              \addplot[red,sharp plot,update limits=false]                 coordinates {(-1,1.00) (34,1.00)}                 node[above] at (axis cs:15,1.00) {};             \legend{Ca2Fe2As2}         \end{axis}     \end{tikzpicture}     \caption{Sequence}     \label{fig:sequenceHeight} \end{figure}   my test.csv file:  sequence,height 1,2.0 2,1.5 3,1.8 4,2.4   Barchart: Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$ I want to solve this system of equations, I have been out of Maths for a long!!  $$\sqrt{x}+y=7$$  $$\sqrt{y}+x=11$$  Just wondering easiest step to find values for $x$ and $y$ from the above equations? Force a Blender operator to always start with default values I'm writing an operator for which it is crucial that every time it gets called all the properties are set to their default values.   The ops.object.delete has 1 param. use_global. It is always false, no matter what the user put in last time it got called. I would like to replicate exactly that.   (For example: I want to modify delta scale. The default val for my_factor is 1, so nothing happens until the user alters that number. If the user calls the operator again, it will have stored whatever number the user has used last, however I want it to always start with 1.) Extremely difficult logic question using conditional statement In this problem, we will only consider (real-valued) functions with domain $\mathbb{R}$. We define two new concepts. Let $f$, and $g$ be two functions.  We say $f$ is a rival of $g$ if   &emsp;&emsp;&emsp;&emsp; $\exists x\in \mathbb{R} \ \text{s.t.} \ \forall y\in \mathbb{R}, x \lt y \implies \lvert f(x)- g(x)\rvert\lt\lvert f(y)-g(y)\rvert$    We say $f$ is a frenemy of $g$ if  &emsp;&emsp;&emsp;&emsp; $\forall x\in\mathbb{R} \ \text{s.t.} \ \  x \lt y \ \ \text{AND} \ \ \lvert f(x)-g(x)\rvert \lt \lvert f(y)-g(y)\rvert$  Below are four claims. Which ones are true and which ones are false? If a claim is true prove it. If a claim is false, show it with a counterexample.  a) If $f$ and $g$ are any two functions and $f$ is a rival of $g$ then $f$ is a frenemy of $g$. b) If $f$ and $g$ are any two functions and $f$ is a frenemy of $g$ then $f$ is a rival of $g$. If $f$ and $g$ are any two functions and $f$ is a rival of $g$, then $g$ is a rival of $f$. Let $f$, $g$, and $h$ be any three functions. If $f$ is a frenemy of $g$ and $g$ is a frenemy of $h$ then $f$ is a frenemy of $h$  What are the four claims stating? Especially for c), the concept is not even present I am having a misunderstanding on how to solve them? Multivariable limit proof: $\lim\limits_{(x,y)\rightarrow (0,0)}\frac{\left|x\right|^a\left|y\right|^b}{\left|x\right|^c + \left|y\right|^d} = 0$ I have some trouble with this: Show that if $a, b \ge 0$, $c, d &gt; 0$ and $\frac{a}{c} + \frac{b}{d} &gt; 1$ then   $$\lim_{(x,y)\rightarrow (0,0)}\frac{\left|x\right|^a\left|y\right|^b}{\left|x\right|^c + \left|y\right|^d} = 0$$  we were not really shown how to evaluate multivariable limits other than trying different paths, which wouldn't work when the limit exists, or by applying the squeeze theorem which is my initial plan. Unfortunately I cannot find a proper bound for the denominator and so I am quite lost right now. Any help would be appreciated. Before:  After:   The pros:  Speeds up the tag synonym voting process. Easier to locate (as opposed to navigating to each individual tag wiki page). Encourages users to vote on more than one at a time.  The cons:  Serial up/downvoting. But this really isn't even an issue because, (1) only experienced users can vote on tag synonyms, and (2) more than one vote is required to approve a tag synonym, so the occasional serial up/downvoter won't significantly impact the overall process. What are the prerequisites to encounter the legendaries in ORAS? There are a lot of legendaries available in ORAS, however a lot of them will only show up/be available if certain prerequisites are met.  I know Groudon/Kyogre are available through the Story, and      Rayquaza &amp; Deoxys   through the postgame Delta episode, but what about the other available legendaries? What do I need to do to catch them? What is the need of exponential generating functions on combinatorial problems? I've been introduced in my last lectures. And for the following problem:     Having 3 different types of books $a,b,c$ in how many ways can we take four different books putting them in a shelf such that the book $a$ can only be taken at most $1$ time, $b$ can be taken at most $3$ times and $c$ can only be taken at most $2$ times?   The ordinary generating function for this problem is:  $\quad \quad \quad \quad \quad \quad \quad $  Looking at the exponential generating function of the same problem, the only part of interest is the coefficient of $x^4$, which is:  $$\left(\frac{ab^3}{1!3!}+\frac{b^3c}{3!1!}+\frac{ab^2c}{1!2!1!}+\frac{b^2c^2}{2!2!}+\frac{abc^2}{1!1!2!}\right)$$ And then multiply it by $4!/4!$. But why do I need an exponential generating function? Isn't only needed to take the coefficient of $x^4$, divide each term of it by $a_1!a_2!\dots a_n!$ with $a_n=\text{power of a,b,c,}\dots$ and then multiply it by $4!4!$? It's not clear why such complication is needed.  By complication I mean to expand that generating function with the factorials, I could expand the ordinary generating function and then artificially add the factorials. Can I apply for a Schengen visa at a country where I'm not a citizen? If so, under what conditions? I'm a citizen of country A who would like to apply for a Schengen visa at a consulate in country B. When is this possible? Continuous v. per-chapter/section numbering of figures, tables, and other document elements Some document elements (e.g., figures in the book class) are numbered per chapter (figure 1.1, 1.2, 2.1, ...). How can I achieve continuous numbering (figure 1, 2, 3, ...)?  And vice versa: Some document elements (e.g., figures in the article class) are numbered continuously. How can I achieve per-section numbering?  \documentclass{book}% for "vice versa" variant, replace `book` with `article`  \begin{document}  \chapter{foo}% for "vice versa" variant, replace `\chapter` with `\section`  \begin{figure} \centering \rule{1cm}{1cm}% placeholder for graphic \caption{A figure} \end{figure}  \end{document}   Bonus question: Is it possible to adjust the numbering of the sectioning headings themselves? Can I, e.g., switch from per-chapter to continuous numbering of sections in the book class? Increase progress bar each second -- error How to preserve non-ASCII characters in ogr2ogr import? public class doublePrecision {     public static void main(String[] args) {          double total = 0;         total += 5.6;         total += 5.8;         System.out.println(total);     } }   The above code prints:  11.399999999999   How would I get this to just print (or be able to use it as) 11.4? I stumbled across this paragraph in a paper:     Hence, user b cannot decrypt C   directly. But using e and d , user b   can quickly factor N.   How is  it possible to speedup the prime factorization when knowing e (public key) and d (private key)?  For clarification:  RSA provides us with these equations:  $n = pq$   $\phi = (p-1)(q-1)$   $gcd(e, \phi) = 1$   $de = 1\pmod{\phi}$   In order to determine $p$ and $q$ an attacker has to factor n which is not feasible. However the paper stated that it is easy to reconstruct $p$ and $q$ when a person knows both (his) private and public keys. How should a physics student study mathematics? According to the Wikipedia article on  the sample SD   $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$$  is a biased estimator of the SD of the population. It states that $E(\sqrt{s^2}) \neq \sqrt{E(s^2)}$.   NB. Random variables are independent and each $x_{i} \sim N(\mu,\sigma^{2})$   My question is two-fold:   What is the proof of the biasedness? How does one compute the expectation of the sample standard deviation    My knowledge of maths/stats is only intermediate. There are several reasons to be looking for a ROM:   Upgrading or downgrading your stock ROM Looking for firmware from a different region/language Flashing a custom ROM for different features, removing bloat, etc.   But where can one find ROMs in the first place?  What sites are likely to have the ROM I need?  ; please add to the wiki answer instead of posting separate answers.  See also: Can't boot without Flash Drive plugged in I had Ubuntu 12.04 Beta installed on my computer. When 12.04 was finally released, I made a bootable USB Flash Drive using Startup Disk Creator.  Then I decided to check if this drive works properly and to reinstall the system on my desktop. I must add, my desktop behaves a bit strange when it comes to bootable USB's, it recognizes them as HDD.  In BIOS I changed priority of boot so USB Flash Drive (recognized as HDD) was first. Successfully booted, I installed Ubuntu. Everything worked fine, but...  Now I cannot boot from my real HDD. Every time I want to boot, I put the USB Flash Drive into my computer, boot, safely remove it and everything works.  What do I need to do to repair boot? Using "on" vs. "in"? Consider the following sentence:     I am developing an application to be installed on Android.   And this:     This has been a major flaw in Android...   To be clear I am unsure of the usage of "in" vs. "on". When should I use "in" and when to use "on"? Why are our planets in the solar system all on the same disc/plane/layer?