I'm studying the basics of quotient groups. I understand that if you build a quotient set from cosets of a group and the subgroup you are using to build them is normal then you end up with a group.  I fail to see why the fact that we can define operations in the quotient set and make it into a group is meaningful.  What is the motivation to do it? Either historical perspective, practical perspective or any other to understand why we care about this is welcome.
The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group, I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question. My Question is :   Why is the name Quotient Group kept,normally in the case of division, let us take the example of $\large \frac{16}{4}$ the Quotient of the Division is '$4$' which means that there are four '$4$'s in $16$, I mean we can find only $4$ elements with value $4$     So how can we apply the same logic in the case of Quotient Groups,like consider the Group $A$ and normal subgroup $B$ of $A$, So if $A/B$ refers to "Quotient group", then does it mean:           Are we finding how many copies of $B$ are present in $A$??, like in the case of normal division, or is it something different ??       I understood the Notion of Cosets and Quotient Groups,b ut I want a different Perspective to add Color to the concept. Can anyone tell me the necessity and background for the invention of Quotient Groups?  Note: I tried my level best in formatting and typing with proper protocol,if in case, any errors still persist, I beg everyone to explain the reason of their downvote (if any), so that I can rectify myself, Thank you.
In my textbook, a relationship is plotted between the resistance of the material and the temperature, and the temperature coefficient is defined as the slope of that graph divided by an arbitrary resistance $R_1$ on the graph. Does that mean that the coefficient varies depending on the temperature I choose to divide the slope by?  If so, how does it stand as a valid reference to the material’s resistance growth rate with temperature?  If not, then where did I go wrong with this train of thought? Does every Abelian group admit a ring structure? Importing single symbol from MnSymbol How to collapse consecutive numbers into ranges? Why is Facebook PHP SDK getUser always returning 0? Doom-style 2.5D Movement Animations I've started working on a demo for my 2.5D game. For a basic scene I figured I would just use the good old fashioned doom sprites textured to a double sided plane. Simple enough. But I've been scratching my head as to how to display the proper sprite animation relative to the player viewing the sprite...if that makes since.  For example suppose I have three players in the scene Players A, B, and C as such:    Player C should see player A as: Here's an example of what I think player A should look like from C's perspective:    While player B would see player A as follows:    My current idea is that for any given player, I will need to know the current direction of each other player, and where they are located/moving to relativly. From this, I could calculate which sprite image is required.  Is this correct? What improvements/optimizations could be made?  ** EDIT ** Total number of directions will be 8 as depicted below:    For each direction there are two sprites (to show walking movement, sprites for the right hand directions are mirrors of the left):    ** EDIT 2 ** Got this working and have a simple javascript/threejs example for anyone interested: How do I blend multiple photos to remove random people from the background? I'm very excited about the new  for Android/iPhone. Let's say you're taking a picture of your friend in front of a landmark with several people walking by. The app allows you to remove the people from the photo, leaving only the background and your friend. The app isn't just doing a clone of the surrounding image to paint over unwanted subjects. It's actually taking multiple images, detecting which objects are moving, and letting you select them for removal. This sounds awesome! But I don't want to be limited to taking pictures with a smart phone.  I would like to take multiple photos of the same scene. (Maybe with a tripod, but it would be better if I didn't have to use a tripod.) Then I would somehow blend them together to remove unwanted elements. How do I do that? I know there are tons of photoshop tutorials out there, but all the ones I find just tell you how to use the clone tool. Let $SL_n(\mathbb{R})$ be the set of $n\times n$ matrix with determinant one. Where can I find a proof that $[SL_n(\mathbb{R}),SL_n(\mathbb{R})]=SL_n(\mathbb{R})$? I've seen this result mentioned in several places, but couldn't find a proof.  For reference, the commutator $[SL_n(\mathbb{R}),SL_n(\mathbb{R})]$ is defined as the group generated by the set of all elements of the form $a^{-1}b^{-1}ab$, with $a,b\in SL_n(\mathbb{R})$. How to restore file copy / transfer status window? I have used so many things like url removal request and deleted the pages etc., but still the errors are shown in my webmaster tools.  How can I remove the urls completely from Google? Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model Is it possible to detect a virus with taskmanager? If I would have a running virus on my system, would I be able to see the process in taskmanager? I mean, would it be possible for a running virus to circumvent the taskmanager so the process doesn't appear in the tasklist of windows7?  Or in other words. If I really now all the processes in taskmanager to be secure, I also know that my PC is clean? I want to prove the definition of limit of a function from definition (1) to definition (2).  (1) A function $f(x)$ is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for every sequence $\{x_n\}$ with limit $x^*$, the limit $\lim_{n-&gt;\infty}f(x_n)$ exists and is equal to $f^*$. (2) A function is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for each $\epsilon &gt;0$, there exists a $\delta_\epsilon &gt;0$ such that $|f(x)-f^*| &lt;\epsilon \forall |x^*-x|&lt;\delta_\epsilon$  I am confused about the steps. Suppose I am given an $\epsilon &gt; 0$ and $x^*$. I have to show the existence of a $\delta_\epsilon &gt;0$ such that for all $x$ satisfying  $|x^*-x|&lt;\delta_\epsilon$ I can construct a sequence $x_n$ which converges to $x^*$ and $x_n=x$ for some $n \geq N$. Then I can say that $f(x_n)-f^*=f(x)-f^* &lt; \epsilon$. I think I may have made some mistake about the steps.  My main confusion is the following : Suppose I choose $\delta_\epsilon=\epsilon$. Then if a sequence $\{x_n\}$ converges to $x^*$ then $x_n=x$ (where $x$ satisfies $|x-x^*|&lt;\epsilon$) for some $n\geq N$. Now, $\lim_{n-&gt;\infty}f(x_n)$ exists implies there exists an $M$ such that $\forall n\geq M$, $f(x_n)-f^* &lt;\epsilon$. Now, I need to show that for some $n \geq M$ indeed $x_n=x$. How do I show that ? This $M$ and $N$ are different. How to show that when the linear transformation $\Phi =\psi \circ \phi : E \rightarrow G$ is injective, then $\phi$ is also injective? How to show that when the linear transformation $\Phi =\psi \circ \phi : E \rightarrow G$  (where $\phi : E \rightarrow F$ and $\psi : F \rightarrow G$) is (1) injective, then $\phi$ is injective and when (2) $\Phi$ is surjective, then $\psi$ is also surjective ?  I've tried to prove it by the definition of injection and surjection, but I'm not convinced myself by it.  For example, in the first case :  If $\Phi$ is injective, $\forall g \in G$, we have $0$ or $1$ $e \in E$ such as $\Phi(e)=g$.  $\Rightarrow \Phi(e) = \psi(\phi(e)) = \psi(f) =g$.   So $\forall f \in F$, we also have $0$ or $1$ $e \in E$ such as $\phi(e)=f \in F$.   $\Rightarrow \phi$ is injective.  Is this right or did I prove nothing at all ? I think it is easier to convince myself/ to understand with surjectivity but still.. I was wondering if there is any rule in English that forbids us to say A Happy New Year when we wish someone Happy New Year? I know that we just say it like that, but I was trying to find the rule which would explain this. Thank you for your help. How do I merge two dictionaries in a single expression (taking union of dictionaries)? I am using tufte-latex package, which uses mathpazo with old style numerals. To make figures (plots) I would like to use lining numbers.  The problem is that the figures are converted from svg to pdf using inkscape and the numbers are not in mathmode (In math mode the numbers are lining numerals by default).  I think of two possible solutions:  a) Is there any command similar to \oldstylenums that does the opposite work?  b) inkscape put the text of the figure in \smash{-22.22} commands. Is any way,  using sed, awk, perl,... to change that to \smash{$-22.22}? Convergence of an alternating series : $ \sum_{n\geq 1} \frac{(-1)^n|\sin n|}{n}$ Study the convergence of $$\displaystyle \sum_{n\geq 1} \frac{(-1)^n|\sin n|}{n}.$$   I am stuck with this series, we need probably some measure of irrationally of $\pi$, unfortunately I am unfamiliar with this. So here is my attempt :  Let $f(x) = \sum \frac{|\sin{n}|}{n} x^n, |x| &lt; 1$  It's not difficult to compute the Fourier series of $|\sin(x)|$ :  $$ \displaystyle|\sin(x)|=\frac{2}{\pi}-\frac{4}{\pi}\sum_{n=1}^{+\infty}\frac{\cos(2nx)}{4n^2-1} $$  Then Fubini's theorem () works very well (because the previous series converges absolutely at $x$ fixed ) and all calculations made, we find that for all $x\in( -1,1)$:  $$ \displaystyle f(x)=\frac{2}{\pi}\sum_{n=1}^{+\infty}\frac{x^n}{n}-\frac{4}{\pi}\sum_{p=1}^{+\infty}\frac{x^2-2x\cos(p)}{(4p^2-1)(x^2-2x\cos(p)+1)} $$  However, the second sum I have not been able to show the convergence. I feel the series diverge because the following series $$ \displaystyle\sum\frac{1}{p^2\sin^2\left(\frac{p}{2}\right)} $$  diverge because  $0$ is an accumulation point of $\displaystyle (n\sin(n))$ sequence.  Any ideas (for the original series) ? On Linux can I keep a program running in background even after I disconnect? If I'm logged into a remote Linux machine, I know I can suspend a process (cntr-z) and move it to the background (bg 1) but if I disconnect from the machine, the process stops.  Is there a way I can leave it in the background and have it there when I log back in? This is a very long running process.  On Ubuntu, logged in via SSH. How can I find the cheapest prepaid SIM card for a given country?