After modifying my profile picture it appears as a black square
Profile picture cannot have transparent background
Order of the Rubik's cube group Associated to the Rubik's cube is a group as described in this : $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise by $90$ degrees.  According to the article the order of the group is: $2^{27} 3^{14} 5^3 7^2 11$.  I have come across several places (for example ) where this order has been found by just counting various ways to arrange edges and such. Is there a nice way to compute the order by appealing to more "group theoretic" tools? Isometry from Manhattan plane to Euclidean plane? Does there exist an isometry from a Manhattan plane $A$ to a Euclidean plane $B$? I.e. a function $\varphi:A \to B$ that suffices $\|\varphi(a)\|_B = \|a\|_A$ for all $a \in A$, where  $\| \cdot \|_A$ is the Manhattan norm and $\| \cdot \|_B$ the Euclidean norm. How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime? I know if $\ i\ $  of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$,  then $\ 12i + 5\ $ must be composite number, because:  $12(3x^2 + (6y-3)x + y - 1) + 5 = 36x^2 + (72y - 36)x + (12y - 7) = (6x + 12y - 7)(6x  + 1)$  How come if $\ i\ $ not of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  then $12i + 5$ must be prime? $\ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$.  For example: $\ 5 = \ 3*1^2 + (6*1-3)*1 - 1\ $ ,when $\ x = y = 1\ $, as proved before,$\ 5*12+5\ $ must be composite number;  $\ 0, \ 1. \ 2.\ 3.\ 4\ $ can't of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  $\ 12*0 + 5 = 5\ $, $\ 5 $ is prime,$\ 12*1 + 5 = 17\ $, $\ 17 $ is prime, and so on, Raising a decimal to a power of decimal? The .net framework provides in the Math class a method for powering double. But by precision requirement I need to raise a decimal to a decimal power [ Pow(decimal a, decimal b) ]. Does the framework have such a function? Does anyone know of a library with this kind of function? Why didn't anybody Summon the Triwizard cup in the maze? What is terminal velocity? I've heard the term especially when the Discovery Channel is covering something about sky diving. Also, it is commonly known that HALO (Hi-Altitude, Lo-Opening) infantry reaches terminal velocity before their chutes open.    Can the terminal velocity be different for one individual weighing 180 pounds versus an individual weighing 250 pounds? I have to submit a manuscript to a journal, and I just realized that they want every single part of the paper to be in 12 pt font. This includes tables text, footnotes, acknowledgements etc. Is there a way to force all text to be the same size? Code breaks when updating jQuery due to changes in toggle Why does Ridders' method work as well as it does? I've just read section 9.2.1 in Numerical Recipes Ed. 3 (Press et al. 2007), which describes Ridders' method of root finding. I understand that allowing for some curvature of the function by linearizing three points with an exponential will probably get you a better approximation and more rapid convergence, but I have no idea how they get that it converges quadratically, and the motivation behind choosing an exponential (rather than e.g. a polynomial) is not clear to me.  So, specifically, I'm wondering (1) why linearize with an exponential, and (2) why the method converges quadratically.  I tried looking for Ridders' original paper "A new algorithm for computing a single root of a real continuous function", but the only copy I could find was in a journal that my school's library does not give me free access to.  UPDATE:  I've realized that the fact that an exponential is strictly positive means that weighting the three points with an exponential won't change their signs. Hence, the root of the linearization will have to land inside the original bracket.  Polynomials, rationals, sinusoids, and logarithms all have positive and negative bits, so using them probably runs the risk of getting bad interpolated roots. One of these squared might be usable, but maybe non-monoticity gives some sort of undesirable effect. Since this is a complete list of all simple functions that I am aware of, exponentials seem to be the only easily available weighting function with this property.  A constant function is also technically single-signed and monotone, but it can't be used to linearize stuff. Now let set $A$ be arbitrary. So set $A$ exists. Using axiom of pairing $\{A\}$ exists. And then using axiom of regularity, we can argue that $A \notin A$. So, there is some $x$ such that $x \notin A$. Is this a valid proof ? Thanks Is there anything beyond subtle hints that Jack becomes the Face of Boe? What do the plus and minus signs mean in Objective-C next to a method? Why is     $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$   Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is     $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$     This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: . Edit source differs from question view? Or are we each other's friends now? Why is causality preserved in special relativity? PART 1: I was reading the article  Wikipedia. I couldn't understand this line:       "if the two events are causally connected ("event A causes event B"), the causal order is preserved (i.e., "event A precedes event B") in all frames of reference."     Is this an assumption or a consequence of STR? Please explain.       Note: My question consists of 2 parts this is the 2nd part. Below is a genral version of my previous question question:.   PART 2: Let there be three events $A$,$B$ and $C$ s.t: $C$ is the result of Simultaneous occurrence of $A$ and $B$. In other words $C$ occurs iff $A$ and $B$ are simultaneous. Now as we know in STR any two events separated in space are not simultaneous in different frames. So In some frames $C$ will occur and in some $C$ will not occur which will cause paradox. I tried many thought experiments to make such a paradox but i failed. In all the experiments that i thought i could not break the causality even by breaking the simultaneity because everytime the fact: "all signals move slower than light" preserved the causality.    So why causlity remains preserved always? Is it due to the fact that nothing can move faster than light? Bounding lines around tax nodes I am trying to draw lines around a number of nodes. I very much like the definition of \convexpath definition given , but it is not working well on my setup (MacTex 2014 with all available updates).   I tried to compile , but what I get is the following, which is clearly not what I should be getting. Any ideas would be much appreciated. Unable to install ECW support on lubuntu 14.04 Complicated FOR loop nesting. Having trouble cleaning up legacy code Given a PRF $F$ , is $G(s) = F_s(1)\|F_s(2)\|\cdots \|F_s(n+1)$ a PRG? If $F$ a PRF, and we construct $G$ using $F$ in the following way:  $$G(s) = F_s(1)\|F_s(2)\|\cdots \|F_s(n+1)$$ where $|s|= n$.   Is $G$ then a PRG? If so how can I prove this? If not how can $G(s)$ be distinguished from a random $r$? And what if $n+1 &lt;$ range of $F_s$? or it could be for any $n &gt; |s|$. Abort Ajax requests using jQuery