I have a statement in my notes that surely must be wrong. If a divides b => a divides b*x for all x in R, R a commutative ring. Surely if R is the real numbers, we can take a = 1, b = 1 and x = 1/2 and the statement fails. So the statement is incorrect, right?
Let us pretend that some set $S$ is closed under exponentiation. There is an identity element $e$ for example. Exponentiation has an inverse operation logorithms
however there is no asociasiativity
does that mean It cant be a group or is there another alegbraic structure it falls under?
Why is this much more difficult than it looks? $$I=\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt{3\vphantom{\large3}}\right)}\right)}dx.$$
Consider a finite group and the sequence (g^n)n∈N. It has infinitely many elements, so they can't all be different. Choose two equal elements with different n's, and ... now you're supposed to get the neutral element somehow, but I don't see how.
@Chris'ssis In case you forgot Compute $$\int^1_{-1}\left(\frac{\sqrt{\frac{3-\cos(2x)\tan^2x-\cos2x+3\tan^2x}{2+2\tan^2x}+\sin^2x-x}}{2\sqrt{\cosh^2(ix)+\frac{1+\cos(\pi+2x)}{2}+x}}-\frac{\sqrt{\cos2x+2\sin^2x-x}\sqrt{\frac{1-\tan^2x+2\sin^2x+2\sin^2(x)\tan^2x+x-x\tan^2x}{1+\tan^2x}}}{2x-2\sin^2x+2\cos^2x}\right)\;\mathrm dx$$
@Charlie :-). Well, I mainly love integral, series, sequences, limits and geometry. However, the latter one I didn't attend too much in the last period of time.
@Alyosha imagine this: One asks you for computing without touching integrals
@Alyosha A non-affine transformation is a not-terribly-simple function from the space to itself. Some such functions are nicer than others, but you'll have to be more specific.
@MarkS. I'll post something on main tomorrow (GMT), but essentially I want to prove problems involving the ratio of the area of a convex quadrilateral to some shape constructed on that quadrilateral (for instance, join the midpoints).
It's easy for parallelograms, you just need to prove it for squares then apply a matrix on it. The problem with non-affine transformations is that the relationships between areas seem more complicated, with some areas of the plane ballooning out and others shrinking.
@Alizter your integral is excellent, I think complex analysis should do it if $a$ is a positive integer, but am not proficient enough at it. Ask on main?
I was just trying to get a handle on @nsanger's question; I don't have much interest in order theory-type stuff. If you want to talk about this, though, go on
@TedShifrin Suppose I have a cylinder $\{x^2+y^2=1:0\leqslant z\leqslant 1\}$ topped with a sphere $\{x^2+y^2+(z-1)^2=1:z\geqslant 1\}$, and I am given this as the outwards pointing normal orientation.
