Logan Maingi
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 Apr 15 comment Does the sequence $x_{n+1} = \frac{1}{2} ( (x_n)^2 + c )$ diverge? The sequence is not necessarily strictly increasing as you claim. Consider $x_0 = \frac32$ and $c= \frac14$, then $x_1 = \frac{11}8 < \frac32$. For these initial conditions the sequence is strictly decreasing and bounded below by 0, so it is convergent. Apr 5 awarded Nice Answer Mar 20 awarded Yearling Mar 1 revised Why aren't integration and differentiation inverses of each other? replaced ~ with \sim since apparently the former doesn't work in MathJax Feb 29 revised Why aren't integration and differentiation inverses of each other? added a couple paragraphs on IVPs Feb 28 comment Why aren't integration and differentiation inverses of each other? @goblin I suppose I was unclear, but I was looking for cases in which the derivative was an automorphism, not merely an isomorphism. You'll notice that both the domain and range of your construction are naturally embedded in the one I did in "option 1" in such a way that they are compatible with the derivative, and indeed this is the smallest such space with that property on which the derivative is invertible. But yes, given the importance of initial value problems, this approach does deserve a mention somewhere, so I'll edit something in soon. Dec 24 awarded Guru Nov 23 comment Weyl transformation of geodesic distance Are you sure you want to look at arbitrary Weyl transformations? If so, then the problem is hopeless as Willie Wong suggests. But If you're doing things related to CFTs, usually the only Weyl transformations which really matter are those induced by conformal maps. That's a much smaller class; arbitrary Weyl transformations form an infinite dimensional group isomorphic to $C^\infty(M)$, but for example on Minkowski space, the "conformal group" (in the standard physics terminology) is 15 dimensional, with a 10-dimensional Poincare subgroup which preserves the metric. Jul 30 awarded Enlightened Jul 30 awarded Nice Answer Apr 12 awarded Enlightened Apr 12 awarded Nice Answer Mar 20 awarded Yearling Jan 5 awarded Nice Answer Jan 2 awarded Good Answer Jan 1 revised Why aren't integration and differentiation inverses of each other? copy-editing Dec 28 awarded Guru Dec 28 awarded Enlightened Dec 28 awarded Nice Answer Dec 28 revised Why aren't integration and differentiation inverses of each other? some additions in part 2