Sridhar Ramesh
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 Feb 9 awarded Yearling Dec 25 awarded Nice Answer Aug 4 comment Derivative in terms of finite differences Thanks for the link! I'll also note the full-text link plouffe.fr/simon/math/…, though perhaps this is of dubious legal status... Aug 4 accepted Derivative in terms of finite differences Aug 4 comment Derivative in terms of finite differences I suppose my example question may boil down to: Is there simple reason by which one could know that the Newton series for $x \mapsto x^p$ based at any point converges to the correct value within a neighborhood of that point? Jul 29 comment Derivative in terms of finite differences I see the identity mentioned in Jordan's "Calculus of Finite Differences", but there does not seem to me to be much discussion of the conditions under which the identity holds, which is really the question I am interested in. (For example, is there simple reason by which one could know a priori that this identity is indeed valid at all points for all functions of the form $f(x) = x^p$, even for non-natural $p$?) Jul 28 revised Derivative in terms of finite differences Fixed starting index Jul 27 revised Derivative in terms of finite differences Minor title change Jul 27 asked Derivative in terms of finite differences Jun 22 awarded Good Question May 24 asked Spectral theorem from Jordan decomposition in infinite dimensions Feb 9 awarded Yearling Nov 21 comment Constructivity of zeros demanded by topological degree If $f$ is of degree 1, it is homotopic to the identity; thus, we can "augment" $g$ so that it is defined on a slightly larger ball than originally, on whose surface it now acts as the identity. And then, considering this augmented $g$, case (A) cannot occur, since surface points are all mapped to different directions. Nov 20 comment Constructivity of zeros demanded by topological degree Ah, this is nice. Still, I wonder if there might be some way to push further! (It irks me that there should be such a nice zero-finding algorithm for the degree 1 case and not others). Does there actually exist, for example, so "pathological" a case as that only finitely many surface starting points work out? Nov 20 revised Constructivity of zeros demanded by topological degree Minor wording change: "disk" -> "ball" Nov 13 asked Constructivity of zeros demanded by topological degree Aug 28 comment Brouwer fixed points via flow For what it's worth, the Hirsch/Kellog-style proof of the Brouwer fixed point theorem turns out to be along the lines I was ultimately hoping for (but, of course, different enough from what I asked about to actually work). Aug 26 comment Rationale behind an requirement on Turing machines Heh, good point. Though we might also consider more sophisticated prefix-free encodings, the simplest is to reserve an "Ok, we're done" character, and that's basically the role the blank symbol plays. Aug 26 comment Rationale behind an requirement on Turing machines One might note that the proposed machine is Turing-complete using any computable prefix-free encoding of the natural numbers. Aug 26 comment Brouwer fixed points via flow Alas, this needn't work even in two dimensions, I realize: Take $B$ to be the complex numbers of at most unit magnitude, and consider $f(x) = (1 + \frac{1}{2}e^{i \pi |x|^2}) x$, which smoothly maps $B$ into itself. The unique fixed point of this is at the origin, but I believe the limiting behavior of the flow starting anywhere on the boundary will be to cycle around the circle of squared radius $1/2$. Alas. Well, I'll leave this question here and perhaps someone still will have a good story to tell about conditions under which we should expect this approach to work.