bio | website | math.berkeley.edu/~sramesh |
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location | Berkeley, CA | |
age | 30 | |
visits | member for | 5 years, 6 months |
seen | 21 hours ago | |
stats | profile views | 3,918 |
I was a graduate student in the Logic program at Berkeley, broadly interested in categorical logic and foundations of mathematics, as well as in applications of category theory to the semantics of programming languages. I work for Google now.
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. |
Aug
26 |
comment |
Brouwer fixed points via flow
@Vidit: Yes, and also the uniqueness. Thus, any solutions defined on an open interval around 0 are compatible, and there is at least one, so we can join them all together and obtain a (unique) solution defined on a maximum open interval around 0 (not necessarily the entire real line), just as claimed, no? |
Aug
26 |
comment |
Brouwer fixed points via flow
@Christian: With the correction, the differential equation becomes $x'(t) = f(x(t)) - x(t) = -2x(t)$ with $x(0) = p$, which has unique solution $x(t) = e^{-2t}p$. The limiting value of this for large $t$ will be $0$, which is a fixed point of $f$. |