bio | website | math.berkeley.edu/~sramesh |
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location | Berkeley, CA | |
age | 29 | |
visits | member for | 4 years, 10 months |
seen | 17 hours ago | |
stats | profile views | 3,633 |
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.
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 |
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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 |
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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 |
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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 $f$. |
Aug 26 |
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Brouwer fixed points via flow
@Christian: $f(x) = 2x$ will not be a map from $B$ to $B$, unless the ball $B$ consists solely of the zero point. |
Aug 26 |
asked | Brouwer fixed points via flow |
Jul 23 |
comment |
Reconstructing the argument that yields Graham's number
@TimothyChow: The upper bound given in the paper is $f^7(12)$ where $f(n) = 2 \mathbin{\uparrow}^{n} 3$. That seems to me about exactly as easy to specify as Graham's number $f^{64}(4)$ where $f(n) = 3 \mathbin{\uparrow}^{n} 3$, so I don't know why the latter ever came up. Perhaps Graham actually tried to outline to Gardner the argument for the upper bound, and found it simpler with the weaker bound. |
Jul 2 |
awarded | Curious |
Jun 22 |
accepted | Relationship of Euler product to coprimality densities for arbitrary sets of primes |
Jun 22 |
asked | Relationship of Euler product to coprimality densities for arbitrary sets of primes |
Jun 22 |
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Additivity of asymptotic density of periodic sets
Ah, nice. Here's a followup question I'm also interested in, which you will perhaps resolve with just as much ease: Suppose now we consider an increasing series of subsets $A_0 \subseteq A_1 \subseteq A_2 \subseteq ...$ of the integers, where each of these is not merely periodic but in fact of the form "Any multiple of a member of F", for some finite set F. Can the density of their union still fail to match the supremum of their individual densities? |
Jun 22 |
accepted | Additivity of asymptotic density of periodic sets |
Jun 22 |
asked | Additivity of asymptotic density of periodic sets |
May 31 |
awarded | Custodian |