bio | website | math.berkeley.edu/~sramesh |
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location | Berkeley, CA | |
age | 29 | |
visits | member for | 5 years, 3 months |
seen | yesterday | |
stats | profile views | 3,789 |
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 |
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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? |
Aug 28 |
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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 |
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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 |
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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 |
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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. |
Jul 23 |
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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. |
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? |
May 14 |
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Commuting limits in relating the harmonic series to coprimality densities
Oh, very nice, thank you. This does settle the question about the existence of the general limit, but, alas, this argument requires that we already know $\prod_{p}(1 - 1/p) = 0$, while my motivating hope remains that there is some way to noncircularly derive this fact from $\prod_{p}(1 - 1/p) = \lim_{q \to \infty} \lim_{n \to \infty} f(n, q)^{-1}$ and $\lim_{n \to \infty}\lim_{q \to \infty} f(n, q)^{-1} = 0$. |
May 14 |
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Commuting limits in relating the harmonic series to coprimality densities
If I'm reading you correctly, you are saying that for sufficiently large $n$ and $q$, we have that $f(n, q)^{-1}$ is at least a quantity which tends to zero. But surely we need to show $f(n, q)^{-1}$ to be at most a quantity which tends to zero, in order to answer the last question? |
May 13 |
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Can the Riemann hypothesis be undecidable?
@Daniel: You're right! I haven't ruled out this possibility! All these years, sitting there unnoticed... That having been said, various sources give Robin's criterion instead as "for all $n > 5040$, $\sigma_1(n) \leq e^{\gamma} n \log \log n$.. To this, exact equality would not serve as a counter-example, and thus falsehood would entail provable falsehood. That having been said, I am not, in fact, familiar enough with the relevant material to verify whether the "$\leq$" form of Robin's criterion is genuine, or, I paranoidly worry, merely the result of careless transcription of the "$<$" form. |
May 24 |
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Avoiding reflexive paradox in set theory
Richard, you are coming now quite close to a principle like "Sets are given by well-founded comprehension"; this is the spirit behind ZF, and once you have that, the rest of its axioms are not far behind [you'll find that the existence of infinite sets, powersets, etc., are not automatically assured, and you may want to assure them]. |
Dec 1 |
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Application of polynomials with non-negative coefficients
Surely, the thing to do is to use q = p(1) + 1 rather than q = p(1), so as not to have to treat the case p(q) = q^n on an ad hoc basis. |
Nov 18 |
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There are two points on the Earth's surface that … ?
What does "separated by the same geodesic distance" mean for two points? |
Nov 17 |
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Tarski's Theorem and Gödel's Second Incompleteness Theorem
Well, one caveat: One needs that Tarski's result is constructive enough that we have not just "For every T, for every model, there is a G such that...", but in fact "For every T, there is a G such that for every model...". But, basically, were Tarski's result phrased as strongly as its proof actually warranted, it would give us everything. |
Nov 17 |
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Tarski's Theorem and Gödel's Second Incompleteness Theorem
I've modified the word of the second paragraph to alleviate this concern. The second incompleteness theorem follows from the simultaneous "external" and "internal" truth of Tarski's theorem (by which I mean, the fact that Tarski's theorem is both true and provable (or, just as well, true inside every model)). |
Nov 2 |
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Entailment and implication
It's also worth noting that the trivial topos, like all toposes, internally considers itself to be non-trivial. It just happens to also consider itself to be trivial... |