bio  website  math.berkeley.edu/~sramesh 

location  Berkeley, CA  
age  29  
visits  member for  4 years, 6 months 
seen  yesterday  
stats  profile views  3,519 
I am 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.
1d

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Brouwer fixed points via flow
For what it's worth, the Hirsch/Kellogstyle 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 prefixfree 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 Turingcomplete using any computable prefixfree 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 
comment 
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 
comment 
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 
May 31 
reviewed  Approve suggested edit on Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem 
May 29 
awarded  Promoter 
May 14 
revised 
Commuting limits in relating the harmonic series to coprimality densities
added 337 characters in body 
May 14 
revised 
Commuting limits in relating the harmonic series to coprimality densities
added 337 characters in body 