bio  website  math.berkeley.edu/~sramesh 

location  Berkeley, CA  
age  30  
visits  member for  5 years, 5 months 
seen  4 hours ago  
stats  profile views  3,875 
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.
2d

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 nonnatural $p$?) 
2d

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 zerofinding 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/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 
comment 
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 
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 $f$. 
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. 