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bio website jamespropp.org
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I am interested in combinatorics, probability, dynamical systems, and various other topics.


Mar
21
comment How exactly does Schützenberger promotion relate to Striker-Williams promotion?
Jessica and I discussed our respective approaches in person yesterday, and we feel that they're quite different. Jessica's answer treats standard tableaux; my answer (which appears below) treats semistandard tableaux. Her tableaux have only two rows; mine can have any number of rows. Her tableaux can be skew; mine must be non-skew, and indeed must be rectangular. She associates tableaux with order ideals in subposets of root posets of type A (triangles); I associate tableaux with points in the order polytope of minuscule posets of type A (rectangles).
Mar
16
revised How exactly does Schützenberger promotion relate to Striker-Williams promotion?
added 1 character in body
Mar
13
revised How exactly does Schützenberger promotion relate to Striker-Williams promotion?
added 763 characters in body
Mar
10
comment slick-proof-of-trick-for-counting-domino-tilings
Nice; I'll definitely include this proof the next time I teach a course on perfect matchings! (Though I am a bit fuzzy on how one would prove the topological lemma that asserts that these four kinds of singularities are all that can happen.)
Mar
9
revised Modeling bubble rafts
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Mar
2
comment Why does the Riemann zeta function have non-trivial zeros?
Pace Nielsen's question deserves a more serious answer. Here is a paraphrase David Hansen might prefer: What concrete number-theoretic phenomena allow one to "see" the influence of 14.1...? Barry Mazur, at the end of "A Lecture on Primes" (msri.org/general_events/20798), alluded to the existence of such phenomena; can anyone provide details?
Mar
2
comment Regularized sums of Mobius sequence
Is there some sort of asymptotic expansion for $M(x)$ in which (a) the constant term is $-2$, (b) the other terms correspond to roots of $\zeta$ on the critical line, and (c) the constant term dominates the other terms when $x >> 10^{-10}$? That would explain why it misleadingly seems to be converging to $-2$, and would reinforce Lucia's point that the zeroes are the reason for the non-convergence.
Mar
2
comment Regularized sums of Mobius sequence
For anyone who's interested in knowing just how small $\Gamma(1/2+14.1\dots i)$ is, Mathematica reports that its real and imaginary parts are on the order of $10^{-10}$.
Mar
2
comment Regularized sums of Mobius sequence
@Lucia: I think your response settles my first two questions, but what about the third? I know that the Dirichlet series converges when Re $s > 1$, but I don't know whether the full domain of convergence includes some values of $s$ with Re $s \leq 1$ (such as real numbers strictly between 0 and 1).
Mar
1
asked Regularized sums of Mobius sequence
Feb
24
comment Distinguishing combinatorial maps by their linearizations
I don't understand how one can get away with ignoring the cycles completely. What if there are nothing but cycles? That is, what if the original combinatorial map is a permutation? Then we know that spectrum of the associated linear map is $GL(n)$-invariant, and the spectrum determines the cycle-structure.
Feb
23
accepted Distinguishing combinatorial maps by their linearizations
Feb
23
comment Distinguishing combinatorial maps by their linearizations
I don't quite see how to reconcile this with the criterion of Saks and Gansner described by Stanley (below). Taking $n=4$ and $j=2$, I find that for the idempotent that maps 1 to 1 and maps 2, 3, and 4 to 2, the largest number of vertices in a union of two paths is 3, while for the idempotent that maps 2 and 4 to 2 and maps 1 and 3 to 1, the largest number of vertices in a union of two paths is $4 \neq 3$. What am I missing?
Feb
22
asked Distinguishing combinatorial maps by their linearizations
Feb
19
revised Is the Ford disk packing optimal?
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Feb
19
accepted Is the Ford disk packing optimal?
Feb
19
answered Is the Ford disk packing optimal?
Feb
18
awarded  Citizen Patrol
Feb
18
asked Modeling bubble rafts
Feb
18
revised Is the Ford disk packing optimal?
added 328 characters in body