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bio website math.harvard.edu/~elkies
location Harvard University, Cambridge, MA 02138
age 48
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2d
comment What's the name of this branched covering?
Note too that Loïc Teyssier's double cover has two branch points (as expected by Riemann-Hurwitz): the involution $(x:y) \leftrightarrow (y:x)$ fixes not just $(1:1)$ but also $(-1:1)$.
May
25
comment What's the name of this branched covering?
You could just call it ${\bf P}^1({\bf C})$ because it's a genus-zero Riemann surface; an explicit coordinate is $(x^2+y^2 : xy)$.
May
24
comment Singularities of Pfaffian hypersurfaces
So presumably that genus 26 curve is isomorphic with the modular curve $X(11)$, right?
May
24
comment Growth of average first derivative of orthogonal polynomials
The average of the derivative $P'$ of any polynomial $P$ on $(a,b)$ is $(P(b)-P(a))/(b-a)$. So for example if $P$ is the Čebyšev polynomial $T_k$ then we get $0$ or $1$ according as $k$ is even or odd. Not sure what you're claiming about the Hermite polynomials: their interval of orthogonality is $\bf R$, on which the average is a divergent integral for $k>0$.
May
21
comment Which domain maximizes the energy of the Lebesgue measure?
It ought to be a circle (or generally a Euclidean sphere) of unit area (volume), right?
May
19
comment Permutations with all cycles odd length and permutations with all cycles even length
In the even case, $1 \cdot 3 \cdot 5 \cdot \cdots \cdot (2n-1)$ is also the number of fixed-point-free involutions, so one could also ask for a bijection between pairs of such involutions and constant-parity permutations of each kind.
May
17
comment An algorithm for Poincare recurrence time
You might get to $10^{-9}$ this way, but never to $10^{-900}$. The approach via simultaneous Diophantine approximation (using the LLL algorithm, or better yet PSLQ) is polynomial in the exponent and readily solve the question for $10^{-900}$ (as was done by O.S.Dawg) and beyond.
May
17
comment how to evaluate the following double summation to infinity without using integration method?
The sum converges fast enough to be computed efficiently to any desired accuracy as it stands. But I suspect that there might be no closed form.
May
16
comment Important open problems that have already been reduced to a finite but infeasible amount of computation
Say that the original problem is "describe all Moore graphs". Then the nontrivial reduction is the use of spectral graph theory to show that the degree is one of $2$, $3$, $7$, or $57$. Each of the first three occurs uniquely (pentagon, Petersen, Hoffman-Singleton) but the last is famously open.
May
14
comment non-intersecting families of subspaces
Do you mean $\omega(k^{m/r})$, rather than the $\omega(k^{m-r})$ attained for $r|m$? (Note too that if $r < m < 2r$ then any two $r$-subspaces intersect so the family cannot be larger than 1.)
May
14
comment Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?
@few_reps the infinite cyclic group is not nearly as interesting as the ring $\bf Z$.
May
13
comment Low height integer points on a rank variety
Are you allowing the degree of the polynomial (i.e. the exponent of $n$) to depend on $k$?
May
12
awarded  Scholar
May
12
comment $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
... and if I had tried 0.3299338011 then I'd have found the Borwein-Straub-Wan-Zudilin paper directly ...
May
12
accepted $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
May
12
comment $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
Hm, so it does; I guess I should have tried both with and without the initial 0...
May
12
comment $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
Thanks, but it seems that the factor of $x$ is missing there, and the integral for a product of $m$ Bessel functions is expressed as a real period of a hypersurface of dimension $m-1$, not $m-3$ as with the "curiosity" from Liviu Nicolaescu (indeed I see that in the next formula (9) Watson evaluates such integrals for $m=4$ in terms of complete elliptic integrals, i.e. real periods of elliptic curves).
May
12
comment $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
(I'll write about your actual answer once I've had the opportunity to look up the references you gave.)
May
12
comment $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
I did give the decimal expansion 0.32993380106006405903979065228695296470 and searched on that (and truncations and simple multiples) before posting the question...
May
12
awarded  Nice Question