1,166 reputation
11029
bio website tcjpn.wordpress.com
location Japan
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visits member for 3 years, 10 months
seen 5 hours ago

4h
comment Why is there a connection between enumerative geometry and nonlinear waves?
Also see "Bernoulli numbers and solitons" by G Rzadkowski tandfonline.com/doi/pdf/10.1142/S1402925110000635 The connections of the Bernoulli numbers to topology are well-known.
14h
comment Hirzebruch's motivation of the Todd class
@John McKay, don't know if you ever followed up on it, but there are connections to the Bose-Einstein distribution. See OEIS-A131758 and use find, or search OEIS for Copeland Bose-Einstein.
15h
comment Hirzebruch's motivation of the Todd class
@Eric, to see the explicit connections of the analyses below to your comment about formal group laws and Hausdorff's result you refer to take a look at my answer to the MOQ I link to in my question and my comment to Stephan's answer there.
15h
comment Hirzebruch's motivation of the Todd class
For the general theory of Hirzebruch, the Lagrange inversion can be extended to formal power series, and, when expressed as such in the indeterminates, the fascinating connections to myriad combinatorial structures are revealed and the general interplay between number theory and topology, as Hirzebruch encourages us to explore as he reminisces. The analytic inverse function theorem then serves as a guide to even more interesting vistas.
16h
comment Hirzebruch's motivation of the Todd class
Insisting in too much rigour leads to a neglect of the inversion formula as a tool for relating formal power series to each other and the connections to Hirzebruch's general theory on genera.
1d
revised Why do Bernoulli numbers arise everywhere?
Hirzebruch connection
1d
revised Why do Bernoulli numbers arise everywhere?
Hirzebruch connection
1d
revised Why do Bernoulli numbers arise everywhere?
Hirzebruch connection
1d
comment Why do Bernoulli numbers arise everywhere?
Define the Bernoulli polynomials as the Appell sequence, i.e., $(B.(0)+x)^n=B_n(x)$, such that $f(B.(x+1))-f(B.(x))={f}'(x)$ when convergent. Then $e^{B.(x+1)t}-e^{B(x)t}=te^{xt}$ implies $e^{B.(x)t}(e^t-1)=te^{xt}$ and the e.g.f. and makes the appearance of $t$ natural. Relates the Bernoulli polynoms to the tangent space.
1d
comment Why do Bernoulli numbers arise everywhere?
Jonah Sinick has some interesting notes on the formula in mathisbeauty.org/ZetaValuesinGeometryandTopology1016.pdf .
1d
comment Why do Bernoulli numbers arise everywhere?
Hausdorff (1906) via Iserles, "Expansions that grow on trees": Matrix diff. eqn. ${Y}'= A(t)\; Y$ with change of variables $Y(t)=e^{\Omega(t)}$ becomes ${\Omega}'=exp[B.\;ad_{\Omega}]A=\frac{ad_{\Omega}}{e^{ad_{\Omega}-1}}A\;.$ citeseerx.ist.psu.edu/viewdoc/…
1d
comment Why do Bernoulli numbers arise everywhere?
For the Harer-Zagier formula the Bernoulli numbers weasel their way in through an asymptotic expansion of the digamma fct. See last page of ocw.mit.edu/courses/mathematics/… .
2d
revised Hirzebruch's motivation of the Todd class
Added link. Corrected a name.
Nov
24
revised Hirzebruch's motivation of the Todd class
more math
Nov
24
comment Hirzebruch's motivation of the Todd class
For notes on Blissard's characterization of $((e^x-1)/x)^{m}$, see "Umbral presentations of polynomial sequences" by B. D. Taylor arxiv.org/abs/math/9908131 .
Nov
24
comment Why do Bernoulli numbers arise everywhere?
There's a fascinating connection between the Bernoulli polynomials and hyperfunctions in "Hyperfunctions, formal groups, and generalied Litschitz summation formulas" by S. Marmi and P. Tempesta: homepage.sns.it/marmi/papers/… . For more on generalized Bernoulli polynomials, cohomology, and homology, and connections to umbral calculus, see papers by Nigel Ray and collaborators.
Nov
23
revised Hirzebruch's motivation of the Todd class
Some algebra
Nov
23
comment Hirzebruch's motivation of the Todd class
see also mathoverflow.net/questions/10630/…
Nov
23
comment Why do Bernoulli numbers arise everywhere?
For the Todd-Chern-Hirzebruch connection see mathoverflow.net/questions/60478/…
Nov
23
revised Hirzebruch's motivation of the Todd class
Elaborated