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# Monthly Archives: September 2019

## Jumpin’ Riemann!…..!..!.!.Mangoldt–da mon–got it!….!..!

The magic of Mangoldt summoning Riemann’s miraculous miniscules-the nontrivial zeros. In response to observations initiated by Matt McIrvin of a sum of exponentials of the imaginary part of the non-trivial zeroes of the Riemann zeta function, assuming the Riemann hypothesis … Continue reading

Posted in Math
Tagged Chebyshev function, Digamma function, Dirac delta function, Euler's product formula. Hadamard's product formula, Fourier transform, Inverse Mellin transform, Landau Xi function, logarithmic derivative, Mangoldt function, Mangoldt summatory function, Riemann jump function, Riemann Xi function, Riemann zeta function nontrivial zeros, Riemann's prime counting function
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## Quotes Post:

Ongoing collection of quotes attributed (mod my lapses in memory) to mathematicians and physicists or about their work: d’Alembert: Go forward, faith will follow!

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## In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms

In the earlier post Compositional Inverse Operators and Sheffer Sequences, I constructed relations among a generic power series, call it , or ordinary generating function (o.g.f.), its compositional inverse and four sets of Sheffer polynomial sequences–two Appell sequences and and … Continue reading

Posted in Math
Tagged Appell polynomial sequences, Associahedra, binomial Sheffer polynomial sequences, complete homogeneous symmetric polynomials, Compositional inversion, conjugation, elementary symmetric polynomials, matrix inverses, multiplicative inversion, Noncrossing partitions, refined Lah polynomials, similarity transformation, Stasheff polytopes, Symmetric polynomials, Umbral composition, Umbral inverse
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## Squaring Triangles

This post illustrates what Feynman praised as a beautiful facet of mathematics–abstraction from the concrete–as well as the fascinating synergy at one of its crossroads–that of algebra and enumerative geometry. One day last fall in a class, several curious 12-th … Continue reading

## Commutators, matrices and an identity of Copeland

The arXiv “Commutators, matrices and an identity of Copeland” by Darij Grinberg proves and extends an identity I proposed for a matrix computation of the partition polynomials generated by iterated multiplication of a tangent vectorwhere and is a function or … Continue reading

Posted in Math
Tagged algebraic combinatorics, Algebraic geometry, Appell polynomial sequences, Associahedra, Butcher series, Combinatorics, commutators, Compositional inversion, convex polytopes, Dyck paths, enumerative geometry, Evolution equations, free cumulants and moments, Free probability, generalized permutahedra, Hirzebruch criterion, Hopf algebra of monoids, inviscid Burgers' equation, iterated convolutions, Kdv equation, Koszul duality, matices, Noncrossing partitions, phylogenetic trees, polygon dissections, Pre-Lie algebra, rooted Cayley trees, Todd class
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