4,459 research outputs found
The combinatorics of associated Hermite polynomials
We develop a combinatorial model of the associated Hermite polynomials and
their moments, and prove their orthogonality with a sign-reversing involution.
We find combinatorial interpretations of the moments as complete matchings,
connected complete matchings, oscillating tableaux, and rooted maps and show
weight-preserving bijections between these objects. Several identities,
linearization formulas, the moment generating function, and a second
combinatorial model are also derived.Comment: [v1]: 18 pages, 16 figures; presented at FPSAC 2007 [v2]: Some minor
errors fixed (thanks Bill Chen, Jang Soo Kim) and text rearranged and cleaned
up; no real content changes [v3]: fixed typos, to appear in European J.
Combinatoric
Ramanujan Graphs in Polynomial Time
The recent work by Marcus, Spielman and Srivastava proves the existence of
bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that
paper did not provide a polynomial time algorithm to actually compute such
graphs. Here, we provide a polynomial time algorithm to compute certain
expected characteristic polynomials related to this construction. This leads to
a deterministic polynomial time algorithm to compute bipartite Ramanujan
(multi)graphs of all degrees and all sizes
Integrability on the Master Space
It has been recently shown that every SCFT living on D3 branes at a toric
Calabi-Yau singularity surprisingly also describes a complete integrable
system. In this paper we use the Master Space as a bridge between the
integrable system and the underlying field theory and we reinterpret the
Poisson manifold of the integrable system in term of the geometry of the field
theory moduli space.Comment: 47 pages, 20 figures, using jheppub.st
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