850,441 research outputs found
On the refined counting of graphs on surfaces
Ribbon graphs embedded on a Riemann surface provide a useful way to describe
the double line Feynman diagrams of large N computations and a variety of other
QFT correlator and scattering amplitude calculations, e.g in MHV rules for
scattering amplitudes, as well as in ordinary QED. Their counting is a special
case of the counting of bi-partite embedded graphs. We review and extend
relevant mathematical literature and present results on the counting of some
infinite classes of bi-partite graphs. Permutation groups and representations
as well as double cosets and quotients of graphs are useful mathematical tools.
The counting results are refined according to data of physical relevance, such
as the structure of the vertices, faces and genus of the embedded graph. These
counting problems can be expressed in terms of observables in three-dimensional
topological field theory with S_d gauge group which gives them a topological
membrane interpretation.Comment: 57 pages, 12 figures; v2: Typos corrected; references adde
Counting free fermions on a line: a Fisher-Hartwig asymptotic expansion for the Toeplitz determinant in the double-scaling limit
We derive an asymptotic expansion for a Wiener-Hopf determinant arising in
the problem of counting one-dimensional free fermions on a line segment at zero
temperature. This expansion is an extension of the result in the theory of
Toeplitz and Wiener-Hopf determinants known as the generalized Fisher-Hartwig
conjecture. The coefficients of this expansion are conjectured to obey certain
periodicity relations, which renders the expansion explicitly periodic in the
"counting parameter". We present two methods to calculate these coefficients
and verify the periodicity relations order by order: the matrix Riemann-Hilbert
problem and the Painleve V equation. We show that the expansion coefficients
are polynomials in the counting parameter and list explicitly first several
coefficients.Comment: 11 pages, minor corrections, published versio
A Point Counting Algorithm for Cyclic Covers of the Projective Line
We present a Kedlaya-style point counting algorithm for cyclic covers over a finite field with not dividing , and
and not necessarily coprime. This algorithm generalizes the
Gaudry-G\"urel algorithm for superelliptic curves to a more general class of
curves, and has essentially the same complexity. Our practical improvements
include a simplified algorithm exploiting the automorphism of ,
refined bounds on the -adic precision, and an alternative pseudo-basis for
the Monsky-Washnitzer cohomology which leads to an integral matrix when . Each of these improvements can also be applied to the original
Gaudry-G\"urel algorithm. We include some experimental results, applying our
algorithm to compute Weil polynomials of some large genus cyclic covers
The number of {1243, 2134}-avoiding permutations
We show that the counting sequence for permutations avoiding both of the
(classical) patterns 1243 and 2134 has the algebraic generating function
supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia
of Integer Sequences.Comment: 7 pages, 1 figur
- …