699 research outputs found
Current Trends in Symmetric Polynomials with their Applications
This Special Issue presents research papers on various topics within many different branches of mathematics, applied mathematics, and mathematical physics. Each paper presents mathematical theories, methods, and their application based on current and recently developed symmetric polynomials. Also, each one aims to provide the full understanding of current research problems, theories, and applications on the chosen topics and includes the most recent advances made in the area of symmetric functions and polynomials
Crystal constructions in Number Theory
Weyl group multiple Dirichlet series and metaplectic Whittaker functions can
be described in terms of crystal graphs. We present crystals as parameterized
by Littelmann patterns and we give a survey of purely combinatorial
constructions of prime power coefficients of Weyl group multiple Dirichlet
series and metaplectic Whittaker functions using the language of crystal
graphs. We explore how the branching structure of crystals manifests in these
constructions, and how it allows access to some intricate objects in number
theory and related open questions using tools of algebraic combinatorics
The Zagier polynomials. Part II: Arithmetic properties of coefficients
The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n}
\binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D.
Zagier in 1998 were recently extended to the polynomial case by replacing
by the Bernoulli polynomials . Arithmetic properties of the
coefficients of these polynomials are established. In particular, the 2-adic
valuation of the modified Bernoulli numbers is determined. A variety of
analytic, umbral, and asymptotic methods is used to analyze these polynomials
Weinstein's functions and the Askey-Gasper identity
In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a
positivity result of special functions which follows from an identity about
Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in
1976.
In 1991 Weinstein presented another proof of the Bieberbach and Milin
conjectures, also using a special function system which (by Todorov and Wilf)
was realized to be the same as de Branges'.
In this article, we show how a variant of the Askey-Gasper identity can be
deduced by a straightforward examination of Weinstein's functions which
intimately are related with a L\"owner chain of the Koebe function, and
therefore with univalent functions
Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function
A common problem in cosmology is to integrate the product of two or more
spherical Bessel functions (sBFs) with different configuration-space arguments
against the power spectrum or its square, weighted by powers of wavenumber.
Naively computing them scales as with the number of
configuration space arguments and the grid size, and they cannot be
done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the
sBFs as sums of products of sine and cosine and then using the product to sum
identities, these integrals can then be performed using 1-D FFTs with scaling. This "rotation" method has the potential to
accelerate significantly a number of calculations in cosmology, such as
perturbation theory predictions of loop integrals, higher order correlation
functions, and analytic templates for correlation function covariance matrices.
We implement this approach numerically both in a free-standing,
publicly-available \textsc{Python} code and within the larger,
publicly-available package \texttt{mcfit}. The rotation method evaluated with
direct integrations already offers a factor of 6-10 speed-up over the
naive approach in our test cases. Using FFTs, which the rotation method
enables, then further improves this to a speed-up of
over the naive approach. The rotation method should be useful in light of
upcoming large datasets such as DESI or LSST. In analysing these datasets
recomputation of these integrals a substantial number of times, for instance to
update perturbation theory predictions or covariance matrices as the input
linear power spectrum is changed, will be one piece in a Monte Carlo Markov
Chain cosmological parameter search: thus the overall savings from our method
should be significant
The large- limit of the 4d superconformal index
We systematically analyze the large- limit of the superconformal index of
superconformal theories having a quiver description. The index
of these theories is known in terms of unitary matrix integrals, which we
calculate using the recently-developed technique of elliptic extension. This
technique allows us to easily evaluate the integral as a sum over saddle points
of an effective action in the limit where the rank of the gauge group is
infinite. For a generic quiver theory under consideration, we find a special
family of saddles whose effective action takes a universal form controlled by
the anomaly coefficients of the theory. This family includes the known
supersymmetric black hole solution in the holographically dual AdS
theories. We then analyze the index refined by turning on flavor chemical
potentials. We show that, for a certain range of chemical potentials, the
effective action again takes a universal cubic form that is controlled by the
anomaly coefficients of the theory. Finally, we present a large class of
solutions to the saddle-point equations which are labelled by group
homomorphisms of finite abelian groups of order into the torus.Comment: 58 pages; v2: minor changes, published versio
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