44,065 research outputs found
Befriending Askey-Wilson polynomials
We recall five families of polynomials constituting a part of the so-called
Askey-Wilson scheme. We do this to expose properties of the Askey-Wilson (AW)
polynomials that constitute the last, most complicated element of this scheme.
In doing so we express AW density as a product of the density that makes
Hermite polynomials orthogonal times a product of four characteristic
function of Hermite polynomials (\ref{fAW}) just pawing the way to a
generalization of AW integral. Our main results concentrate mostly on the
complex parameters case forming conjugate pairs. We present new fascinating
symmetries between the variables and some newly defined (by the appropriate
conjugate pair) parameters. In particular in (\ref% {rozwiniecie1}) we
generalize substantially famous Poisson-Mehler expansion formula (\ref{PM}) in
which Hermite polynomials are replaced by Al-Salam-Chihara polynomials.
Further we express Askey-Wilson polynomials as linear combinations of
Al-Salam-Chihara (ASC) polynomials. As a by-product we get useful identities
involving ASC polynomials. Finally by certain re-scaling of variables and
parameters we reach AW polynomials and AW densities that have clear
probabilistic interpretation.Comment: 2
Preconditioning complex symmetric linear systems
A new polynomial preconditioner for symmetric complex linear systems based on
Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear
systems is herein presented. It applies to Conjugate Orthogonal Conjugate
Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative
solvers and does not require any estimation of the spectrum of the coefficient
matrix. An upper bound of the condition number of the preconditioned linear
system is provided. Moreover, to reduce the computational cost, an inexact
variant based on incomplete Cholesky decomposition or orthogonal polynomials is
proposed. Numerical results show that the present preconditioner and its
inexact variant are efficient and robust solvers for this class of linear
systems. A stability analysis of the method completes the description of the
preconditioner.Comment: 26 pages, 4 figures, 4 table
Polynomials with symmetric zeros
Polynomials whose zeros are symmetric either to the real line or to the unit
circle are very important in mathematics and physics. We can classify them into
three main classes: the self-conjugate polynomials, whose zeros are symmetric
to the real line; the self-inversive polynomials, whose zeros are symmetric to
the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric
by an inversion with respect to the unit circle followed by a reflection in the
real line. Real self-reciprocal polynomials are simultaneously self-conjugate
and self-inversive so that their zeros are symmetric to both the real line and
the unit circle. In this survey, we present a short review of these
polynomials, focusing on the distribution of their zeros.Comment: Keywords: Self-inversive polynomials, self-reciprocal polynomials,
Pisot and Salem polynomials, M\"obius transformations, knot theory, Bethe
equation
Unquenched QCD dirac operator spectra at nonzero baryon chemical potential
The microscopic spectral density of the QCD Dirac operator at nonzero baryon chemical potential for an arbitrary number of quark flavors was derived recently from a random matrix model with the global symmetries of QCD. In this paper we show that these results and extensions thereof can be obtained from the replica limit of a Toda lattice equation. This naturally leads to a factorized form into bosonic and fermionic QCD-like partition functions. In the microscopic limit these partition functions are given by the static limit of a chiral Lagrangian that follows from the symmetry breaking pattern. In particular, we elucidate the role of the singularity of the bosonic partition function in the orthogonal polynomials approach. A detailed discussion of the spectral density for one and two flavors is given
On matrix model partition functions for QCD with chemical potential
Partition functions of two different matrix models for QCD with chemical potential are computed for an arbitrary number of quark and complex conjugate anti-quark flavors. In the large-N limit of weak nonhermiticity complete agreement is found between the two models. This supports the universality of such fermionic partition functions, that is of products of characteristic polynomials in the complex plane. In the strong nonhermiticity limit agreement is found for an equal number of quark and conjugate flavours. For a general flavor content the equality of partition functions holds only for small chemical potential. The chiral phase transition is analyzed for an arbitrary number of quarks, where the free energy presents a discontinuity of first order at a critical chemical potential. In the case of nondegenerate flavors there is first order phase transition for each separate mass scale
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