716 research outputs found
Spectral density of generalized Wishart matrices and free multiplicative convolution
We investigate the level density for several ensembles of positive random
matrices of a Wishart--like structure, , where stands for a
nonhermitian random matrix. In particular, making use of the Cauchy transform,
we study free multiplicative powers of the Marchenko-Pastur (MP) distribution,
, which for an integer yield Fuss-Catalan
distributions corresponding to a product of independent square random
matrices, . New formulae for the level densities are derived
for and . Moreover, the level density corresponding to the
generalized Bures distribution, given by the free convolution of arcsine and MP
distributions is obtained. We also explain the reason of such a curious
convolution. The technique proposed here allows for the derivation of the level
densities for several other cases.Comment: 10 latex pages including 4 figures, Ver 4, minor improvements and
references updat
Dobinski-type relations and the Log-normal distribution
We consider sequences of generalized Bell numbers B(n), n=0,1,... for which
there exist Dobinski-type summation formulas; that is, where B(n) is
represented as an infinite sum over k of terms P(k)^n/D(k). These include the
standard Bell numbers and their generalizations appearing in the normal
ordering of powers of boson monomials, as well as variants of the "ordered"
Bell numbers. For any such B we demonstrate that every positive integral power
of B(m(n)), where m(n) is a quadratic function of n with positive integral
coefficients, is the n-th moment of a positive function on the positive real
axis, given by a weighted infinite sum of log-normal distributions.Comment: 7 pages, 2 Figure
Dobinski-type relations: Some properties and physical applications
We introduce a generalization of the Dobinski relation through which we
define a family of Bell-type numbers and polynomials. For all these sequences
we find the weight function of the moment problem and give their generating
functions. We provide a physical motivation of this extension in the context of
the boson normal ordering problem and its relation to an extension of the Kerr
Hamiltonian.Comment: 7 pages, 1 figur
Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem
We consider the numbers arising in the problem of normal ordering of
expressions in canonical boson creation and annihilation operators. We treat a
general form of a boson string which is shown to be associated with
generalizations of Stirling and Bell numbers. The recurrence relations and
closed-form expressions (Dobiski-type formulas) are obtained for these
quantities by both algebraic and combinatorial methods. By extensive use of
methods of combinatorial analysis we prove the equivalence of the
aforementioned problem to the enumeration of special families of graphs. This
link provides a combinatorial interpretation of the numbers arising in this
normal ordering problem.Comment: 10 pages, 5 figure
Factors that determine the effectiveness of peer interventions in prisons in England and Wales
Epidemiological assessment of the prison population globally shows undeniable health need, with research evidence consistently demonstrating that the prevalence of ill health is higher than rates reported in the wider community. Since a meeting convened by the World Health Organisation in the mid-1990s, prisons have been regarded as legitimate settings for health promotion and a myriad of interventions have been adopted to address prisoners’ health and social need. Peer-based approaches have been a common health intervention used within the prison system, but despite their popularity little evidence exists on the approach. This paper presents findings from an expert symposium – part of a wider study which included a systematic review – designed to gather expert opinion on whether and how peer–based approaches work within prisons and if they can contribute to improving the health of prisoners. Experts were selected from various fields including the prison service, academic research and third sector organisations. Expert evidence suggested that the magnitude of success of peer interventions in prison settings is contingent on understanding the contextual environment and a recognition that peer interventions are co-constructed with prison staff at all levels of the organisation. Implications for developing peer-based interventions in prison are given which assist in developing the concept, theory and practice of the health promoting prison
From Quantum Mechanics to Quantum Field Theory: The Hopf route
We show that the combinatorial numbers known as {\em Bell numbers} are
generic in quantum physics. This is because they arise in the procedure known
as {\em Normal ordering} of bosons, a procedure which is involved in the
evaluation of quantum functions such as the canonical partition function of
quantum statistical physics, {\it inter alia}. In fact, we shall show that an
evaluation of the non-interacting partition function for a single boson system
is identical to integrating the {\em exponential generating function} of the
Bell numbers, which is a device for encapsulating a combinatorial sequence in a
single function. We then introduce a remarkable equality, the Dobinski
relation, and use it to indicate why renormalisation is necessary in even the
simplest of perturbation expansions for a partition function. Finally we
introduce a global algebraic description of this simple model, giving a Hopf
algebra, which provides a starting point for extensions to more complex
physical systems
Hierarchical Dobinski-type relations via substitution and the moment problem
We consider the transformation properties of integer sequences arising from
the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form
exp(x (a*)^r a), r=1,2,..., under the composition of their exponential
generating functions (egf). They turn out to be of Sheffer-type. We demonstrate
that two key properties of these sequences remain preserved under
substitutional composition: (a)the property of being the solution of the
Stieltjes moment problem; and (b) the representation of these sequences through
infinite series (Dobinski-type relations). We present a number of examples of
such composition satisfying properties (a) and (b). We obtain new Dobinski-type
formulas and solve the associated moment problem for several hierarchically
defined combinatorial families of sequences.Comment: 14 pages, 31 reference
A generic Hopf algebra for quantum statistical mechanics
In this paper, we present a Hopf algebra description of a bosonic quantum
model, using the elementary combinatorial elements of Bell and Stirling
numbers. Our objective in doing this is as follows. Recent studies have
revealed that perturbative quantum field theory (pQFT) displays an astonishing
interplay between analysis (Riemann zeta functions), topology (Knot theory),
combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure).
Since pQFT is an inherently complicated study, so far not exactly solvable and
replete with divergences, the essential simplicity of the relationships between
these areas can be somewhat obscured. The intention here is to display some of
the above-mentioned structures in the context of a simple bosonic quantum
theory, i.e. a quantum theory of non-commuting operators that do not depend on
space-time. The combinatorial properties of these boson creation and
annihilation operators, which is our chosen example, may be described by
graphs, analogous to the Feynman diagrams of pQFT, which we show possess a Hopf
algebra structure. Our approach is based on the quantum canonical partition
function for a boson gas.Comment: 8 pages/(4 pages published version), 1 Figure. arXiv admin note: text
overlap with arXiv:1011.052
Some useful combinatorial formulae for bosonic operators
We give a general expression for the normally ordered form of a function
F(w(a,a*)) where w is a function of boson annihilation and creation operators
satisfying [a,a*]=1. The expectation value of this expression in a coherent
state becomes an exact generating function of Feynman-type graphs associated
with the zero-dimensional Quantum Field Theory defined by F(w). This enables
one to enumerate explicitly the graphs of given order in the realm of
combinatorially defined sequences. We give several examples of the use of this
technique, including the applications to Kerr-type and superfluidity-type
hamiltonians.Comment: 8 pages, 3 figures, 17 reference
Exponential Operators, Dobinski Relations and Summability
We investigate properties of exponential operators preserving the particle
number using combinatorial methods developed in order to solve the boson normal
ordering problem. In particular, we apply generalized Dobinski relations and
methods of multivariate Bell polynomials which enable us to understand the
meaning of perturbation-like expansions of exponential operators. Such
expansions, obtained as formal power series, are everywhere divergent but the
Pade summation method is shown to give results which very well agree with exact
solutions got for simplified quantum models of the one mode bosonic systems.Comment: Presented at XIIth Central European Workshop on Quantum Optics,
Bilkent University, Ankara, Turkey, 6-10 June 2005. 4 figures, 6 pages, 10
reference
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