787 research outputs found
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
Hopf Algebras in General and in Combinatorial Physics: a practical introduction
This tutorial is intended to give an accessible introduction to Hopf
algebras. The mathematical context is that of representation theory, and we
also illustrate the structures with examples taken from combinatorics and
quantum physics, showing that in this latter case the axioms of Hopf algebra
arise naturally. The text contains many exercises, some taken from physics,
aimed at expanding and exemplifying the concepts introduced
A product formula and combinatorial field theory
We treat the problem of normally ordering expressions involving the standard boson operators a, ay where [a; ay] = 1. We show that a simple product formula for formal power series | essentially an extension of the Taylor expansion | leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions | in essence, a combinatorial eld theory. We apply these techniques to some examples related to specic physical Hamiltonians
Challenges and opportunities on lipid metabolism disorders diagnosis and therapy: Novel insights and future perspective
Dyslipidemia has been globally recognized, for almost seven decades, as one of the most important risk factors for the development and complications of atherosclerotic cardiovascular disease (ASCVD) [...]
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
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
Laguerre-type derivatives: Dobinski relations and combinatorial identities
We consider properties of the operators D(r,M)=a^r(a^\dag a)^M (which we call
generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and
a^\dag are boson annihilation and creation operators respectively, satisfying
[a,a^\dag]=1. We obtain explicit formulas for the normally ordered form of
arbitrary Taylor-expandable functions of D(r,M) with the help of an operator
relation which generalizes the Dobinski formula. Coherent state expectation
values of certain operator functions of D(r,M) turn out to be generating
functions of combinatorial numbers. In many cases the corresponding
combinatorial structures can be explicitly identified.Comment: 14 pages, 1 figur
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
Operator solutions for fractional Fokker-Planck equations
We obtain exact results for fractional equations of Fokker-Planck type using
evolution operator method. We employ exact forms of one-sided Levy stable
distributions to generate a set of self-reproducing solutions. Explicit cases
are reported and studied for various fractional order of derivatives, different
initial conditions, and for different versions of Fokker-Planck operators.Comment: 4 pages, 3 figure
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