Combinatorics and Boson normal ordering: A gentle introduction

We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick's theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.Comment: 8 pages, 1 figur

Euler Polynomials and Identities for Non-Commutative Operators

Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt, expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, due to J.-C. Pain, links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.Comment: 20 page

Heisenberg-Weyl algebra revisited: Combinatorics of words and paths

The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and applications.Comment: 5 pages, 3 figure

Bernoulli type polynomials on Umbral Algebra

The aim of this paper is to investigate generating functions for modification of the Milne-Thomson's polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. By applying the Umbral algebra to these generating functions, we provide to deriving identities for these polynomials

Roles of Intra-fruit Oxygen and Carbon Dioxide in Controlling Pepper (Capsicum annuum L.) Seed Development and Storage Reserve Deposition

Seeds developing within a locular space inside hollow fruit experience chronic exposure to a unique gaseous environment. Using two pepper cultivars, `Triton\u27 (sweet) and `PI 140367\u27 (hot), we investigated how the development of seeds is affected by the gases surrounding them. The atmospheric composition of the seed environment was characterized during development by analysis of samples withdrawn from the fruit locule with a gas-tight syringe. As seed weight plateaued during development, the seed environment reached its lowest O2 concentration (19%) and highest CO2 concentration (3%). We experimentally manipulated the seed environment by passing different humidified gas mixtures through the fruit locule at a rate of 60 to 90 mL·min-1. A synthetic atmosphere containing 3% CO2, 21% O2, and 76% N2 was used to represent a standard seed environment. Seeds developing inside locules supplied with this mixture had enhanced average seed weight, characterized by lower variation than in the no-flow controls due to fewer low-weight seeds. The importance of O2 in the seed microenvironment was demonstrated by reduction in seed weight when the synthetic atmosphere contained only 15% O2 and by complete arrest of embryo development when O2 was omitted from the seed atmosphere. Removal of CO2 from the synthetic atmosphere had no effect on seed weight, however, the CO2-free treatment accelerated fruit ripening by 4 days in the hot pepper. In the sweet peppers, fruit wall starch and sucrose were reduced by the CO2-free treatment. The results demonstrate that accretionary seed growth is being limited in pepper by O2 availability and suggest that variation in seed quality is attributable to localized limitations in O2 supply

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

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

Monomiality principle, Sheffer-type polynomials and the normal ordering problem

We solve the boson normal ordering problem for $(q(a^\dag)a+v(a^\dag))^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^\dag$ are boson annihilation and creation operators, satisfying $[a,a^\dag]=1$. This consequently provides the solution for the exponential $e^{\lambda(q(a^\dag)a+v(a^\dag))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics "Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005), Myczkowce, Poland. 13 pages, 31 reference

A differential identity for Green functions

If P is a differential operator with constant coefficients, an identity is derived to calculate the action of exp(P) on the product of two functions. In many-body theory, P describes the interaction Hamiltonian and the identity yields a hierarchy of Green functions. The identity is first derived for scalar fields and the standard hierarchy is recovered. Then the case of fermions is considered and the identity is used to calculate the generating function for the Green functions of an electron system in a time-dependent external potential.Comment: 14 page

Marine fisheries and future ocean conflict

Conflict over marine fishery resources is a growing security concern. Experts expect that global changes in our climate, food systems and oceans may spark or exacerbate resource conflicts. An initial scan of 803 relevant papers and subsequent intensive review of 31 fisheries conflict studies, focused on subnational and international conflicts, suggests that four substantial scientific gaps need addressing to improve our understanding of the nature and drivers of fisheries conflict. First, fisheries conflict and levels of conflict intensity are not precisely defined. Second, complex adaptive systems thinking is underutilized but has the potential to produce more realistic causal models of fishery conflict. Third, comparative large‐scale data and suitably integrative methodologies are lacking, underscoring the need for a standardized and comparable database of fisheries conflict cases to aid extrapolation beyond single case‐studies. Fourth, there is room for a more widespread application of higher order concepts and associated terminology. Importantly, the four gaps highlight the homogenized nature of current methodological and theoretical approaches to understanding fishery conflict, which potentially presents us with an oversimplified understanding of these conflicts. A more nuanced understanding of the complex and dynamic nature of fishery conflict and its causes is not only scientifically critical, but increasingly relevant for policymakers and practitioners in this turbulent world