Hopf algebras: motivations and examples

This paper provides motivation as well as a method of construction for Hopf algebras, starting from an associative algebra. The dualization technique involved relies heavily on the use of Sweedler's dual

The Algebra of Binary Search Trees

We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson-Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of Planar Binary Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric Functions and Free Symmetric Functions. We briefly explain how the main known properties of the Loday-Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees.Comment: 49 page

Endohedral Metallofullerene Derivatives

Trimetallic nitride endohedral metallofullerene derivatives and their preparation are described. The trimetallic nitride endohedral metallofullerene derivatives have the general formula A(sub 3-n)X(sub n)@C(sub m)(R) where n ranges from 0 to 3, A and X may be trivalent metals and may be either rare earth metal or group IIIB metals, m is between about 60 and about 200, and R is preferably an organic group. Derivatives where the R group forms cyclized derivatives with the fullerene cage are also described

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

Environments for sonic ecologies

This paper outlines a current lack of consideration for the environmental context of Evolutionary Algorithms used for the generation of music. We attempt to readdress this balance by outlining the benefits of developing strong coupling strategies between agent and en- vironment. It goes on to discuss the relationship between artistic process and the viewer and suggests a placement of the viewer and agent in a shared environmental context to facilitate understanding of the artistic process and a feeling of participation in the work. The paper then goes on to outline the installation ‘Excuse Me and how it attempts to achieve a level of Sonic Ecology through the use of a shared environmental context

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

An individual-based model to explore the impacts of lesser-known social dynamics on wolf populations

The occurrence of wolf populations in human-dominated landscapes is challenging worldwide because of conflicts with human activities. Modeling is an important tool to project wolf dynamics and expansion, and help in decision making concerning management and conservation. However, some individual behaviors and pack dynamics of the wolf life cycle are still unclear to ecologists. Here we present an individual-based model (IBM) to project wolf populations while exploring the lesser-known processes of the wolf life cycle. IBMs are bottom-up models that simulate the fate of individuals interacting with each other, with population-level properties emerging from the individual-level simulations. IBMs are particularly adapted to represent social species such as the wolf that exhibits complex individual interactions. Our IBM projects wolf demography including fine-scale individual behavior and pack dynamics based on up-to-date scientific literature. We explore four processes of the wolf life cycle whose consequences on population dynamics are still poorly understood: the pack dissolution following the loss of a breeder, the adoption of young dispersers by packs, the establishment of new packs through budding, and the different breeder replacement strategies. While running different versions of the IBM to explore these processes, we also illustrate the modularity and flexibility of our model, an asset to model wolf populations experiencing different ecological and demographic conditions. The different parameterization of pack dissolution, territory establishment by budding, and breeder replacement processes influence the projections of wolf populations. As such, these processes require further field investigation to be better understood. The adoption process has a lesser impact on model projections. Being coded in R to facilitate its understanding, we expect that our model will be used and further adapted by ecologists for their own specific applications

Role of thyroid hormones in early postnatal development of skeletal muscle and its implications for undernutrition

Published online by Cambridge University Press 09 Mar 2007Energy intake profoundly influences many endocrine axes which in turn play a central role in development. The specific influence of a short period of mild hypothyroidism, similar to that induced by undernutrition, in regulating muscle development has been assessed in a large mammal during early postnatal life. Hypothyroidism was induced by providing methimazole and iopanoic acid in the feed of piglets between 4 and 14 d of age, and controls were pair-fed to the energy intake of their hypothyroid littermates. Thyroid status was evaluated, and myofibre differentiation and cation pump concentrations were then assessed in the following functionally distinct muscles: longissimus dorsi (l. dorsi), soleus and rhomboideus. Reductions in plasma concentrations of thyroxine (T4; 32%, P < O·Ol), triiodothyronine (T3;48%, P < 0·001), free T3, (58%, P < 0·001)and hepatic 5'-monodeiodinase (EC 1.11.1.8) activity (74%, P < 0·001) occurred with treatment. Small, although significant, increases in the proportion of type I slow-twitch oxidative fibres occurred with mild hypothyroidism, in l. dorsi (2%, P < 0·01) and soleus(7%, P < 0·01). Nuclear T3-receptor concentration in l. dorsi of hypothyroid animals compared with controls increased by 46% (P < 0·001), a response that may represent a homeostatic mechanism making muscle more sensitive to low levels of circulating thyroid hormones. Nevertheless, Na+, K+-ATPase (EC 3.6.1.37) concentration was reduced by 15–16% in all muscles (l.dorsi P< 0·05,soleus P < 0·001, rhomboideus P < 0·05), and Ca2+-ATPase (EC 3.6.1.38) concentration was significantly reduced in the two slow-twitch muscles: by 22% in rhomboideus (P < 0·001) and 23% in soleus (P < 0·05). It is concluded that during early postnatal development of large mammals a period of mild hypothyroidism, comparable with that found during undernutrition, induces changes in myofibre differentiation and a down-regulation of cation pumps in skeletal muscle. Such changes would result in slowness of movement and muscle weakness, and also reduce ATP hydrolysis with a concomitant improvement in energetic efficiency.A. P. Harrison, D. R. Tivey, T. Clausen, C. Duchamp and M. J. Daunce

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