Hofstadter butterflies of carbon nanotubes: Pseudofractality of the magnetoelectronic spectrum

The electronic spectrum of a two-dimensional square lattice in a perpendicular magnetic field has become known as the Hofstadter butterfly [Hofstadter, Phys. Rev. B 14, 2239 (1976).]. We have calculated quasi-one-dimensional analogs of the Hofstadter butterfly for carbon nanotubes (CNTs). For the case of single-wall CNTs, it is straightforward to implement magnetic fields parallel to the tube axis by means of zone folding in the graphene reciprocal lattice. We have also studied perpendicular magnetic fields which, in contrast to the parallel case, lead to a much richer, pseudofractal spectrum. Moreover, we have investigated magnetic fields piercing double-wall CNTs and found strong signatures of interwall interaction in the resulting Hofstadter butterfly spectrum, which can be understood with the help of a minimal model. Ubiquitous to all perpendicular magnetic field spectra is the presence of cusp catastrophes at specific values of energy and magnetic field. Resolving the density of states along the tube circumference allows recognition of the snake states already predicted for nonuniform magnetic fields in the two-dimensional electron gas. An analytic model of the magnetic spectrum of electrons on a cylindrical surface is used to explain some of the results.Comment: 14 pages, 12 figures update to published versio

Upper Bounds on the Bisection Width of 3- and 4-regular Graphs

We derive new upper bounds on the bisection width of graphs which have a regular vertex degree. We show that the bisection width of large 3-regular graphs with |V| vertices is at most 1/6 |V|. For the bisection width of large 4-regular graphs we show an upper bound of 2/5 |V|

Who is not developing open source software? non-users, users, and developers

The development of knowledge requires investment, which may be made in terms of financial resources or time. Open source software (OSS) has challenged much of the traditional reasoning by suggesting that individuals behave altruistically and contribute to a public good, despite the opportunity to free-ride. The lion's share of the existing literature on OSS examines communities, that is, those individuals whom are already part of the OSS community. In contrast, this paper starts from users with the requisite skill to use and develop OSS. This group of skilled individuals could potentially invest into the development of OSS knowledge, but they may or may not do so in actuality. This paper, therefore, explores three issues, which have not been extensively explored in the literature, namely, (1) how frequently a group of skilled people use OSS, (2) reasons for differences among users and non-users in terms of use and attitudes, and (3) how frequently, and why, some users contribute to OSS projects (and thereby become developers). In doing so, we consider the opportunity costs of use and development of OSS, which has been largely neglected in the literature. We find that the individuals have a rather pragmatic attitude to firms and that many are active in both firms and OSS community, which raises many questions for future research about the role and influence of firms on the development and diffusion of OSS.Opportunity cost, Motivation, Diffusion, Knowledge development, Public goods, Open source software,