Power-law energy level-spacing distributions in fractals

In this article we investigate the energy spectrum statistics of fractals at the quantum level. We show that the energy-level distribution of a fractal follows a power-law behaviour, if its energy spectrum is a limit set of piece-wise linear functions. We propose that such a behaviour is a general feature of fractals, which can not be described properly by random matrix theory. Several other arguments for the power-law behaviour of the energy level-spacing distributions are proposed

Step-wise development of resilient ambient campus scenarios

This paper puts forward a new approach to developing resilient ambient applications. In its core is a novel rigorous development method supported by a formal theory that enables us to produce a well-structured step-wise design and to ensure disciplined integration of error recovery measures into the resulting implementation. The development method, called AgentB, uses the idea of modelling database to support a coherent development of and reasoning about several model views, including the variable, event, role, agent and protocol views. This helps system developers in separating various modelling concerns and makes it easier for future tool developers to design a toolset supporting this development. Fault tolerance is systematically introduced during the development of various model views. The approach is demonstrated through the development of several application scenarios within an ambient campus case study conducted at Newcastle University (UK) as part of the FP6 RODIN project. © 2009 Springer Berlin Heidelberg

Strong enhancement of superconductivity in finitely ramified fractal lattices

Using the Sierpinski gasket (triangle) and carpet (square) lattices as examples, we theoretically study the properties of fractal superconductors. For that, we focus on the phenomenon of $s$-wave superconductivity in the Hubbard model with attractive on-site potential and employ the Bogoliubov-de Gennes approach and the theory of superfluid stiffness. For the case of the Sierpinski gasket, we demonstrate that fractal geometry of the underlying crystalline lattice can be strongly beneficial for superconductivity, not only leading to a considerable increase of the critical temperature $T_c$ as compared to the regular triangular lattice but also supporting macroscopic phase coherence of the Cooper pairs. In contrast, the Sierpinski carpet geometry does not lead to pronounced effects, and we find no substantial difference as compared with the regular square lattice. We conjecture that the qualitative difference between these cases is caused by different ramification properties of the fractals.Comment: 9 pages, 6 figure

Hall conductivity of Sierpinski carpet

We calculate the Hall conductivity of a Sierpinski carpet using Kubo-Bastin formula. The quantization of Hall conductivity disappears when we increase the depth of the fractal. The Hall conductivity is no more proportional to the Chern number. Nevertheless, these quantities behave in a similar way showing some reminiscence of a topological nature of the Hall conductivity. We also study numerically the bulk-edge correspondence and find that the edge states become less manifested when the depth of Sierpinski carpet is increased

Linearized spectral decimation in fractals

In this article we study the energy level spectrum of fractals which have block-hierarchical structures. We develop a method to study the spectral properties in terms of linearization of spectral decimation procedure and verify it numerically. Our approach provides qualitative explanations for various spectral properties of self-similar graphs within the theory of dynamical systems, including power-law level-spacing distribution, smooth density of states and effective chaotic regime

A Survey on Event-B Decomposition

Model decomposition is a powerful tool to scale the design of large and complex systems. It enables developers to separate components development from the concerns of their integration and orchestration. Event-B is a refinementbased formal method, equipped with three decomposition styles that come with solid semantic foundations and strong tool support. This paper intends to give some useful insights and modelling guidelines for using these decomposition styles, illustrated by an actual development of a master data updating system

Level-spectra Statistics in Planar Fractal Tight-Binding Models

In this communication, we study the level-spectra statistics when a noninteracting electron gas is confined in \textit{Sierpi\'{n}ski Carpet} (\textit{SC}) lattices. These \textit{SC} lattices are constructed under two representative patterns of the $self$ and $gene$ patterns, and classified into two subclass lattices by the area-perimeter scaling law. By the singularly continuous spectra and critical traits using two level-statistic tools\iffalse the nearest spacing distribution and alternative gap-ratio distribution\fi, we ascertain that both obey the critical phase due to broken translation symmetry and the long-range order of scaling symmetry. The Wigner-like conjecture is confirmed numerically since both belong to the Gaussian orthogonal ensemble. An analogy was observed in a quasiperiodic lattice~\cite{Zhong1998Level}. In addition, this critical phase isolates the crucial behavior near the metal-insulator transition edge in Anderson model. The lattice topology of the self-similarity feature can induce level clustering behavior.Comment: 11 pages, 6 figures, 2 Table