Continuous cellular automata on irregular tessellations : mimicking steady-state heat flow

Leaving a few exceptions aside, cellular automata (CA) and the intimately related coupled-map lattices (CML), commonly known as continuous cellular automata (CCA), as well as models that are based upon one of these paradigms, employ a regular tessellation of an Euclidean space in spite of the various drawbacks this kind of tessellation entails such as its inability to cover surfaces with an intricate geometry, or the anisotropy it causes in the simulation results. Recently, a CCA-based model describing steady-state heat flow has been proposed as an alternative to Laplace's equation that is, among other things, commonly used to describe this process, yet, also this model suffers from the aforementioned drawbacks since it is based on the classical CCA paradigm. To overcome these problems, we first conceive CCA on irregular tessellations of an Euclidean space after which we show how the presented approach allows a straightforward simulation of steady-state heat flow on surfaces with an intricate geometry, and, as such, constitutes an full-fledged alternative for the commonly used and easy-to-implement finite difference method, and the more intricate finite element method

Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance

We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as nϵ for ϵ>0 and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate k/n of 13/72=0.180… and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime

Design and Implementation of a Framework for the Interconnection of Cellular Automata in Software and Hardware

There has been a move recently in academia, industry, and the consumer space towards the use of unsupervised parallel computation and distributed networks (i.e., networks of computing elements working together to achieve a global outcome with only local knowledge). To fully understand the types of problems that these systems are applied to regularly, a representative member of this group of unsupervised parallel and distributed systems is needed to allow the development of generalizable results. Although not the only potential candidate, the field of cellular automata is an excellent choice for analyzing how these systems work as it is one of the simplest members of this group in terms of design specification. The current ability of the field of cellular automata to represent the realm of unsupervised parallel and distributed systems is limited to only a subset of the possible systems, which leads to the main goal of this work of finding a method of allowing cellular automata to represent a much larger range of systems. To achieve this goal, a conceptual framework has been developed that allows the definition of interconnected systems of cellular automata that can represent most, if not all, unsupervised parallel and distributed systems. The framework introduces the concept of allowing the boundary conditions of a cellular automaton to be defined by a separately specified system, which can be any system that is capable of producing the information needed, including another cellular automaton. Using this interconnection concept, two forms of computational simplification are enabled: the deconstruction of a large system into smaller, modular pieces; and the construction of a large system built from a heterogeneous set of smaller pieces. This framework is formally defined using an interconnection graph, where edges signify the flow of information from one node to the next and the nodes are the various systems involved. A library has been designed which implements the interconnection graphs defined by the framework for a subset of the possible nodes, primarily to allow an exploration of the field of cellular automata as a potential representational member of unsupervised parallel and distributed systems. This library has been developed with a number of criteria in mind that will allow it to be instantiated on both hardware and software using an open and extendable architecture to enable interaction with external systems and future expansion to take into account novel research. This extendability is discussed in terms of combining the library with genetic algorithms to find an interconnected system that will satisfy a specific computational goal. There are also a number of novel components of the library that further enhance the capabilities of potential research, including methods for automatically building interconnection graphs from sets of cellular automata and the ability to skip over static regions of a given cellular automaton in an intelligent way to reduce computation time. With a particular set of cellular automaton parameters, the use of this feature reduced the computation time by 75%. As a demonstration of the usefulness of both the library and the framework that it implements, a hardware application has been developed which makes use of many of the novel aspects that have been introduced to produce an interactive art installation named 'Aurora'. This application has a number of design requirements that are directly achieved through the use of library components and framework definitions. These design requirements included a lack of centralized control or data storage, a need for visibly dynamic behaviour in the installation, and the desire for the visitors to the installation to be able to affect the visible movement of patterns across the surface of the piece. The success of the library in this application was heavily dependent on its instantiation on a mixture of hardware and software, as well as the ability to extend the library to suit particular needs and aspects of the specific application requirements. The main goal of this thesis research, finding a method that allows cellular automata to represent a much larger range of unsupervised parallel and distributed systems, has been partially achieved in the creation of a novel framework which defines the concept of interconnection, and the design of an interconnection graph using this concept. This allows the field of cellular automata, in combination with the framework, to be an excellent representational member of an extended set of unsupervised parallel and distributed systems when compared to the field alone. A library has been developed that satisfies a broad set of design criteria that allow it to be used in any future research built on the use of cellular automata as this representational member. A hardware application was successfully created that makes use of a number of novel aspects of both the framework and the library to demonstrate their applicability in a real world situation

Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance

We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, polynomial scaling distance and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate~k/n of~13/72 = 0.180... and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime.Comment: 15 pages, 6 figure

Computer modelling of complex systems with applications in physical and related areas

Computational modelling techniques have been applied in physics, biology and other fields for decades to investigate the scale-invariant properties in non-equilibrium complex (many-cell) systems. Specific examples have been considered to underpin the simulation of cellular systems, le sandpiles as simple models of transport phenomena, and soap froths as models of many-cell cellular networks. A number of characteristic properties have been investigated to explore common features of complex systems. Particularly interesting for the simple sandpile automaton is the achievement of the critical state through the phenomenon known as self-organised criticality (SOC). Various simulation algorithms eg cellular automata, direct simulation and Monte Carlo have been used to model the sandpile and froth systems respectively. The studies of a directed and dissipative CML sandpile model provide evidence for the occurrence of SOC, with the system characterised by simple power-law distributions. For the soap froth model, the effect on the evolution of the presence of defects is investigated, together with the impressions of varying the amount of disorder Scaling properties obtained, for various initial conditions, are given in detail. The improvements on methods of computational modelling, and the limitations of software and hardware implementation are also briefly discussed

Proceedings of the COST Action TU1403 : Adaptive Facades Network Final Conference

info:eu-repo/semantics/publishedVersio

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition