125 research outputs found
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Mini-Workshop: Dynamics of Stochastic Systems and their Approximation
The aim of this workshop was to bring together specialists in the area of stochastic dynamical systems and stochastic numerical analysis to exchange their ideas about the state of the art of approximations of stochastic dynamics. Here approximations are considered in the analytical sense in terms of deriving reduced dynamical systems, which are less complex, as well as in the numerical sense via appropriate simulation methods. The main theme is concerned with the efficient treatment of stochastic dynamical systems via both approaches assuming that ideas and methods from one ansatz may prove beneficial for the other. A particular goal was to systematically identify open problems and challenges in this area
Complex and Adaptive Dynamical Systems: A Primer
An thorough introduction is given at an introductory level to the field of
quantitative complex system science, with special emphasis on emergence in
dynamical systems based on network topologies. Subjects treated include graph
theory and small-world networks, a generic introduction to the concepts of
dynamical system theory, random Boolean networks, cellular automata and
self-organized criticality, the statistical modeling of Darwinian evolution,
synchronization phenomena and an introduction to the theory of cognitive
systems.
It inludes chapter on Graph Theory and Small-World Networks, Chaos,
Bifurcations and Diffusion, Complexity and Information Theory, Random Boolean
Networks, Cellular Automata and Self-Organized Criticality, Darwinian
evolution, Hypercycles and Game Theory, Synchronization Phenomena and Elements
of Cognitive System Theory.Comment: unformatted version of the textbook; published in Springer,
Complexity Series (2008, second edition 2010
A study of poststenotic shear layer instabilities
Imperial Users onl
Dedication to Professor Michael Tribelsky
Professor Tribelsky's accomplishments are highly appreciated by the international community. The best indications of this are the high citation rates of his publications, and the numerous awards and titles he has received. He has made numerous fundamental contributions to an extremely broad area of physics and mathematics, including (but not limited to) quantum solid-state physics, various problems in light–matter interaction, liquid crystals, physical hydrodynamics, nonlinear waves, pattern formation in nonequilibrium systems and transition to chaos, bifurcation and probability theory, and even predictions of the dynamics of actual market prices. This book presents several extensions of his results, based on his inspiring publications
Nonlocal Models in Biology and Life Sciences: Sources, Developments, and Applications
Nonlocality is important in realistic mathematical models of physical and
biological systems at small-length scales. It characterizes the properties of
two individuals located in different locations. This review illustrates
different nonlocal mathematical models applied to biology and life sciences.
The major focus has been given to sources, developments, and applications of
such models. Among other things, a systematic discussion has been provided for
the conditions of pattern formations in biological systems of population
dynamics. Special attention has also been given to nonlocal interactions on
networks, network coupling and integration, including models for brain dynamics
that provide us with an important tool to better understand neurodegenerative
diseases. In addition, we have discussed nonlocal modelling approaches for
cancer stem cells and tumor cells that are widely applied in the cell migration
processes, growth, and avascular tumors in any organ. Furthermore, the
discussed nonlocal continuum models can go sufficiently smaller scales applied
to nanotechnology to build biosensors to sense biomaterial and its
concentration. Piezoelectric and other smart materials are among them, and
these devices are becoming increasingly important in the digital and physical
world that is intrinsically interconnected with biological systems.
Additionally, we have reviewed a nonlocal theory of peridynamics, which deals
with continuous and discrete media and applies to model the relationship
between fracture and healing in cortical bone, tissue growth and shrinkage, and
other areas increasingly important in biomedical and bioengineering
applications. Finally, we provided a comprehensive summary of emerging trends
and highlighted future directions in this rapidly expanding field.Comment: 71 page
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Unconventional computing platforms and nature-inspired methods for solving hard optimisation problems
The search for novel hardware beyond the traditional von Neumann architecture has given rise to a modern area of unconventional computing requiring the efforts of mathematicians, physicists and engineers. Many analogue physical systems, including networks of nonlinear oscillators, lasers, condensates, and superconducting qubits, are proposed and realised to address challenging computational problems from various areas of social and physical sciences and technology. Understanding the underlying physical process by which the system finds the solutions to such problems often leads to new optimisation algorithms. This thesis focuses on studying gain-dissipative systems and nature-inspired algorithms that form a hybrid architecture that may soon rival classical hardware.
Chapter 1 lays the necessary foundation and explains various interdisciplinary terms that are used throughout the dissertation. In particular, connections between the optimisation problems and spin Hamiltonians are established, their computational complexity classes are explained, and the most prominent physical platforms for spin Hamiltonian implementation are reviewed.
Chapter 2 demonstrates a large variety of behaviours encapsulated in networks of polariton condensates, which are a vivid example of a gain-dissipative system we use throughout the thesis. We explain how the variations of experimentally tunable parameters allow the networks of polariton condensates to represent different oscillator models. We derive analytic expressions for the interactions between two spatially separated polariton condensates and show various synchronisation regimes for periodic chains of condensates. An odd number of condensates at the vertices of a regular polygon leads to a spontaneous formation of a giant multiply-quantised vortex at the centre of a polygon. Numerical simulations of all studied configurations of polariton condensates are performed with a mean-field approach with some theoretically proposed physical phenomena supported by the relevant experiments.
Chapter 3 examines the potential of polariton graphs to find the low-energy minima of the spin Hamiltonians. By associating a spin with a condensate phase, the minima of the XY model are achieved for simple configurations of spatially-interacting polariton condensates. We argue that such implementation of gain-dissipative simulators limits their applicability to the classes of easily solvable problems since the parameters of a particular Hamiltonian depend on the node occupancies that are not known a priori. To overcome this difficulty, we propose to adjust pumping intensities and coupling strengths dynamically. We further theoretically suggest how the discrete Ising and -state planar Potts models with or without external fields can be simulated using gain-dissipative platforms. The underlying operational principle originates from a combination of resonant and non-resonant pumping. Spatial anisotropy of pump and dissipation profiles enables an effective control of the sign and intensity of the coupling strength between any two neighbouring sites, which we demonstrate with a two dimensional square lattice of polariton condensates. For an accurate minimisation of discrete and continuous spin Hamiltonians, we propose a fully controllable polaritonic XY-Ising machine based on a network of geometrically isolated polariton condensates.
In Chapter 4, we look at classical computing rivals and study nature-inspired methods for optimising spin Hamiltonians. Based on the operational principles of gain-dissipative machines, we develop a novel class of gain-dissipative algorithms for the optimisation of discrete and continuous problems and show its performance in comparison with traditional optimisation techniques. Besides looking at traditional heuristic methods for Ising minimisation, such as the Hopfield-Tank neural networks and parallel tempering, we consider a recent physics-inspired algorithm, namely chaotic amplitude control, and exact commercial solver, Gurobi. For a proper evaluation of physical simulators, we further discuss the importance of detecting easy instances of hard combinatorial optimisation problems. The Ising model for certain interaction matrices, that are commonly used for evaluating the performance of unconventional computing machines and assumed to be exponentially hard, is shown to be solvable in polynomial time including the Mobius ladder graphs and Mattis spin glasses.
In Chapter 5 we discuss possible future applications of unconventional computing platforms including emulation of search algorithms such as PageRank, realisation of a proof-of-work protocol for blockchain technology, and reservoir computing
Analog Photonics Computing for Information Processing, Inference and Optimisation
This review presents an overview of the current state-of-the-art in photonics
computing, which leverages photons, photons coupled with matter, and
optics-related technologies for effective and efficient computational purposes.
It covers the history and development of photonics computing and modern
analogue computing platforms and architectures, focusing on optimization tasks
and neural network implementations. The authors examine special-purpose
optimizers, mathematical descriptions of photonics optimizers, and their
various interconnections. Disparate applications are discussed, including
direct encoding, logistics, finance, phase retrieval, machine learning, neural
networks, probabilistic graphical models, and image processing, among many
others. The main directions of technological advancement and associated
challenges in photonics computing are explored, along with an assessment of its
efficiency. Finally, the paper discusses prospects and the field of optical
quantum computing, providing insights into the potential applications of this
technology.Comment: Invited submission by Journal of Advanced Quantum Technologies;
accepted version 5/06/202
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