Quantum computation of multifractal exponents through the quantum wavelet transform

We study the use of the quantum wavelet transform to extract efficiently information about the multifractal exponents for multifractal quantum states. We show that, combined with quantum simulation algorithms, it enables to build quantum algorithms for multifractal exponents with a polynomial gain compared to classical simulations. Numerical results indicate that a rough estimate of fractality could be obtained exponentially fast. Our findings are relevant e.g. for quantum simulations of multifractal quantum maps and of the Anderson model at the metal-insulator transition.Comment: 9 pages, 9 figure

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

A quasi-dimensional spark ignition two stroke engine model

Despite challenges with poor emissions and fuel economy, gasoline two stroke engines continue to be developed for a number of applications. The primary reasons for the choice of a gasoline two stroke engine includes its low cost, mechanical simplicity and high specific power output. Some applications for the gasoline two stroke engine include small capacity motorcycles and scooters, off road recreational vehicles, hand held power tools and unmanned aerial vehicles. New technologies, which are already mature in four stroke engines, are now being applied to two stroke engines. Such technologies include direct fuel injection, electronic engine management and exhaust gas after treatment. To implement these new technologies computation models are being continuously developed to improve the design process of engines. Multi-dimensional computational fluid dynamics modelling is now commonly applied to engine research and development, it is a powerful tool that can give great insight into the thermofluid working of an engine. Multi-dimensional tools are however computationally expensive and quasi-dimensional modelling methods are often better suited for the analysis of an engine, for example in transient engine simulation. This thesis reports the development of a new quasi-dimensional combustion model for a loop scavenged two stroke engine. The model differs from other quasi-dimensional models available in the literature as it accounts for a bulk motion of the flame front due to the tumble motion created by the loop scavenge process. In this study the tumble motion is modelled as an ellipsoid vortex and the size of the vortex is defined by the combustion chamber height and a limiting elliptical aspect ratio. The limiting aspect ratio has been observed in experimental square piston compression machines and optical engines. The new model also accounts for a wrinkled flame brush thickness and its effects on the interaction between flame front and combustion chamber. The new combustion model has been validated against experimental engine tests in which the flame front propagation was measured using ionization probes. The probes were able determine the flame front shape, the bulk movement of the flame front due to tumble and also the wrinkled flame brush thickness

Thermomechanical analysis of rock asperity in fractures of enhanced geothermal systems

Enhanced Geothermal Systems (EGS) offer great potential for dramatically expanding the use of geothermal energy and become a promising supplement for fossil energy. The EGS is to extract heat by creating a subsurface system to which cold water can be added through injection wells. Injected water is heated by contact with rock and returns to the surface through production well. Fracture provides the primary conduit for fluid flow and heat transfer in natural rock. Fracture is propped by fracture roughness with varying heights which is called asperity. The stability of asperity determines fracture aperture and hence imposes substantial effect on hydraulic conductivity and heat transfer efficiency in EGS. Firstly, two rough fracture surfaces are characterized by statistical method and fractal analysis. The asperity heights and enclosed aperture heights are described by probability density function before cold water is pumped into fracture. Secondly, when water injection and induced cooling occurs, the thermomechanical analysis of single asperity is studied by establishing an un-symmetric damage mechanics model. The deformation curve of asperity under thermal stress is determined. Thirdly, deformation of fracture with various asperities on it in response to thermal stress is analyzed by a new stratified continuum percolation model. This model incorporates the fracture surface characteristics and preceding deformation curve of asperity. The fracture closure and fracture stiffness can be accurately quantified by this model. In addition, the scaling invariance and multifractal parameters in this process are identified and validated with Monte Carlo simulation --Abstract, page iii

p-Adic Mathematical Physics

A brief review of some selected topics in p-adic mathematical physics is presented.Comment: 36 page

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized Criticality and Turbulence" (2012, 2013, Bern, Switzerland

Liouville Quantum Gravity and KPZ

Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.Comment: 56 pages. Revised version contains more details. To appear in Inventione