Descriptive set theory and the ergodic theory of countable groups

The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel reducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of residually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Abért and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirming a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how approximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group's reduced C^*-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions.

Characteristic Functions for Ergodic Tuples

Gohm, Rolf; Dey, S., 'Characteristic function for ergodic tuples', Integral Equations and Operator Theory 58(1) pp.43-63 RAE2008

An analysis of VUV C (IV) emission from the JET divertor giving measurements of electron temperatures

A description of the radiation emitted by impurities from within a plasma is crucial if spectral line intensities are to be used in detailed studies, such as the analysis of impurity transport. The simplest and most direct check that can be made on measurements of line intensities is to analyse their ratios with other lines from the same ion. This avoids uncertainties in determining the volume of the emitting plasma and the absolute sensitivity calibration of the spectrometer and, in some cases, the need even for accurate measurements of parameters such as electron density. Consistency is required between the measured line intensity ratios and the theoretical values. The expected consistency has not been found for radiation emitted from the JET scrape-off layer (e.g. Lawson et al 2009a JINST 4 P04013), meaning that the description of the spectral line intensities of impurity emission from the plasma edge is incomplete. In order to gain further understanding of the discrepancies, an analysis has been carried out for emission from the JET divertor plasma and this is reported in this paper. Carbon was the main low Z intrinsic impurity in JET and an analysis of spectral line intensity ratios has been made for the C (IV) radiation emitted from the JET divertor. In this case, agreement is found between the measured and theoretical ratios to a very high accuracy, namely to within the experimental uncertainty of similar to +/- 10%. This confirms that the description of the line intensities for the present observations is complete. For some elements and ionization stages, an analysis of line intensity ratios can lead to the determination of parameters such as the electron temperature of the emitting plasma region and estimates of the contribution of recombination to the electron energy level populations. This applies to C (IV) and, to show the value and possibilities of the spectral measurements, these parameters have been calculated for a database of Ohmic and additionally heated phases of a large number of pulses. The importance of dielectronic, radiative and charge-exchange recombination as well as ionization has been investigated. In addition, the development of T-e throughout two example discharges is illustrated. The presented results indicate a number of areas for further investigation.

Generic Ergodic Capacity Bounds for Fixed-Gain AF Dual-Hop Relaying Systems

This paper elaborates on the ergodic capacity of fixed-gain amplify-and-forward (AF) dual-hop systems, which have recently attracted considerable research and industry interest. In particular, two novel capacity bounds that allow for fast and efficient computation and apply for nonidentically distributed hops are derived. More importantly, they are generic since they apply to a wide range of popular fading channel models. Specifically, the proposed upper bound applies to Nakagami-m, Weibull, and generalized-K fading channels, whereas the proposed lower bound is more general and applies to Rician fading channels. Moreover, it is explicitly demonstrated that the proposed lower and upper bounds become asymptotically exact in the high signal-to-noise ratio (SNR) regime. Based on our analytical expressions and numerical results, we gain valuable insights into the impact of model parameters on the capacity of fixed-gain AF dual-hop relaying systems. © 2011 IEEE.

Ergodic Capacity of Cognitive TAS/GSC Relaying in Nakagami-m Fading Channels

We examine the impact of transmit antenna selection with receive generalized selection combining (TAS/GSC) for cognitive decode-and-forward (DF) relaying in Nakagami-m fading channels. We select a single transmit antenna at the secondary transmitter which maximizes the receive signal-to-noise ratio (SNR) and combine a subset of receive antennas with the largest SNRs at the secondary receiver. In an effort to assess the performance, we first derive the probability density function and cumulative distribution function of the end-to-end SNR using the moment generating function. We then derive new exact closed-form expression for the ergodic capacity. More importantly, by deriving the asymptotic expression for the high SNR approximation of the ergodic capacity, we gather deep insights into the high SNR slope and the power offset. Our results show that the high SNR slope is 1/2 under the proportional interference power constraint. Under the fixed interference power constraint, the high SNR slope is zero.

ELM-induced transient tungsten melting in the JET divertor

The original goals of the JET ITER-like wall included the study of the impact of an all W divertor on plasma operation (Coenen et al 2013 Nucl. Fusion 53 073043) and fuel retention (Brezinsek et al 2013 Nucl. Fusion 53 083023). ITER has recently decided to install a full-tungsten (W) divertor from the start of operations. One of the key inputs required in support of this decision was the study of the possibility of W melting and melt splashing during transients. Damage of this type can lead to modifications of surface topology which could lead to higher disruption frequency or compromise subsequent plasma operation. Although every effort will be made to avoid leading edges, ITER plasma stored energies are sufficient that transients can drive shallow melting on the top surfaces of components. JET is able to produce ELMs large enough to allow access to transient melting in a regime of relevance to ITER.

Transient W melt experiments were performed in JET using a dedicated divertor module and a sequence of I-P = 3.0 MA/B-T = 2.9 T H-mode pulses with an input power of P-IN = 23 MW, a stored energy of similar to 6 MJ and regular type I ELMs at Delta W-ELM = 0.3 MJ and f(ELM) similar to 30 Hz. By moving the outer strike point onto a dedicated leading edge in the W divertor the base temperature was raised within similar to 1 s to a level allowing transient, ELM-driven melting during the subsequent 0.5 s. Such ELMs (delta W similar to 300 kJ per ELM) are comparable to mitigated ELMs expected in ITER (Pitts et al 2011 J. Nucl. Mater. 415 (Suppl.) S957-64).

Although significant material losses in terms of ejections into the plasma were not observed, there is indirect evidence that some small droplets (similar to 80 mu m) were released. Almost 1 mm (similar to 6 mm(3)) of W was moved by similar to 150 ELMs within 7 subsequent discharges. The impact on the main plasma parameters was minor and no disruptions occurred. The W-melt gradually moved along the leading edge towards the high-field side, driven by j x B forces. The evaporation rate determined from spectroscopy is 100 times less than expected from steady state melting and is thus consistent only with transient melting during the individual ELMs. Analysis of IR data and spectroscopy together with modelling using the MEMOS code Bazylev et al 2009 J. Nucl. Mater. 390-391 810-13 point to transient melting as the main process. 3D MEMOS simulations on the consequences of multiple ELMs on damage of tungsten castellated armour have been performed.

These experiments provide the first experimental evidence for the absence of significant melt splashing at transient events resembling mitigated ELMs on ITER and establish a key experimental benchmark for the MEMOS code.

Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of ℝ d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called quenched random Lorentz tube. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.

Ergodic properties of triangle partitions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Ergodic crossover in partially self-avoiding stochastic walks

Consider a one-dimensional environment with N randomly distributed sites. An agent explores this random medium moving deterministically with a spatial memory μ. A crossover from local to global exploration occurs in one dimension at a well-defined memory value μ1=log2N. In its stochastic version, the dynamics is ruled by the memory and by temperature T, which affects the hopping displacement. This dynamics also shows a crossover in one dimension, obtained computationally, between exploration schemes, characterized yet by the trajectory size (Np) (aging effect). In this paper we provide an analytical approach considering the modified stochastic version where the parameter T plays the role of a maximum hopping distance. This modification allows us to obtain a general analytical expression for the crossover, as a function of the parameters μ, T, and Np. Differently from what has been proposed by previous studies, we find that the crossover occurs in any dimension d. These results have been validated by numerical experiments and may be of great value for fixing optimal parameters in search algorithms. © 2013 American Physical Society.