We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, for a general attractive probabilistic cellular automata on SZd, where S is finite, we prove that a condition (A) is equivalent to the (time-) convergence towards equilibrium of this Markovian parallel dynamics, in the uniform norm, exponentially fast. This condition (A) means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite ‘box’-volume. For a class of reversible PCA dynamics on {−1, +1}Zd / with a naturally associated Gibbsian potential ϕ, we prove that a Weak Mixing condition for ϕ implies the validity of the assumption (A); thus the ‘exponential ergodicity’ of the dynamics towards the unique Gibbs measure associated to ϕ holds. On some particular examples of this PCA class, we verify that our assumption (A) is weaker than the Dobrushin-Vasershtein ergodicity condition. For some special PCA, the ‘exponential ergodicity’ holds as soon as there is no phase transition.

The Qweak experiment at Thomas Jefferson National Accelerator Facility (Jefferson Lab) will measure the proton's weak charge by measuring the parity violating asymmetry in elastic electron-proton scattering at very low momentum transfer, with the aim of determining the proton's weak charge with 4% combined statistical and systematic errors. The experimental apparatus includes a longitudinally polarized electron beam, a liquid hydrogen target, a room temperature toroidal magnetic spectrometer, and a set of precision detectors for the scattered electrons. The toroidal magnetic spectrometer, which will deflect away the inelastic scattered electrons and focus the elastic scattered electrons onto the detectors, plays a crucially important role in the experiment. In this thesis, in order to meet the requirements for the installation and calibration of the toroidal magnetic spectrometer, the numerical simulation of the spectrometer's magnetic field based on a realistic magnet model is discussed, a precise 3D field mapping is introduced, and some simulation results are provided. The zero-crossing analysis technique, which can be used to precisely infer the individual coil locations of the toroidal magnet, is presented and explored in detail. / October 2007

The Qweak experiment at Thomas Jefferson National Accelerator Facility (Jefferson Lab) will measure the proton's weak charge by measuring the parity violating asymmetry in elastic electron-proton scattering at very low momentum transfer, with the aim of determining the proton's weak charge with 4% combined statistical and systematic errors. The experimental apparatus includes a longitudinally polarized electron beam, a liquid hydrogen target, a room temperature toroidal magnetic spectrometer, and a set of precision detectors for the scattered electrons. The toroidal magnetic spectrometer, which will deflect away the inelastic scattered electrons and focus the elastic scattered electrons onto the detectors, plays a crucially important role in the experiment. In this thesis, in order to meet the requirements for the installation and calibration of the toroidal magnetic spectrometer, the numerical simulation of the spectrometer's magnetic field based on a realistic magnet model is discussed, a precise 3D field mapping is introduced, and some simulation results are provided. The zero-crossing analysis technique, which can be used to precisely infer the individual coil locations of the toroidal magnet, is presented and explored in detail.

The Qweak experiment at Thomas Jefferson National Accelerator Facility (Jefferson Lab) will measure the proton's weak charge by measuring the parity violating asymmetry in elastic electron-proton scattering at very low momentum transfer, with the aim of determining the proton's weak charge with 4% combined statistical and systematic errors. The experimental apparatus includes a longitudinally polarized electron beam, a liquid hydrogen target, a room temperature toroidal magnetic spectrometer, and a set of precision detectors for the scattered electrons. The toroidal magnetic spectrometer, which will deflect away the inelastic scattered electrons and focus the elastic scattered electrons onto the detectors, plays a crucially important role in the experiment. In this thesis, in order to meet the requirements for the installation and calibration of the toroidal magnetic spectrometer, the numerical simulation of the spectrometer's magnetic field based on a realistic magnet model is discussed, a precise 3D field mapping is introduced, and some simulation results are provided. The zero-crossing analysis technique, which can be used to precisely infer the individual coil locations of the toroidal magnet, is presented and explored in detail.

Neutral current Drell-Yan production in proton-proton collisions at the LHC was studied with the ATLAS detector. The 20.1 inverse femtobarn data set used in this precision measurement was collected in 2012 during which the LHC collided protons at a centre-of-mass energy of 8 TeV. The production rate or differential cross-section was measured in three-dimensions: invariant mass, absolute rapidity, and cosine of the polar angle in the Collins-Soper frame. A measurement of the forward-backward asymmetry was obtained from the differential cross-section by summing over the forward and the backward events and taking their difference. The three-dimensional differential cross-section measurement presented in this dissertation can be used to constrain the invariant mass- and rapidity-dependent parton distribution functions of the proton and the forward-backward asymmetry results can be used to extract a measurement of the weak mixing angle. / Graduate

The Qweak experiment has taken data to make a 2.5 percent measurement of parity violating elastic ep asymmetry in the four momentum transfer region of 0.0250 (GeV/c)2. This asymmetry is proportional to the weak charge of the proton, which is related to the weak mixing angle, sin2(thetaW). The final Qweak measurement will provide the most precise measurement of the weak mixing angle below the Z° pole to test the Standard Model prediction. A description of the experimental apparatus is provided in this dissertation. The experiment was carried out using a longitudinally polarized electron beam of up to 180 microampere on a 34.5 cm long unpolarized liquid hydrogen target. The Qweak target is not only the world’s highest cryogenic target ever built for a parity experiment but also is the least noisy target. This dissertation provides a detailed description of this target and presents a thorough analysis of the target performance. Statistical analysis of Run 1 data, collected between Feb - May 2011, is done to extract a blinded parity violating asymmetry of size--299.7 +/- 13.4 (stat.) +/- 17.2 (syst.) +/- 68 (blinding) parts-per-billion. This resulted in a preliminary proton’s weak charge of value 0.0865 +/- 0.0085, a 9% measurement. Based on this blinded asymmetry, the weak mixing angle was determined to be sin2(thetaW) = 0.23429 +/- 0.00211.

The use of Hilbert space theory became an important tool for ergodic theoreticians ever since John von Neumann proved the fundamental Mean Ergodic theorem in Hilbert space. Recurrence is one of the corner stones in the study of dynamical systems. In this dissertation some extended ideas besides those of the basic, well-known recurrence results are investigated. Hilbert space theory proves to be a very useful approach towards the solution of multiple recurrence problems in ergodic theory. Another very important use of Hilbert space theory became evident only relatively recently, when it was realized that non-commutative dynamical systems become accessible to the ergodic theorist through the important Gelfand-Naimark-Segal (GNS) representation of C*-algebras as Hilbert spaces. Through this construction we are enabled to invoke the rich catalogue of Hilbert space ergodic results to approach the more general, and usually more involved, non-commutative extensions of classical ergodic-theoretical results. In order to make this text self-contained, the basic, standard, ergodic-theoretical results are included in this text. In many instances Hilbert space counterparts of these basic results are also stated and proved. Chapters 1 and 2 are devoted to the introduction of these basic ergodic-theoretical results such as an introduction to the idea of measure-theoretic dynamical systems, citing some basic examples, Poincairé’s recurrence, the ergodic theorems of Von Neumann and Birkhoff, ergodicity, mixing and weakly mixing. In Chapter 2 several rudimentary results, which are the basic tools used in proofs, are also given. In Chapter 3 we show how a Hilbert space result, i.e. a variant of a result by Van der Corput for uniformly distributed sequences modulo 1, is used to simplify the proofs of some multiple recurrence problems. First we use it to simplify and clarify the proof of a multiple recurrence result by Furstenberg, and also to extend that result to a more general case, using the same Van der Corput lemma. This may be considered the main result of this thesis, since it supplies an original proof of this result. The Van der Corput lemma helps to simplify many of the tedious terms that are found in Furstenberg’s proof. In Chapter 4 we list and discuss a few important results where classical (commutative) ergodic results were extended to the non-commutative case. As stated before, these extensions are mainly due to the accessibility of Hilbert space theory through the GNS construction. The main result in this section is a result proved by Niculescu, Ströh and Zsidó, which is proved here using a similar Van der Corput lemma as in the commutative case. Although we prove a special case of the theorem by Niculescu, Ströh and Zsidó, the same method (Van der Corput) can be used to prove the generalized result. Copyright 2004, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. Please cite as follows: Beters, FJC 2004, A Hilbert space approach to multiple recurrence in ergodic theory, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-02222006-104936 / > / Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

The Q-weak experiment at Thomas Jefferson National Accelerator Facility (USA) will make a precision determination of the proton weak charge with approximately 4% combined statistical and systematic uncertainties via a measurement of the parity violating asymmetry in elastic electron-proton scattering at very low momentum transfer and forward angle. This will allow an extraction of the weak mixing angle at Q^2=0.026 (GeV/c)^2 to approximately 0.3%. The weak mixing angle is a fundamental parameter in the Standard Model of electroweak interactions. At the proposed accuracy, a measured deviation of this parameter from the predicted value would indicate new physics beyond what is currently described in the Standard Model. Without deviation from the predicted value, this measurement would place stringent limits on possible extensions to the Standard Model and constitute the most precise measurement of the proton's weak charge to date. The key experimental apparatus include a liquid hydrogen target, a toroidal magnetic spectrometer and a set of eight Cerenkov detectors. The Cerenkov detectors form the main detector system for the Q-weak experiment and are used to measure the parity violating asymmetry during the primary Q-weak production runs. The Cerenkov detectors form the main subject of this thesis. Following a brief introduction to the experiment, the design, development, construction, installation, and testing of this detector system will be discussed in detail. This is followed by a detailed discussion of detector diagnostic data analysis and the corresponding detector performance. The experiment has been successfully constructed and commissioned, and is currently taking data. The thesis will conclude with a discussion of the preliminary analysis of a small portion of the liquid hydrogen data.

The goal of the Q-weak experiment is to make a measurement of the proton's weak charge ($Q^p_W = 1-4\sin^2\theta_W$) to an accuracy of ~4%. This would represent a ~0.3% determination of the weak mixing angle ($\sin^2\theta_W$) at low energy. The measurement may be used for a precision test of the Standard Model (SM) prediction on the running of $\sin^2\theta_W$ with energy scale. The Q-weak experiment operates at Thomas Jefferson National Accelerator Facility (Jefferson Lab). The experiment determines the proton's weak charge by measuring the parity violating asymmetry in elastic electron-proton scattering at low momentum transfer $Q^2 = 0.026 (GeV/c)^2$ and forward angles (~8 degree). The anticipated size of the asymmetry, based on the SM, is about 230 parts per billion (ppb). With the proposed accuracy, the experiment may probe new physics beyond Standard Model at the TeV scale. This thesis focuses on my contributions to the experiment, including track reconstruction for momentum transfer determination of the scattering process, and the focal plane scanner, a detector I designed and built to measure the flux profile of scattered electrons on the focal plane of the Q-weak spectrometer to assist in the extrapolation of low beam current tracking results to high beam current. Preliminary results from the commissioning and the first run period of the Q-weak experiment are reported and discussed.

The Q-weak experiment at Thomas Jefferson National Accelerator Facility (USA) will make a precision determination of the proton weak charge with approximately 4% combined statistical and systematic uncertainties via a measurement of the parity violating asymmetry in elastic electron-proton scattering at very low momentum transfer and forward angle. This will allow an extraction of the weak mixing angle at Q^2=0.026 (GeV/c)^2 to approximately 0.3%. The weak mixing angle is a fundamental parameter in the Standard Model of electroweak interactions. At the proposed accuracy, a measured deviation of this parameter from the predicted value would indicate new physics beyond what is currently described in the Standard Model. Without deviation from the predicted value, this measurement would place stringent limits on possible extensions to the Standard Model and constitute the most precise measurement of the proton's weak charge to date. The key experimental apparatus include a liquid hydrogen target, a toroidal magnetic spectrometer and a set of eight Cerenkov detectors. The Cerenkov detectors form the main detector system for the Q-weak experiment and are used to measure the parity violating asymmetry during the primary Q-weak production runs. The Cerenkov detectors form the main subject of this thesis. Following a brief introduction to the experiment, the design, development, construction, installation, and testing of this detector system will be discussed in detail. This is followed by a detailed discussion of detector diagnostic data analysis and the corresponding detector performance. The experiment has been successfully constructed and commissioned, and is currently taking data. The thesis will conclude with a discussion of the preliminary analysis of a small portion of the liquid hydrogen data.