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This thesis describes an experimental and analytic study of the effects of magnetic non-linearity and finite length on the loss and field distribution in solid iron due to a travelling mmf wave. In the first half of the thesis, a two-dimensional solution is developed which accounts for the effects of both magnetic non-linearity and eddy-current reaction; this solution is extended, in the second half, to a three-dimensional model. In the two-dimensional solution, new equations for loss and flux/pole are given; these equations contain the primary excitation, the machine parameters and factors describing the shape of the normal B-H curve. The solution applies to machines of any air-gap length. The conditions for maximum loss are defined, and generalised torque/frequency curves are obtained. A relationship between the peripheral component of magnetic field on the surface of the iron and the primary excitation is given. The effects of magnetic non-linearity and finite length are combined analytically by introducing an equivalent constant permeability into a linear three-dimensional analysis. The equivalent constant permeability is defined from the non-linear solution for the two-dimensional magnetic field at the axial centre of the machine to avoid iterative solutions. In the linear three-dimensional analysis, the primary excitation in the passive end-regions of the machine is set equal to zero and the secondary end faces are developed onto the air-gap surface. The analyses, and the assumptions on which they are based, were verified on an experimental machine which consists of a three-phase rotor and alternative solid iron stators, one with copper end rings, and one without copper end rings j the main dimensions of the two stators are identical. Measurements of torque, flux /pole, surface current density and radial power flow were obtained for both stators over a range of frequencies and excitations. Comparison of the measurements on the two stators enabled the individual effects of finite length and saturation to be identified, and the definition of constant equivalent permeability to be verified. The penetration of the peripheral flux into the stator with copper end rings was measured and compared with theoretical penetration curves. Agreement between measured and theoretical results was generally good.

The overall aim of this project was to study 'functionalised' calcium carbonates (FCCs) for use as a carrier for the controlled release of 'actives,' by permeation and diffusion, and is being proposed as an environmentally friendly and non-toxic pharmaceutical excipient, nutraceutical, and flavour carrier. The delivery of a drug to its target site in the appropriate amount and time-frame in order for it to have a controlled release effect whilst achieving the maximum therapeutic effect remains a topic of design and development for novel drug delivery systems. FCCs encompass a family of new pharmaceutical excipients in which the conditions of manufacture follow strict process regulations with respect to the grade of reagents that are employed and the microbiological environment under which they are produced, and include freedom from organic polymers. Adjustments to the FCC production process can be used to produce a wide range of different morphologies, and raise the possibility of tailoring the void structures of the particles to provide controlled release delivery vehicles for actives across many fields, including drugs and flavours. However, such tailoring can only be fully optimised by a fundamental characterisation of the way in which a drug, loaded into an FCC, then flows and diffuses out over a period of time to provide the delayed release. It was found that adsorption on the FCC surface is selective, for example, saccharin does not become adsorbed from 4-(2-hydroxyethyl)piperazine-1-ethanesulfonic acid (HEPES) buffer solution, and neither does anethole from ethanol. FCC also does not adsorb the cationic probe benzyltrimethylammonium bromide (BTMAB) or the anionic probe sodium 2-naphthalenesulphonate (Na2NS). However, it was found that vanillin adsorbs onto the FCC in an amount of 2.00 ± 0.59 mg g^-1. Aspirin and vanillin adsorption from ethanolic solutions with various additions of water onto FCC TP was investigated and fitted with the Tóth isotherm. It was estimated that vanillin adsorbed onto around 17 %, and aspirin onto around 39 %, of the overall FCC TP surface area without the addition of any water. An equation was formulated in order to approximate the adsorption as a function of the FCC's surface coverage by the water. This is discussed in Chapter 4 and has also been published in a peer-reviewed academic journal (Levy et al., 2017). Chapter 5 discusses the preliminary steps of the loading of vanillin and saccharin into FCC, and the results were inconclusive for a majority of samples, concluding that the loading and analysis methods need refining. The modelling of the diffusion profiles of vanillin loaded FCC S07 and S10 was successful, and resulted in diffusion coefficients of 231.9 x 10^-16 m^2 s^-1 and 248.44 x 10^-16 m^ s^-1, respectively. This is outlined in Chapter 6. Chapter 7 describes the 'zero length column' (ZLC) technique, which was used as a way to characterise the diffusivity of the intraparticle pores of each FCC grade. However, it was established that there are many experimental artefacts present with such a method. This work outlines the development of the novel 'finite length column' (FLC), which was developed as a means to overcome the limitations of the ZLC (Levy et al., 2015). Effective diffusivity coefficients in the long-term region of the diffusion curves of the FCC samples range from 1.06-106 x 10 ^-16 m ^2 s^-1. The FLC was then used in preliminary trials to dilute FCC with an inert solid in order to further refine the ZLC technique, and is discussed in Chapter 8. Two mathematical methods were also developed to aid in the refinement. The reported effective diffusivity coefficient for FCC 03 in the long-term region of the diffusion curve is 49.5 x 10^-16 m^2 s^-1. In conclusion, this work confirms that FCC has potential for use as a carrier for the controlled release of 'actives' by diffusion. The utilisation of mathematical modelling in conjunction with experimental methods in the study of drug release and delivery is steadily increasing due to its enormous future potential; it will enable the optimisation of novel dosage forms and the elucidation of release mechanisms at a major reduction in cost and time compared with the number of experimental studies required to do so.

The structure of free resolutions of finite length modules over regular local rings has long been a topic of interest in commutative algebra. Conjectures by Buchsbaum-Eisenbud-Horrocks and Avramov-Buchweitz predict that in this setting the minimal free resolution of the residue field should give, in some sense, the smallest possible free resolution of a finite length module. Results of Tate and Shamash describing the minimal free resolution of the residue field over a local hypersurface ring, together with the theory of matrix factorizations developed by Eisenbud and Eisenbud-Peeva, suggest analogous lower bounds for the size of free resolutions of finite length modules of infinite projective dimension over such rings. In this dissertation we describe both positive and negative results pertaining to these lower bounds. By refining an argument of Charalambous, we show that the lower bounds hold in certain multigraded settings. We are also able to obtain results for finite free resolutions of multigraded modules, recovering results of Charalambous and Santoni. For the local case, however, we use a construction of Iyengar-Walker to provide examples showing that the lower bounds do not always hold. In order to accomplish this, we make use of the theory of higher matrix factorizations developed by Eisenbud-Peeva to investigate the structure of free resolutions over complete intersections of arbitrary codimension. Adviser: Mark E. Walker

This Dissertation consists of solutions to two major problems. Chapter 2 presents theory and numerical calculations of electrostatic Bernstein modes in an inhomogeneous cylindrical plasma column. These modes rely on finite Larmor radius (FLR) effects to propagate radially across the column until they are reflected when their frequency matches the upper hybrid frequency. This reflection sets up an internal normal mode on the column, and also mode-couples to the electrostatic surface cyclotron wave (which allows the normal mode to be excited and observed using external electrodes). Numerical results predicting the mode spectra, using a novel linear Vlasov code on a cylindrical grid, are presented and compared to an analytic WKB theory. A previous version of the theory[6] expanded the plasma response in powers of 1/B, approximating the local upper hybrid frequency, and consequently its frequency predictions are spuriously shifted with respect to the numerical results presented here. A new version of the WKB theory avoids this approximation using the exact cold fluid plasma response and does a better job of reproducing the numerical frequency spectrum. The effect of multiple ion species on the mode spectrum is also considered, to make contact with experiments that observe cyclotron modes in a multi-species pure ion plasma.[1]Chapter 3 presents theory and numerical calculation for the finite-length diocotron mode frequency with arbitrary azimuthal mode number. The numerical calculation solves a bounce-averaged version of the Vlasov equation to determine the perturbed potential in the presence of the mode, along with its frequency. The analytic theory is also obtained from the bounce averaged Vlasov equation, but we derive a theorem that allows us to obtain an effective fluid theory consistent with the full Vlasov theory, which integrates out the surface phase space associated with the bouncing motion, considerably simplifying the analysis. We use this effective fluid theory to derive frequency shifts for finite-length cylindrical plasmas, and find good agreement with experiment and with our numerical bounce-averaged Vlasov theory.

Low-density parity-check (LDPC) codes in conjunction with iterative decoding based on message-passing algorithms have been shown to achieve excellent performance over a variety of communication channels. In this dissertation, we investigate LDPC codes under two scenarios which present different challenges for providing robust communications. First, LDPC codes are analyzed and designed for time-selective, frequency-nonselective complex-fading channels where both the amplitude and the phase of the transmitted signal are altered by the channel. To combat the fading, we use a pilot symbol-assisted scheme where known pilot symbols are added to the LDPC code and used to help estimate the channel at the receiver. Using asymptotic, infinite-length analysis, we investigate several iterative, message-passing, joint-estimation-and-decoding strategies; optimal energy distribution between pilot and data symbols; and LDPC code design. Several interesting results are obtained regarding unification of analysis and code design. Next, we address the issue of achieving very low error rates with practical, finite-length LDPC codes for binary erasure channels. Finite-length LDPC codes suffer from error floors which limit the achievable error rate, and this error-floor performance is determined by stopping sets. Asymptotic analysis of weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with lower error floors but does not reflect the behavior of sublinearly-sized stopping sets, which can dominate the iterative-decoding, error-floor performance. Thus, we provide a perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, advantages of protograph structure, and effects of precoding. We show for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly-growing stopping sets, and this polynomial relationship can be approximately captured by a single parameter. Further, we begin to address the question, Given finite stopping-set sizes and finite codelengths, do the stopping sets follow the behavior predicted by linear or sublinear analysis?

Parallel concatenated convolutional codes, a.k.a. turbo codes, have attracted much attention since their appearance in 1993. Turbo codes are decoded by iterative (turbo) decoders in which a component decoder for each component code exists. Turbo decoders perform quite well with respect to bit and packet error rates. The code structure coupled with the iterative decoding scheme provides performance close to the fundamental communication limits predicted by Shannon with reasonable complexity. In this thesis we investigate the performance of finite length turbo codes. Due to the ad hoc nature of turbo decoding, error rate performance analysis is difficult. In the first part of this thesis, we classify the different characteristics of turbo decoding based on empirical results and analyses available in the literature. This classification helps distinguish two modes of turbo decoding behavior: optimal mode and decoding failures. The performance in the optimal mode is affected by the code structure of a turbo code. The interleaver in turbo code construction makes it difficult to find the code structure for a particular interleaver. In order to overcome this difficulty, we concentrate on special error events of the component convolutional codes. We propose an efficient method to estimate the code structure. By this estimate, an approximation to the error rate performance can be obtained for the optimal mode. Decoding failures occur due to nonoptimal operation of a turbo decoder. Therefore, analysis of iterative decoding is of interest with regard to decoding failures. Iterative decoding analysis methods are available for infinite length turbo codes. By investigating the properties of the component decoders, a probabilistic model is developed for finite lengths. Using this model, we find approximations to probability of decoding failure. Combining the analysis methods for the two modes of a turbo decoder yields an accurate overall approximation to the error rate performance. Numerical results are presented to verify the accuracy of the approximation.

Transition to turbulent flow in curved pipe has been well studied through experiments and numerical simulations. Numerical simulations often use helical pipe geometry with infinite length such that the inlet and outlet boundary conditions can be modeled as periodic which reduces computational time. In the present study, we examined a finite length curved pipe with a Poiseuille flow imposed at the inlet and a stress-free boundary condition at the outlet. Direct numerical simulation of the Navier-Stokes equations for rigid walls and a Newtonian fluid was performed using nek5000. Straight extensions were added to the inlet and the outlet such to diminish the impact of boundary conditions on the flow field in the region with curvature. The examined model has a pipe radius of curvature that is three times that of the pipe radius. The model has over 300 million nodes and required an order of magnitude greater computational time when compared to the infinite length curved pipe. Results show that the critical Reynolds number (initiation of instabilities) is greater compared to a straight pipe and occurs near Re=5000-5200. This Re is also larger than the critical Reynolds number typically reported for an infinite length curved pipe (Re= 4200-4300). As expected, flow patterns in the finite length curved pipe were shown to be evolving through the curvature as opposed to that of an infinite length curved pipe where it remains constant. In addition, the initial instabilities observed in the flow did not originate from a Dean flow instability, initiated through secondary flow, but rather were first observed near the outer wall.

We study graded modules of finite length over the weighted polynomial ring R=k[x_{1},...,x_{n}], k any field, having a certain strongly selfdual resolution. We give a construction method of these Gorenstein modules via symmetric matrices in divided powers. Our main result is the following equivalence: Let n be an odd integer. A graded R-module of finite length has a selfdual minimal free resolution with a symmetric respectively skew symmetric middle matrix if and only if it can be defined by a symmetric respectively skew symmetric matrix in divided powers. The correspondence depends on the parity of (n-1)/2. We give applications, such as a proof of a conjecture of Eisenbud and Schreyer: Let R be trivially weighted. The monoid of Betti tables of free resolutions of graded Cohen-Macaulay modules over R depends on the characteristic of the base field k.

The volume of data continues to rapidly grow as information pours from various platforms. The huge amount of data needs to be transferred and stored with extremely high reliability. The error correcting codes (ECCs) are an integral part of modern-day communication, computation, and data storage systems in order to safeguard data against the adverse effects of noise and interference. The spatially-coupled (SC) codes are a class of graph-based ECCs that have recently emerged as an excellent choice for error correction in modern data storage and communication due to their outstanding performance, low decoding latency, and simple implementation. An SC code is constructed by coupling several instances of a block code into a single coupled chain. In the asymptotic limit of large code lengths, SC codes enjoy capacity achieving performance. Due to simplifying assumptions and averaging effects, results from the asymptotic domain are not readily translatable to the practical, finite-length setting. Despite this chasm, finite-length analysis of SC codes is still largely unexplored. We tackle the problem of finite-length optimized design of SC codes in the context of various channel models.First, we present a systematic framework with low computational complexity for designing finite-length SC codes with superior error floor performance. Next, we tailor our design method for various channel models by targeting the combinatorial objects in the graph of SC codes that are detrimental over these settings. Then, we generalize our framework for the finite-length analysis and design of irregular SC codes. Finally, we increase the coupling dimensionality, and we present a novel systematic framework to efficiently connect several SC codes and construct multi-dimensional spatially-coupled (MD-SC) codes.In this research, we use advanced mathematical techniques from algebra, combinatorics, graph theory, probability theory, and optimization theory to develop algorithms and design frameworks with affordable complexity. Our frameworks are especially beneficial for modern storage applications, e.g. magnetic-recording and Flash memories.

This dissertation discusses the effects of collisions and finite plasma length on Trivelpiece-Gould waves on a magnetized, single-species plasma column. Starting from Poisson's equation and a drift-kinetic equation with an energy- and momentum-conserving Fokker-Planck collision term, a dispersion equation is obtained for an azimuthally symmetric wave on an infinitely long column. The dispersion relation includes the effect of velocity- scattering collisions with impact parameters less than the cyclotron radius and recovers Landau damping as collisionality approaches zero. For wavenumbers such that --where is the total wavenumber, and are the wavenumbers along and transverse to the magnetic field, and is the Debye length--Landau damping is exponentially small, and the complex frequency of the wave is approximately where is the plasma frequency, is a collisionality parameter, and is the collision frequency. When the Debye length is larger than the cyclotron radius, long-range interactions between particles on different field lines are also significant but cannot be treated by a Fokker-Planck collision operator. Fluid theory provides a simpler context for incorporating these long-range interactions, since their primary effect is the enhancement of transport across the magnetic field. Fluid analysis reveals that the damping rate obtained from kinetic theory corresponds to bulk viscosity and that viscous relaxation of radial shear in the parallel flow, due to long-range collisions, gives an important additional contribution to the damping rate. Lastly, azimuthally symmetric normal modes are calculated for a cold, finite-length plasma column. The dispersion equation for Trivelpiece-Gould waves on a cold, strongly magnetized plasma has the property that two waves with wavenumbers and have the same frequency if Such degenerate waves are mixed upon reflection at the ends of the plasma column, and consequently each normal mode involves many such waves. The modes often exhibit sharp features along resonance cones with slope

The performance of random error control codes approaches the Shannon capacity limit as the code length goes to infinity. When the code length is finite, then the code will be unable to achieve arbitrarily low error probability and a nonzero codeword error rate is inevitable. Information-theoretic bounds on codeword error rate may be found as a function of length through traditional methods such as sphere packing. Alternatively, the behavior of finite-length codes can be characterized in terms of an information-outage probability. The information-outage probability is the probability that the mutual-information rate, which is a random variable, is less than the code rate.;In this thesis, a Gaussian approximation is proposed that accurately models the information-outage probability for codewords of moderately short length. The information-outage probability is related to several previously derived bounds, including Shannon's sphere-packing and random coding bounds, as well as a bound on maximal error probability known as Feinstein's lemma. It is shown that the information-outage probability is a useful predictor of achievable error rate.

Due to their remote location, some applications of structures in space environments require that the structures themselves monitor their own loading history and structural integrity. Thus, an integrated sensor and structural element is sought that exhibits a high sensitivity to loading while preserving the stiffness and strength of the structural loadbearing member. The use of finite length displacement sensors permits the development of transducers whose sensitivity scales with the geometry of the instrument rather than with the maximum strain in the transducer thus allowing the instrument to act as a load-bearing member. For example, the design of a six-resultant-load component transducer is described that, within the confines of linear elastostatics, produces zero cross-talk between load components while retaining the geometric scaling property. The above measurement of the torsion component is examined in detail to demonstrate practical considerations in the manufacturing of such an instrument. In order to validate the analytical description of the behavior of the finite length displacement sensor presented, a torsion transducer was manufactured and evaluated. Its performance is compared to an error analysis based upon certain known manufacturing errors and demonstrates sensing behavior as predicted by the analytical model. Minimizing the sensitivity to the considered manufacturing errors is also discussed. In addition, a technique is presented to monitor the in-situ structural integrity of prismatic structures. Since finite gauge length sensors integrate deformations over a relatively long path length this technique requires only a few sensors to monitor an entire structural volume and probable locations of damage do not have to be known a-priori. The technique acts as a "balance" device, indicating the presence of a flaw by deviation from a "null" signal, requiring little data processing. The monitoring of a solid circular cylinder with a spherical cavity is presented in detail to demonstrate the behavior of this technique. The extension of this technique to anisotropic, nonhomogeneous materials is also presented. Finally, speculation is made as to further applications of finite length displacement sensors.

Coherent transceivers are the key elements of fibre-optical communication systems enabling high-speed transmission. These transceivers utilize state-of-the-art electrical and optical components, digital signal processing (DSP), and advanced coded modulation (CM) schemes. This thesis focuses on transceiver performance optimization techniques - in particular, two aspects are covered. The first aspect is related to transceiver impairments characterization and mitigation via application of advanced DSP techniques. Transceiver performance can be detrimentally affected by various imperfections in its optical and electrical components and, therefore, it is important to mitigate the impact of those imperfections. Two techniques for characterization and compensation of transceiver skews were proposed and investigated -calibration techniques based on the Gardner timing error detector and signal image spectrum measurement. Also, the impact of various transmitter in-phase/quadrature (IQ) impairments was considered-specifically, modulation impairments, skews, electrical IQ cross-talk and frequency response mismatch. Post-compensation based on advanced post-equalizer and multiple-input multiple-output (MIMO) pre-emphasis approaches were proposed and investigated. The second aspect is related to application of advanced CM techniques and optimization of the signaling scheme. Conventional uniform signaling based on quadrature amplitude modulation (QAM) is a sub-optimal solution in terms of linear and nonlinear performance, and advanced signaling schemes can be used to improve the overall system performance. Finite-length probabilistic constellation shaping, specifically, sphere shaping architecture was considered and optimized for improved linear and nonlinear performance of the system. The performance of the system employing finite-length sphere shaping architecture was extensively studied in comparison with uniform signaling and infinite-length Maxwell-Boltzmann (MB) shaping. Optimal shaping regimes were identified for long-haul multi-span and extended-reach single-span transmission links.

A theoretical model is derived to study the point-wise stress variations which occur in the constituents of a hybrid 3-D short fiber composite subjected to arbitrary homogeneous loading conditions. The model includes the capability to analyze composites containing different types of fibers, different aspect ratios of fibers (as well as continuous fibers), and different fiber orientations. The composite’s stiffness tensor is developed by volume averaging the point-wise stress field in each constituent present in the material system. Validation of the model is accomplished by comparing predicted stiffness properties to experimental data and other accepted models presently available in the literature for PMC’s, MMC’s, and BMC’s. A derivation of a theoretical model describing the resulting point-wise stress redistribution which occurs in the matrix and the fibrous regions caused by fiber-fiber interaction at the ends of finite length fibers or fractured fibers is also presented. This theoretical development includes the significant dependence of stress redistribution on fiber volume fraction, constituent properties, and crack size. Therefore, its use is not limited to polymeric composites but is also applicable to metal matrix and ceramic matrix systems. The model is extended to include one of the first quantitative analyses of variable fiber spacing which occurs in virtually every composite manufactured. A novel fiber discount method is proposed to study multiple fiber fractures which are of extreme importance when attempting to predict tensile strength of fiber dominated composite laminates. A test methodology employing a macro-model composite with embedded strain gauges is presented which can be used to validate (or invalidate) micro-mechanical models currently being developed and used by the scientific community. Results obtained with the embedded resistance gauges and the embedded fiber optic strain sensors (FP-FOSS) are validated with classical test and analytical techniques. These techniques include model composites subjected to thermal effects and mechanical loading sequences. The ability to vary specific physical parameters in the experimental model, such as fiber aspect ratio, fiber volume fraction, interphase/interface, and constituent properties (i.e. model PMC’s and MMC’s), in a systematic fashion enables this technique to study various physical aspects present in actual composite systems. The capability to initiate a fiber fracture at a specific location and load level is demonstrated. It is revealed that significantly different strain concentration exists in PMC composites which contain different fiber volume fractions and crack sizes. By varying fiber spacing between neighbors, a study is initiated on composites containing eccentrically located fibers. These results demonstrate that an asymmetric stress state exists in composites containing variable fiber spacing and fiber fractures. The fact that multiple fiber fracture is achieved in a methodical fashion demonstrates the versatility of the model. These studies show that this experimental technique can model various physical phenomena which occur in actual composite systems.

Until this present work, the acoustics of waveguides has been divided into two broadly distinct fields---linear acoustics in ducts of complex geometry such as those with curvature or varying width, and nonlinear acoustics restricted to simple geometry ducts without curvature or flare. This PhD unites these distinct branches to give a complete mathematical description of weakly nonlinear wave propagation in a general shaped duct in both two and three dimensions. Such ducts have important applications---the clearest example is that of brass instruments, where it has been demonstrated that nonlinear wave steepening gives rise to the characteristic 'brassy' sounds of, for example, the trombone. As the ducts of these instruments have a very complicated geometry involving curvature, torsion and varying width, the goal of the PhD is to address what effect, if any, such changes in duct geometry have on the acoustic properties of such instruments. Other potential applications include the study of acoustics in curved aircraft engine intakes and even the nonlinear sound propagation through the trunk of an elephant. The first results chapter is focused on the exposition of the method used for the remainder of the paper, with the introduction of a new ``nonlinear admittance term'' as well as the associated algebra for it. An elegant notation for the nonlinear algebra is also developed, greatly simplifying the equations. The method is applied to one and two dimensional ducts and some analytical results are derived relating the work to previously published results. Numerical results are also presented and compared to other sources. The concept of nonlinear reflectance is also introduced---illustrating the effect of wave amplitude on the amount of energy reflected in a duct. The next results chapter builds on this work extending it to three dimensions. Numerical results are presented for three characteristic ducts---a curved duct, a horn and a helical duct, being one of the first works to study acoustics in helical pipes for both linear and nonlinear sound propagation. The final results chapter, utilising all of the previous work, addresses the problem of an open ended duct of finite length with nonlinear effects included. Results are compared with the linear results from the Wiener-Hopf method and new results are presented illustrating the effect of geometry and nonlinearity on the resonances of finite length waveguides culminating in the study of the resonances of a trombone.

Low-density parity-check (LDPC) codes can approach the Shannon limit performance closely, and are becoming one of the most promising channel codes in the Error Control Coding area. The performance of an LDPC code with given length is mainly affected by the sizes of some combinatorial characteristics of its corresponding bipartite graph and the distribution of variable degrees and check node degrees. With the help of the density evolution algorithm, randomly designed LDPC codes, with carefully chosen degree distribution pairs, have been shown to achieve better Shannon capacity performance than their regular counterparts when decoded using the iterative belief propagation (BP) decoding algorithm. However, regular LDPC codes outperform their irregular counterparts in term of their error floors. Therefore, construction of LDPC codes which have both attractive error rate performance and low error floor performance has become an attractive topic.Currently, optimization of variable node degree distribution and check node degree distribution for an LDPC code with given length, which could help the code to achieve good Shannon limit performance, has been extensively studied. In this dissertation, we demonstrate some approaches of improving the error floor performance for a given LDPC code with desired degree distribution pair. The contribution mainly includes three parts. In the first part, the relationship between cycles and unfavorable combinatorial characteristics (or trapping set for codes over AWGN channels and stopping set for codes over BEC channels) of LDPC codes is analyzed. The analysis indicates that large girths of LDPC codes could lead to low error floors. Based on this analysis, an approach of designing any individual irregular LDPC code with girth of six is proposed. The second part presents a novel algorithm of enumerating the worst unfavorable combinatorial characteristics with the help of building a searching tree. With the help of the enumeration results of this novel algorithm, the worst unfavorable combinatorial characteristics can be eliminated by refining the parity-check matrices of LDPC codes, which results in the improvement of the error floor performance of LDPC codes based on our simulation results. The above mentioned methods focus on designing LDPC codes with low error floors from the encoder side of the LDPC codes. Another possible approach of lowering the error floors is from the decoder side, which is studied in the third part of this dissertation. In the third part, the trapping sets of LDPC codes are extensively analyzed and a concept of pseudo-cycle is proposed. Based on the analysis, we present an improved decoder, a two-stage decoder, for LDPC codes to enable dealing with the negative influence caused by those unfavorable combinatorial properties of LDPC codes. The simulation results show that the error floors of LDPC codes can be lowered by more than one order of magnitude.Unlike the current methods for lowering the error floors of LDPC codes, all the approaches proposed in this dissertation can be applied to any individual LDPC code. Based on our simulation results, these approaches can effectively decrease the errorfloors for any specific LDPC code.

One-dimensional (1D) nanomaterials are of great importance for a number of potential applications. However, in order to realize this potential a thorough understanding of the charge-carrier dynamics in these materials is required, since these largely determine the optoelectronic properties of the materials in question. This thesis investigates the charge-carrier dynamics of two 1D nanomaterials, single-walled carbon nanotubes (CNTs) and tungsten-oxide nanowires (WOxNWs), with the goal of better understanding the nature of their optoelectronic responses, and how nanomaterial geometry and morphology influence these responses. We do this using terahertz time-domain spectroscopy (THz-TDS) and optical pump - terahertz probe time-domain spectroscopy (OPTP). Firstly, we discuss how to properly analyse and interpret the data obtained from these experiments when measuring 1D nanomaterials. While the data obtained from THz-TDS is fairly straight-forward to analyse, OPTP experimental data can be far from trivial. Depending on the relative size of the sample geometry compared to the probe wavelength, various approximations can be used to simplify the extraction of their ultrafast response. We present a general method, based on the transfer matrix method, for evaluating the applicability of these approximations for a given multilayer structure, and show the limitations of the most commonly used approximations. We find that these approximations are only valid in extreme cases where the thickness of the sample is several orders of magnitude smaller or larger than the wavelength, which highlight the danger originating from improper use of these approximations. We then move on to investigate how the charge-carrier dynamics of our CNTs is influenced by nanotube length and density. This is done through studying the nature of the broad THz resonance observed in finite-length CNTs, and how the nanotube length and density affects this resonance. We do this by measuring the conductivity spectra of thin films comprising bundled CNTs of different average lengths in the frequency range 0.3-1000 THz and temperature interval 10-530 K. From this we show that the observed temperature-induced changes in the terahertz conductivity spectra depend strongly on the average CNT length, with a conductivity around 1 THz that increases/decreases as the temperature increases for short/long tubes. This behaviour originates from the temperature dependence of the electron scattering rate, which results in a subsequent broadening of the observed THz conductivity peak at higher temperatures and a shift to lower frequencies for increasing CNT length. Finally, we show that the change in conductivity with temperature depends not only on tube length, but also varies with tube density. We record the effective conductivities of composite films comprising mixtures of WS2 nanotubes and CNTs vs CNT density for frequencies in the range 0.3-1 THz, finding that the conductivity increases/decreases for low/high density films as the temperature increases. This effect arises due to the density dependence of the effective length of conducting pathways in the composite films, which again leads to a shift and temperature dependent broadening of the THz conductivity peak. Next, we investigate the conflicting reports regarding the ultrafast photoconductive response of films of CNTs, which apparently exhibit photoconductivities that can vastly differ, even in sign. Here we observe explicitly that the THz photoconductivity of CNT films is a highly variable quantity which correlates with the length of the CNTs, while the specific type of CNT has little influence. Moreover, by comparing the photo-induced change in THz conductivity with heat-induced changes, we show that both occur primarily due to heat-generated modification of the Drude electron relaxation rate, resulting in a broadening of the plasmonic resonance present in finite-length metallic and doped semiconducting CNTs. This clarifies the nature of the photo-response of CNT films and demonstrates the need to carefully consider the geometry of the CNTs, specifically the length, when considering them for application in optoelectronic devices. We then move on to consider our WOxNWs. We measure the terahertz conductivity and photoconductivity spectra of thin films compromising tungsten-oxide (WOx) nanowires of average diameters 4 nm and 100 nm, and oxygen deficiencies WO2.72 and WO3 using THz-TDS and OPTP. From this we present the first experimental evidence of a metal-to-insulator transition in WOx nanowires, which occurs when the oxygen content is increased from x=2.72 -> 3 and manifests itself as a massive drop in the THz conductivity due to a shift in the Fermi level from the conduction band down into the bandgap. Furthermore we present the first experimental measurements of the photoexcited charge-carrier dynamics of WOx nanowires on a picosecond timescale and map the influence of oxygen-content and nanowire diameter. From this we show that the decay-dynamics of the nanowires is characterized by a fast decay of < 1 ps, followed by slow decay of 3-10 ps, which we attribute to saturable carrier trapping at the surface of the nanowires.

A systematic study of high power backward wave oscillators (BWOs) was performed using the Sinus-6 repetitively-pulsed relativistic electron beam accelerator. Peak output power of up to 550 MW was measured in a frequency range of 9.2 to 9.8 GHz with pulse lengths up to 10 ns. The experimental results were compared with numerical simulations using TWOQUICK, a 2.5 D, relativistic, fully electromagnetic, particle-in-cell (PIC) code. It was observed that a significant increase in microwave efficiency can be obtained by using a BWO with a spatially varying coupling impedance instead of the traditional uniform distribution. Based on experiments using variations of a two stage nonuniform amplitude BWO in conjunction with TWOQUICK simulations, it was determined that the efficiency enhancement can be attributed to the prebunching of the electron beam.

Applications of Item Response Theory, which depend upon its parameter invariance property, require that parameter estimates be unbiased. All current estimation methods produce statistically biased estimates of both item and ability parameters. A new method, Weighted Likelihood Estimation (WLE), is derived, and proved to be less biased than Maximum Likelihood Estimation (MLE) with the same asymptotic variance and normal distribution. WLE removes the first order bias term from MLE. Two Monte Carlo studies compare WLE with MLE and Bayesian Model Estimation (BME) of ability in conventional tests and tailored tests. The Monte Carlo studies favor WLE over MLE and BME on several criteria over a wide range of the ability scale.

When a digital process is realized on a general-purpose computer or a special-purpose hardware, errors due to finite register length are introduced. These errors are due primarily to arithmetic roundoff, coefficient quantization, and scaling rules. This paper addresses the effects of finite word length on a direct-form implementation of a high order H (z) transfer function. The development and analysis of a modified direct-form realization suggested by Dr. Fred O. Simons, are carried out via FORTRAN emulation of a fourth-order low-pass Butterworth filter. The results are presented as a parametric tradeoff of signal-to-noise ratio at the filter output versus word length. Conclusions are drawn by comparing the modified direct-form with the canonic direct-form. The analysis presented here is intended to illustrate how a high order transfer function can be realized directly without decomposing into a group of low-order subfilters.

This thesis details an investigation into the effect of a duct on acoustic source radiation. The duct is assumed to be hard-walled, hollow, have a constant cross-section and have no axial flow. This study was motivated by the increasing tendency for ducted propellers to be located close to the open end relative to the acoustic wavelength. Investigations are conducted using the widely accepted semi-infinite duct model, formed from the Wiener-Hopf technique, as well as a new finite length duct formulation. In this study, the conditions are established under which duct effects on source radiation are considerable, and the duct plays a governing role on the overall acoustic radiation of the source. The duct acoustic model assumes the acoustic pressure field can be reconstructed from a summation of propagating cut-on and evanescent cut-off modes, which have largely been unstudied. Systematic studies of the radiation characteristics of cut-off modes are documented in this thesis, detailing properties such as their modal directivity and radiation efficiency. It is found that cut-off modal radiation is predominantly directed towards the rear-arc with an effect of up to 5 dB at certain angles and a more general increase of 3 dB in the rear-arc far-field radiation. In many cases, sources close to the duct open end relative to the acoustic wavelength are shown to be largely unaffected by the duct, causing equal free-field and ducted radiation. The effect of source axial position and varying duct length are shown to have a diminishing effect on the ducted source radiation as frequency is increased. At high frequency the duct has a relatively small effect on the acoustic radiation. Dipole sources located near the duct rim are shown to have a significant increase in ducted radiation, causing an omni-directional directivity - which can largely be attributed to cut-off modal radiation. Dipole sources located exactly on the duct rim are shown to have directivity characteristics solely governed by modes which are cut-off and have almost all radiation directed towards the rear-arc.

If C is a convex body in R^n and X is a k-dimensional linear subspace of R^n, we denote by S(C,X) the shadow boundary of C over X which is defined as the collection of all points which belong to C and to one of its tangent (n-k)-flats orthogonal to X. For almost all directions in R^3, the shadow boundary is a curve encompassing the body C. It has been established long ago by G. Ewald, D.G. Larman and C.A. Rogers [11] that, for every given C, S(C,X) is almost always a topological (k-1)-sphere. As a follow on from this result, in 1974 Peter McMullen asked whether most of these shadow boundaries would have finite “length” [15]. This is already shown to be true for polytopes and also true for general convex bodies when the dimension of the subspace X is 1 or n-1. Here we show that almost all shadow boundaries have finite “length” whatever the dimension k, 0< k< n, of the subspace X. The set of shadow boundaries of infinite “length” has also been considered in the context of Baire category. In 1989, P. Gruber and H. Sorger proved that, in the Baire category sense, most pairs (C,X), where C is a convex body in R^n and X an (n-1)-dimensional subspace of R^n, produce shadow boundaries S(C,X) of infinite length. Here we show that this result also holds for pairs (C,X) where X is a k-dimensional subspace, 0< k< n. We also consider the length of increasing paths in the 1-skeleton of a convex body. We conclude with observations and open questions arising from the work on shadow boundaries of the first two chapters.

Effective transmission of data over a noisy wireless channel is a vital part of today's high speed technology driven society. In a wireless cell network, information is sent from mobile users to base stations. The information being transmitted is protected by error-control codes. In a conventional architecture the signal processing, including error-control decoding, is performed locally at each base station. Recently, a new architecture has emerged called Centralized Radio Access Network (C-RAN), which involves the centralized processing of the signals in a computing cloud. Using a computing cloud allows computational resources to be pooled, which improves utilization and efficiency. When the computational resources are finite and when the computational load varies over time, then there is a chance that the load exceeds the available resources. This situation creates a so-called computational outage, which has characteristics that are similar to outages caused by channel fading or interference. In this report, the computational complexity is quantified for a common class of error-correcting codes known as low-density parity check (LDPC) codes. To make the analysis tractable, a binary erasure channel is assumed. The concept of density evolution is used to obtain the complexity as a function of the code design parameters and the signal-to-interference-plus-noise ratio (SINR) of the channel. The analysis shows that there is a trade-off in that aggressively signaling at a high data rate causes high computational demands, while conservatively backing off on the rate can dramatically reduce the computational demand. Motivated by this trade-off, a scheduling algorithm is developed that balances the demands for high throughput and low computational outage rates.

With the advent of power electronics, inverter-driven induction motor are finding increased use in industries because of applications that demand variable speed operations and because of the potential savings in energy usage. However, these drives sometimes produce unacceptably high levels in vibration and acoustic noise. A literature survey has revealed that while there has been intensive research on the design of inverters to minimize acoustic noise radiation from these drives, the vibro-acoustic behaviour of an induction motor structure has received relatively little attention. The primary objective of this research project, therefore, is to develop a general strategy/algorithm for estimating the acoustic noise radiated from inverter-driven induction motors. By using a three-phase, 2.2 kW induction motor, the vibration modes due to various structural components (such as the rotor, the stator/casing, the endshields and the base plate) of the motor structure were analysed by experimental modal testing. Results indicate that the vibration modes due to the rotor are only important at low frequencies. It has been found that the power injection method gives more accurate measurement of the damping of a motor structure than the modal testing and the time decay methods. If a point force excitation is used, then it is more accurate to measure the sound radiation efficiency than the power conversion efficiency for motor structures. The effect of three different inverter designs (an ideal ‘almost sinusoidal’ controller and two commercially available PWM inverters) on the radiated acoustic power were assessed for both no-load and load conditions using sound intensity measurements conducted in an anechoic room. The results indicate that although the sound power level due to aerodynamic and mechanical noise increases at a rate of 12 dB per doubling of the motor speed, the electromagnetic noise dominates at low motor speeds and is still a significant noise source even at high motor speeds. For inverters with low switching frequencies, the radiated sound power level is almost 15 dB higher than the ideal case at low speeds and is relatively insensitive to the motor speed. For inverters that implement the random modulation technique, the change in the total sound power level with the level of the random modulation is very small but the tonal nature of the noise is greatly reduced. The vibration behaviour of a motor structure was modeled using the finite element method (FEM) and validated using the experimental modal testing results. It has been found that it is essential to model the laminated stator as an orthotropic structure. While the details of other structural components (such as the endshields, the teeth in the stator and the windings) are not so important, it is essential that they are incorporated into the structural model as simplified structures to account for their mass, stiffness and boundary conditions imposed on the motor structure. Based on this structural model, the radiated acoustic power for various operating conditions has been predicated using the boundary element (BEM) and the electromagnetic force calculated from an electromagnetic finite element model. The predicted results agree reasonably well with experimental measurements. Despite the success of the FEM/BEM approaches, they can be prohibitively expensive (in terms of computer resources required) to apply to large motors and high frequencies. Thus the feasibility of using a statistical method, namely, the statistical energy analysis (SEA), to estimate the radiated acoustic sound power from an inverter-driven induction motor has been examined. In order to carry out this analysis, analytical expressions for calculating the natural frequencies and radiation efficiency of finite length circular cylindrical shells (which are simplified models of the stator and casing of a motor structure) were firstly derived. The internal loss factors and coupling loss factors of the motor structure were determined experimentally using the power injection method. Then by introducing an equivalent surface mobility of circular cylindrical shells for the electromagnetic force, the vibration response and the acoustic noise radiated from each part of the motor structure were estimated. Results indicate that SEA method is potentially an efficient and effective tool in estimating the noise radiated from inverter-driven induction motors.

Existing theories for the mechanical response of particle fluid mixtures have been reviewed and extended. They are made appropriate to geometries in which the dominant loading is a shearing one. The theories are then applied to the description of filtration experiments (these were performed by researchers dn a parallel research programme at Loughborough University). Two limits are distinguished: one in which particles experience a strong double layer interaction and one in which the particles are neutral and the fluid is the only significant force-mediating medium. The existing theories that have been reviewed and used are the quasi-static two-phase continuum mechanics framework (including seepage effects), the granular temperature theory for neutral particles and the common consolidation theory for strongly interacting particles. To extend these general theories - and especially to enter reliable constitutive relations - a micromechanical analysis is carried out and methods are developed to arrive at expressions for bulk properties. An analysis is performed of the response to a small localised fluctuation in either stress or solidosity of a particle-fluid mixture under arbitrary mean loading conditions. This analysis leads to a condition for stability of a mixture in terms of the solidosity sensitivity of the particle pressure and the solidosity sensitivity of the viscous constitutive parameters of the mixture given a mean loading regime. In the analysis it is recognised that a slurry in motion (especially shear) will always experience fluctuations. Two applications of the stability analysis are then presented. First it is recognised that homogenisation is impossible when the system is unstable. Second the border between a stable (packed) region and a free flowing region is defined by the edge of the stability condition, as made appropriate to the prevailing loading conditions. This piece of fundamental analysis is then used to describe filtration experiments, notably ones in which shear plays a distinctive role - these are torsion shear filtration and crossflow filtration. In order to analyse torsion shear filtration a calculation is carried out of a Newtonian fluid in a cylindrical vessel, loaded at the top by a rotating piston. A range of result is obtained: flow in an infinite cylinder, flow in a cylinder of finite length and flow in a finite cylinder with two immiscible fluids, occupying various sections of the cylindrical domain. The latter problem is particularly relevant to the torsion shear filtration problem as it shows that no significant shearing stress reaches, the cake until the fluid region near the piston is of the order of magnitude of the particle size of the mixture. Once shear can penetrate the cake the effects of it are noticed in that in a stable heterogeneous medium structures formation takes place in the direction of the major principal stress, implying that the greater the shear that is applied the greater the angle at which structures form. Then a calculation is presented to demonstrate the reduction in uniaxial stiffnes due to structures formation and the experimental result is recovered that for neutral particles cakes becoine denser when the shear is increased. This result is qualitative, though quantitative formulas are presented. The latter require parameters such as an estimate of the magnitude of the stiffness fluctuations that are hard to determine from current experiments. For double layer interacting particles the effects of shear are noticed at an earlier stage in the filtration process as particle interactions transmit the forces exerted externally on the mixture. The overall stiffness due to shearing is then estimated (stability is here required) and it is shown that the normal stress on the medium is reduced due to the fluctuations induced by the shearing. A lattice-Boltzmann, simulation of the same configuration confirms this interesting result. A crossflow setup has been analysed. A somewhat simplified one dimensional investigation is presented. The key point is that the edge of the cake near the septum is defined by the edge of stability analysis and this piece of information enables' a full survey of experimental results with a wide range of process paraméters (feed solidosity, crossflow velocity, crossflow pressure, particle type, pH). Two key experimental parameters are predicted: the end of filtration filtrate flow and time constant with which this end stage is reached. Double layer interacting particles and neutral particles have been explored. Some key findings pertaining especially to cases of thisn cakes are as follows. Double layer interacting particles: the end of filtration filtrate flux depends on the ratio of the crossflow velocity and feed solidosity only. The time constant depends Existing theories for the mechanical response of particle fluid mixtures have been reviewed and extended. They are made appropriate to geometries in which the dominant loading is a shearing one. The theories are then applied to the description of filtration experiments (these were performed by researchers dn a parallel research programme at Loughborough University). Two limits are distinguished: one in which particles experience a strong double layer interaction and one in which the particles are neutral and the fluid is the only significant force-mediating medium. The existing theories that have been reviewed and used are the quasi-static two-phase continuum mechanics framework (including seepage effects), the granular temperature theory for neutral particles and the common consolidation theory for strongly interacting particles. To extend these general theories - and especially to enter reliable constitutive relations - a micromechanical analysis is carried out and methods are developed to arrive at expressions for bulk properties. An analysis is performed of the response to a small localised fluctuation in either stress or solidosity of a particle-fluid mixture under arbitrary mean loading conditions. This analysis leads to a condition for stability of a mixture in terms of the solidosity sensitivity of the particle pressure and the solidosity sensitivity of the viscous constitutive parameters of the mixture given a mean loading regime. In the analysis it is recognised that a slurry in motion (especially shear) will always experience fluctuations. Two applications of the stability analysis are then presented. First it is recognised that homogenisation is impossible when the system is unstable. Second the border between a stable (packed) region and a free flowing region is defined by the edge of the stability condition, as made appropriate to the prevailing loading conditions. This piece of fundamental analysis is then used to describe filtration experiments, notably ones in which shear plays a distinctive role - these are torsion shear filtration and crossflow filtration. In order to analyse torsion shear filtration a calculation is carried out of a Newtonian fluid in a cylindrical vessel, loaded at the top by a rotating piston. A range of result is obtained: flow in an infinite cylinder, flow in a cylinder of finite length and flow in a finite cylinder with two immiscible fluids, occupying various sections of the cylindrical domain. The latter problem is particularly relevant to the torsion shear filtration problem as it shows that no significant shearing stress reaches, the cake until the fluid region near the piston is of the order of magnitude of the particle size of the mixture. Once shear can penetrate the cake the effects of it are noticed in that in a stable heterogeneous medium structures formation takes place in the direction of the major principal stress, implying that the greater the shear that is applied the greater the angle at which structures form. Then a calculation is presented to demonstrate the reduction in uniaxial stiffnes due to structures formation and the experimental result is recovered that for neutral particles cakes becoine denser when the shear is increased. This result is qualitative, though quantitative formulas are presented. The latter require parameters such as an estimate of the magnitude of the stiffness fluctuations that are hard to determine from current experiments. For double layer interacting particles the effects of shear are noticed at an earlier stage in the filtration process as particle interactions transmit the forces exerted externally on the mixture. The overall stiffness due to shearing is then estimated (stability is here required) and it is shown that the normal stress on the medium is reduced due to the fluctuations induced by the shearing. A lattice-Boltzmann simulation of the same configuration confirms this interesting result. A crossflow setup has been analysed. A somewhat simplified one dimensional investigation is presented. The key point is that the edge of the cake near the septum is defined by the edge of stability analysis and this piece of information enables a full survey of experimental results with a wide range of process parameters (feed solidosity, crossflow velocity, crossflow pressure, particle type, pH).

Abstract: In this dissertation we present new methods for designing efficient Raptor codes in finite and practical block lengths. First we propose an extension of Raptor codes which keeps all the desirable properties, including the linear complexity of encoding and decoding per information bit, and improves the performance in terms of the reception rate. Our simulations show a 10% reduction in the required overhead at the benchmark block length of 64,520 bits, and with the same complexity per information bit. Second, we consider the practical setting with short block lengths of 1000

The adaptation of Tollmien-Schlichting waves as they propagate over the leading and trailing edges of finite-length compliant panels is investigated by means of numerical simulation. The behaviour so determined is pertinent to the application of compliant walls for transition delay. We consider a model problem where the compliant panels form a section in the walls bounding a plane channel fluid flow. The results obtained are encouraging. They indicate that compliant panels with lengths comparable to the Tollmien-Schlichting wavelength can have a significant stabilising effect. In some instances, the passage of a Tollmien-Schlichting wave over the edges of a compliant panel leads to the excitation of stable flow-induced surface waves. The presence of such additional waves does not appear to be associated with any adverse effect upon the stability of the Tollmien-Schlichting wave. The numerical scheme used for the simulations derives from a mixed finite-difference/spectral discretisation of the linearised two-dimensional Navier-Stokes equations, which were taken in a vorticity-velocity formulation. Numerical stability problems were overcome by treating the inertia of the compliant wall and the fluid together when imposing the boundary conditions. This allowed the interactively coupled fluid and the wall motion to be computed without any prior restriction on the form taken by the disturbances. An investigation was also carried out into the linear stability of plane channel flow bounded by compliant walls throughout. In the case of the Tollmien-Schlichting mode this relied, for the most part, on the determination of numerical solutions to the Orr-Sommerfeld equation. Flow-induced surface waves could be studied more readily using an approximate analytic theory. Good agreement was achieved between the predictions of the analytic theory and numerical results obtained directly from the Orr-Sommerfeld equation, particularly for travelling wave flutter. The linear stability results for the wholly compliant-walled channel were used to analyse the behaviour displayed by Tollmien-Schlichting waves in the numerical simulations that were conducted with finite-length compliant panels.

Codebook design, generation, and adaptation, based on matching to stochastic source examples or prior knowledge of source distribution, has played a central role in many applications of source coding. The original iterative ''natural type selection'' (NTS) algorithm performs stochastic codebook generation of memoryless sources, and achieves the rate-distortion bound, as it asymptotically converges to the optimal codebook reproduction distribution, Q*. However, these optimality results are subject to significant limitations that compromise the practical applicability of NTS, namely: i) the string length L is required to go to infinity at the outset, before NTS iterations begin, whereas the iteration complexity is exponential in L, and ii) it is only applicable to discrete and memoryless sources, thus precluding a vast portion of important lossy coding applications. This thesis offers means to eliminate or circumvent these critical shortcomings by proposing new and enhanced NTS algorithms, complemented by optimality proofs that are not subject to the above limitations. To circumvent the need to start with asymptotically large string length, L, the approach leverages a maximum likelihood framework to estimate, at each NTS iteration n, the reproduction distribution most likely to generate the sequence of K length-L codewords that respectively and independently “d-match” (i.e., are within distortion d from) a sequence of K length-L source words. The reproduction distribution estimated at iteration n is used to regenerate the codebook for iteration n+1. The sequence of reproduction distributions estimated by NTS is shown to converge, asymptotically in K, n, and L (in this order), to the optimal distribution that achieves the rate-distortion bound. Thus, the string length L is the last parameter to be sent to infinity. Moreover, it is established that, for finite length L, the new NTS algorithm converges to the best achievable distribution, i.e., as constrained by the string length L, and details are provided for various types of sources, where numerical simulations show that the algorithm rate of convergence in n for finite length L is at least as fast as convergence in n with infinite L. To handle sources with memory, NTS is further generalized by considering source sub-vectors or ''super-symbols'', of memory depth M, during d-match search in the codebook, maximum likelihood estimation of reproduction distribution, and codebook regeneration. Asymptotic convergence, in L and M, to the optimal reproduction distribution is also established for sources with memory. As for, perhaps the more challenging, sources over continuous alphabet spaces, which are inconsistent with the traditional concept of ''type'' or ''typical sequence'', in the proposed asymptotically optimal approach, we employ empirical probability measures for codebook reproduction distribution estimation. Methodologies for optimal codebook generation and adaptation are further developed and employed in two promising example applications in the areas of i) wireless communications and ii) machine learning. In particular, for 5G cellular systems and next generation wireless local area networks, directional beamforming with large antenna arrays is key to mitigating the substantial signal loss experienced at the millimeter wave frequency band, where it entails a significant increase in the number of beams required to maintain cell coverage, and hence an increase in the beam management overhead necessary to maintain link with mobile users. This observation, in turn, suggests that the underlying problem of finding the optimal set of beam steering directions will benefit from fundamental signal processing and codebook design methodologies, and specifically from basic principles and algorithms for cluster analysis. This part of the thesis establishes and exploits the equivalence between the problem of optimizing a set of beam steering directions and the classical problems of clustering in pattern recognition and codebook design in data compression, albeit with an unusual distortion measure. Subsequently, a global optimization approach within the deterministic annealing framework is derived, to circumvent poor local optima that may riddle the cost surface under the classical gradient descent clustering techniques. System simulation results show that the proposed approaches deliver considerable gains, over the baseline beam steering techniques, in terms of average signal-to-noise ratio. The third part of the thesis is concerned with codebook design and adaptation for machine learning or artificial intelligence. Machine learning applications have exploded in recent years due to the availability of huge data sets, as well as advances in computational and storage capabilities. Although successful methods have been proposed to reduce learning system complexity while maintaining required accuracy levels, theoretical understanding of the underlying trade-offs remains elusive. In this work, the classical supervised learning problem is reformulated within a rate-distortion framework. It provides insights into the underlying accuracy-complexity trade-offs, by considering the overall learning system as consisting of two components. The first is tasked with extracting (learning) from the source the minimal number of information bits necessary to ultimately achieve the prescribed output accuracy. The learned bits are then used to retrieve the desired output from the second component, an appropriately designed codebook. The premise here is that an optimal system is characterized by having to learn the minimum amount of information from the source, just sufficient to yield the system output at the desired precision, which implies efficiency in terms of system complexity, generalization and training data requirements. The design and training of such a reformulated system is detailed, and asymptotically optimal performance that achieves the rate-distortion bound is established.

It is well known that any finitely generated Z-module is a direct sum of a projective (in fact a free) module and a Noetherian module (in fact a module of finite length) (for example see [Fu]). More generally, [Sm1] proved that if R is a right Noetherian ring with maximal Artinian right ideal A, then every finitely generated right R-module is the direct sum of a projective module and a module of finite length if and only if the ideal A = eR for some idempotent e in R and the ring R/A is a left and right hereditary left and right Noetherian semiprime ring (see [Sm1, Theorem 3.3]). It was left open in [Sm1] whether the assumption that R be right Noetherian is necessary. In fact, it is not, as Chatters [Ch] showed, by proving that if R is a ring such that every cyclic right R-module is the direct sum of a projective module and a Noetherian module, then R is a right Noetherian ring (see [Ch, Theorem 3.1]). Chatters [Ch, Theorem 4.1] also proved that if a is an ordinal and R a ring such that every cyclic right R-module is the direct sum of a projective module and a module of Krull dimension at most a, then the right R-module R has Krull dimension at most ? + 1. Van Huynh and Dan [HD] have considered rings with the property that every cyclic right module is the direct sum of a projective module and an Artinian module, or the property that every cyclic right module is the direct sum of a projective module and a semisimple module. This led to the investigations in [SHD] and [Sm2]. The following terminology was introduced. Let X be a class of modules. Then hX is defined to be the class of modules M such that for each submodule N of H, M/N belongs to X. Moreover, dX is defined to be the class of modules M such that for each submodule N of M, there exists a direct summand K of M such that N ? K and K/N belongs to X. Finally, eX is defined to be the class of modules M such that for each essential submodule E of M, M/E belongs to X. It is proved in [Sm2] that when X is the class U: the class of modules with finite uniform dimension, then a module M belongs to eU if and only if M/N belongs to hU for some semisimple submodule N of M (Theorem 1.2.1). This fact led [Sm2] to prove that a module M belongs to dU if and only if M = M1 ⊕ M2 where is a semisimple module and M2 belongs to hU (Theorem 1.2.3). Moreover, [Sm2] proved that when X is the class N; the class of Noetherian modules, or when X is the class K: the class of modules which have Krull dimension then a module M belongs to dN (respectively, dK) if and only if M = M1 ⊕ M2 where M1 is semisimple and M2 belongs to N (respectively, K) (Theorem 1.2.4). In the first two sections of chapter I of this thesis, we present all of the background material from [SHD] and [Sm2] and, for completeness, we include the proofs. In the third section, we prove a generalization of Theorem 1.2.4. i.e. we prove that when X is the class of modules with dual Krull dimension at most alpha, for some ordinal alpha > 0, then a module M belongs to dX if and only if M = M1 ⊕ M2 where M, is a semisimple module and M2 belongs to X (Theorem 1.3.11). In section 2.1, we define the property (P) : a module M satisfies (P) provided that for any submodule N of M, there exists a direct summand K of M such that Soc K ⊆ N ⊆ K. We prove that a module M is the direct sum of modules with (P) and M is eventually semisimple if and only if M = M1 ⊕ M2 ⊕ M3 where M1 is a semisimple module, M2 a finite direct sum of uniform modules and M3 has finite uniform dimension and zero socle (Theorem 2.1.5). In section 2.2, we define the property (P*): a module M satisfies (P*) provided that for any submodule N of M, there exists a direct summand K of M with K ⊆ N and N/K ⊆ Rad M/K. We prove that a module M is a direct sum of modules satisfying (P*) and the radical of M has finite uniform dimension if and only if M= M1 ⊕ M2 ⊕ M3 where M, is a semisimple module M2 is a radical module with finite uniform dimension and M3 is a finite direct sum of local submodules and has finite uniform dimension (Theorem 2.2.8). In chapter III, we define h*X (respectively, e*X) to be the class of modules M such that every (small) submodule of M belongs to X. Moreover, we define d*X to be the class of modules M such that for each submodule N of M, N contains a direct siimmand K of M such that N/K belongs to X.

In this thesis, we study the mathematical solution of wave scattering problems which describe the behaviour of waves incident on obstacles and are highly relevant to a raft of applications in the aerospace industry. The techniques considered in the present work can be broadly classed into two categories: analytically based methods which use special transforms and functions to provide a near-complete mathematical description of the scattering process, and numerical techniques which select an approximate solution from a general finite-dimensional space of possible candidates. The first part of this thesis addresses an analytical approach to the scattering of acoustic and vortical waves on an infinite periodic arrangement of finite-length flat blades in parallel mean flow. This geometry serves as an unwrapped model of the fan components in turbo-machinery. Our contributions include a novel semi-analytical solution based on the Wiener-Hopf technique that extends previous work by lifting the restriction that adjacent blades overlap, and a comprehensive study of the composition of the outgoing energy flux for acoustic wave scattering on this array of blades. These results provide an insight into the importance of energy conversion between the unsteady vorticity shed from the trailing edges of the cascade blades and the acoustic field. Furthermore, we show that the balance of incoming and outgoing energy fluxes of the unsteady field provides a convenient tool for understanding several interesting scattering symmetries on this geometry. In the second part of the thesis, we focus on numerical techniques based on the boundary integral method which allows us to write the governing equations for zero mean flow in the form of Fredholm integral equations. We study the solution of these integral equations using collocation methods for two-dimensional scatterers with smooth and Lipschitz boundaries. Our contributions are as follows: Firstly, we explore the extent to which least-squares oversampling can improve collocation. We provide rigorous analysis that proves guaranteed convergence for small amounts of oversampling and shows that superlinear oversampling can ensure faster asymptotic convergence rates of the method. Secondly, we examine the computation of the entries in the discrete linear system representing the continuous integral equation in collocation methods for hybrid numerical-asymptotic basis spaces on simple geometric shapes in the context of high-frequency wave scattering. This requires the computation of singular highly-oscillatory integrals and we develop efficient numerical methods that can compute these integrals at frequency-independent cost. Finally, we provide a general result that allows the construction of recurrences for the efficient computation of quadrature moments in a broad class of Filon quadrature methods, and we show how this framework can also be used to accelerate certain Levin quadrature methods.

Fan noise is one of the dominant sources of aircraft engine noise, both at approach and at take-off. Improved attenuation of fan noise with acoustic liners and the reduction of fan noise at source remain key technology challenges for the foreseeable future. Over- Tip-Rotor (OTR) acoustic treatments have been investigated experimentally during the last decade and significant fan noise reductions have been measured, most recently at NASA by using a rotor-alone rig and multiple lined circumferential grooves. This thesis aims at improving our understanding of the noise reduction mechanisms of OTR liners through the development of an analytical prediction model. The fan noise is modelled with point or distributed monopole and dipole sources based on Green's functions for infinite hard or lined cylindrical ducts containing uniform mean flow; these are combined with an anechoic or unflanged inlet termination and an embedded lined section of finite length representing the 'rotor-alone' source with its OTR liner. Key aspects of the propagation/ attenuation and liner impedance modelling are cross verified with reference FEM solutions. OTR predictions of liner insertion loss are obtained for comparison with the NASA experimental data, which entailed coupling of the propagation model with that of a partially non-locally reacting impedance model, and these yielded peak broadband in-duct noise reductions of up to 4 dB, in line with the measurements. To complement the analytical work, wind tunnel experiments have been conducted at ECL on a simplified over-tip liner configuration. The rotor and OTR liner were represented by a static aerofoil with its tip located over a flat plate containing a flush-mounted liner insert and separated from the airfoil tip by a small gap. The measured sound spectra exhibit peak broadband gap noise reductions of 5-10 dB due to the OTR liner and noise reduction in trailing edge noise of up to 5 dB in the absence of a gap.

In this thesis, the studies of diffusion and the electrochemical systems based on the modelling and simulation of chronoamperometry and cyclic voltammetry for various novel electrochemical systems are reported and discussed. In Chapter 1, fundamental and essential concepts and theories in electrochemistry are introduced to provide key background information and to assist the understanding of the later chapters. Similarly, Chapter 2 outlines the introductory aspects for the methodologies of electrochemical modelling and simulation by using the finite difference method to discretise the diffusion equations, together with the methods used for validating and testing the simulations. Chapter 3 and 4 present the introduction and application of a new physicochemical parameter characterising mass transport, the "diffusion indicator", which offers sensitive and comparable information about the changes and trends in chronoamperometric current responses in a simple and straightforward form by distinguishing the relative contributions from linear and convergent diffusion. Based on the analytical and modelling studies of electrochemical systems with four different electrode geometries (spheres, discs, cylinders, and bands), the concept of the diffusion indicator is developed and discussed in Chapter 3. Chapter 4 applies this new parameter to chronoamperometric studies of several cylindrical and ring electrodes, where the comparative studies of the diffusive flux of the analyte toward electrodes with similar but different geometries from different spatial locations are presented. Chapter 5 discusses the theoretical studies for a particular system regarding the nano-impact method, which involves the electrochemical process of depleting the dopant inside a spherical particle after the particle collides with the electrode and is adsorbed on the electrode surface. Two different theoretical models are adopted and utilised, and a dimensional analysis is presented. Chapter 6 investigates the cyclic voltammetry of a reversible one-electron-transfer reaction at electrochemical systems with different electrode geometries. A polynomial equation is introduced to describe the forward peak potentials at different scan rates for an 'infinitely long cylinder' electrode. Together with literature equations, this introduced equation enables us to produce an evaluation of infinite cylinder approximation for other cylinder-like electrodes of finite length and where the cylinder ends contribute significantly to the net flux. Chapter 7 applies the diffusion-equation-based simulations to an adsorbed-enzyme-mediated electrochemical system together with the theoretical analysis of cyclic voltammetric current responses on both flat and porous electrodes to understand the role of the Michaelis-Menten kinetics under both diffusional regimes and, in particular, to explore the role of nano-structure in the enzyme-based catalytic reaction. In this study, literature reports of the apparent potential dependent Michaelis constants are explained. Four classes of voltammograms are identified corresponding to qualitatively different current responses. The conditions under which porosity can usefully aid the electro-catalysis are determined providing a basis for scale-up of the enzyme-catalysed redox processes.