This paper is a review of the major literature dealing with transformations of random variates which achieve variance stabilization and approximate normalization. The subject can be said to have been initiated by a genetical paper of R. A. Fisher (1922) which uses the angular transformation Φ = 2 arcsin√p to deal with the analysis of proportions p with E(p) = P. Here it turns out that Var Φ is almost independent of P and so stabilizes the variance. Some fourteen years later Bartlett introduced the so-called square-root transformation which achieves variance stabilization for variates following a Poisson distribution. These two transformations and their ramifications in theory and application are fully discussed. along with refinements introduced by later writers, notably Curtiss (1943) and Anscombe (1948). Another important transformation discussed is one which refers to an analysis of observations on to a logarithmic scale, and here there are uses in analysis of variance situations and theoretical problems in the field of estimation: in the case of the latter, the work of D. J. Finney (1941) is considered in some detail. The asymptotic normality of the transformation is also considered. Transformations primarily designed to bring about ultimate normality in distribution are also included. In particular, there is reference to work on the chi-square probability integral (Fisher), (Wilson and Hilferty (1931)) and the logarithmic transformation of a correlation coefficient (Fisher (1921)). Other miscellaneous topics referred include i. the probability integral transformation (Probits), with applications in bioassay: ii. applications of transformation theory to set up approximate confidence intervals for distribution parameters (BIom (1954)): iii. transformations in connection with the interpretation of so-called 'ranked' data. / M.S.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. 27 January 2016. / The direct and inverse spectral problems for an inhomogeneous string with an interval of zero density and a concentrated mass at the end point moving with damping are investigated. The partial differential equation is mapped into an ordinary differential equation using separation of variables which in turn is transformed into a Sturm-Liouville differential equation with boundary conditions depending on these parathion variable. The Marchenko approach is employed in the inverse problem to recover the potential, density and other parameters from the knowledge of the two spectra and length of the string.

M.Sc., Faculty of Science, University of the Witwatersrand, 2011 / Continued fractions have been extensively studied in number-theoretic ways. In this text, we will illuminate some of the geometric properties of contin- ued fractions by considering them as compositions of MÄobius transformations which act as isometries of the hyperbolic plane H2. In particular, we examine the geometry of simple continued fractions by considering the action of the extended modular group on H2. Using these geometric techniques, we prove very important and well-known results about the convergence of simple con- tinued fractions. Further, we use the Farey tessellation F and the method of cutting sequences to illustrate the geometry of simple continued fractions as the action of the extended modular group on H2. We also show that F can be interpreted as a graph, and that the simple continued fraction expansion of any real number can be can be found by tracing a unique path on this graph. We also illustrate the relationship between Ford circles and the action of the extended modular group on H2. Finally, our work will culminate in the use of these geometric techniques to prove well-known results about the relationship between periodic simple continued fractions and quadratic irrationals.

University of Macau / Faculty of Science and Technology. / Department of Computer and Information Science

In the construction of Canonical Loop Quantum Gravity, General Relativity is rewritten in terms of the Ashtekar variables to simplify its quantisation. They consist of a densitised triad and a connection terms. The latter depends by definition and by construction on a free parameter β, called the Barbero–Immirzi parameter. This freedom is passed on to the quantum theory as it appears in the expressions of certain operators. Their discreet spectra depend on the arbitrary value of this parameter β, meaning that the scale of those spectra is not uniquely defined. To get around this ambiguity, we propose to consider a theory of Conformal Loop Quantum Gravity, by imposing a local conformal symmetry through the addition of a scalar field. We construct our theory starting from the usual Einstein–Hilbert action for General Relativity to which we add the action for the massless scalar field and rewrite it in terms of a new set of Ashtekar-like variables. They are constructed through a set of canonical transformations, which allow to move the Barbero–Immirzi parameter from the connection to the scalar variable. We then show that the theory can be quantised by fulfilling the conditions for a Dirac quantisation. Finally, we present some first elements of the quantum formalism. It is expected that with such a scalar-tensor theory, the quantum operators should not depend on the free parameter directly but rather on the dynamical scalar field, solving therefore the ambiguity.

by Chan Kwong Wing Raymond. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 115-123). / Acknowledgments --- p.i / Table of Contents --- p.ii-v / List of Tables --- p.vi / List of Figures --- p.vii / Abstract --- p.1 / Chapter Chapter 1 : --- Introduction / Chapter 1.1 --- An Introduction to Video Compression --- p.3 / Chapter 1.2 --- Overview of Problems --- p.4 / Chapter 1.2.1 --- Analog Video and Digital Problems --- p.4 / Chapter 1.2.2 --- Low Bit Rate Application Problems --- p.4 / Chapter 1.2.3 --- Real Time Video Compression Problems --- p.5 / Chapter 1.2.4 --- Source Coding and Channel Coding Problems --- p.6 / Chapter 1.2.5 --- Bit-rate and Quality Problems --- p.7 / Chapter 1.3 --- Organization of the Thesis --- p.7 / Chapter Chapter 2 : --- Background and Related Work / Chapter 2.1 --- Introduction --- p.9 / Chapter 2.1.1 --- Analog Video --- p.9 / Chapter 2.1.2 --- Digital Video --- p.10 / Chapter 2.1.3 --- Color Theory --- p.10 / Chapter 2.2 --- Video Coding --- p.12 / Chapter 2.2.1 --- Predictive Coding --- p.12 / Chapter 2.2.2 --- Vector Quantization --- p.12 / Chapter 2.2.3 --- Subband Coding --- p.13 / Chapter 2.2.4 --- Transform Coding --- p.14 / Chapter 2.2.5 --- Hybrid Coding --- p.14 / Chapter 2.3 --- Transform Coding --- p.15 / Chapter 2.3.1 --- Discrete Cosine Transform --- p.16 / Chapter 2.3.1.1 --- 1-D Fast Algorithms --- p.16 / Chapter 2.3.1.2 --- 2-D Fast Algorithms --- p.17 / Chapter 2.3.1.3 --- Multidimensional DCT Algorithms --- p.17 / Chapter 2.3.2 --- Quantization --- p.18 / Chapter 2.3.3 --- Entropy Coding --- p.18 / Chapter 2.3.3.1 --- Huffman Coding --- p.19 / Chapter 2.3.3.2 --- Arithmetic Coding --- p.19 / Chapter Chapter 3 : --- Existing Compression Scheme / Chapter 3.1 --- Introduction --- p.20 / Chapter 3.2 --- Motion JPEG --- p.20 / Chapter 3.3 --- MPEG --- p.20 / Chapter 3.4 --- H.261 --- p.22 / Chapter 3.5 --- Other Techniques --- p.23 / Chapter 3.5.1 --- Fractals --- p.23 / Chapter 3.5.2 --- Wavelets --- p.23 / Chapter 3.6 --- Proposed Solution --- p.24 / Chapter 3.7 --- Summary --- p.25 / Chapter Chapter 4 : --- Fast 3D-DCT Algorithms / Chapter 4.1 --- Introduction --- p.27 / Chapter 4.1.1 --- Motivation --- p.27 / Chapter 4.1.2 --- Potentials of 3D DCT --- p.28 / Chapter 4.2 --- Three Dimensional Discrete Cosine Transform (3D-DCT) --- p.29 / Chapter 4.2.1 --- Inverse 3D-DCT --- p.29 / Chapter 4.2.2 --- Forward 3D-DCT --- p.30 / Chapter 4.3 --- 3-D FCT (3-D Fast Cosine Transform Algorithm --- p.30 / Chapter 4.3.1 --- Partitioning and Rearrangement of Data Cube --- p.30 / Chapter 4.3.1.1 --- Spatio-temporal Data Cube --- p.30 / Chapter 4.3.1.2 --- Spatio-temporal Transform Domain Cube --- p.31 / Chapter 4.3.1.3 --- Coefficient Matrices --- p.31 / Chapter 4.3.2 --- 3-D Inverse Fast Cosine Transform (3-D IFCT) --- p.32 / Chapter 4.3.2.1 --- Matrix Representations --- p.32 / Chapter 4.3.2.2 --- Simplification of the calculation steps --- p.33 / Chapter 4.3.3 --- 3-D Forward Fast Cosine Transform (3-D FCT) --- p.35 / Chapter 4.3.3.1 --- Decomposition --- p.35 / Chapter 4.3.3.2 --- Reconstruction --- p.36 / Chapter 4.4 --- The Fast Algorithm --- p.36 / Chapter 4.5 --- Example using 4x4x4 IFCT --- p.38 / Chapter 4.6 --- Complexity Comparison --- p.43 / Chapter 4.6.1 --- Complexity of Multiplications --- p.43 / Chapter 4.6.2 --- Complexity of Additions --- p.43 / Chapter 4.7 --- Implementation Issues --- p.44 / Chapter 4.8 --- Summary --- p.46 / Chapter Chapter 5 : --- Quantization / Chapter 5.1 --- Introduction --- p.49 / Chapter 5.2 --- Dynamic Ranges of 3D-DCT Coefficients --- p.49 / Chapter 5.3 --- Distribution of 3D-DCT AC Coefficients --- p.54 / Chapter 5.4 --- Quantization Volume --- p.55 / Chapter 5.4.1 --- Shifted Complement Hyperboloid --- p.55 / Chapter 5.4.2 --- Quantization Volume --- p.58 / Chapter 5.5 --- Scan Order for Quantized 3D-DCT Coefficients --- p.59 / Chapter 5.6 --- Finding Parameter Values --- p.60 / Chapter 5.7 --- Experimental Results from Using the Proposed Quantization Values --- p.65 / Chapter 5.8 --- Summary --- p.66 / Chapter Chapter 6 : --- Entropy Coding / Chapter 6.1 --- Introduction --- p.69 / Chapter 6.1.1 --- Huffman Coding --- p.69 / Chapter 6.1.2 --- Arithmetic Coding --- p.71 / Chapter 6.2 --- Zero Run-Length Encoding --- p.73 / Chapter 6.2.1 --- Variable Length Coding in JPEG --- p.74 / Chapter 6.2.1.1 --- Coding of the DC Coefficients --- p.74 / Chapter 6.2.1.2 --- Coding of the DC Coefficients --- p.75 / Chapter 6.2.2 --- Run-Level Encoding of the Quantized 3D-DCT Coefficients --- p.76 / Chapter 6.3 --- Frequency Analysis of the Run-Length Patterns --- p.76 / Chapter 6.3.1 --- The Frequency Distributions of the DC Coefficients --- p.77 / Chapter 6.3.2 --- The Frequency Distributions of the DC Coefficients --- p.77 / Chapter 6.4 --- Huffman Table Design --- p.84 / Chapter 6.4.1 --- DC Huffman Table --- p.84 / Chapter 6.4.2 --- AC Huffman Table --- p.85 / Chapter 6.5 --- Implementation Issue --- p.85 / Chapter 6.5.1 --- Get Category --- p.85 / Chapter 6.5.2 --- Huffman Encode --- p.86 / Chapter 6.5.3 --- Huffman Decode --- p.86 / Chapter 6.5.4 --- PutBits --- p.88 / Chapter 6.5.5 --- GetBits --- p.90 / Chapter Chapter 7 : --- "Contributions, Concluding Remarks and Future Work" / Chapter 7.1 --- Contributions --- p.92 / Chapter 7.2 --- Concluding Remarks --- p.93 / Chapter 7.2.1 --- The Advantages of 3D DCT codec --- p.94 / Chapter 7.2.2 --- Experimental Results --- p.95 / Chapter 7.1 --- Future Work --- p.95 / Chapter 7.2.1 --- Integer Discrete Cosine Transform Algorithms --- p.95 / Chapter 7.2.2 --- Adaptive Quantization Volume --- p.96 / Chapter 7.2.3 --- Adaptive Huffman Tables --- p.96 / Appendices: / Appendix A : The detailed steps in the simplification of Equation 4.29 --- p.98 / Appendix B : The program Listing of the Fast DCT Algorithms --- p.101 / Appendix C : Tables to Illustrate the Reording of the Quantized Coefficients --- p.110 / Appendix D : Sample Values of the Quantization Volume --- p.111 / Appendix E : A 16-bit VLC table for AC Run-Level Pairs --- p.113 / References --- p.115

Spectral methods have been used for many applications in digital logic design, digital signal processing and telecommunications. In digital logic design they are implemented for testing of logical networks, multiplexer-based logic synthesis, signal processing, image processing and pattern analysis. New developments of more efficient algorithms for spectral transformations (Rademacher-Walsh, Generalized Reed-Muller, Adding, Arithmetic, multiple-valued Walsh and multiple-valued Generalized Reed- Muller) their implementation and applications will be described.

The use of efficient basis functions to model and represent flows with internal sharp velocity gradients, such as shocks or eddy microfronts, are investigated. This is achieved by analysing artificial data, observed atmospheric turbulence data and by the use of a Burgers' equation based spectral model. The concept of an efficient decomposition of a function into a basis set is presented and alternative analysis methods are investigated. The development of a spectral model using a generalized basis for the Burgers' equation is presented and simulations are performed using a modified Walsh basis and compared with the Fourier (trigonometric) basis and finite difference techniques. The wavelet transform is shown to be superior to the Fourier transform or the windowed Fourier transform in terms of defining the predominant scales in time series of turbulent shear flows and in 'zooming in' on local coherent structures associated with sharp edges. Disadvantages are found to be its inability to provide clear information on the scale of periodicity of events. Artificial time series of varying amounts of noise added to structures of different scales are analyzed using different wavelets to show that the technique is robust and capable of detecting sharp edged coherent structures such as those found in shear driven turbulence. The Haar function is used as a wavelet to detect ubiquitous zones of concentrated shear in turbulent flows sometimes referred to as microfronts. The location and organization of these shear zones suggest that they may be edges of larger scale eddies. A wavelet variance of the wavelet phase plane is defined to detect and highlight events and obtain measures of predominant scales of coherent structures. Wavelet skewness is computed as an indicator of the systematic sign preference of the gradient of the transition zone. Inverse wavelet transforms computed at the dilation corresponding to the peak wavelet variance are computed and shown to contain a significant fraction of the total energy contained in the record. The analysis of data and the numerical simulation results are combined to propose that the sharp gradients normally found in shear induced turbulence significantly affect the nature of the turbulence and hence the choice of the basis set used for the simulation of turbulence. / Graduation date: 1991

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 123-125).

published_or_final_version / abstract / Electrical and Electronic Engineering / Master / Master of Philosophy