Let K be a real quadratic number field. Let p be a prime which is inert in K. We denote the completion of K at the place p by Kp. Let ƒ > 1 be a positive integer coprime to p. In this thesis we give a p-adic construction of special elements u(r, ??) ∈ Kxp for special pairs (r, ??) ∈ (ℤ/ƒℤ)x x Hp where Hp = ℙ¹(ℂp) ℙ¹(ℚp) is the so called p-adic upper half plane. These pairs (r, ??) can be thought of as an analogue of classical Heegner points on modular curves. The special elements u(r, ??) are conjectured to be global p-units in the narrow ray class field of K of conductor ƒ. The construction of these elements that we propose is a generalization of a previous construction obtained in [DD06]. The method consists in doing p-adic integration of certain ℤ-valued measures on ??=ℤpxℤp pℤpxpℤp . The construction of those measures relies on the existence of a family of Eisenstein series (twisted by additive characters) of varying weight. Their moments are used to define those measures. We also construct p-adic zeta functions for which we prove an analogue of the so called Kronecker's limit formula. More precisely we relate the first derivative at s = 0 of a certain p-adic zeta function with -logₚ NKp/Qp u(r, ??). Finally we also provide some evidence both theoretical and numerical for the algebraicity of u(r, ??). Namely we relate a certain norm of our p-adic invariant with Gauss sums of the cyclotomic field Q (zetaf, zetap). The norm here is taken via a conjectural Shimura reciprocity law. We also have included some numerical examples at the end of section 18.

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Electronic solid state circuitry controlled by an SDK-85 microcomputer forms the basis of a digitally controlled Fourier synthesizer module. The hardware involved in this module is the major thrust of this particular research work. Ultimately, once properly machine programmed, the microcomputer will control the base frequency, the number of harmonics present in the output waveshape, and the relative amplitudes and phases of these harmonics.The first phase of the hardware interface entails the conversion of square waves (from 8253 timers via the microcomputer) to triangle waveforms (via LM741 integrators). The triangle waves are then sine shaped by special clipping circuitry. It is the resultant sine waves that are then digitally controlled by machine software. The first stage of the above process eliminates the problem of frequency dependence and gain dropoff of the LM741 operational amplifiers. The relative amplitude of the triangle wave output of the LM741 is compensated for such that the sine shaping circuitry receives a constant amplitude waveform. The second phase of the hardware interface uses digital control of the final sine waveform by means of a Voltage Controlled Amplifier (VCA) and a Digital to Analog Converter (DAC). The VCA acts as the actual interface between the computer and the module.Finally, the output of the VCA'a are mixed and sent through an audio amplifier or to an oscilloscope for user inspection.

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Bibliography: leaves 121-123 / viii, 123 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1985

Thesis (Ph. D.)--University of Newcastle, 2000. / Department of Civil, Surveying and Environmental Engineering. Includes bibliographical references (leaves R-1-R-9). Also available online.

Thesis (M. Sc.)--School of Mathematics and Statistics, Faculty of Science, University of Sydney, 2004. / Bibliography: leaves 129-135.

Diss. Stockholm : Handelshögskolan, 2006.