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An overview of the statistical procedures used in the analysis of electrocardiogram traces is presented in this thesis. / The purpose of these procedures is twofold: (i) they may suggest underlying mechanisms that influence heart rate (ii) they may be used as a means of classifying one or more patients into disease categories, by using objective criteria rather than the subjective approaches prevalent in current practice. / In an attempt to apply the methods discussed in this thesis, a selected group of patients was analyzed using spectral analysis. / Lack of information and of control of the patients' activities while they were being monitored precluded the possibility of obtaining definitive results.

The product of independent random variables is studied through products of Mellin transforms. Although the frequency function tor the product had often been calculated, the use of Mellin transforms was first introduced by Epstein. In the present essay the theory of this application is developed and many examples are discussed. The theory, originally developed for positive random variables only, is made to include any real variable.

The Lame polynomials naturally arise when separating variables in Laplace's equation in elliptic-spherical coordinates. The products of these polynomials form a class of spherical harmonics, which are the joint eigenfunctions of a quantum completely integrable system of commuting, second-order differential operators P0 = DSN , P2,..., PN -1 acting on Cinfinity( SN ). These operators depend on parameters and thus constitute an ensemble. / In the main result presented in this thesis, we compute the limiting mean level spacings distribution for the zeroes of Lame polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters. As an application, we compute the limiting level spacings distribution of the zeroes of Van Vleck polynomials.

The scientific aim of computational neuroanatomy using magnetic resonance imaging (MRI) is to quantify inter- and intra-subject morphological variabilities. A unified statistical framework for analyzing temporally varying brain morphology is presented. Based on the mathematical framework of differential geometry, the deformation of the brain is modeled and key morphological descriptors such as length, area, volume dilatation and curvature change are computed. To increase the signal-to-noise ratio, Gaussian kernel smoothing is applied to 3D images. For 2D curved cortical surface, diffusion smoothing, which generalizes Gaussian kernel smoothing, has been developed. Afterwards, statistical inference is based on the excursion probability of random fields defined on manifolds. / This method has been applied in localizing the regions of brain tissue growth and loss in a group of 28 normal children and adolescents. It is shown that children's brains change dramatically in localized areas even after age 12.

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