Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Bayesian methods for estimation, optimization and experimental design

Mathematical models often contain parameters that need to be calibrated from measured data. The emergence of efficient Markov Chain Monte Carlo (MCMC) methods has made the Bayesian approach a standard tool in quantifying the uncertainty in the parameters. With MCMC, the parameter estimation problem can be solved in a fully statistical manner, and the whole distribution of the parameters can be explored, instead of obtaining point estimates and using, e.g., Gaussian approximations. In this thesis, MCMC methods are applied to parameter estimation problems in chemical reaction engineering, population ecology, and climate modeling. Motivated by the climate model experiments, the methods are developed further to make them more suitable for problems where the model is computationally intensive. After the parameters are estimated, one can start to use the model for various tasks. Two such tasks are studied in this thesis: optimal design of experiments, where the task is to design the next measurements so that the parameter uncertainty is minimized, and model-based optimization, where a model-based quantity, such as the product yield in a chemical reaction model, is optimized. In this thesis, novel ways to perform these tasks are developed, based on the output of MCMC parameter estimation. A separate topic is dynamical state estimation, where the task is to estimate the dynamically changing model state, instead of static parameters. For example, in numerical weather prediction, an estimate of the state of the atmosphere must constantly be updated based on the recently obtained measurements. In this thesis, a novel hybrid state estimation method is developed, which combines elements from deterministic and random sampling methods.

Differentiation of Metabolically Distinct Areas within Head and Neck Region using Dynamic 18F-FDG Positron Emission Tomography Imaging

Positron Emission Tomography (PET) using ¹⁸F-FDG is playing a vital role in the diagnosis and treatment planning of cancer. However, the most widely used radiotracer, ¹⁸F-FDG, is not specific for tumours and can also accumulate in inflammatory lesions as well as normal physiologically active tissues making diagnosis and treatment planning complicated for the physicians. Malignant, inflammatory and normal tissues are known to have different pathways for glucose metabolism which could possibly be evident from different characteristics of the time activity curves from a dynamic PET acquisition protocol. Therefore, we aimed to develop new image analysis methods, for PET scans of the head and neck region, which could differentiate between inflammation, tumour and normal tissues using this functional information within these radiotracer uptake areas. We developed different dynamic features from the time activity curves of voxels in these areas and compared them with the widely used static parameter, SUV, using Gaussian Mixture Model algorithm as well as K-means algorithm in order to assess their effectiveness in discriminating metabolically different areas. Moreover, we also correlated dynamic features with other clinical metrics obtained independently of PET imaging. The results show that some of the developed features can prove to be useful in differentiating tumour tissues from inflammatory regions and some dynamic features also provide positive correlations with clinical metrics. If these proposed methods are further explored then they can prove to be useful in reducing false positive tumour detections and developing real world applications for tumour diagnosis and contouring.

Adaptive Markov chain Monte Carlo and Bayesian filtering for state space models

This thesis is concerned with the state and parameter estimation in state space models. The estimation of states and parameters is an important task when mathematical modeling is applied to many different application areas such as the global positioning systems, target tracking, navigation, brain imaging, spread of infectious diseases, biological processes, telecommunications, audio signal processing, stochastic optimal control, machine learning, and physical systems. In Bayesian settings, the estimation of states or parameters amounts to computation of the posterior probability density function. Except for a very restricted number of models, it is impossible to compute this density function in a closed form. Hence, we need approximation methods. A state estimation problem involves estimating the states (latent variables) that are not directly observed in the output of the system. In this thesis, we use the Kalman filter, extended Kalman filter, Gauss–Hermite filters, and particle filters to estimate the states based on available measurements. Among these filters, particle filters are numerical methods for approximating the filtering distributions of non-linear non-Gaussian state space models via Monte Carlo. The performance of a particle filter heavily depends on the chosen importance distribution. For instance, inappropriate choice of the importance distribution can lead to the failure of convergence of the particle filter algorithm. In this thesis, we analyze the theoretical Lᵖ particle filter convergence with general importance distributions, where p ≥2 is an integer. A parameter estimation problem is considered with inferring the model parameters from measurements. For high-dimensional complex models, estimation of parameters can be done by Markov chain Monte Carlo (MCMC) methods. In its operation, the MCMC method requires the unnormalized posterior distribution of the parameters and a proposal distribution. In this thesis, we show how the posterior density function of the parameters of a state space model can be computed by filtering based methods, where the states are integrated out. This type of computation is then applied to estimate parameters of stochastic differential equations. Furthermore, we compute the partial derivatives of the log-posterior density function and use the hybrid Monte Carlo and scaled conjugate gradient methods to infer the parameters of stochastic differential equations. The computational efficiency of MCMC methods is highly depend on the chosen proposal distribution. A commonly used proposal distribution is Gaussian. In this kind of proposal, the covariance matrix must be well tuned. To tune it, adaptive MCMC methods can be used. In this thesis, we propose a new way of updating the covariance matrix using the variational Bayesian adaptive Kalman filter algorithm.

Statistical segmentation methods and color variance analysis of retinal images

In this research, the effectiveness of Naive Bayes and Gaussian Mixture Models classifiers on segmenting exudates in retinal images is studied and the results are evaluated with metrics commonly used in medical imaging. Also, a color variation analysis of retinal images is carried out to find how effectively can retinal images be segmented using only the color information of the pixels.

Contribution of matrix vesicles and alkaline phosphatase to ectopic bone formation

Endochondral calcification involves the participation of matrix vesicles (MVs), but it remains unclear whether calcification ectopically induced by implants of demineralized bone matrix also proceeds via MVs. Ectopic bone formation was induced by implanting rat demineralized diaphyseal bone matrix into the dorsal subcutaneous tissue of Wistar rats and was examined histologically and biochemically. Budding of MVs from chondrocytes was observed to serve as nucleation sites for mineralization during induced ectopic osteogenesis, presenting a diameter with Gaussian distribution with a median of 306 ± 103 nm. While the role of tissue-nonspecific alkaline phosphatase (TNAP) during mineralization involves hydrolysis of inorganic pyrophosphate (PPi), it is unclear how the microenvironment of MV may affect the ability of TNAP to hydrolyze the variety of substrates present at sites of mineralization. We show that the implants contain high levels of TNAP capable of hydrolyzing p-nitrophenylphosphate (pNPP), ATP and PPi. The catalytic properties of glycosyl phosphatidylinositol-anchored, polidocanol-solubilized and phosphatidylinositol-specific phospholipase C-released TNAP were compared using pNPP, ATP and PPi as substrates. While the enzymatic efficiency (k cat/Km) remained comparable between polidocanol-solubilized and membrane-bound TNAP for all three substrates, the k cat/Km for the phosphatidylinositol-specific phospholipase C-solubilized enzyme increased approximately 108-, 56-, and 556-fold for pNPP, ATP and PPi, respectively, compared to the membrane-bound enzyme. Our data are consistent with the involvement of MVs during ectopic calcification and also suggest that the location of TNAP on the membrane of MVs may play a role in determining substrate selectivity in this micro-compartment.

Spatial variability of soil fertility and its relationship with seed physiological potential in a soybean production area

The correlation of soil fertility x seed physiological potential is very important in the area of seed technology but results published with that theme are contradictory. For this reason, this study to evaluate the correlations between soil chemical properties and physiological potential of soybean seeds. On georeferenced points, both soil and seeds were sampled for analysis of soil fertility and seed physiological potential. Data were assessed by the following analyses: descriptive statistics; Pearson's linear correlation; and geostatistics. The adjusted parameters of the semivariograms were used to produce maps of spatial distribution for each variable. Organic matter content, Mn and Cu showed significant effects on seed germination. Most variables studied presented moderate to high spatial dependence. Germination and accelerated aging of seeds, and P, Ca, Mg, Mn, Cu and Zn showed a better fit to spherical semivariogram: organic matter, pH and K had a better fit to Gaussian model; and V% and Fe showed a better fit to the linear model. The values for range of spatial dependence varied from 89.9 m for P until 651.4 m for Fe. These values should be considered when new samples are collected for assessing soil fertility in this production area.

Fluorescence spectra and ab initio theoretical studies of HCOOH /

A fluorescence excitation spectrum of formic acid monomer (HCOOH) , has been recorded in the 278-246 nm region and has been attributed to an n >7r* electron promotion in the anti conformer. The S^< S^ electronic origins of the HCOOH/HCOOD/DCOOH/DCOOD isotopomers were assigned to weak bands observed at 37431.5/37461.5/37445.5/37479.3 cm'''. From a band contour analysis of the 0°^ band of HCOOH, the rotational constants for the excited state were estimated: A'=1.8619, B'=0.4073, and C'=0.3730 cm'\ Four vibrational modes, 1/3(0=0), j/^(0-C=0) , J/g(C-H^^^) and i/,(0-H^yJ were observed in the spectrum. The activity of the antisymmetric aldehyde wagging and hydroxyl torsional modes in forming progressions is central to the analysis, leading to the conclusion that the two hydrogens are distorted from the molecular plane, 0-C=0, in the upper S. state. Ab initio calculations were performed at the 6-3 IG* SCF level using the Gaussian 86 system of programs to aid in the vibrational assignments. The computations show that the potential surface which describes the low frequency OH torsion (twisting motion) and the CH wagging (molecular inversion) motions is complex in the S^ excited electronic state. The OH and CH bonds were calculated to be twisted with respect to the 0-C=0 molecular frame by 63.66 and 4 5.76 degrees, respectively. The calculations predicted the existence of the second (syn) rotamer which is 338 cm'^ above the equilibrium configuration with OH and CH angles displaced from the plane by 47.91 and 41.32 degrees.

Low frequency Raman scattering in amorphous materials: fused quartz, "pyrex" boro-silicate glass and soda-lime silicate glass

Raman scattering in the region 20 to 100 cm -1 for fused quartz, "pyrex" boro-silicate glass, and soft soda-lime silicate glass was investigated. The Raman spectra for the fused quartz and the pyrex glass were obtained at room temperature using the 488 nm exciting line of a Coherent Radiation argon-ion laser at powers up to 550 mW. For the soft soda-lime glass the 514.5 nm exciting line at powers up to 660 mW was used because of a weak fluorescence which masked the Stokes Raman spectrum. In addition it is demonstrated that the low-frequency Raman coupling constant can be described by a model proposed by Martin and Brenig (MB). By fitting the predicted spectra based on the model with a Gaussian, Poisson, and Lorentzian forms of the correlation function, the structural correlation radius (SCR) was determined for each glass. It was found that to achieve the best possible fit· from each of the three correlation functions a value of the SCR between 0.80 and 0.90 nm was required for both quartz and pyrex glass but for the soft soda-lime silicate glass the required value of the SCR. was between 0.50 and 0.60 nm .. Our results support the claim of Malinovsky and Sokolov (1986) that the MB model based on a Poisson correlation function provides a universal fit to the experimental VH (vertical and horizontal polarizations) spectrum for any glass regardless of its chemical composition. The only deficiency of the MB model is its failure to fit the experimental depolarization spectra.

Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes.

In this paper, we develop finite-sample inference procedures for stationary and nonstationary autoregressive (AR) models. The method is based on special properties of Markov processes and a split-sample technique. The results on Markovian processes (intercalary independence and truncation) only require the existence of conditional densities. They are proved for possibly nonstationary and/or non-Gaussian multivariate Markov processes. In the context of a linear regression model with AR(1) errors, we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations. This transformation leads to a new model which has the form of a two-sided autoregression to which standard classical linear regression inference techniques can be applied. We show how to derive tests and confidence sets for the mean and/or autoregressive parameters of the model. We also develop a test on the order of an autoregression. We show that a combination of subsample-based inferences can improve the performance of the procedure. An application to U.S. domestic investment data illustrates the method.

Finite-Sample Inference Methods for Simultaneous Equations and Models with Unobserved and Generated Regressors

We propose finite sample tests and confidence sets for models with unobserved and generated regressors as well as various models estimated by instrumental variables methods. The validity of the procedures is unaffected by the presence of identification problems or \"weak instruments\", so no detection of such problems is required. We study two distinct approaches for various models considered by Pagan (1984). The first one is an instrument substitution method which generalizes an approach proposed by Anderson and Rubin (1949) and Fuller (1987) for different (although related) problems, while the second one is based on splitting the sample. The instrument substitution method uses the instruments directly, instead of generated regressors, in order to test hypotheses about the \"structural parameters\" of interest and build confidence sets. The second approach relies on \"generated regressors\", which allows a gain in degrees of freedom, and a sample split technique. For inference about general possibly nonlinear transformations of model parameters, projection techniques are proposed. A distributional theory is obtained under the assumptions of Gaussian errors and strictly exogenous regressors. We show that the various tests and confidence sets proposed are (locally) \"asymptotically valid\" under much weaker assumptions. The properties of the tests proposed are examined in simulation experiments. In general, they outperform the usual asymptotic inference methods in terms of both reliability and power. Finally, the techniques suggested are applied to a model of Tobin’s q and to a model of academic performance.

Testing Mean-Variance Efficiency in CAPM with Possibly Non-Gaussian Errors : An Exact Simulation-Based Approach

In this paper we propose exact likelihood-based mean-variance efficiency tests of the market portfolio in the context of Capital Asset Pricing Model (CAPM), allowing for a wide class of error distributions which include normality as a special case. These tests are developed in the frame-work of multivariate linear regressions (MLR). It is well known however that despite their simple statistical structure, standard asymptotically justified MLR-based tests are unreliable. In financial econometrics, exact tests have been proposed for a few specific hypotheses [Jobson and Korkie (Journal of Financial Economics, 1982), MacKinlay (Journal of Financial Economics, 1987), Gib-bons, Ross and Shanken (Econometrica, 1989), Zhou (Journal of Finance 1993)], most of which depend on normality. For the gaussian model, our tests correspond to Gibbons, Ross and Shanken’s mean-variance efficiency tests. In non-gaussian contexts, we reconsider mean-variance efficiency tests allowing for multivariate Student-t and gaussian mixture errors. Our framework allows to cast more evidence on whether the normality assumption is too restrictive when testing the CAPM. We also propose exact multivariate diagnostic checks (including tests for multivariate GARCH and mul-tivariate generalization of the well known variance ratio tests) and goodness of fit tests as well as a set estimate for the intervening nuisance parameters. Our results [over five-year subperiods] show the following: (i) multivariate normality is rejected in most subperiods, (ii) residual checks reveal no significant departures from the multivariate i.i.d. assumption, and (iii) mean-variance efficiency tests of the market portfolio is not rejected as frequently once it is allowed for the possibility of non-normal errors.

Exact Skewness-Kurtosis Tests for Multivariate Normality and Goodness-of-fit in Multivariate Regressions with Application to Asset Pricing Models

We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross-equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared to simulation-based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal, Student t; normal mixtures and stable error models. In the Gaussian case, finite-sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi-stage Monte Carlo test methods. For non-Gaussian distribution families involving nuisance parameters, confidence sets are derived for the the nuisance parameters and the error distribution. The procedures considered are evaluated in a small simulation experi-ment. Finally, the tests are applied to an asset pricing model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995.

Short run and long run causality in time series: Inference

We propose methods for testing hypotheses of non-causality at various horizons, as defined in Dufour and Renault (1998, Econometrica). We study in detail the case of VAR models and we propose linear methods based on running vector autoregressions at different horizons. While the hypotheses considered are nonlinear, the proposed methods only require linear regression techniques as well as standard Gaussian asymptotic distributional theory. Bootstrap procedures are also considered. For the case of integrated processes, we propose extended regression methods that avoid nonstandard asymptotics. The methods are applied to a VAR model of the U.S. economy.

Probe-level linear model fitting and mixture modeling results in high accuracy detection of differential gene expression

Affiliation: Institut de recherche en immunologie et en cancérologie, Université de Montréal