The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts. Part A: Spectral properties of the Schrödinger equation This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained. Part B: Graph monomials and sums of squares In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares. Part C: Eigenvalue asymptotics of banded Toeplitz matrices This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above. Part D: Stretched Schur polynomials This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients. / <p>At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript</p>

We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.

By means of the cluster expansion method we show that a recent result of Poghosyan and Ueltschi (2009) combined with a result of Nehring (2012) yields a construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in R^d of Ginibre's Fermi-Dirac gas of such loops. The latter will be identified as a Gibbs perturbation of the ideal Fermi gas. On generalizing these considerations we will obtain the existence of a large class of Gibbs perturbations of the so-called KMM-processes as they were introduced by Nehring (2012). Moreover, it is shown that certain "limiting Gibbs processes" are Gibbs in the sense of Dobrushin, Lanford and Ruelle if the underlying potential is positive. And finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.

Differential-algebraic equations (DAEs) result in many areas of science and engineer- ing. In this thesis, numerical methods for solving DAEs are compared for two prob- lems, energy-economic models and traffic flow models. An energy-economic model is presented based on the Hubbert model of oil production and is extended to include economic factors for the first time. Using numerical methods to simulate the DAE model, the resulting graphs break the symmetry of the traditional Hubbert curve. For the traffic flow models, a numerical method is developed to solve the steady-state flow pattern including the linearly unstable regime, i.e. solutions which cannot be found with an initial value solver. / UOIT

Structural equation models are systems of simultaneous linear equations that are generalizations of linear regression, and have many applications in the social, behavioural and biological sciences. A serious barrier to applications is that it is easy to specify models for which the parameter vector is not identifiable from the distribution of the observable data, and it is often difficult to tell whether a model is identified or not. In this thesis, we study the most straightforward method to check for identification – solving a system of simultaneous equations. However, the calculations can easily get very complex. Grobner basis is introduced to simplify the process. The main idea of checking identification is to solve a set of finitely many simultaneous equations, called identifying equations, which can be transformed into polynomials. If a unique solution is found, the model is identified. Grobner basis reduces the polynomials into simpler forms making them easier to solve. Also, it allows us to investigate the model-induced constraints on the covariances, even when the model is not identified. With the explicit solution to the identifying equations, including the constraints on the covariances, we can (1) locate points in the parameter space where the model is not identified, (2) find the maximum likelihood estimators, (3) study the effects of mis-specified models, (4) obtain a set of method of moments estimators, and (5) build customized parametric and distribution free tests, including inference for non-identified models.

A parallel implicit adaptive-mesh-refinement scheme is proposed for the solution of the Navier-Stokes equations as applied to two-dimensional steady-state hypersonic laminar flows in conjunction with an equilibrium high-temperature equation of state. A finite-volume discretization is applied to the governing equations. Limited piecewise-linear solution reconstruction and Riemann solvers (Roe and HLLE, both modified for a general equation of state) are used to evaluate the inviscid fluxes. The gradients in the viscous fluxes are calculated using diamond-path reconstruction. The system of non-linear algebraic equations resulting from the finite-volume discretization are solved using an inexact Newton method with GMRES to solve the update step of the Newton method. GMRES is preconditioned with Schwarz preconditioning with local block-fill incomplete lower-upper factorization. Multigrid and pseudo-transient continuation are used for startup. Numerical results, including flows at Mach numbers of 7.0, are discussed and demonstrate the validity and efficiency of the scheme.

A parallel implicit adaptive-mesh-refinement scheme is proposed for the solution of the Navier-Stokes equations as applied to two-dimensional steady-state hypersonic laminar flows in conjunction with an equilibrium high-temperature equation of state. A finite-volume discretization is applied to the governing equations. Limited piecewise-linear solution reconstruction and Riemann solvers (Roe and HLLE, both modified for a general equation of state) are used to evaluate the inviscid fluxes. The gradients in the viscous fluxes are calculated using diamond-path reconstruction. The system of non-linear algebraic equations resulting from the finite-volume discretization are solved using an inexact Newton method with GMRES to solve the update step of the Newton method. GMRES is preconditioned with Schwarz preconditioning with local block-fill incomplete lower-upper factorization. Multigrid and pseudo-transient continuation are used for startup. Numerical results, including flows at Mach numbers of 7.0, are discussed and demonstrate the validity and efficiency of the scheme.

Structural equation models are systems of simultaneous linear equations that are generalizations of linear regression, and have many applications in the social, behavioural and biological sciences. A serious barrier to applications is that it is easy to specify models for which the parameter vector is not identifiable from the distribution of the observable data, and it is often difficult to tell whether a model is identified or not. In this thesis, we study the most straightforward method to check for identification – solving a system of simultaneous equations. However, the calculations can easily get very complex. Grobner basis is introduced to simplify the process. The main idea of checking identification is to solve a set of finitely many simultaneous equations, called identifying equations, which can be transformed into polynomials. If a unique solution is found, the model is identified. Grobner basis reduces the polynomials into simpler forms making them easier to solve. Also, it allows us to investigate the model-induced constraints on the covariances, even when the model is not identified. With the explicit solution to the identifying equations, including the constraints on the covariances, we can (1) locate points in the parameter space where the model is not identified, (2) find the maximum likelihood estimators, (3) study the effects of mis-specified models, (4) obtain a set of method of moments estimators, and (5) build customized parametric and distribution free tests, including inference for non-identified models.

Problem: Many individuals with chronic illnesses such as stroke and ongoing activity limitations report self-perceived health related quality of life (HRQL) that is similar to that of healthy individuals. This phenomenon is termed response shift (RS). RS describes how people change: internal standards in assessing HRQL (recalibration), values (reprioritization), or how they define HRQL (reconceptualization), after an event such as stroke. Changes in HRQL post stroke may be inaccurate if RS is not taken into account. Increased knowledge of RS may affect the way in which HRQL measures are used, both clinically and in research. The overall objective was to assess RS in construct specific HRQL models post stroke: physical function, mental health, and participation. Methods: Data were analysed from the longitudinal study “Understanding Quality of Life Post-Stroke: A Study of Individuals and their Caregivers”. Six-hundred and seventy- eight persons with stroke at 1, 3, 6, and 12 months post stroke participated. Generic and stroke specific HRQL measures were collected. Descriptive analysis was completed with SAS, and identification of RS utilized structural equation modeling with LISREL. Results: Mean age of participants was 67 years (SD 14.8), and 45% were female. RS was identified in mental health using a framework which was developed for identifying RS statistically with multiple time points. RS was also identified in physical function where it had not been expected, possibly due to the self perceived nature of the response options. The effect size of change in physical function was affected by the presence of RS. The timing of RS in mental health and physical function was primarily around the 12 month time period, and predominantly recalibration RS. RS was also identified in participation. Conclusions: The framework that was developed was useful in identifying RS and incorporated important issues such as multiple testing and validation of the model. The presence of RS affects measurement of HRQL constructs post stroke; recalibration RS can be measured clinically with specific methods to account for RS. RS should also be measured in research studies to ensure accurate measurement of change. Future research should evaluate additional models in stroke and other populations. / October 2008

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