The subject of this thesis is the fractional order Sobolev space, H[superscript]r[subscript]p, as considered by Nikol'skii; the goal is to demonstrate an imbedding theorem for H[superscript]r[subscript]p analogous to the classical imbedding theorem for W[superscript]m[subscript]p which was first shown by Sobolev. The properties established here for spaces H[superscript]r[subscript]p defined over all of Rn, including completeness and imbedding theorems, are demonstrated by a technique involving the approximation of functions in those spaces by entire functions of the exponential type. Properties of such entire functions, which are of interest in theire own right, are developed in a separate chapter. An extension theorem for differentiable developed in a separate chapter. An extension theorem for differentiable functions defined over an open subset of Rn is also proved. / Science, Faculty of / Mathematics, Department of / Graduate

The 584 Å photoelectron spectra from eight polyatomic molecules (CH₃I, CH₃Cl, CH₃Br, CH₃CHO, CH₃COCH₃, SF₆, CH₃CN and C₂H₅CN) are described and shown to give all the ionization potentials less than 21.21 eV in each case. The results are interpreted in terms of the electronic structures of these molecules as given by molecular orbital theory. They are compared with results from other sources, and agreements and differences explained. A brief account of other existing methods used for the determination of ionization potentials with their advantages and disadvantages is given. The major components of the instruments are briefly discussed, and use of a Single-Grid Photoelectron Spectrometer in the detection of fine structure in the photoelectron spectra is pointed out. / Science, Faculty of / Chemistry, Department of / Graduate

This thesis deals with the identification of parameters in distributed parameter systems. Two sensitivity methods, namely; Meissinger's method and the method of structural sensitivity are extended to obtain the sensitivity coefficients of discretized distributed parameter systems. The method of Bingulac and Kokotovic is extended to identify parameters in the one and the two dimensional parabolic differential equations. CSMP (continuous system modeling programme) is used throughout to simulate the systems. Results for both sensitivity schemes are obtained, and it is found that although structural sensitivity is advantageous for parameter identification in ordinary differential equations, this is not the case for partial differential equations. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

In this thesis we are concerned with obtaining an integral representation of a class of nonlinear additive and biadditive functionals on function spaces of measurable functions and on L[superscript] p-spaces, p > 0 . The associated measure space is essentially atom-free finite and o-finite. Also we are concerned to the extend the presence of atoms in a measure space complicates the representation theory for functionals of the type under consideration here. A class of nonlinear transformations on L[superscript] p-spaces, 1 ≤ p ≤ ∞, called Urysohn operators. [11] taking measurable functions to measurable functions is studied and we describe an integral representation for this class when the associated measure space is an arbitrary 0-finite measure space and this characterization extends our previous results where the measure space considered was atom-free. / Science, Faculty of / Mathematics, Department of / Graduate

In this research experiments were carried out to estimate hydrophobic and electrostatic interactions in soluble proteins. Five proteins, lysozyme, lactalbumin, ovalbumin, myoglobin and ribonuclease-A were chromatographed isocratically on a HIC column at several molalities (0.3-1.3m) of each of three different neutral salts, ferrous sulfate, ammonium sulfate and sodium sulfate. The calculated retention coefficients were then fitted to a recently developed chromatographic model in two ways. a) Multiple regression analysis was conducted to estimate C values according to the non-linear model (log k = A + B log m + C m) . b) Simple regression analysis was conducted to estimate C′ values according to the linear model (log k = A′ + C′m) at higher salt concentrations (above 0.3m). Results indicated that C′ values better estimate the hydrophobic interactions than C values, in experiments conducted only at higher salt concentrations. The comparison of C and C′ values with ANS, CPA, and Bigelow's average hydrophobicity indices showed no clear correlations. But, omission of ovalbumin improved the correlation coefficient of C′ with ANS. Both parameters indicated straight line relationships with molal surface tension increment of salts. Further, the same model was used to estimate the hydrophobic and electrostatic interactions in protein-protein interactions. Lysozyme and avidin were chromatographed on a lysozyme immobilized affinity column. Lysozyme-lysozyme interaction showed more affinity for hydrophobic interactions at low pH values. Avidin-lysozyme interaction showed both hydrophophobic and electrostatic interactions. Both interactions showed a greater change in the strength of hydrophobic interaction rather than the surface area of interaction, to changing pH. / Land and Food Systems, Faculty of / Graduate

In experimental or quasi-experimental studies in which a repeated measures design is used, it is common to obtain scores on several dependent variables on each measurement occasion. Multivariate mixed model (MMM) analysis of variance (Thomas, 1983) is a recently developed alternative to the MANOVA procedure (Bock, 1975; Timm, 1980) for testing multivariate hypotheses concerning effects of a repeated factor (called occasions in this study) and interaction between repeated and non-repeated factors (termed group-by-occasion interaction here). If a condition derived by Thomas (1983), multivariate multi-sample sphericity (MMS), regarding the equality and structure of orthonormalized population covariance matrices is satisfied (given multivariate normality and independence for distributions of subjects' scores), valid likelihood-ratio MMM tests of group-by-occasion interaction and occasions hypotheses are possible. To date, no information has been available concerning actual (empirical) levels of significance of such tests when the MMS condition is violated. This study was conducted to begin to provide such information. Departure from the MMS condition can be classified into three types— termed departures of types A, B, and C respectively: (A) the covariance matrix for population ℊ (ℊ = 1,...G), when orthonormalized, has an equal-diagonal-block form but the resulting matrix for population ℊ is unequal to the resulting matrix for population ℊ' (ℊ ≠ ℊ'); (B) the G populations' orthonormalized covariance matrices are equal, but the matrix common to the populations does not have equal-diagonal-block structure; or (C) one or more populations has an orthonormalized covariance matrix which does not have equal-diagonal-block structure and two or more populations have unequal orthonormalized matrices. In this study, Monte Carlo procedures were used to examine the effect of each type of violation in turn on the Type I error rates of multivariate mixed model tests of group-by-occasion interaction and occasions null hypotheses. For each form of violation, experiments modelling several levels of severity were simulated. In these experiments: (a) the number of measured variables was two; (b) the number of measurement occasions was three; (c) the number of populations sampled was two or three; (d) the ratio of average sample size to number of measured variables was six or 12; and (e) the sample size ratios were 1:1 and 1:2 when G was two, and 1:1:1 and 1:1:2 when G was three. In experiments modelling violations of types A and C, the effects of negative and positive sampling were studied. When type A violations were modelled and samples were equal in size, actual Type I error rates did not differ significantly from nominal levels for tests of either hypothesis except under the most severe level of violation. In type A experiments using unequal groups in which the largest sample was drawn from the population whose orthogonalized covariance matrix has the smallest determinant (negative sampling), actual Type I error rates were significantly higher than nominal rates for tests of both hypotheses and for all levels of violation. In contrast, empirical levels of significance were significantly lower than nominal rates in type A experiments in which the largest sample was drawn from the population whose orthonormalized covariance matrix had the largest determinant (positive sampling). Tests of both hypotheses tended to be liberal in experiments which modelled type B violations. No strong relationships were observed between actual Type I error rates and any of: severity of violation, number of groups, ratio of average sample size to number of variables, and relative sizes of samples. In equal-groups experiments modelling type C violations in which the orthonormalized pooled covariance matrix departed at the more severe level from equal-diagonal-block form, actual Type I error rates for tests of both hypotheses tended to be liberal. Findings were more complex under the less severe level of structural departure. Empirical significance levels did not vary with the degree of interpopulation heterogeneity of orthonormalized covariance matrices. In type C experiments modelling negative sampling, tests of both hypotheses tended to be liberal. Degree of structural departure did not appear to influence actual Type I error rates but degree of interpopulation heterogeneity did. Actual Type I error rates in type C experiments modelling positive sampling were apparently related to the number of groups. When two populations were sampled, both tests tended to be conservative, while for three groups, the results were more complex. In general, under all types of violation the ratio of average group size to number of variables did not greatly affect actual Type I error rates. The report concludes with suggestions for practitioners considering use of the MMM procedure based upon the findings and recommends four avenues for future research on Type I error robustness of MMM analysis of variance. The matrix pool and computer programs used in the simulations are included in appendices. / Education, Faculty of / Educational and Counselling Psychology, and Special Education (ECPS), Department of / Graduate

We study a hydrodynamic evolution of a non-spherical core-collapse supernova in multidimensions. We begin our study from the moment of shock revival and continue for the first week after explosion when expansion of the supernova ejecta becomes homologous. We observe growth and interaction of Richtmyer-Meshkov, Rayleigh-Taylor, and Kelvin- Helmholtz instabilities resulting in an extensive mixing of the heavy elements throughout the ejecta. We obtain a series of models at progressively higher resolution and provide preliminary discussion of numerical convergence. Unlike in the previous studies, our computations are performed in a single domain. Periodic mesh mapping is avoided. This is made possible by employing an adaptive mesh refinement strategy in which computational workload (defined as a product of the total number of computational cells and the length of the time step) is monitored and, if necessary, limited. Our results are in overall good agreement with the simulations reported by Kifonidis et al. We demonstrate, however, that the amount of mixing and kinematic properties of radioactive species (i.e. 56Ni) is extremely anisotropic. In particular, we find that the model displays a strong tendency to expand laterally away from the equatorial plane toward the poles. Although this behavior is usually attributed to numerical artifacts characteristic of computations with assumed symmetry (axis-effect), the observed behavior can be attributed to a large heat content of the equatorial regions of the explosion model. Future studies are needed to verify that this explosion model property does not have a systematic character. / A Thesis submitted to the Department of ScientifiC Computing in partial fulfillment of the requirements for the degree of Master of Science. / Fall Semester, 2009. / July 29, 2009. / Non-Spherical, Core-Collapse Supernova AMR / Includes bibliographical references. / Tomasz Plewa, Professor Directing Thesis; Peter Hoeflich, Committee Member; Gordon Erlebacher, Committee Member.

Heteroepitaxial thin films are essential components in many technological applications including optical, electronic and other functional devices. These films are also becoming important in the coating technologies for high-temperature materials applications. Typical heteroepitaxial systems involve one or more solid phases deposited on support structure called the substrate. Often the lattice and thermal mismatch in these systems results in significant elastic strains that, under the appropriate temperature conditions, drive mass transport by diffusion. Surface diffusion in these systems is usually a dominant mass transport mechanism that leads to morphological evolution of the surface. This evolution is called stress-driven morphological growth, and it has received much attention by materials modelers. In the current work, the problem of stress-driven morphological evolution in strained thin films is revisited; we develop a generalized formulation of this problem in the non-linear regime based upon a curvilinear coordinate formalism and finite element solution of the elastic sub-problem. This combination of methods facilitates the analysis of the onset of the instability and the early stage temporal evolution of the film surface. We apply our numerical scheme to surface wave, dot, pit, and ring morphologies and demonstrate the effects of model parameters on the incipient instabilities. / A Thesis submitted to the Department of ScientifiC Computing in partial fulfillment of the requirements for the degree of Master of Science. / Summer Semester, 2010. / June 28, 2010. / Thin Films, Instability, Epitaxy / Includes bibliographical references. / Anter El-Azab, Professor Directing Thesis; Gordon Erlebacher, Committee Member.

The primary hydrodynamic flow feature of early explosion phases of a core-collapse supernova is a spherical shock. This shock is born deep in the central regions of the collapsing stellar core, stalls shortly afterward, and in case of a successful explosion is revived and becomes the supernova shock. The revival process involves a standing accretion shock instability, SASI. This shock instability is considered the key processes aiding the core-collapse supernova (ccSN) explosion. The aim of our study is to identify feasible conditions and parameters for an experimental system that is able to capture the essential characteristics of SASI. We use analytic methods and high-resolution hydrodynamic simulations in multidimensions to investigate a possible experimental design on the National Ignition Facility. The experimental configuration involves a steady, spherical shock. We explore a viable region of parameters and obtain limits on the shocked flow geometry. We study the stability properties of the shock and its post-shock region. We discuss key differences between the experimental setup and astrophysical environment. The obtained flowfield closely resembles conveging nozzle flow. The post-shock region, in contrast to the supernova setting, is found to be stably stratified and insensitive to perturbations upstream of the shock. We conclude that it is not possible to capture the characteristics of the supernova SASI for the converging shocked flow configuration considered here. However, such configuration offers a very stable setting for precision studies of shocked, dense, high temperature plasmas requiring finely-controlled conditions. / A Thesis submitted to the Department of ScientifiC Computing in partial fulfillment of the requirements for the degree of Master of Science. / Fall Semester, 2011. / October 21, 2011. / hydrodynamics, laboratory astrophysics, shockwaves, supernovae / Includes bibliographical references. / Tomasz Plewa, Professor Directing Thesis; Gordon Erlebacher, Committee Member; Michael Navon, Committee Member.

Advances in computational power have lead to many developments in science and en- tertainment. Powerful simulations which required expensive supercomputers can now be carried out on a consumer personal computer and many children and young adults spend countless hours playing sophisticated computer games. The focus of this research is the development of tools which can help bring the entertaining and appealing traits of video games to scientific education. Video game developers use many tools and programming languages to build their games, for example the Blender 3D content creation suite. Blender includes a Game Engine that can be used to design and develop sophisticated interactive experiences. One important tool in computer graphics and animation is the particle system, which makes simulated effects such as fire, smoke and fluids possible. The particle system available in Blender is unfortunately not available in the Blender Game Engine because it is not fast enough to run in real-time. One of the main factors contributing to the rise in computational power and the increas- ing sophistication of video games is the Graphics Processing Unit (GPU). Many consumer personal computers are equipped with powerful GPUs which can be harnassed for general purpose computation. This thesis presents a particle system library is accelerated by the GPU using the OpenCL programming language. The library integrated into the Blender Game Engine providing an interactive platform for exploring fluid dynamics and creating video games with realistic water effects. The primary system implemented in this research is a fluid sim- ulator using the Smoothed Particle Hydrodynamics technique for simulating incompressible fluids such as water. The library created for this thesis can simulate water using SPH at 40fps with upwards x  of 100,000 particles on an NVIDIA GTX480 GPU. The fluid system has interactive features such as object collision, and the ability to add and remove particles dynamically. These features as well as phsyical properties of the simulation can be controlled intuitively from the user interface of Blender. / A Thesis submitted to the Department of ScientifiC Computing in partial fulfillment of the requirements for the degree of Master of Science. / Fall Semester, 2011. / August 24, 2011. / Game Design, GPU, OpenCL, SPH / Includes bibliographical references. / Gordon Erlebacher, Professor Directing Thesis; Tomasz Plewa, Committee Member; Anter El-Azab, Committee Member.