Measurement invariance across time and reporter is rarely reported in the literature for measures of youth disclosure, even though it is often necessary to establish at least strong invariance before proceeding to further analyses such as comparing means across time or reporter. Measurement invariance was examined across time (ages 11, 14, and 17) and across reporter (youth report of disclosure to mother, youth report of disclosure to father, mother report of youth disclosure, father report of youth disclosure) with a sample of 348 youth and their parents. Youth report of disclosure to mothers demonstrated strong invariance across ages 11-14 and 14-17, but strong equivalence was not found for mother report over time across any age. Youth report of disclosure to mothers and fathers demonstrated strong equivalence at ages 11, 14, 17, and across ages 11-14-17. Mother and father reports also demonstrated strong equivalence at ages 11, 14, and 17. The item "I talk with my parent about how I am doing with school work" had lower factor loadings and higher intercepts at age 11 than at other ages for fathers and mothers and compared to youth report. Implications for youth disclosure theory and construct development are discussed.

This dissertation investigates the interaction of population invariance, equating assumptions, and equating accuracy with group differences. In addition, matched samples equating methods are considered as a possible way to improve equating accuracy with large group differences. Data from one administration of four mixed-format Advanced Placement (AP) Exams were used to create pseudo old and new forms sharing common items. Population invariance analyses were conducted based on levels of examinee parental education using a single group equating design. Old and new form groups with common item effect sizes (ESs) ranging from 0 to 0.75 were created by sampling examinees based on their level of parental education. Equating was conducted for four common item nonequivalent group design equating methods: frequency estimation, chained equipercentile, IRT true score, and IRT observed score. Additionally, groups with ESs greater than zero were matched using three different matching techniques including exact matching on parental education level and propensity score matching with several other background variables. The accuracy of equating results was evaluated by comparing each equating relationship with an ES greater than zero to the equating relationship where the ES equaled zero. Differences between comparison and criterion equating relationships were quantified using the root expected mean squared difference (REMSD) statistic, classification consistency, and standard errors of equating (SEs). The accuracy of equating results and the adequacy of equating assumptions was compared for unmatched and matched samples. As ES increased, equating results tended to become less accurate and less consistent across equating methods. However, there was relatively little population dependence of equating results, despite large subgroup performance differences. Large differences between criterion and comparison equating relationships appeared to be caused instead by violations of equating assumptions. As group differences increased, the degree to which frequency estimation and chained equipercentile assumptions held decreased. In addition, all four AP Exams showed some evidence of multidimensionality. Because old and new form groups were selected to differ in terms of their respective levels of parental education, the matching methods that included parental education appeared to improve equating accuracy and the degree to which equating assumptions held, at least for very large ESs.

In this work, we demonstrate that the Leray-Schauder topological degree theory can be used for the development of a topological degree theory for maximal monotone perturbations of demicontinuous operators of type (S+) in separable reflexive Banach spaces. This is an extension of Berkovits’ degree development for operators as the perturbations above. Berkovits has developed a topological degree for demicontinuous mappings of type (S+), and has shown that the degree mapping is unique under the assumption that it satisfies certain general properties. He proved that if f is a bounded demicontinous mapping of type (S+), G is an open bounded subset of X, and 0 ∈/ f(∂G), then there exists ε0 > 0 such that for every ε ∈ (0, ε0) we have 0 ∈/ (I+ (1/ε)QQ∗ (f))(∂G). Here, Q is a compact linear injection from a Hilbert space H into X, such that Q(H) is dense in X, and Q∗ its adjoint. The map I+ 1 εQQ∗ (f) is a compact displacement of the identity, for which the Leray-Schauder degree is well defined. The Berkovits degree is obtained as the limit of this Leray-Schauder degree as ε tends to zero. We utilize a demicontinuous (S+)-approximation of the form Tt + f, where Tt is the Yosida approximant of T. Namely, we show that if G is an open bounded set in X and 0 ∈/ (T + f)(∂G), then there exist ε0 > 0, t0 > 0, such that for every ε ∈ (0, ε0), t ∈ (0, t0), we have 0 ∈/ (I + (1/ε)QQ∗ (Tt + f))(∂G). Our degree is the limit of the Leray-Schauder degree of the compact displacement of the identity I + (1/ε)QQ∗ (Tt + f) as ε, t → 0. Various extension of the degree has been considered. Finally some properties and applications in invariance of domain, eigenvalue and surjectivity results have also been discussed.

Factor mixture modeling (FMM) has been increasingly used to investigate unobserved population heterogeneity. This Monte Carlo simulation study examined the issue of measurement invariance testing with FMM when there are covariate effects. Specifically, this study investigated the impact of excluding and misspecifying covariate effects on the class enumeration and measurement invariance testing with FMM. Data were generated based on three FMM models where the covariate had impact on the latent class membership only (population model 1), both the latent class membership and the factor (population model 2), and the latent class membership, the factor, and one item (population model 3). The number of latent classes was fixed at two. These two latent classes were distinguished by factor mean difference for conditions where measurement invariance held in the population, and by both factor mean difference and intercept differences across classes when measurement invariance did not hold in the population. For each of the population models, different analysis models that excluded or misspecified covariate effects were fitted to data. Analyses consisted of two parts. First, for each analysis model, class enumeration rates were examined by comparing the fit of seven solutions: 1-class, 2-class configural, metric, and scalar, and 3-class configural, metric, and scalar. Second, assuming the correct solution was selected, the fit of analysis models with the same solution was compared to identify a best-fitting model. Results showed that completely excluding the covariate from the model (i.e., the unconditional model) would lead to under-extraction of latent classes, except when the class separation was large. Therefore, it is recommended to include covariate in FMM when the focus is to identify the number of latent classes and the level of invariance. Specifically, the covariate effect on the latent class membership can be included if there is no priori hypothesis regarding whether measurement invariance might hold or not. Then fit of this model can be compared with other models that included covariate effects in different ways but with the same number of latent classes and the same level of invariance to identify a best-fitting model.

Collection of data through survey-type measurements and analysis contributes rich, meaningful information to the chemical education research enterprise. This dissertation reports two strands of research that each contribute a “snapshot” of the state of chemical education on two different levels. The first uses survey research methods, collecting data from faculty members to learn about postsecondary chemistry education across the United States. The second uses survey instruments of student achievement emotions within the organic chemistry classroom, collecting longitudinal data to learn about the relationships of emotions with achievement over time. Both areas are of interest because chemical education research produces evidence-based instructional practices as well as survey instruments of student characteristics, many of which are ready to be used in classroom, yet there is a recognized disconnect between development of these products and enacted practices. The research in this dissertation improves upon previous methodology in both strands of research included while reporting data with implications for instructional, research, and policy matters. A national survey of postsecondary chemistry faculty uses a stratified sampling procedure to gather information about the state of education in chemistry classrooms. The use of the teacher-centered systemic reform model of educational change enables us to use the data collected in the survey to gather empirical support of the relationship between faculty members’ beliefs about how students learn chemistry more effectively, faculty members’ self-efficacy for instruction and chemistry content, and the instructional practices that they utilize in the course for which they felt they had the most influence. This information is paramount for the developers of evidence-based instructional practices as well as parties interested in determining the methods best suited to the dissemination of these tools. Professional development activities designed to inspire the use of evidence-based instructional tools or techniques must acknowledge the belief systems of faculty members and the need for change in these beliefs prior to the incorporation of new methods. These results present a call for reform efforts on fostering change from its core, i.e., the beliefs of those who ultimately adopt evidence-based instructional practices. Dissemination and design should incorporate training and materials that highlight the process by which faculty members interpret reformed practices within their belief system, and explore belief change in the complex context of education reform. Another example of the use of national survey data is the determination of the niche distribution of classroom response systems, also known as clickers. It is determined in this study that clickers are used more often in large courses taught at the lower level across the United States. This niche is deemed a more suitable situation for the use of clickers than others. This information is important for researchers developing tools intended for use within the classroom. Despite the possibility for use in all contexts, the national population of faculty members will adopt tools in the contexts which are deemed most suitable; the niche markets of educational tools can provide insight in to best development practices also well as direction for the optimization of the experience for the most frequent users of these tools. The other set of studies in this dissertation utilize the control-value theory of achievement emotions in the postsecondary organic chemistry context to explore nuanced relationships of affect with achievement. These studies utilize a longitudinal panel data collection mechanism, enhancing our ability to understand relationships. The control-value theory posits that there are a set of nine achievement emotions, dictated by control and value, which influence achievement. Two of these achievement emotions, anxiety and enjoyment, are determined in one study to fluctuate over the semester of organic chemistry and significantly influence achievement as measured by examination scores. These are supported by their theoretical interpretation as activating emotions, and when experienced, inspire students to take measures that ultimately either increase or reduce their success. A deactivating emotion, boredom, is measured in another study and found to also hold a reciprocal relationship with achievement when measured over time. In both studies, results show that the reciprocal causation model with an exam snowballing effect best fits data among the alternative models. There is a small and significant negative relationship between anxiety and performance contrasted with a positive relationship between enjoyment and performance throughout the semester. Negative relationships were observed between boredom and examination performance across the term. In addition, relationships were observed to be stronger at the beginning of the course term. Future research should consider achievement emotions in light of educational reforms to ensure that innovative curricula or pedagogies are functioning in the classroom as intended.

La description multifractale des signaux a été initiée au cours des vingt dernières années, notamment dans le domaine de la turbulence pleinement développée. Les propriétés de régularité ponctuelle des signaux étudiés sont caractérisées à l'aide d'un spectre de singularités. L'analyse multifractale de ces signaux consiste à mesurer ce spectre de singularités, à l'aide de formalismes multifractals. L'apparition des transformées en ondelette, à la même époque, a permis d'affiner la pratique de l'analyse multifractale, sans pour autant toujours reposer sur des bases mathématiques solides. S. Jaffard a récemment introduit les coefficients dominants, qui permettent de construire un formalisme multifractal mathématiquement bien fondé, et au cadre d'application large : il rend possible la mesure de l'ensemble du spectre de singularités, et reste valide lorsque les signaux analysés contiennent des singularités oscillantes. Ce nouvel outil est pour la première fois mis en oeuvre, numériquement caractérisé et appliqué à des signaux de vitesse turbulente. La question du bon usage pratique des formalismes multifractals, qui reposent sur la mesure d'exposants de fonctions de structure, est essentielle. Le travail présenté se propose d'y apporter des éléments de réponse. Une étude numérique, sur un large panel de processus multifractals synthétiques, a permis d'illustrer et de caractériser un aspect essentiel de l'analyse multifractale pratique, l'existence d'un ordre critique. Un estimateur de cet ordre critique est construit et numériquement caractérisé. Une relecture des résultats obtenus en turbulence est alors effectuée. Enfin, la question de l'universalité des exposants des fonctions de structure en turbulence pleinement développée est abordée. Une modélisation de l'exposant de la fonction de structure d'ordre trois est proposée et comparée à des résultats expérimentaux, mettant en évidence le caractère non universel de sa valeur.

Depuis près d'un demi-siècle, les ingénieurs savent décrire et prévoir la propagation d'une fissure dans un milieu élastique homogène modèle. Le cas des matériaux réels est beaucoup plus complexe. En effet, on ne sait pas relier leur durée de vie ou leur résistance à leur microstructure. Passage obligé avant de telles prédictions, il est nécessaire de comprendre comment le désordre structural du matériau influe sur le comportement d'une fissure. Dans cette optique, les surfaces de rupture représentent un champ d'investigation très prometteur. A travers une étude portant sur divers matériaux hétérogènes, nous caractérisons les propriétés statistiques de leur rugosité et déterminons dans quelle mesure elles sont indépendantes du matériau. Nous montrons notamment que les surfaces de rupture présentent des propriétés d'invariance d'échelle anisotropes, caractérisées par deux exposants universels. Etudiant ensuite une céramique de verre, matériau hétérogène modèle dont on peut contrôler la microstructure, on montre qu'il existe une seconde classe de surfaces de rupture caractérisée par la même structure anisotrope mais présentant des exposants plus faibles. Utilisant enfin des outils théoriques issus de la physique statistique hors équilibre combinés avec la mécanique de la rupture, nous établissons le lien entre ces propriétés e! t les mécanismes généraux de rupture à l'échelle microscopique. Cette étude nous permet notamment d'associer les deux classes de surfaces de rupture à un processus de fissuration mettant en jeux de l'endommagement pour l'un et à une rupture parfaitement fragile pour l'autre.

Ziel dieser Arbeit ist die Überwindung einer Differenz, die zwischen der Theorie der Phase bzw. der Phasendynamik und ihrer Anwendung in der Zeitreihenanalyse besteht: Während die theoretische Phase eindeutig bestimmt und invariant unter Koordinatentransformationen bzw. gegenüber der jeweils gewählten Observable ist, führen die Standardmethoden zur Abschätzung der Phase aus gegebenen Zeitreihen zu Resultaten, die einerseits von den gewählten Observablen abhängen und so andererseits das jeweilige System keineswegs in eindeutiger und invarianter Weise beschreiben. Um diese Differenz deutlich zu machen, wird die terminologische Unterscheidung von Phase und Protophase eingeführt: Der Terminus Phase wird nur für Variablen verwendet, die dem theoretischen Konzept der Phase entsprechen und daher das jeweilige System in invarianter Weise charakterisieren, während die observablen-abhängigen Abschätzungen der Phase aus Zeitreihen als Protophasen bezeichnet werden. Der zentrale Gegenstand dieser Arbeit ist die Entwicklung einer deterministischen Transformation, die von jeder Protophase eines selbsterhaltenden Oszillators zur eindeutig bestimmten Phase führt. Dies ermöglicht dann die invariante Beschreibung gekoppelter Oszillatoren und ihrer Wechselwirkung. Die Anwendung der Transformation bzw. ihr Effekt wird sowohl an numerischen Beispielen demonstriert - insbesondere wird die Phasentransformation in einem Beispiel auf den Fall von drei gekoppelten Oszillatoren erweitert - als auch an multivariaten Messungen des EKGs, des Pulses und der Atmung, aus denen Phasenmodelle der kardiorespiratorischen Wechselwirkung rekonstruiert werden. Abschließend wird die Phasentransformation für autonome Oszillatoren auf den Fall einer nicht vernachlässigbaren Amplitudenabhängigkeit der Protophase erweitert, was beispielsweise die numerischen Bestimmung der Isochronen des chaotischen Rössler Systems ermöglicht. / The aim of this work is to bridge the gap between the theoretical description of the phase dynamics of coupled oscillators and the application of the theory to model reconstruction from time series analysis. In the theory, the phase of a self-sustained oscillator is defined in an unambiguous way, whereas the standard techniques used to estimate phases from given time series provide observabledependent results, so that generally these estimates deviate from the true phase. To stress this crucial issue, we term the observable-dependent phase-like variables as protophases. The main goal of this work is to develop a deterministic transformation from arbitrary protophases to the true, unique phase of the selfsustained oscillator. This approach allows us to obtain an invariant description of coupled oscillators and of their interaction. The application of the transformation and its efficiency are illustrated by means of numerical examples, as well as by the reconstruction of phase models of the cardiorespiratory interaction from multivariate time series of ECG, pulse and respiration. Next, the transformation from protophases to phases is extended for the case of three coupled oscillators. Finally, we go beyond the phase approximation and extend the phase transformation for autonomous oscillators to the case when the amplitude dynamics cannot be neglected. This technique for example allows us to compute numerically the isochrones of the chaotic Roessler system.

We consider a statistical theory as being invariant when the results of two statisticians' independent data analyses, based upon the same statistical theory and using effectively the same statistical ingredients, are the same. We discuss three aspects of invariant statistical theories. Both model checking and checking for prior-data conflict are assessments of single null hypothesis without any specific alternative hypothesis. Hence, we conduct these assessments using a measure of surprise based on a discrepancy statistic. For the discrete case, it is natural to use the probability of obtaining a data point that is less probable than the observed data. For the continuous case, the natural analog of this is not invariant under equivalent choices of discrepancies. A new method is developed to obtain an invariant assessment. This approach also allows several discrepancies to be combined into one discrepancy via a single P-value. Second, Bayesians developed many noninformative priors that are supposed to contain no information concerning the true parameter value. Any of these are data dependent or improper which can lead to a variety of difficulties. Gelman (2006) introduced the notion of the weak informativity as a comprimise between informative and noninformative priors without a precise definition. We give a precise definition of weak informativity using a measure of prior-data conflict that assesses whether or not a prior places its mass around the parameter values having relatively high likelihood. In particular, we say a prior Pi_2 is weakly informative relative to another prior Pi_1 whenever Pi_2 leads to fewer prior-data conflicts a priori than Pi_1. This leads to a precise quantitative measure of how much less informative a weakly informative prior is. In Bayesian data analysis, highest posterior density inference is a commonly used method. This approach is not invariant to the choice of dominating measure or reparametrizations. We explore properties of relative surprise inferences suggested by Evans (1997). Relative surprise inferences which compare the belief changes from a priori to a posteriori are invariant under reparametrizations. We mainly focus on the connection of relative surprise inferences to classical Bayesian decision theory as well as important optimalities.

This thesis studies the Reach Control Problem (RCP) for affine systems defined on simplices. The thesis focuses on cases when it is known that the problem is not solvable by continuous state feedback. Previous work has proposed (discontinuous) piecewise affine feedback to resolve the gap between solvability by open-loop controls and solvability by feedbacks. The first results on solvability by time-varying feedback are presented. Time-varying feedback has the advantage to be more robust to measurement errors circumventing problems of discontinuous controllers. The results are theoretically appealing in light of the strong analogies with the theory of stabilization for linear control systems. The method is shown to solve RCP for all cases in the literature where continuous state feedback fails, provided it is solvable by open loop control. Textbook examples are provided. The motivation for studying RCP and its relevance to complex control specifications is illustrated using a material transfer system.