The moduli space of stable quotients, introduced by Marian, Oprea, and Pandharipande, provides a nonsingular compactification of the moduli space of degree d maps from smooth genus 1 curves into projective space ℙⁿ. This is done by allowing the domain curve to have nodal singularities and by admitting certain rational maps. The rational maps are introduced in the following way: A map to projective space can be defined by a quotient bundle of the trivial bundle on the domain curve; in the compactification, the quotient bundle is replaced by a sheaf which may not be locally free. The boundary is filtered by the degree of the torsion subsheaf of the quotient. Yijun Shao has defined a similar compactification of the moduli space of degree d maps from ℙ¹ into a Grassmannian. A blow-up process is carried out on the compactification in order to produce a boundary which is a simple normal crossings divisor: The closed subschemes in the filtration of the boundary are blown up in order of decreasing torsion. In this thesis, we carry out an analogous blow-up process on the moduli space of stable quotients. We show that the end result is a nonsingular compactification which has as its boundary a simple normal crossings divisor.

The relationship between temperature and soil respiration has been well explored although uncertainties remain. This thesis examined the relationship between temperature and rates of heterotrophic respiration in soils from three adjacent sub-alpine Australian vegetation types; woodland, shrubland and grassland. Temperature sensitivity of soil (Q10) has recently been a hotly debate topic, one side concluding that decomposition of recalcitrant, less labile components of soil organic matter are insensitive to temperature. Whilst others argue that there is no difference in the temperature sensitivities of labile and recalcitrant carbon pools. Robust modeling of rates of soil respiration requires characterization of the temperature response of both labile and recalcitrant pools. Laboratory incubation provides a means of characterizing the temperature response of rates of respiration whilst reducing the confounding effects encountered in the field, such as seasonal fluctuations in temperature, moisture and substrate supply. I used a novel system that allowed laboratory measurement of gas exchange in soils over a range of temperatures under controlled conditions. Measurements included CO2 efflux and O2 uptake over a range of temperatures from 5 to 40oC, characterization of temperature response and sensitivity, and respiratory quotients. Rates of heterotrophic respiration fitted both exponential and Arrhenius functions and temperature sensitivity varied and depended on the model used, vegetation type and depth in the soil profile. Long-term incubation indicated both labile and resistant pools of carbon had similar temperature sensitivities. Respiratory quotients provided a strongly predictive measure of the potential rate of decomposition of soil C, independent of the temperature response of respiration, providing a tool that may be used alongside derived parameters to help understand shifts in microbial use of C substrates. Vegetation type influenced soil chemical properties and rates of heterotrophic respiration. Rates of respiration correlated well with concentrations of carbon and nitrogen as has been previously observed, unlike previous studies however a positive correlation was observed between indices of plant available phosphorus and respiration. The soils examined were from three adjacent vegetation types formed on common geology, I concluded that vegetation type had a significant influence on soil, in contrast to the commonly held view by ecologists that soil type drives patterns in vegetation. Climatic effects such as longer, dryer hotter summer, reduced snow cover and increased incidence of extreme weather events such as frosts and bushfire are likely to drive patterns in vegetation in this region and therefore have a significant impact on carbon cycling in Sub-alpine Australian soils.

The purpose of this study is to determine the relationships between various factors and the academic achievement of the children in the sunshine Room established at Diamond Hill, Fort Worth, Texas. The factors under consideration are health improvements, behavior ratings obtained from the use of a rating scale and intelligence quotients.

A space of pseudoquotients can be described as a space of either single term quotients (the injective case) or the quotient of sequences (the non-injective case) where the parent sets for the numerator and the denominator satisfy particular conditions. The first part of this project is concerned with the minimal of conditions required to have a well-defined set of pseudoquotients. We continue by adding more structure to our sets and discuss the effect on the resultant pseudoquotient. Pseudoquotients can be thought of as extensions of the parent set for the numerator since they include a natural embedding of that set. We answer some questions about the extension properties. One family of these questions involves assuming a structure (algebraic or topological) on a set and asking if the set of pseudoquotients generated has the same structure. A second family of questions looks at maps between two sets and asks if there is an extension of that map between the corresponding pseudoquotients? If so, do the properties of the original map survive the extension? The result of our investigations on the abstract setting will be compared with some well-known spaces of pseudoquotients and Boehmians (a particular case of non-injective pseudoquotients). We will show that the conditions discussed in the first part are satisfied and we will use that to reach conclusions about our extension spaces and the extension maps. The Fourier transform is one of the maps that we will continuously revisit and discuss. Finally many spaces of Boehmians have been introduced where the initial set is a particular class of functions on either locally compact groups R and or a compact group such as a sphere. The natural question is, can we generalize the construction to any locally compact group. In some previous work such construction is discussed, however here we go further; we use characters and define the Fourier transform of integrable and square integrable Boehmians on a locally compact group. Then we discuss the properties of such transform.

As a sixth grade teacher in Wolfin School, Amarillo, Texas the writer is interested in determining to what extent academic success my be predicted by the use of intelligence tests.

Initiée indépendamment par Beilinson et Bernstein et par Brylinski et Kashiwara, l'étude des opérateurs différentiels sur les variétés de drapeaux complets a permis de répondre à une conjecture de Kazhdan et Lusztig. Ayant été poursuivie notamment par les travaux de Borho et Brylinski, cette étude a mis à jour plusieurs propriétés intéressantes sur les opérateurs différentiels sur les variétés de drapeaux. Cependant, en dehors du cas des variétés de drapeaux et du cas des variétés toriques projectives, qui a été étudié de manière combinatoire, les opérateurs différentiels sont plutôt mal compris sur les variétés projectives.Dans cette thèse, nous nous pencherons sur le cas de certaines compactifications magnifiques Y d'espaces symétriques G/H de petit rang, et nous comparerons les résultats obtenus avec ceux connus sur les variétés de drapeaux. Nous allons commencer par construire un opérateur différentiel global sur Y qui ne provient pas de l'action infinitésimale de l'algèbre de Lie de G, ce qui constitue une différence avec le cas des variétés de drapeaux.Ensuite, nous nous intéresserons à trois cas particulier que nous exprimerons comme des quotients GIT d'une certaine grassmannienne X. Grâce à cette description, nous verrons plusieurs similitudes avec le cas des variétés de drapeaux : nous montrerons que l'algèbre des opérateurs globaux sur Y est de type fini, et que pour tout faisceau inversible L sur Y, ses sections globales forment un module simple pour l'algèbre des opérateurs différentiels globaux de Y tordus par L. Enfin, en utilisant des arguments de cohomologie locale, nous montrerons que c'est également le cas pour les groupes de cohomologie supérieurs / Started independently by Beilinson and Bernstein, and by Brylinski and Kashiwara, the study of global differential operators on complete flag varieties has been very useful to answer a conjecture of Kazhdan and Lusztig. In their subsequent work, Borho and Brylinski have discovered many interesting properties on differential operators on flag varieties. But apart from the case of flag varieties, and the case of projective toric varieties, which has been investigated with combinatorial methods, differential operators on projective varieties are rather badly known.In this thesis, we will investigate the case of some wonderful compactifications Y of symmetric spaces G/H of small rank, and we will compare our results with what is known in the case of flag varieties. We will first construct a differential operator on Y which does not come from the infinitesimal action of G, which is different from the case of flag varieties.We will then look at three particular cases, which will be expressed as GIT quotients of some Grassmannian X. With this description, we will find some similarities with the case of flag varieties : we will show that the algebra of global differential operators is of finite type, and that for each invertible sheaf L on Y, the module of its global sections is simple as a module over the algebra of global differential operators of Y twisted by L. Finally, using arguments of local cohomology, we will show that it is still the case for higher cohomology groups

This thesis comes in three main parts. In the first of these (comprising chapters 2 - 6), the basic theory of Riemannian and conformal submersions is described and the relevant geometric machinery explained. The necessary Clifford algebra is established and applied to understand the relationship between the spinor bundles of the base, the fibres and the total space of a submersion. O'Neill-type formulae relating the covariant derivatives of spinor fields on the base and fibres to the corresponding spinor field on the total space are derived. From these, formulae for the Dirac operators are obtained and applied to prove results on Dirac morphisms in cases so far unpublished. The second part (comprising chapters 7-9) contains the basic theory and known classifications of G2-structures and Spin+ 7 -structures in seven and eight dimensions. Formulae relating the covariant derivatives of the canonical forms and spinor fields are derived in each case. These are used to confirm the expected result that the form and spinorial classifications coincide. The mean curvature vector of associative and Cayley submanifolds of these spaces is calculated in terms of naturally-occurring tensor fields given by the structures. The final part of the thesis (comprising chapter 10) is an attempt to unify the first two parts. A certain `7-complex' quotient is described, which is analogous to the well-known hyper-Kahler quotient construction. This leads to insight into other possible interesting quotients which are correspondingly analogous to quaternionic-Kahler quotients, and these are speculated upon with a view to further research.

Autism spectrum disorders (ASDs) are a biologically-based and behaviorally-defined spectrum of conditions which impact development. These conditions affect and are diagnosed based on features in three psychological and behavioral domains: social interaction, communication, and repetitive behaviors. Developing better ways to identify early signs of autism, whether through behavioral or other types of measures, is important because it will allow children to gain access to interventions and treatments earlier, which has demonstrated positive outcomes. Over the past 10 years, the prevalence of reported autism cases has increased. As a result, much research has focused on the etiology and phenotype of autism. Investigations seeking early signs of autism have generally studied vulnerable populations, particularly infants with an older sibling diagnosed with autism. Aside from observable behavioral differences, biological abnormalities, often within the gastrointestinal and immune systems as well as endocrine, autonomic and other systems, have been observed in a significant number of children diagnosed with autism. These features raise the possibility that cellular and tissue change in body and brain may be altering brain function such that behaviors emerge later and downstream of these cellular and tissue problems. However, research on the pathophysiology underlying these medical features, and particularly regarding how they develop in infancy, has received almost no attention. Such investigation would require measuring pathophysiological and medical features alongside current standard measures of behavioral and phenotypical presentations of autism. This thesis describes a study, funded by the Department of Defense Autism Research Program and carried out at the Massachusetts General Hospital Lurie Center, that proposed to look for early markers of autism in the pathophysiological domains in high risk infants and place them into developmental context by correlating these observations (some of which might potentially become early markers) with well-established neurocognitive measures. The goal of the study is to find biomarkers of clinical importance that reflect the pathophysiologial development of autism which might substantially precede behavioral changes that are currently used as a standard of diagnosis, but are not developmentally apparent or reliably measurable until well into the second or third year of life. While the overall scope of the study encompassed a range of systemic and nervous system measures as well as neurocognitive assessments, the focus of this thesis is mainly on a subset of the behavioral and neurocognitive measures collected through the study, specifically the Autism Diagnostic Observation Scale (ADOS), Autism Observational Scale for Infants (AOSI), Mullen Scales of Early Learning (MSEL) and Vineland Adaptive Behavior Scales (VABS). Subject development was tracked and assessed through developmental quotients (DQs) and then correlated to measures designed to identify autistic-like features. Results demonstrate that verbal development was the most significant indicator for autism. Additionally, delay in communication preceded problems with socialization. The analysis and information used for this thesis will contribute to the infrastructure utilized by the investigators for assessing further behavioral data. In addition, this behavioral data and the metrics generated in these analyses will be analyzed in relation to physiological data (e.g. brain, autonomic, metabolic, immune, and microbiome data). Tracking early biomedical development, especially alongside the current standard of observing behavioral development, has the potential of offering more comprehensive understanding of the brain-behavior-body relationship in children diagnosed with ASD, which can hopefully contribute to a non-invasive, more accurate, and earlier method of diagnosis and to the development of more treatment options.

This thesis consists of a summary and three papers, concerning some aspects of representation theory for complex finite dimensional semi-simple Lie algebras with focus on the BGG-category O. Paper I is motivated by the many useful properties of functors on category O given by tensoring with finite dimensional modules, such as projective functors and translation functors. We study properties of functors on O given by tensoring with arbitrary (possibly infinite dimensional) modules. Such functors give rise to a faithful action of O on itself via exact functors which preserve tilting modules, via right exact functors which preserve projective modules, and via left exact functors which preserve injective modules. Papers II and III both deal with Kostant's problem. In Paper II we establish an effective criterion equivalent to the answer to Kostant's problem for simple highest weight modules, in the case where the Lie algebra is of type A. Using this, we derive some old and new results which answer Kostant's problem in special cases. An easy sufficient condition derived from this criterion using Kazhdan-Lusztig combinatorics allows for a straightforward computational check using a computer, by which we get a complete answer for simple highest weight modules in the principal block of O for algebras of rank less than 5. In Paper III we relate the answer to Kostant's problem for certain modules to the answer to Kostant's problem for a module over a subalgebra. We also give a new description of a certain quotient of the dominant Verma module, which allows us to give a bound on the multiplicities of simple composition factors of primitive quotients of the universal enveloping algebra.

Dans cette thèse, on introduit la notion de revêtement potentiellement inséparable et on se propose de développer une théorie de la ramification pour ces derniers. Le langage utilisé est celui des schémas en groupoïdes. Après avoir établi quelques résultats préliminaires au chapitre 1, on prouve au chapitre 2 un théorème de quotient d'un schéma en groupoïdes par un sous-groupoïde. Au chapitre 3, on utilise ces résultats pour entreprendre l'étude générale du formalisme des revêtements inséparables. Enfin, au chapitre 4, on spécialise au cas des revêtements sous un schéma en groupes diagonalisable et on étudie en détail la structure de ces derniers. En particulier, on exprime le lieu Gorenstein de ces morphismes en fonction des constantes de structure du revêtement et on prouve dans ce cadre une formule analogue à la formule de Riemann-Hurwitz des revêtements ramifiés classiques. / In this thesis, we introduce the notion of inseparable coverings and we try to develop a ramification theory for such objects. We make use of the groupoid scheme formalism. In section 1, we establish preliminary results on scheme epimorphisms. We use these results in the next section to prove a quotient theorem for groupoid schemes.Then in section 3 we introduce the general formalism of inseparable coverings.Finally, in the last section we consider in greater details inseparable coverings given by the action of a diagonalizable group scheme. We compute the Gorenstein locus of these morphisms and we prove a formula analogous to the classical Riemann-Hurwitz formula.