Philosophiae Doctor - PhD / Categorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting.

As teaching is a highly complex activity, so too is learning to teach. One pedagogy which has been shown to promote teacher learning is the use of small group discussion. This thesis examines the experiences of seven preservice secondary teachers at a New Zealand university who met weekly during their second practicum to discuss their experiences at their placement schools. Individual interviews conducted with five of the participants revealed that students felt positively about the weekly meetings. The preservice teachers appreciated 1) being able to hear about the experiences of other preservice teachers 2) tell others about their teaching 3) being able to seek advice and potential solutions to problems 4) the sense of personal connection and emotional support they gained during the weekly sessions. The students reported that the weekly meetings allowed them to think about their teaching from the perspective of others, and gave them time to reflect about their experiences while on practicum. This study situates these findings within the literature on initial teacher education and offers suggestions for future research using this pedagogy.

Talk is the vehicle of exchange in language classrooms when communicative competence is being developed. Turns of talk then facilitate the meaning-making process as students and teachers collaboratively come to understand the discourse of knowledge they are co-constructing. During the pivotal third turn in the essential teaching exchange, there is potential for teachers to realise productive pedagogies as they facilitate their students' organisational and pragmatic skills in the foreign language. This study brings a lens to Initiation-Response-Evaluation (IRE) triadic dialogue, which has been criticised for its monological overuse and limitation of students' language production. Analysis of the third turn shows the uptake to be an implicit move in the exchange. Teachers appear not to be aware of the potential benefit it offers them for co-constructed language use at that point in teacher-student interactions. Teachers draw on students' background knowledge and experiential learning in the four domains of productive pedagogies (intellectual quality, supportive classroom environment, recognition of difference and connectedness) when they engage them through an authentic use of language. For this study, potential for productive pedagogy was investigated in the classroom talk of two teachers of Japanese at year 10 level. In a case study, six transcribed and translated lessons were subjected to conversational and membership categorisation analyses using Bachman's (1990) communicative language ability framework to describe language production around the third turn and to hypothesise its effectiveness in providing opportunities for students to generate output in the target language.

Public media and previous research have focused mainly on listing day returns of initial public offers (IPOs) by new economy companies in specific periods such as before April 2000, without examining any subsequent equity offers by new economy companies. This study addresses the issue of multiple equity offers and provides additional understanding of new economy initial and seasoned equity offers (SEOs). Without, a priori, favouring any existing explanation of initial and long-term share returns, this research tests a wide range of theories in order to provide insight into share returns of equity offerings by new economy companies listed on the Australian Stock Exchange between 1994 and 2004. In general, this thesis documents the ability of publicly available information (obtained from offer documents and company announcements to the market) to explain the returns of equity-issuing new economy companies in Australia. In other words, how useful is public information in the valuation of initial and seasoned equity offers of new economy stocks? Specifically, the thesis seeks to examine the ability of public information to explain (a) listing day and long-term returns subsequent to initial public offers by new economy companies, and the probability of IPO withdrawal, (b) announcement period and long-term returns of seasoned equity offers by new economy companies, and (c) the relationships between the initial and any subsequent equity offers by new economy companies (within three years of listing) in terms of probability of seasoned equity offer, duration between the IPO and the first SEO, and frequency of seasoned equity offers within the first three years of IPO. First, the thesis finds that public information is used by investors to value new economy stocks on listing day and in the long run. The negative effect of withdrawal probability on listing day returns of successful IPOs is confirmed in this thesis in the context of the fixed-price offer process in the new economy sector in Australia. While new economy equity-issuing companies have inferior long-term returns compared to the market index and the small capitalisation stock index, they do not underperform relative to their respective industry index returns. Second, this study also finds that public information can explain new economy stock returns around the announcements of seasoned offers and in the long run. Third, the results reveal that publicly available information can be used to explain the incidence and to estimate the probability of seasoned equity offers by recent new economy IPOs. Furthermore, it is found that public information has the ability to explain the duration between the IPO and the first seasoned offer, as well as the frequency of seasoned offers in the first three years after listing. The results of the study support the theoretical predictions about the effects of public information (representing IPO characteristics) and the incidence of a seasoned equity offer. In particular, IPO quality signalling by retained ownership and by underpricing, and the market feedback effect of post-IPO returns have been confirmed for new economy equity offers in Australia. Underpriced new economy IPOs and those with greater proportion of ownership retained after the offer are significantly more likely to have a seasoned equity offer within three years of listing. Likewise, new economy IPOs with superior aftermarket returns are significantly more likely to have a seasoned equity offer. The implication of this research is that public information contained in offer documents and in company announcements is important to valuation of the Australian Stock Exchange listed new economy companies. Thus, the regulators and the Stock Exchange should continue to insist on a high level of information disclosure prior to equity offers in order to enable investors to properly value companies within the new economy sector.

In General Relativity, the Einstein field equations allow us to study the evolution of a spacelike 3-manifold, provided that its metric and extrinsic curvature satisfy a system of geometric constraint equations. The so-called Einstein constraint equations, arise as a consequence of the fact that the 3-manifold in question is necessarily a submanifold of the spacetime its evolution defines. This thesis is devoted to a study of the structure of the Einstein constraint system in the special case when the spacelike 3-manifold also satisfies the quasispherical ansatz of Bartnik [B93]. We make no mention of the generality of this gauge; the extent to which the quasispherical ansatz applies remains an open problem. After imposing the quasispherical gauge, we give an argument to show that the resulting Einstein constraint system may be viewed as a coupled system of partial differential equations for the parameters describing the metric and second fundamental form. The hencenamed quasisperical Einstein constraint system, consists of a parabolic equation, a first order elliptic system and (essentially) a system of ordinary differential equations. The question of existence of solutions to this system naturally arises and we provide a partial answer to this question. We give conditions on the initial data and prescribable fields under which we may conclude that the quasispherical Einstein constraint system is uniquley solvable, at least in a region surrounding the unit sphere. The proof of this fact is centred on a linear iterative system of partial differential equations, which also consist of a parabolic equation, a first order elliptic system and a system of ordinary differential equations. We prove that this linear system consistently defines a sequence, and show via a contraction mapping argument, that this sequence must converge to a fixed point of the iteration. The iteration, however, has been specifically designed so that any fixed point of the iteration coincides with a solution of the quasispherical Einstein constraints. The contraction mapping argument mentioned above, relies heavily on a priori estimates for the solutions of linear parabolic equations. We generalise and extend known results 111 concerning parabolic equations to establish special a priori estimates which relate a useful property: the L2-Sobolev regularity of the solution of a parabolic equation is greater than that of the coefficients of the elliptic operator, provided that the initial data is sufficiently regular. This 'smoothing' property of linear parabolic equations along with several estimates from elliptic and ordinary differential equation theory form the crucial ingredients needed in the proof of the existence of a fixed point of the iteration. We begin in chapter one by giving a brief review of the extensive literature concerning the initial value problem in General Relativity. We go on, after mentioning two of the traditional methods for constructing spacetime initial data, to introduce the notion of a quasispherical foliation of a 3-manifold and present the Einstein constraint system written in terms of this gauge. In chapter two we introduce the various inequalities and tracts of analysis we will make use of in subsequent chapters. In particular we define the, perhaps not so familiar, complex differential operator 9 (edth) of Newman and Penrose. In chapter three we develop the appropriate Sobolev-regularity theory for linear parabolic equations required to deal with the quasispherical initial data constraint equations. We include a result due to Polden [P] here, with a corrected proof. This result was essential for deriving the results contained in the later chapters of [P], and it is for this reason we include the result. We don't make use of it explicitly when considering the quasispherical Einstein constraints, but the ideas employed are similar to those we use to tackle the problem of existence for the quasispherical constraints. Chapter four is concerned with the local existence of quasispherical initial data. We firstly consider the question of existence and uniqueness when the mean curvature of the 3-manifold is prescribed, then after introducing the notion of polar curvature, we also present another quasispherical constraint system in which we consider the polar curvature as prescribed. We prove local existence and uniqueness results for both of these alternate formulations of the quasispherical constraints. This thesis was typeset using LATEXwith the package amssymb.

<p>This thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.</p>

Localised optimal initial perturbations are studied to gain an understanding of the global stability properties of streamwise corner-flow. A self-similar and a modified base-flow are considered. The latter mimics a characteristic deviation from the self-similar solution, commonly observed in experiment. Poweriterations in terms of subsequent direct and adjoint linearised Navier-Stokes solution sweeps are employed to converge optimal solutions for two optimisation times. The optimal response manifests as a wave packet that initially gains energy through the Orr mechanism and continues growing exponentially thereafter. The study at hand represents the first global stability analysis of streamwise corner-flow and confirms key observations made in theoretical and/or experimental work on the subject. Namely, the presence of an inviscid instability mechanism in the near-corner region and a destabilising effect of the characteristic mean-flow deformation found in experiment. / <p>QC 20130604</p>

Syfte: Syftet med denna studie var att jämföra den initiala komforten mellan sfäriska och asfäriska stabila linser då uppfattningen har varit att asfäriska linser (med sin mer följsamma passform) borde vara bekvämare. Metoder: Studien omfattade 30 deltagare med en medelålder på 26 år. Med hjälp av topografens (Topcon CA-100) kontaktlinstillpassningsprogramvara valdes en sfärisk A- design lins (från Nordiska Lins) och en asfärisk A90 lins (från Expert Optik, Sverige) ut med passande parametrar. Linserna sattes i vardera öga utan att försökspersonen visste vilken lins som var vilken. Efter 5 respektive 15 minuter fick patienten, med hjälp av en modifierad VAS-skala, avgöra hur linserna kändes både i höger och vänster öga. Resultat: Ingen signifikant skillnad i komforten mellan de asfäriska och sfäriska linserna varken observerades efter 5 (p=0,78) eller 15 (p=0,84) minuter. Överlag ökade komforten både för de asfäriska och för de sfäriska linserna efter att de suttit i en stund; komforten ökade med 0,9 enheter för de asfäriska respektive 1,1 enheter för de sfäriska linserna mellan 5 och 15 minuters bärtid. Det fanns en signifikant skillnad i komfort för den enskilda asfäriska linsen mellan första och andra graderingstillfället (p=0,015) och samma gällde de sfäriska (p=0,0001). Slutsats: Resultatet visar ingen markant skillnad i initial komfort på stabila linser med dessa olika typer av designer; en multikurvig asfärisk samt en trekurvig sfärisk lins. Med detta i åtanke behöver man inte, ur ett komfortmässigt perspektiv, ta hänsyn till den specifika designen vid tillpassning med moderna stabila linser.

Previous studies have suggested that iron PRBs, receiving high concentrations of inorganic constituents in groundwater, may experience passivation because of the accumulation of inorganic precipitates. In an iron PRB containing more highly reactive material, even though the initial contaminant removal rate is faster than for less reactive material, a faster migration of the contaminant removal front may occur due to the greater reactivity loss, caused by faster accumulation of secondary precipitates. In contrast, an iron PRB containing less reactive material may show a slower accumulation of precipitates, and thus will show a slower migration of the contaminant removal front over time. Thus, it is hypothesized that an iron material having moderate initial reactivity may be more advantageous than material having a higher reactivity in terms of long-term performance. The objective of this study was to test this hypothesis by evaluating the changes of the reactivities of different iron materials in the presence of dissolved CaCO₃. <br /><br /> Four different iron materials (Connelly, G-M, Ispat and Peerless) were selected for the column experiments. The changes in reactivities of the iron and formation of secondary precipitates over time were assessed, primarily by the iron corrosion rates, calculated from the hydrogen gas generation rates, by the cis-DCE removal rates and by the alkalinity profiles. The accumulation of precipitates in the four columns caused passivation of the iron. The passivation of the iron in turn resulted in migration of the mineral precipitation fronts as well as profiles of cis-DCE, TCE, VC, alkalinity, Eh, pH, and chloride. Connelly and G-M had longer periods of operation than Ispat and Peerless and thus their performance was the primary test of the hypothesis. G-M iron, which had the higher initial corrosion rate, compared to Connelly, showed a faster accumulation of precipitates near the influent end. The difference in accumulated precipitates resulted in a difference in the leading edge of the organic profiles and a significant difference in the pattern of passivation, indicating a faster passivation in the region near the influent end for G-M. <br /><br /> Model simulations were performed using the same fitting parameters but with different initial corrosion rate constants to further test the hypothesis. The model provided a reasonable representation of changing reactivities of the columns, being consistent with the observed data. In the simulation for long-term prediction, the cases of higher corrosion rates showed earlier breakthroughs and steeper curves than those of lower corrosion rates. Also, the predictions showed greater porosity loss for the case of higher corrosion rate. Thus, long-term predictions support the hypothesis. Accurate determination of model parameters such as cis-DCE degradation rate constants and iron corrosion rates are required for better predictions of long-term performance.

Previous studies have suggested that iron PRBs, receiving high concentrations of inorganic constituents in groundwater, may experience passivation because of the accumulation of inorganic precipitates. In an iron PRB containing more highly reactive material, even though the initial contaminant removal rate is faster than for less reactive material, a faster migration of the contaminant removal front may occur due to the greater reactivity loss, caused by faster accumulation of secondary precipitates. In contrast, an iron PRB containing less reactive material may show a slower accumulation of precipitates, and thus will show a slower migration of the contaminant removal front over time. Thus, it is hypothesized that an iron material having moderate initial reactivity may be more advantageous than material having a higher reactivity in terms of long-term performance. The objective of this study was to test this hypothesis by evaluating the changes of the reactivities of different iron materials in the presence of dissolved CaCO₃. <br /><br /> Four different iron materials (Connelly, G-M, Ispat and Peerless) were selected for the column experiments. The changes in reactivities of the iron and formation of secondary precipitates over time were assessed, primarily by the iron corrosion rates, calculated from the hydrogen gas generation rates, by the cis-DCE removal rates and by the alkalinity profiles. The accumulation of precipitates in the four columns caused passivation of the iron. The passivation of the iron in turn resulted in migration of the mineral precipitation fronts as well as profiles of cis-DCE, TCE, VC, alkalinity, Eh, pH, and chloride. Connelly and G-M had longer periods of operation than Ispat and Peerless and thus their performance was the primary test of the hypothesis. G-M iron, which had the higher initial corrosion rate, compared to Connelly, showed a faster accumulation of precipitates near the influent end. The difference in accumulated precipitates resulted in a difference in the leading edge of the organic profiles and a significant difference in the pattern of passivation, indicating a faster passivation in the region near the influent end for G-M. <br /><br /> Model simulations were performed using the same fitting parameters but with different initial corrosion rate constants to further test the hypothesis. The model provided a reasonable representation of changing reactivities of the columns, being consistent with the observed data. In the simulation for long-term prediction, the cases of higher corrosion rates showed earlier breakthroughs and steeper curves than those of lower corrosion rates. Also, the predictions showed greater porosity loss for the case of higher corrosion rate. Thus, long-term predictions support the hypothesis. Accurate determination of model parameters such as cis-DCE degradation rate constants and iron corrosion rates are required for better predictions of long-term performance.