This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups. In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd. In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective convergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give necessary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic. Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions. / QC 20100705

Denna avhandling handlar om barns och elevers individuella olikheter och är en del av projektet "Pedagogik för elever med förmåga och fallenhet för matematik” vid Växjö universitet, finansierat av Vetenskapsrådet. Syftet är att studera elever med särskilda förmågor i matematik och den pedagogiska praktik som är deras vardag. Hur kan dessa elever och deras omgivning beskrivas och hur upptäcker, identifierar och bemöter lärare dessa elever? Två empiriska studier har genomförts, en fallstudie där vi får följa två elever genom deras senare år i grundskolan samt en enkätstudie med 180 lärare i grundskolansom fått beskriva sin undervisning i matematik och sin bild av elever med särskilda förmågor i matematik. Fallstudien visar att det finns både gemensamma egenskaper och olikheter när det gäller personlighet och uttryck för den matemtiska förmågan hos de elever som deltog i studien, variationer som behöver mötas med varierade åtgärder. Enkätstudien visar på en snäv syn hos lärare när det gäller bedömning av matematisk förmåga, de elever som enligt lärarna utmärker sig som förmågor gör det genom att arbeta snabbt, tänka snabbt, de är oftast aktiva och självständiga på lektionerna och skriver bra resultat på proven. Denna syn på förmåga kan kopplas samman med den undervisningsmodell som dominerar i grundskolan idag, tyst matematik med hjälp av läromedel. Studien visar att en sådan undervisning inte ger elever med särskilda förmågor i matematik det stöd och den stimulans de är i ehov av för att utvecklas efter sina förutsättningar. / This study concerns students’ individual differences and is part of a research project Gifted Education in Mathematics financedby the Swedish Research Foundation. The aim is to study students with high abilities in mathematics in their daily learning environments: How can these students and their teaching‐learning situations be described and how do teachers experience, identify and treat these students? Two empirical studies were carried out, a casestudy of two pupils who were followed through their later years in compulsory school, and a survey of 180 compulsoryschool teachers covering questions about their teaching practices in mathematics and their views on highly able students. The case studies show that there are both individual differences and common traits among the talented students who took part i the study, variations which call for a varied provision for these students. However, the survey shows that teacher have narrow views of what characterises talented students as hardworking, high achieving etc, views closely related to the dominant model for classroom provision as individualised teaching highly dependent on textbooks. The study shows that such teaching does not give the talented students the support and encouragement that they need in order t develop according to their needs.

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We prove equidistribution results for (pieces of) closed horospheres in cofinite hyperbolic manifolds of dimension n+1, using spectral methods. This extends earlier results by Hejhal [Hej2] and Str¨ombergsson [St1] in dimension 2.

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This thesis consists of two papers. 1. Betty Nannyonga, D.J.T. Sumpter, J.Y.T. Mugisha and L.S. Luboobi: The Dynamics,causes and possible prevention of Hepaititis E outbreaks. 2. Betty Nannyonga, D.J.T. Sumpter, andStam Nicolis: A dynamical systems approach tosocial and economic development. The first paper deals with a deterministic approach of modelling a Hepatitis E outbreak whenmalaria is endemic in a population. We design three models based on the epidemiology ofHepatitis E, malaria, and the co-infection of both diseases. We t our designed models to datathat was collected in a Hepatitis E outbreak in Kitgum district, Uganda, to estimate parameterssuch as the transmission rate, basic reproduction number and recovery rate of those aected. Inthe tting we pursue two approaches, the logistic approach when the natural mortality is zero,and a detailed tting using PottersWheel Toolbox, when natural mortality is not equal to zero.In both cases, we seek to explore how endemic malaria could aect a Hepatitis E outbreak, andsuch a disease’s ability to persist in a population over a long period of time. As a measure ofthe eect of malaria on Hepatitis E transmission, we use a modication parameter such thatwhen the estimated value is greater than unity, then malaria favours Hepatitis E, otherwisewe conclude that it inhibits its spread. In the same paper we attempt to estimate the levelof sanitation required to prevent future outbreaks, in terms of availability of latrines and safedrinking water.In the second paper, we look at the eects of child mortality and average child per woman(fertility rates) on economic development (demographic transition). We use data that is readilyavailable from Gapminder, to extract two dynamical systems, one for child mortality and grossdomestic product (GDP), and the second for child mortality, gross domestic product and averagechild per woman. The models obtained are analyzed numerically for existence and stability. Weuse the Gapminder data to obtain a model that comforms to the demographic transition. Ratherthan using data to justify the assumptions of our models, we use data directly to propose dynamicmodels for the economy. The major question is then, how can we use the model to determine thebest strategy to maximize development? We answer this question by setting constraints, wherewe assume that the economy can improve by 3% while the empirical value for child mortality istwice reduced. Then, we determine the time taken to reach the desired gross domestic product,set to that of a developed economy with low child mortality rates. These approximations makeit possible to draw some conclusions about the best strategy to invest: either directly into theeconomy, or indirectly through child health care. From the simulations we can also determinethe point at which to switch the investment strategy. We end this paper by including averagechild per woman and construct and study the model for the three variables.

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The subject of the thesis is foundational aspects of partial functions (Papers 1, 2 & 4) and some choice principles (Papers 3 & 4) in the context of constructive mathematics. Paper 1 studies the inversion functions of commutative rings. The foundational problem of having them only partially defined is overcome by extending them to total functions. This cannot be done constructively unless the rings themselves are extended at the same time. We study such extensions, called wheels. It is investigated how identities for wheels relate to identities for commutative rings. Paper 2 studies the foundations of partial functions in Martin-Löf's type theory according to the view of subsets as propositional functions, in particular in connection with equivalence relations that the functions are supposed to preserve. The first and second isomorphism theorems of algebra are verified, showing that our approach is flexible enough for some natural mathematical proofs to be carried out. Paper 3 shows that the difference between the principles of intensional and extensional choice can be described as the principle of excluded middle plus a certain mild extensionality principle, which follows from the principle that functions are identical if they are identical at every point. Paper 4 studies a constructive calculus of indefinite and definite descriptions. It has the property that it can be interpreted straightforwardly in type theory with all terms referring to individuals. In this respect it differs from other constructive calculi of descriptions, which are known to be conservative extensions of description-free calculi but for which descriptions cannot be interpreted as referring to individuals in general. The appendix includes a predicative version of Birkhoff's theorem. It states that if a class of algebras is closed under homomorphic images, subalgebras and products and contains a set-indexed family of algebras that satisfies the same identities as the class, then the class can be axiomatized by a set of equations.