<p>How can one model, characterize and analyse systems consisting of a large number of interacting units? This thesis addresses various related aspects of this issue using dynamical systems theory. The thesis is constituted by four papers presenting the main findings as well as an introductory part giving the context of the work.</p><p>In Paper I, we use evolutionary game theory to study a social dilemma where a group can obtain a collective benefit, but only if a threshold on the number of members willing to cooperate at a personal cost is met. The replicator dynamics of our model has interesting features, including catastrophic events. We also generalize the analysis to other n-player games.</p><p>In Paper II, we introduce several new concepts and principles for describing and studying hierarchicalmultilevel systems with interactions both between groups and among individual units. We illustrate these concepts through defining a class of systems called higher order cellular automata, and discuss new phenomena and patterns of behaviour found in these systems.</p><p>In Paper III, we introduce a local information measure for one-dimensional lattice systems in order to characterize the coherent structures emerging from the dynamics of higher order cellular automata as well as ordinary cellular automata. A further objective is to investigate to what extent information can be viewed as a local quantity in such systems. We demonstrate the applicability of our informationmeasure to these problems.</p><p>Paper IV is a continuation of Paper III. We show that local information is a locally conserved quantity in the important class of surjective cellular automata and provide bounds on the information transport.</p><p>The novel concepts and findings of the thesis may lead to an improved understanding of various phenomena found in systems of interacting units, and in particular systems with amultilevel dynamics.</p> / Paper I: reprinted with kind permission of Elsevier, sciencedirect. com Paper II: reprinted from "Advances in Complex Systems", Vol. 8, nos. 2-3. Baas, Na and Helvik T. Higher Order Cellular Automata, 169-192, 2005, with permission from World Scientific Publishing Co. Pte. Ltd. Singapore.

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<p>Dyscalculia and General Mathematical Dificulties</p>

<p>Abstract</p><p>My essay is about pupils with difficulties in mathematics. I have choosed to do a literaturestudy and interviews of teachers to get some answers to my framing of the questions.</p><p>· What is mathematical difficulties?</p><p>· How do we discover pupils with mathematical difficulties?</p><p>· What can a teather do to facilitate for pupils with mathematical difficulties?</p><p>· What resources is there to help pupils with mathematical difficulties?</p><p>I have done my investigationin two schools to be abel to see if the schools are working in the same way and have the same prerequisite of pupils with mathematical difficulties.</p><p>By doing an interview with both teachers and remedial teachers I have got a better insight how to help pupils in the best way. Both of this schools are putting the pupils in the middle and give them wath they need to get to the destinations that claims. The procedure are not the same between the schools. The bigger school have more resources while the smaller school have more material.</p>

<p>Let C and C′ be two smooth self transverse immersions of S1 into R2. BothC and C′ divide the plane into a number of disks and one unbounded component.An isotopy of the plane which takes C to C′ induces a 1-1 correspondence betweenthe disks of C and C′. An obvious necessary condition for there to exist an areapreserving isotopy of the plane taking C to C′ is that there exists an isotopy forwhich the area of every disk of C has the same area as the corresponding disk ofC′. In this paper we show that this is also a sufficient condition.</p>

<p>Model theory and combinatorial pregeometries are closely related throughthe so called algebraic closure operator on strongly minimal sets. Thestudy of projective and ane pregeometries are especially interestingsince they have a close relation to vectorspaces. In this thesis we willsee how the relationship occur and how model theory can concludea very strong classi cation theorem which divides pregeometries withcertain properties into projective, ane and degenerate (trivial) cases.</p> / <p>Modellteori är ett ämne som är starkt relaterat till studien av kombinatoriska pregeometrier, detta genom den algebraiska tillslutningsoperatorn som agerar på starkt minimala mängder. Studien av projektivaoch affina pregeometrier är speciellt intressant genom dessas relation till vektorrum. I den här uppsatsen kommer vi att se hur denna relation uppstår och hur modellteori kan förklara en väldigt stark klassifikationssats, som delar upp pregeometrier med speciella egenskaper i projektiva, affina och degenererade (triviala) fall.</p>

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