<p>Matematik är ett av de viktigaste ämnena i dagens skola, men det är också ett av de ämnena som eleverna ofta tycker är tråkigast. På senare år har undersökningar visat på brister i elevernas matematikkunskaper och skolans matematik har blivit beskylld för att inte vara tillräckligt tydligt kopplad till verkligheten.</p><p>Genom att ta del av tidigare forskning kom jag fram till sex riktlinjer för att utforma ett bättre läromedel i matematik. Jag bestämde mig således för att designa ett läromedel: med öppna lösningar där eleverna själva fick formulera problemen, ett läromedel som appellerade till flera sinnen, som var konkret utan abstrakta termer, som gjorde det möjligt för eleverna att använda sin kreativitet, som inbjöd till diskussioner och kommunikation och som fångade elevernas engagemang.</p><p>Jag bestämde mig för att utforma mitt läromedel kring Felicia, hennes detektivbyrå och fallet med stölderna på hotell Citronella.</p>

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<p>This PhD-thesis consists of the five papers</p><p>- On the Hochschild (co)homology of quantum exterior algebras, to appear in Comm. Algebra,</p><p>-Complexity and periodicity, Coll. Math. 104 (2006), no. 2, 169-191,</p><p>-Twisted support varieties,</p><p>-Modules with reducible complexity, to appear in J. Algebra,</p><p>- On support varieties for modules over complete intersections, to appear in Proc. Amer. Math. Soc.</p><p>These papers are roughly divided into two groups; the ¯rst three study modules over Artin algebras using techniques from Hochschild cohomology, whereas the last two papers study modules over commutative Noetherian local rings, in particular modules over complete intersections. </p>

<p>In this thesis we investigate necessary and su±cient conditions for an n-dimensional space, <em>n</em> ≥ 4, to be locally conformal to an Einstein space. After reviewing the classical results derived in tensors we consider the four-dimensional spinor result of Kozameh, Newman and Tod. The involvement of the four-dimensional Bach tensor (which is divergence-free and conformally well-behaved) in their result motivates a search for an n-dimensional generalization of the Bach tensor Bab with the same properties. We strengthen a theorem due to Belfagón and Jaén and give a basis (U<em>_ab, V _ab</em> and <em>W_ab</em>) for all n-dimensional symmetric, divergence-free 2-index tensors quadratic in the Riemann curvature tensor. We discover the simple relationship B_ab = 1/2U_ab + 1/6V _ab and show that the Bach tensor is the unique tensor with these properties in four dimensions. Unfortunately we have to conclude, in general that there is no direct analogue in higher dimension with all these properties.</p><p>Nevertheless, we are able to generalize the our-dimensional results due to Kozameh, Newman and Tod to <em>n</em> dimensions. We show that a generic space is conformal to an Einstein space if and only if there exists a vector field satisfying two conditions. The explicit use of dimensionally dependent identities (some of which are newly derived in this thesis) is also exploited in order to make the two conditions as simple as possible; explicit examples are given in five and six dimensions using these tensor identities. For n dimensions, we define the tensors babc and Bab, and we show that their vanishing is a conformal invariant property which guarantees that the space with non-degenerate Weyl tensor is a conformal Einstein space.</p>

<p>In this thesis we use the method of Semantic Mirrors to create a graph of words that are semantically related to a seed word. Spectral graph partitioning methods are then used to partition the graph into subgraphs, and thus dividing the words into different word senses. These methods are applied to a bilingual lexicon of English and Swedish adjectives. A panel of human evaluators have looked at a few examples, and evaluated consistency within the derived senses and synonymy with the seed word.</p>

<p>Authentication is an indispensable part of Quantum Cryptography, which is an unconditionally secure key distribution technique based on the laws of nature. Without proper authentication, Quantum Cryptography is vulnerable to “man-in-the-middle” attacks. Therefore, to guarantee unconditional security of any Quantum Cryptographic protocols, the authentication used must also be unconditionally secure. The standard in Quantum Cryptography is to use theWegman-Carter authentication, which is unconditionally secure and is based on the idea of universal hashing.</p><p>In this thesis, we first investigate properties of a Strongly Universal hash function family to facilitate understanding the properties of (classical) authentication used in Quantum Cryptography. Then, we study vulnerabilities of a recently proposed authentication protocol intended to rule out a "man-in-the-middle" attack on Quantum Cryptography. Here, we point out that the proposed authentication primitive is not secure when used in a generic Quantum Cryptographic protocol. Lastly, we estimate the lifetime of authentication using encrypted tags when the encryption key is partially known. Under simplifying assumptions, we derive that the lifetime is linearly dependent on the length of the authentication key. Experimental results that support the theoretical results are also presented.</p>

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<p>Syftet med detta examensarbete är att urskilja de faktorer som är särskilt viktiga och även avgörande för ett främjande eller hindrande av elevens motivation för matematik. Som grund för arbetet finns en litteraturstudie som behandlar begreppet motivation i undervisningen. Därefter följer en kvalitativ intervjustudie där fyra utvalda lärares erfarenhet av motivationshöjande undervisning i matematik beskrivs.</p><p>Resultatet av undersökningen visar att lärarna genom ett gediget intresse samt en drivkraft för vidare utveckling inom matematiken lägger stor vikt vid att förmedla sin egen kunskap samt entusiasm för ämnet till eleven. Detta förhållningssätt beskrivs som den centrala faktorn för elevens motivation för matematik. Den ligger till grund för övriga faktorer: lärarens förmåga att variera undervisningen samt lärarens förmåga att individualisera undervisningen utifrån varje elevs behov. Detta krävs enligt både informanter och litteraturen för att se meningen med matematiken, få förståelse för matematiska strategier i diskussioner med andra samt slutligen för att nå målen beskrivna i styrdokument och kursplaner med bibehållen motivationen för matematik. Som hindrande faktorer för motivation framhålls föräldrars och klasskamraters negativa inställning till matematik, grupptryck, ökande av skriftliga uträkningar samt stora klasser med kortare tid till varje elev för läraren. Samtliga hinder gör det svårare för läraren att fokusera på främjande faktorer trots att informanterna är eniga om att det på olika sätt går att få eleverna motiverade för matematik i årskurs 4-6.</p>

<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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