Svenska elever har presterat dåligt i internationella undersökningar en längre tid när det gäller algebraområdet i matematik. Elevernas begreppsförståelse har pekats ut som en faktor som spelar in i de dåliga resultaten för svenska elever del. Syftet med denna studie har därför varit att ta reda på den roll som begrepp och begreppsförmåga spelar vid inlärning av algebra samt vilken begreppsförståelse elever i årskurs 4-6 har. Genom en systematisk litteraturstudie har frågeställningarna besvarats. Resultaten visar att brister i begreppsförståelse i algebra också leder till brister i kunskap i algebra. Undervisning med fokus på begrepp leder till bättre förståelse för begrepp samtidigt som det även leder till procedurell kunskap. Elever i årskurs 4-6 kan hantera variabler och använda dem i matematiska uttryck. Fördelar med en tidig introduktion av variabelbegreppet är att elever bygger en bättre förståelse för begreppet.

Breast cancer is one of the most frequently occurring cancers amongst women throughout the world. After the diagnosis of the disease, monitoring its progression is important in predicting the chances of long term survival of patients. The Nottingham Prognostic Index (NPI) is one of the most common indices used to categorise the patients into different groups depending upon the severity of the disease. One of the key factors of this index is cancer grade which is determined by pathologists who examine cell samples under a microscope. This manual method has a higher chance of false classification and may lead to incorrect treatment of patients. There is a need to develop automated methods that employ advanced computational methods to help pathologists in making a decision regarding the classification of breast cancer grade. Fourier transform infra-red spectroscopy (FTIR) is one of the relatively new techniques that has been used for diagnosis of various cancer types with advanced computational methods in the literature. In this thesis we examine the use of advanced fuzzy methods with the FTIR spectral data sets to develop a model prototype that can help clinicians with breast cancer grading. Initial work is focussed on using the commonly used clustering algorithms k-means and fuzzy c-means with principal component analysis on different cancer spectral data sets to explore the complexities within them. After that, a novel model based on Type-II fuzzy logic is developed for use on a complex breast cancer FTIR spectral data set that can help clinicians classify breast cancer grades. The data set used for the purpose consists of multiple cases of each grade. We consider two types of uncertainty, one within the spectra of a single case of a grade (intra -case) and other when comparing it with other cases of same grade (inter-case). Features have been extracted in terms of interval data from various peaks and troughs. The interval data from the features has been used to create Type-I fuzzy sets for each case. After that the Type-I fuzzy sets are combined to create zSlices based General Type-II fuzzy sets for each feature for each grade. The created benchmark fuzzy sets are then used as prototypes for classification of unseen spectral data. Type-I fuzzy sets are created for unseen spectral data and then compared against the benchmark prototype Type-II fuzzy sets for each grade using a similarity measure. The best match based on the calculated similarity scores is assigned as the resultant grade. The novel model is tested on an independent spectral data set of oral cancer patients. Results indicate that the model was able to successfully construct prototype fuzzy sets for the data set, and provide in-depth information regarding the complexities of the data set as well as helping in classification of the data.

This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes. 1. The comparison of the mean-periodicity properties of zeta functions with the much better known, but nonetheless conjectural, automorphicity properties of Hasse–Weil L functions. The latter of the two is a widely believed aspect of the Langlands program. In somewhat vague language, the two notions are dual to each other. One route to this result is broadly comparable to the Rankin-Selberg method, in which Fesenko’s “boundary function” plays the role of an Eisenstein series. 2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective theory of GL1(A(S)) zeta-integrals, where S is an arithmetic surface. When combined with adelic duality, we see that mean-periodicity may be accessible through further developments in higher dimensional adelic analysis. The results of the first flavour have some bearing on questions asked first by Langlands, and those of the second kind are an extension of the ideas of Tate for Hecke L-functions. The theorems proved here directly extend those of Fesenko and Suzuki on two-dimensional adelic analysis and the interplay between mean-periodicity and automorphicity.

In this thesis we study some new classes of nonassociative division algebras. First we introduce a generalisation of both associative cyclic algebras and of Waterhouse's nonassociative quaternions. An important aspect of these algebras is the simplicity of their construction, which is a modification of the classical definition of associative cyclic algebras. By taking the parameter used in the classical definition from a larger field, we lose the property of associativity but gain many new examples of division algebras. This idea is also applied to obtain a generalisation of the first Tits construction. We go on to study constructions of Menichetti, Knuth, and Hughes and Kleinfeld, which have previously only been considered over finite fields. We extend these definitions to infinite fields and get new examples of division algebras, including some over the real numbers. Recently, both associative and nonassociative division algebras have been applied to the theory of space-time block coding. We explore this connection and show how the algebras studied in this thesis can be used to construct space-time block codes.

This thesis is divided into two parts. The rst part studies homological algebra of quasihereditary algebras, with the underlying theme being to understand properties of the Yoneda algebra of standard modules. We will rst show how homological properties of a quasi-hereditary algebra are carried over to its tensor products and wreath products. We then determine the extgroups between indecomposable standard modules of a Cubist algebra of Chuang and Turner. We will also determine generators, hence the quiver, of the Yoneda algebra of standard modules for the rhombal algebras of Peach. We also obtain a higher multiplication vanishing condition for certain rhombal algebras. The second part of this thesis studies the notion of simple-minded systems, introduced by Koenig and Liu. Such systems were designed to generate the stable module categories of artinian algebras by extension, in the same way as the sets of simple modules. We classify simple-minded systems for representation- nite self-injective algebras, and establish connections of them to various notions in combinatorics and related derived categories. We also look at the notion of simple-minded systems de ned on triangulated categories, and obtain some classi cation results using a connection between the simple-minded systems of a triangulated category and of its orbit category.

The family of planar nearrings enjoys quite a few geometric and combinatoric properties. Circular planar nearrings are members of this family which have the character of circles of the complex plane. On the other hand, they also have some properties which one may not find among the circles of the complex plane. In this dissertation, we first review the definition and characterization of a planar nearring, and some various ways of constructing planar nearrings, as well as various ways of constructing BIBD's from a planar nearring. Circularity of a planar nearring is then introduced, and examples of circularity planar nearrings are given. Then, some nonisomorphic BIBD's arising from the same additive group of a planar nearring are examined. To provide examples of nonabelian planar nearrings, the structures of Frobenius groups with kernel of order 64 are completely determined and described. On the other hand, examples of Ferrero pairs (N, Φ)'s with nonabelian Φ, which produce circular planar nearrings, are provided. Finally, we study the structures of circular planar nearrings generated from the finite prime fields from geometric and combinatoric points of view. This study is then carried back to the complex plane. In turn, it gives a good reason for calling a block from a circular planar nearring a "circle."

published_or_final_version / Mathematics / Master / Master of Science

Working with examination timetabling is an extremely challenging task due to the difficulty of finding good quality solutions. Most of the studies in this area rely on improvement techniques to enhance the solution quality after generating an initial solution. Nevertheless, the initial solution generation itself can provide good solution quality even though the ordering strategies often using graph colouring heuristics, are typically quite simple. Indeed, there are examples where some of the produced solutions are better than the ones produced in the literature with an improvement phase. This research concentrates on constructive approaches which are based on squeaky wheel optimisation i.e. the focus is upon finding difficult examinations in their assignment and changing their position in a heuristic ordering. In the first phase, the work is focused on the squeaky wheel optimisation approach where the ordering is permutated in a block of examinations in order to find the best ordering. Heuristics are alternated during the search as each heuristic produces a different value of a heuristic modifier. This strategy could improve the solution quality when a stochastic process is incorporated. Motivated by this first phase, a squeaky wheel optimisation concept is then combined with graph colouring heuristics in a linear form with weights aggregation. The aim is to generalise the constructive approach using information from given heuristics for finding difficult examinations and it works well across tested problems. Each parameter is invoked with a normalisation strategy in order to generalise the specific problem data. In the next phase, the information obtained from the process of building an infeasible timetable is used. The examinations that caused infeasibility are given attention because, logically, they are hard to place in the timetable and so they are treated first. In the adaptive decomposition strategy, the aim is to automatically divide examinations into difficult and easy sets so as to give attention to difficult examinations. Within the easy set, a subset called the boundary set is used to accommodate shuffling strategies to change the given ordering of examinations. Consequently, the graph colouring heuristics are employed on those constructive approaches and it is shown that dynamic ordering is an effective way to permute the ordering. The next research chapter concentrates on the improvement approach where variable neighbourhood search with great deluge algorithm is investigated using various neighbourhood orderings and initialisation strategies. The approach incorporated with a repair mechanism in order to amend some of infeasible assignment and at the same time aiming to improve the solution quality.

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