| 1. |
- Eriksen, Niklas, 1974-, et al.
(författare)
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Diagonal checker-jumping and Eulerian numbers for color-signed permutations
- 2000
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Ingår i: The Electronic Journal of Combinatorics. - Clemson, USA : Electronic Journal of Combinatorics. - 1097-1440 .- 1077-8926. ; 7
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Tidskriftsartikel (refereegranskat)abstract
- We introduce color-signed permutations to obtain a very explicit combinatorial interpretation of the q-Eulerian identities of Brenti and some generalizations. In particular, we prove an identity involving the golden ratio, which allows us to compute upper bounds on how high a checker can reach in a classical checker-jumping problem, when the rules are relaxed to allow also diagonal jumps.
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| 2. |
- Bergwall, Andreas, 1972-
(författare)
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Proof-related reasoning in upper secondary mathematics textbooks : Characteristics, comparisons, and conceptualizations
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Proofs and proving are difficult to learn and difficult to teach. A common problem is that many students use specific examples as evidence for general statements. Difficulties with proofs are also part of the transition problems that exist between secondary and tertiary schooling in mathematics. As mathematics teaching often follows a textbook, the design of textbooks has been pointed out as one possible cause of the problems, and international textbook research suggests that proofs often have only a marginal place in textbooks.This thesis focuses on proofs and proving in upper secondary mathematics textbooks. It also addresses theoretical and methodological questions about what marks an opportunity to develop proving competence, and which properties of such opportunities are relevant to investigate and characterize. The thesis is based on data from four Swedish and Finnish textbook series for upper secondary school, and focuses on sections on logarithms, primitive functions, definite integrals, and combinatorics. It examines how addressed mathematical principles are justified, and whether the textbooks’ exercises offer opportunities to develop proof-related skills such as formulating and investigating hypotheses, developing and evaluating arguments, identifying and correcting errors, and finding counterexamples.The results show that just over half of the mathematical principles addressed in the analyzed textbook material are justified, and that only half of the justifications are general proofs. Few exercises are proof-related (10%), and those that include reasoning about general cases even fewer. General proofs are more common in the Finnish books, but proof-related tasks are more common and of a more varied nature in the Swedish ones. The most common form of proofs are direct derivations of calculation formulas, while reasoning about existence and uniqueness is unusual, as are contrapositive proofs and proofs by contradiction.Based on the results, explicit suggestions are offered as to what teaching can pay more attention to. For the analysis and design of proof-related activities, a framework consisting of four main categories is proposed: develop a statement, investigate a statement, develop an argument, and investigate an argument. Several properties that such activities may have, regardless of which category they belong to, are discussed. Finally, three areas for future research are suggested: how worked examples can support students’ learning of proof, how textbooks can be designed to stimulate formulation as well as the formal proving of hypotheses, and mapping of differences regarding proof between upper secondary and university textbooks.
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| 3. |
- Dalevi, Daniel A., et al.
(författare)
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Measuring genome divergence in bacteria : A case study using Chlamydian data
- 2002
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Ingår i: Journal of Molecular Evolution. - New York, USA : Springer-Verlag New York. - 0022-2844 .- 1432-1432. ; 55, s. 24-36
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Tidskriftsartikel (refereegranskat)abstract
- We have studied the relative contribution of inversions, transpositions, deletions, and nucleotide substitutions to the evolution of Chlamydia trachomatis and Chlamydia pneumoniae. The minimal number of rearrangement events required for converting the gene order structure of one genome into that of the other was estimated to 59 6 events, including 13% inversions, 38% short inversions, and 49% transpositions. In contrast to previous findings, no examples of horizontal gene transfer subsequent to species divergence were identified, nor any evidence for an excessive number of tandem gene duplications. A statistical model was used to compare nucleotide frequencies for a set of genes uniquely present in one species to a set of orthologous genes present in both species. The two data sets were not significantly different, which is indicative of a low frequency of horizontal gene transfer events. This is based on the assumption that a foreign gene of different nucleotide content will not have become completely ameliorated, as verified by simulations of the amelioration rate at twofold and fourfold degenerate codon sites. The frequencies of nucleotide substitutions at twofold and fourfold degenerate sites, deletions, inversions, and translocations were estimated to 1.42, 0.62, 0,18, 0.01, and 0.01 per site, respectively.
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