Permutation classes, sorting algorithms, and lattice paths

A permutation is said to avoid a pattern if it does not contain any subsequence which is order-isomorphic to it. Donald Knuth, in the first volume of his celebrated book "The art of Computer Programming", observed that the permutations that can be computed (or, equivalently, sorted) by some particular data structures can be characterized in terms of pattern avoidance. In more recent years, the topic was reopened several times, while often in terms of sortable permutations rather than computable ones. The idea to sort permutations by using one of Knuth’s devices suggests to look for a deterministic procedure that decides, in linear time, if there exists a sequence of operations which is able to convert a given permutation into the identical one. In this thesis we show that, for the stack and the restricted deques, there exists an unique way to implement such a procedure. Moreover, we use these sorting procedures to create new sorting algorithms, and we prove some unexpected commutation properties between these procedures and the base step of bubblesort. We also show that the permutations that can be sorted by a combination of the base steps of bubblesort and its dual can be expressed, once again, in terms of pattern avoidance. In the final chapter we give an alternative proof of some enumerative results, in particular for the classes of permutations that can be sorted by the two restricted deques. It is well-known that the permutations that can be sorted through a restricted deque are counted by the Schrӧder numbers. In the thesis, we show how the deterministic sorting procedures yield a bijection between sortable permutations and Schrӧder paths.

Elites or middle classes? : lessons from transnational research for the study of social stratification in Africa

In diesem Arbeitspapier will ich zur künftigen Forschung über soziale Stratifikation in Afrika beitragen, indem ich die theoretischen Implikationen und empirischen Herausforderungen der Konzepte "Elite" und "Mittelklasse" untersuche. Diese Konzepte stammen aus teilweise miteinander konkurrierenden Theorietraditionen. Außerdem haben Sozialwissenschaftler und Historiker sie zu verschiedenen Zeiten und mit Bezug auf verschiedene Regionen unterschiedlich verwendet. So haben Afrikaforscher und -forscherinnen soziale Formationen, die in anderen Teilen der Welt als Mittelklasse kategorisiert wurden, meist als Eliten aufgefasst und tun dies zum Teil noch heute. Elite und Mittelklasse sind aber nicht nur Begriffe der sozialwissenschaftlichen Forschung, sondern zugleich Kategorien der sozialen und politischen Praxis. Die Art und Weise, wie Menschen diese Begriffe benutzen, um sich selbst oder andere zu beschreiben, hat wiederum Rückwirkungen auf sozialwissenschaftliche Diskurse und umgekehrt. Das Arbeitspapier setzt sich mit beiden Aspekten auseinander: mit der Geschichte der theoretischen Debatten über Elite und Mittelklasse und damit, was wir aus empirischen Studien über die umstrittenen Selbstverortungen sozialer Akteure lernen können und über ihre sich verändernden Auffassungen und Praktiken von Elite- oder Mittelklasse-Sein. Weil ich überzeugt bin, dass künftige Forschung zu sozialer Stratifikation in Afrika außerordentlich viel von einer historisch und regional vergleichenden Perspektive profitieren kann, analysiert dieses Arbeitspapier nicht nur Untersuchungen zu afrikanischen Eliten und Mittelklassen, sondern auch eine Fülle von Studien zur Geschichte der Mittelklassen in Europa und Nordamerika sowie zu den neuen Mittelklassen im Globalen Süden.

On Groups with Two Isomorphism Classes of Derived Subgroups

The structure of groups which have at most two isomorphism classes of derived subgroups (D-2-groups) is investigated. A complete description of D-2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem. In addition, detailed information is obtained about soluble D-2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble D-2-groups is found, and the locally free D-2-groups are characterized.

Teaching and aligning upper level mathematics classes to Michigan high school content expectations in an alternative education setting : approaches and issues

After teaching regular education secondary mathematics for seven years, I accepted a position in an alternative education high school. Over the next four years, the State of Michigan adopted new graduation requirements phasing in a mandate for all students to complete Geometry and Algebra 2 courses. Since many of my students were already struggling in Algebra 1, getting them through Geometry and Algebra 2 seemed like a daunting task. To better instruct my students, I wanted to know how other teachers in similar situations were addressing the new High School Content Expectations (HSCEs) in upper level mathematics. This study examines how thoroughly alternative education teachers in Michigan are addressing the HSCEs in their courses, what approaches they have found most effective, and what issues are preventing teachers and schools from successfully implementing the HSCEs. Twenty-six alternative high school educators completed an online survey that included a variety of questions regarding school characteristics, curriculum alignment, implementation approaches and issues. Follow-up phone interviews were conducted with four of these participants. The survey responses were used to categorize schools as successful, unsuccessful, and neutral schools in terms of meeting the HSCEs. Responses from schools in each category were compared to identify common approaches and issues among them and to identify significant differences between school groups. Data analysis showed that successful schools taught more of the HSCEs through a variety of instructional approaches, with an emphasis on varying the ways students learned the material. Individualized instruction was frequently mentioned by successful schools and was strikingly absent from unsuccessful school responses. The main obstacle to successful implementation of the HSCEs identified in the study was gaps in student knowledge. This caused pace of instruction to also be a significant issue. School representatives were fairly united against the belief that the Algebra 2 graduation requirement was appropriate for all alternative education students. Possible implications of these findings are discussed.

PARTITIONING THE BLOCKS OF A STEINER TRIPLE SYSTEM INTO PARTIAL PARALLEL CLASSES

Does there exist a Steiner Triple System on v points, whose blocks can be partitioned into partial parallel classes of size m, where m ≤ [v⁄3], m | b and b is the number of blocks of the STS(v)? We give the answer for 9 ≤ v ≤ 43. We also show that whenever 2|b, v ≡ 3 (mod 6) we can find an STS(v) whose blocks can be partitioned into partial parallel classes of size 2, and whenever 4|b , v ≡ 3 (mod 6), there exists an STS(v) whose blocks can be partitioned into partial parallel classes of size 4.