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.

Family business: Multiple members of major phytohormone classes orchestrate plant stress responses

Low-molecular-weight compounds such as jasmonic, abscisic and salicylic acids are commonly thought to be regulators of plant stress responses. However, it is becoming clear that these molecules, often referred to as phytohormones, are only a part of bigger groups of compounds with biological activity. We propose that the concept of "hormone families" may help to better understand plant physiological responses by taking into account not only the alleged main regulators, but also their precursors, conjugates and catabolites. Novel approaches to profile potentially active compounds in plants are discussed.

Classes of multichannel EEG microstates in light and deep hypnotic conditions

The study assessed the brain electric mechanisms of light and deep hypnotic conditions in the framework of EEG temporal microstates. Multichannel EEG of healthy volunteers during initial resting, light hypnosis, deep hypnosis, and eventual recovery was analyzed into temporal EEG microstates of four classes. Microstates are defined by the spatial configuration of their potential distribution maps ([Symbol: see text]potential landscapes') on the head surface. Because different potential landscapes must have been generated by different active neural assemblies, it is reasonable to assume that they also incorporate different brain functions. The observed four microstate classes were very similar to the four standard microstate classes A, B, C, D [Koenig, T. et al. Neuroimage, 2002;16: 41-8] and were labeled correspondingly. We expected a progression of microstate characteristics from initial resting to light to deep hypnosis. But, all three microstate parameters (duration, occurrence/second and %time coverage) yielded values for initial resting and final recovery that were between those of the two hypnotic conditions of light and deep hypnosis. Microstates of the classes B and D showed decreased duration, occurrence/second and %time coverage in deep hypnosis compared to light hypnosis; this was contrary to microstates of classes A and C which showed increased values of all three parameters. Reviewing the available information about microstates in other conditions, the changes from resting to light hypnosis in certain respects are reminiscent of changes to meditation states, and changes to deep hypnosis of those in schizophrenic states.