Computing Generators of Free Modules over Orders in Group Algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ.

Computations in Relative Algebraic K-Groups

Let G be finite group and K a number field or a p-adic field with ring of integers O_K. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K_0(O_K[G],K) as an abstract abelian group. We solve the discrete logarithm problem, both in K_0(O_K[G],K) and the locally free class group cl(O_K[G]). All algorithms have been implemented in MAGMA for the case K = \IQ. In the second part of the manuscript we prove formulae for the torsion subgroup of K_0(\IZ[G],\IQ) for large classes of dihedral and quaternion groups.

Evaluation von SINUS-Hessen 2005 - 2007

Unter dem Namen SINUS werden seit über 10 Jahren bundesweit erfolgreiche Projekte zur Weiterentwicklung des mathematisch-naturwissenschaftlichen Unterrichts durchgeführt. Das Projekt SINUS-Quest, dessen Abschlussbericht hier vorgelegt wird, entstand aus dem Anliegen der Projektleitung von SINUS-Hessen, eine eigene Evaluation des hessischen Projektes SINUS-Transfer (2005 – 2007) durchzuführen. Die Evaluation sollte nicht nur summativ sein, sondern den SINUS-Prozess selber mit beeinflussen. Dazu sollten schulspezifische Befragungsergebnisse an die einzelnen Schulen zurückgemeldet werden, und zwar unter Bezugnahme auf den hessischen Durchschnitt, um die Stärken und den Entwicklungsbedarf einzelner Schulteams gezielt identifizieren und bei der Weiterentwicklung berücksichtigen zu können. Im Jahre 2005 wurde die Projektgruppe SINUS-Quest für die Konzipierung und die Durchführung des Evaluationsprojektes gegründet, und zwar als Kooperationsprojekt zwischen der SINUS-Projektleitung, dem Institut für Qualitätsentwicklung (IQ) in Wiesbaden, vertreten durch die Arbeitseinheit für „Empirische Fundierung der Schulentwicklung und Qualitätssicherung der Evaluation“ und der mathematikdidaktischen Arbeitsgruppe von Prof. Dr. Rolf Biehler an der Universität Kassel. An der Vorbefragung haben ca. 2000 hessische Lehrerinnen und Lehrer teilgenommen, an der Nachbefragung ca. 1200. Ihnen allen sei an dieser Stelle für die aktive Mitarbeit herzlich gedankt. Wir bedanken uns besonders herzlich bei den Set-Koordinatoren und Koordinatorinnen und den SINUS-Schulprojektleitungen, ohne die der sehr gute Rücklauf unserer Fragebögen nicht zustande gekommen wäre. Ein herzlicher Dank geht auch an das Leibniz-Institut für die Pädagogik der Naturwissenschaften (IPN) in Kiel, das als SINUS-Projektträger SINUS-Quest finanziell gefördert hat. Kassel, im September 2009 Rolf Biehler, Pascal Fischer, Christoph Maitzen, Carmen Maxara, Tanja Nieder

Double burden malnutrition of preschool children and its association with brain development and milk consumption

Protecting the quality of children growth and development becomes a supreme qualification for the betterment of a nation. Double burden child malnutrition is emerging worldwide which might have a strong influence to the quality of child brain development and could not be paid-off on later life. Milk places a notable portion during the infancy and childhood. Thus, the deep insight on milk consumption pattern might explain the phenomenon of double burden child malnutrition correlated to the cognitive impairments. Objective: Current study is intended (1) to examine the current face of Indonesian double burden child malnutrition: a case study in Bogor, West Java, Indonesia, (2) to investigate the association of this phenomenon with child brain development, and (3) to examine the contribution of socioeconomic status and milk consumption on this phenomenon so that able to formulate some possible solutions to encounter this problem. Design: A cross-sectional study using a structured coded questionnaire was conducted among 387 children age 5-6 years old and their parents from 8 areas in Bogor, West-Java, Indonesia on November 2012 to December 2013, to record some socioeconomic status, anthropometric measurements, and history of breast feeding. Diet and probability of milk intake was assessed by two 24 h dietary recalls and food frequency questionnaire (FFQ). Usual daily milk intake was calculated using Multiple Source Method (MSM). Some brain development indicators (IQ, EQ, learning, and memory ability) using Projective Multi-phase Orientation method was also executed to learn the correlation between double burden child malnutrition and some brain development indicator. Results and conclusions: A small picture of child double burden malnutrition is shown in Bogor, West Java, Indonesia, where prevalence of Severe Acute Malnutrition (SAM) is 27.1%, Moderate Acute Malnutrition (MAM) is 24.9%, and overnutrition is 7.7%. This phenomenon proves to impair the child brain development. The malnourished children, both under- and over- nourished children have significantly (P-value<0.05) lower memory ability compared to the normal children (memory score, N; SAM = 45.2, 60; MAM = 48.5, 61; overweight = 48.4, 43; obesity = 47.9, 60; normal = 52.4, 163). The plausible reasons behind these evidences are the lack of nutrient intake during the sprout growth period on undernourished children or increasing adiposity on overnourished children might influence the growth of hippocampus area which responsible to the memory ability. Either undernutrition or overnutrition, the preventive action on this problem is preferable to avoid ongoing cognitive performance loss of the next generation. Some possible solutions for this phenomenon are promoting breast feeding initiation and exclusive breast feeding practices for infants, supporting the consumption of a normal portion of milk (250 to 500 ml per day) for children, and breaking the chain of poverty by socioeconomic improvement. And, the national food security becomes the fundamental point for the betterment of the next. In the global context, the causes of under- and over- nutrition have to be opposed through integrated and systemic approaches for a better quality of the next generation of human beings.