Fourier-Matrizen und Ringe mit Basis

Bei der Bestimmung der irreduziblen Charaktere einer Gruppe vom Lie-Typ entwickelte Lusztig eine Theorie, in der eine sogenannte Fourier-Transformation auftaucht. Dies ist eine Matrix, die nur von der Weylgruppe der Gruppe vom Lie-Typ abhängt. Anhand der Eigenschaften, die eine solche Fourier- Matrix erfüllen muß, haben Geck und Malle ein Axiomensystem aufgestellt. Dieses ermöglichte es Broue, Malle und Michel füur die Spetses, über die noch vieles unbekannt ist, Fourier-Matrizen zu bestimmen. Das Ziel dieser Arbeit ist eine Untersuchung und neue Interpretation dieser Fourier-Matrizen, die hoffentlich weitere Informationen zu den Spetses liefert. Die Werkzeuge, die dabei entstehen, sind sehr vielseitig verwendbar, denn diese Matrizen entsprechen gewissen Z-Algebren, die im Wesentlichen die Eigenschaften von Tafelalgebren besitzen. Diese spielen in der Darstellungstheorie eine wichtige Rolle, weil z.B. Darstellungsringe Tafelalgebren sind. In der Theorie der Kac-Moody-Algebren gibt es die sogenannte Kac-Peterson-Matrix, die auch die Eigenschaften unserer Fourier-Matrizen besitzt. Ein wichtiges Resultat dieser Arbeit ist, daß die Fourier-Matrizen, die G. Malle zu den imprimitiven komplexen Spiegelungsgruppen definiert, die Eigenschaft besitzen, daß die Strukturkonstanten der zugehörigen Algebren ganze Zahlen sind. Dazu müssen äußere Produkte von Gruppenringen von zyklischen Gruppen untersucht werden. Außerdem gibt es einen Zusammenhang zu den Kac-Peterson-Matrizen: Wir beweisen, daß wir durch Bildung äußerer Produkte von den Matrizen vom Typ A(1)1 zu denen vom Typ C(1) l gelangen. Lusztig erkannte, daß manche seiner Fourier-Matrizen zum Darstellungsring des Quantendoppels einer endlichen Gruppe gehören. Deswegen ist es naheliegend zu versuchen, die noch ungeklärten Matrizen als solche zu identifizieren. Coste, Gannon und Ruelle untersuchen diesen Darstellungsring. Sie stellen eine Reihe von wichtigen Fragen. Eine dieser Fragen beantworten wir, nämlich inwieweit rekonstruiert werden kann, zu welcher endlichen Gruppe gegebene Matrizen gehören. Den Darstellungsring des getwisteten Quantendoppels berechnen wir für viele Beispiele am Computer. Dazu müssen unter anderem Elemente aus der dritten Kohomologie-Gruppe H3(G,C×) explizit berechnet werden, was bisher anscheinend in noch keinem Computeralgebra-System implementiert wurde. Leider ergibt sich hierbei kein Zusammenhang zu den von Spetses herrührenden Matrizen. Die Werkzeuge, die in der Arbeit entwickelt werden, ermöglichen eine strukturelle Zerlegung der Z-Ringe mit Basis in bekannte Anteile. So können wir für die meisten Matrizen der Spetses Konstruktionen angeben: Die zugehörigen Z-Algebren sind Faktorringe von Tensorprodukten von affinen Ringe Charakterringen und von Darstellungsringen von Quantendoppeln.

Conceptual knowledge discovery and data analysis

In this paper, we discuss Conceptual Knowledge Discovery in Databases (CKDD) in its connection with Data Analysis. Our approach is based on Formal Concept Analysis, a mathematical theory which has been developed and proven useful during the last 20 years. Formal Concept Analysis has led to a theory of conceptual information systems which has been applied by using the management system TOSCANA in a wide range of domains. In this paper, we use such an application in database marketing to demonstrate how methods and procedures of CKDD can be applied in Data Analysis. In particular, we show the interplay and integration of data mining and data analysis techniques based on Formal Concept Analysis. The main concern of this paper is to explain how the transition from data to knowledge can be supported by a TOSCANA system. To clarify the transition steps we discuss their correspondence to the five levels of knowledge representation established by R. Brachman and to the steps of empirically grounded theory building proposed by A. Strauss and J. Corbin.

Is Māori food sovereignty affected by adherence, or lack thereof, to Te Tiriti O Waitangi?

Māori food sovereignty was and continues to be severely affected by British colonisation and influence. This situation and its causes, both past and present, will be examined in the context of Māori culture, especially Te Tiriti, a founding document of New Zealand. The general concept of food sovereignty is introduced, and its key principle: having power over one's own food system. A brief history of Māori food systems is necessary as context–their adaptation from Polynesia to New Zealand and later, to European settlement. Te Tiriti and the Treaty of Waitangi are explained: how they were created and why Te Tiriti is authoritative. Then, different aspects of Te Tiriti are linked to various parts of Māori food sovereignty and traditional Māori concepts. Through breaching tuku whenua in Te Tiriti, land issues have arisen: the methods used to confiscate Māori land and the effects this had on traditional food, health and urbanisation are discussed. Culture as taonga is examined: the ways in which losing a food system leads to losing traditions, and how this breaches Te Tiriti. Then the (mostly negative) effects of this shift in food systems on the environment is examined, and how this relates back to taonga as well as the traditional duty of kaitiakitanga and, again, the breaching of Te Tiriti. Lastly, the underlying issue of power is examined in relation to rangatiratanga and overall sovereignty–this is also linked to Te Tiriti.

Drinking water issues in Rural India: Need for stakeholders’ participation in Water resources management

Water is the very essential livelihood for mankind. The United Nations suggest that each person needs 20-50 litres of water a day to ensure basic needs of drinking, cooking and cleaning. It was also endorsed by the Indian National Water Policy 2002, with the provision that adequate safe drinking water facilities should be provided to the entire population both in urban and in rural areas. About 1.42 million rural habitations in India are affected by chemical contamination. The provision of clean drinking water has been given priority in the Constitution of India, in Article 47 conferring the duty of providing clean drinking water and improving public health standards to the State. Excessive dependence of ground water results in depletion of ground water, water contamination and water borne diseases. Thus, access to safe and reliable water supply is one of the serious concerns in rural water supply programme. Though government takes certain serious steps in addressing the drinking water issues in rural areas, still there is a huge gap between demand and supply. The Draft National Water Policy 2012 also states that Water quality and quantity are interlinked and need to be managed in an integrated manner and with Stakeholder participation. Water Resources Management aims at optimizing the available natural water flows, including surface water and groundwater, to satisfy competing needs. The World Bank also emphasizes on managing water resources, strengthening institutions, identifying and implementing measures of improving water governance and increasing the efficiency of water use. Therefore stakeholders’ participation is viewed important in managing water resources at different levels and range. This paper attempts to reflect up on portray the drinking water issues in rural India, and highlights the significance of Integrated Water Resource Management as the significant part of Millennium Development Goals, and Stakeholders’ participation in water resources management.