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.

Violence and social orders: a conceptual framework for interpreting recorded human history

Violencia y órdenes sociales: un marco conceptual para la interpretación de la historia humana registrada

UK construction orders and output: A comparison of government and commercially available data 1995-2001

Because of the importance and potential usefulness of construction market statistics to firms and government, consistency between different sources of data is examined with a view to building a predictive model of construction output using construction data alone. However, a comparison of Department of Trade and Industry (DTI) and Office for National Statistics (ONS) series shows that the correlation coefcient (used as a measure of consistency) of the DTI output and DTI orders data and the correlation coefficient of the DTI output and ONS output data are low. It is not possible to derive a predictive model of DTI output based on DTI orders data alone. The question arises whether or not an alternative independent source of data may be used to predict DTI output data. Independent data produced by Emap Glenigan (EG), based on planning applications, potentially offers such a source of information. The EG data records the value of planning applications and their planned start and finish dates. However, as this data is ex ante and is not correlated with DTI output it is not possible to use this data to describe the volume of actual construction output. Nor is it possible to use the EG planning data to predict DTI construc-tion orders data. Further consideration of the issues raised reveal that it is not practically possible to develop a consistent predictive model of construction output using construction statistics gathered at different stages in the development process.

War, Peace and World Orders from Antiquity until the 20th century

Chapters contributed by experts on each period examine how world views have determined the view of war and peace, and the conduct of war, throughout European history.