Explicit Construction of Self-Dual Integral Normal Bases for the Square-Root of the Inverse Different

Let $K$ be a finite extension of \Q_p, let $L/K$ be a finite abelian Galois extension of odd degree and let \bo_L be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional \bo_L-ideal with square equal to the inverse different of $L/K$. For $p$ an odd prime and L/\Q_p contained in certain cyclotomic extensions, Erez has described integral normal bases for A_{L/\Q_p} that are self-dual with respect to the trace form. Assuming K/\Q_p to be unramified we generate odd abelian weakly ramified extensions of $K$ using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions

Self-Dual Integral Normal Bases and Galois Module Structure

Let $N/F$ be an odd degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak O}_N$ and {\mathfrak O}_F=\bo respectively. Let $\mathcal{A}$ be the unique fractional ${\mathfrak O}_N$-ideal with square equal to the inverse different of $N/F$. Erez has shown that $\mathcal{A}$ is a locally free ${\mathfrak O}[G]$-module if and only if $N/F$ is a so called weakly ramified extension. There have been a number of results regarding the freeness of $\mathcal{A}$ as a $\Z[G]$-module, however this question remains open. In this paper we prove that $\mathcal{A}$ is free as a $\Z[G]$-module assuming that $N/F$ is weakly ramified and under the hypothesis that for every prime $\wp$ of ${\mathfrak O}$ which ramifies wildly in $N/F$, the decomposition group is abelian, the ramification group is cyclic and $\wp$ is unramified in F/\Q. We make crucial use of a construction due to the first named author which uses Dwork's exponential power series to describe self-dual integral normal bases in Lubin-Tate extensions of local fields. This yields a new and striking relationship between the local norm-resolvent and Galois Gauss sum involved. Our results generalise work of the second named author concerning the case of base field \Q

Rynki pracy przyszłości: wyzwanie dla polityki edukacyjnej

An analysis of the relationships between new technologies and „human resources” requires the simultaneous application of many methodologies. This approach is imposed by the complexity of the phenomenon. As a rule, the micro level is discussed in the rich literature of the subject matter: enterprises and sectors, especially changes of a quantitative nature. The author approaches the problem from two aspects: micro and macro, but conclusions relating to the wider dimension dominate. The article focuses on the aspect of changes and relationships between supply and demand. In the final part the author describes the main determinants of the educational policy of governments in this area and indicates the need for wider access to data for a better understanding of the problem. This is closely connected with work on this problem taken up by the Centre for Educational Research and Innovation (CERI) from the Organisation for Economic Co-operation and Development (OECD).Analiza relacji występujących między nowymi technologiami i „zasobami ludzkimi” wymaga równoczesnego stosowania wielu metodologii. Potrzebę tę wymusza złożoność zjawiska. Bogata literatura przedmiotu dotyczy głównie poziomu mikro: przedsiębiorstw i sektorów gospodarki, przede wszystkim zaś - zmian o charakterze ilościowym. Artykuł ujmuje problem z dwóch perspektyw: mikro i makro, lecz dominują wnioski odnoszące się do szerszego planu. Skoncentrowano się w nim na aspekcie zmian i związków występujących między podażą i popytem. W końcowej części zostały określone główne wyznaczniki polityki edukacyjnej rządów w tym zakresie, ze wskazaniem potrzeby rozszerzenia dostępności danych w celu lepszego zrozumienia problemu. Wiąże się to ściśle z pracami nad tym zagadnieniem, podjętymi przez Centrum Badań Edukacyjnych i Innowacji (Centre for Educational Research and lnnovation - CERI), działającym przy Organizacji Współpracy Ekonomicznej i Rozwoju (Organisation for Economic Co-operation and Development - OECD)

Modelling Vertical Migration:Combined Effects of Environment and Animal

Vertical migration is one of the most important types of animal migrations, encompassing most zooplankton and many fish species in the world’s oceans

Family formation and structure in Finland

The article is an overview of the changes in family formation and structure in Finland during the last few decades. The period examined extends from the 1950s until the beginning of the 1990s with the emphasis on the current situation. The article starts with a look on the changes in union establishment and shows that, on the whole, the age at starting the union has been unchanged if one takes into consideration that a union today starts with premarital cohabitation. There are more divorces than earlier. The dissolution frequency is higher among consensual unions than in marriages. The most common type of family is a family consisting of married parents with children. Living in marriage has decreased among young persons while consensual unions have become more common. The proportion of one-parent families has been almost unchanged during the last two decades. In the 1990s there has been a weak tendency to a growing proportion of families with three children. The most apparent trend during the following decades will be the rapidly growing number of families without children

The Mind of Messiaen

In the course of this study I have accomplished my initial goal of looking into the mind of Mesiaen. It is a fascinating and colorful mind, filled with beliefs and theories which are evident throughout all his works. It appears that his surroundings influenced his believe system, and that it was, in turn, his believe system which influenced his incredible compositional style. Some aspects, such as that of sound-color association, appear to be aspects that he was born with, while other aspects, such as that of rhythm, are aspects which he absorbed in the process of his studies and in the attempting to assert himself as an original composer. Aspects such as bird calls and his religious faith came from the deeply rooted loves that he developed as a child, and all of these mixed together in an occupation which was, as far as he was concerned, his calling in life. It is understandable that such a devoted person would mix all of his experiences together in pursuit of that which he believes to be his life\u27s plan and purpose, and when this is understood, the rich and exotic style in which Messiaen wrote can also become clearly understood

Making the right decisions?

This theory testing study investigates the decision-making of the member states in the Dispute Settlement Body of the WTO, using a game theoretical approach. Game theory has its origins in political philosophy and can thus be valuable when analysing the behaviour of states in international regimes. Through using the specific game theory model Stag Hunt, this study aims to examine if such a model can be used to explain the behaviour of states in dispute settlement situations, if there is a difference in strategies between the member states depending on their economic status and if there is a difference in strategies between member states before and after the Great Recession of 2008-2009. To answer this, a quantitative analysis of the disputes is first made, to which a game theoretic model later is applied. When analysing the implications of the model on the dispute settlement, it is concluded that using the Stag Hunt game, many segments of the strategies of the member states can be explained using the model. However, there is a need for further development of the model to give a more complete and fair image of the states’ strategies in dispute settlement situations. The result from the study also shows that there is a difference in behaviour depending on the economic status of the countries. However, there is no shown change in behaviour by the states after the financial crisis