Fields in Mathematical Logic

Chapter 1, Computable fields. We define computable fields, using the natural numbers NN. We show that if K is a computable field, then its algebraic closure K' is a computable field. We show that if K is a computable field such that the equality relation is computably enumerable, then its algebraic closure K' is a computable field such that the equality relation is computably enumerable. We show that if K is a computable field such that the equality relation is computable, then its algebraic closure K' is a computable field such that the equality relation is computable. With this definition of a computable field, we show that there is however a computable field with a non-computable property. Chapter 2, Model theory of fields. We use first order logic to look at the (+,.,0,1)-structures called fields. Let TF be the theory of fields. The complete extensions of TF gives the best descriptions of fields. Let Th K be the complete theory of a field K. We consider the isomorphism classes of models of a theory T. If K is a finite field, then Th K has just 1 isomorphism class. If K is an infinite field, then Th K has an isomorphism class of cardinality k, for any infinite cardinality k. Hence we look at the isomorphism classes with the smallest infinite cardinality, the countable cardinality w. We try to find the number of such classes of a theory T. Let n(T) be the number of isomorphism classes of models of cardinality w, of a theory T. We show that if K is a finite field, then n(Th K) = 0. We show that if K is a infinite field, then n(Th K) is not one of 0, 1, 2. We show that if K is an infinite field which is algebraically closed, then n(Th K) = w. We show that if K is an infinite field of characteristic 0 with finite algebraic degree over its prime subfield, then n(Th K) > w. From observations we make, we consider the following possible pattern: if K is a finite field then n(Th K) = 0, if K is an infinite field which is algebraically closed then n(Th K) = w, if K is an infinite field which is not algebraically closed then n(Th K) > w. I do not have a proof of this pattern. We consider the (+,.,0,1,<=)-structures called ordered fields, and show that the order relation (<=) of the rational numbers QQ can be defined by a (+,.,0,1)-formula. At the end we show that QQ and QQ extended with a transcendental element c are not elementary equivalent (+,.,0,1)-structures

Das Studium der Reden Ciceros in Spanien im fünfzehnten und sechzehnten Jahrhundert

Ausgehend von den zahlreichen frühen Übersetzungen antiker griechischer und römischer Autoren ins Katalanische und Kastilische sucht der vorliegende Aufsatz die Eigenart der humanistischen Bewegung in Spanien im fünfzehnten und sechzehnten Jahrhundert anhand der Beschäftigung mit den Werken Ciceros, vor allem der Reden herauszuarbeiten. Sie erweisen sich als geprägt von zwei Faktoren, von dem Interesse an sprachlichen Phänomenen, deren Klärung Nebrija und andere besonders für die eigene Sprache fruchtbar machen, und von dem Interesse an rhetorischen Aspekten, die vor allem für die Verkündigung der christlichen Botschaft genutzt werden, während nur wenige, vor allem im Ausland (Italien) tätige Spanier sich durch Ciceroimitation, Beobachtungen zum Text oder Untersuchungen von Sachproblemen auszeichnen. Appendices geben einen knappen Überblick über die Entwicklung der Folgezeit, nicht zuletzt den Einfluß der Jesuiten.Starting from the numerous early translations of ancient Greek and Roman authors, this paper investigates the elements peculiar to the humanistic movement in Spain in the fifteenth and sixteenth centuries on the basis of the interest shown in Cicero's works, especially his speeches. This interest concentrates on the one hand on linguistic phenomena the clarification of which Nebrija and others make use of for their own language, and on the other hand it turns to rhetorical aspects from which authors of preaching manuals try to benefit. Only a few Spaniards, especially those living ouside the country (e.g. in Italy) endeavour to imitate Cicero's style or concern themselves with problems of the text or antiquarian issues. The later development (including the special role of the Jesuits) is briefly illustrated by appendices

The role of the wingless gene in the control of growth and pattern formation during drosophila wing development

Recent work on Drosophila limb development has indicated that short-range interactions between distinctly specified populations of cells (compartments) establish organizing centers at compartment boundaries. These organizing centers direct pattern formation and growth in the developing limbs. In the Drosophila wing imaginal disc, there are at least two such organizing centers, located at the anterior/posterior (A/P) and dorsal/ventral (D/V) compartment boundaries. The genetic hierarchies which establish these organizers are starting to be understood, and it also appears that the key mediators of some of the organizers have been identified. Thus Decapentaplegic (Dpp, a secreted signalling molecule of the TGF-B family) is the mediator of the A/P organizer, while Wingless (DWnt-l, a secreted molecule of the Wnt family) is a key mediator of the D/V organizer. In this thesis, several aspects of Wingless function in the wing imaginal disc are examined. Two regulatory mutations, spadeflag (spdfg) and Sternopleural (Sp), that affect Wingless expression in the wing imaginal disc are characterized. The analysis of the mutation spdfg, together with other data, identifies a role of Wingless as a localized mitogen in the developing wing hinge, and also indicates that cells in different regions of the wing disc respond very differently to the Wingless signal. The mutations spdfg and Sp are also among the tools used to examine the position of Wingless in the genetic hierarchy that establishes and mediates the activity of the D/V organizer. These experiments extend the evidence suggesting that Wingless mediates both short-range and long-range effects of the D/V organizer. Wg does so by controlling the expression domains of different target genes, including the acheate-scute genes, Distal-less and vestigial. Finally, the mechanism by which Wingless mediates the activity of the D/V organizer is examined. The results obtained suggest that Wingless functions as a long-range morphogen

Henry Wells Lawrence Memorial Lectures, Number 2

Second, third, and fourth lectures in a series established in honor of Henry Wells Lawrence, Professor of History at Connecticut College. Contents of this volume are: The Pattern of Democratic Change in the United States: Andrew Jackson and Franklin Delano Roosevelt, Arthur Meier Schlesinger, Jr., Associate Professor of History, Harvard University The Problem of the Democratization of Germany, Carl Joachim Friedrich, Professor of Government, Harvard University Variations on the Liberal Theme, Alpheus Thomas Mason, McCormick Professor of Jurisprudence, Princeton Universityhttps://digitalcommons.conncoll.edu/ccbooks/1008/thumbnail.jp