Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms. As a consequence, algorithmic number theory has gradually emerged as an important and distinct field with connections to computer science and cryptography as well as other areas of mathematics. This text provides a comprehensive introduction to algorithmic number theory for beginning graduate students, written by the leading experts in the field. It includes several articles that cover the essential topics in this area, and in addition, there are contributions pointing in broader directions, including cryptography, computational class field theory, zeta functions and L-series, discrete logarithm algorithms, and quantum computing.

Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms. As a consequence, algorithmic number theory has gradually emerged as an important and distinct field with connections to computer science and cryptography as well as other areas of mathematics. This 2008 text provides a comprehensive introduction to algorithmic number theory for beginning graduate students, written by the leading experts in the field. It includes several articles that cover the essential topics in this area, and in addition, there are contributions pointing in broader directions, including cryptography, computational class field theory, zeta functions and L-series, discrete logarithm algorithms, and quantum computing.

On March 8/9, 1976 a workshop on "Mathematical Models in Medicine" was held at Mainz (German Federal Republic) by the group of "Mathe- matical Models" of the Deutsche Gesellschaft fUr Medizinische Doku- mentation, Informatik und Statistik. Purpose of this conference was to bring together experts from the GFR and neighbouring countries working in this field to evaluate possibilities and limits of this area of research in discussions with interested participants. This issue of Lecture Notes contains the invited contributions as well as the relevan~ remarks made by the discussants. Corresponding to the aims of the workshop the contributors had been encouraged to demon- strate their mathematical models in the light of actual applied examples. It had been our intention to restrict attention to a small number of specific areas in order to achieve a concentrated in depth treatment in these restricted areas. The areas chosen contain two - Epidemio- logy and Cell Models - which in the organisers feeling are not yet as well established in Continental Europe and one - Pharmacokinetic- with a more direct appeal to applied workers. While in the Epidemio- logy of infectious and parasitic diseases today strategies of control and eradication are gaining importance, the cell models are concerned with explaining the modes of genesis of cancerous growth and the kinetics and interactions within multi-cell structures.