This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as - via extended case studies - carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.
This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to or...
The history of reciprocity laws is a history of algebraic number theory. This is a book on reciprocity laws, and our introductory remark is placed at the beginning as a warning: in fact a reader who is acquainted with little more than a course in elementary number theory may be surprised to learn that quadratic reciprocity does - in a sense that we will explain - belong to the realm of algebraic number theory. Heeke { 348, p. 59]) has formulated this as follows: Von der Entdeckung des Reziprozitiitsgesetzes kann man die mo derne Zahlentheorie datieren. Seiner Form nach gehort es noch der...
The history of reciprocity laws is a history of algebraic number theory. This is a book on reciprocity laws, and our introductory remark is placed at ...
This unique book on the subject addresses fundamental problems and will be the standard reference for a long time to come. The authors have different scientific origins and combine these successfully, creating a text aimed at graduate students and researchers that can be used for courses and seminars.
This unique book on the subject addresses fundamental problems and will be the standard reference for a long time to come. The authors have differe...
1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's Elemen- ta, where we find a proof of their infinitude, now regarded as canonical. One feels that Euclid's argument has its place in The Book, often quoted by the late Paul ErdOs, where the ultimate forms of mathematical arguments are preserved. Proofs of most other results on prime number distribution seem to be still far away from their optimal form and the aim of this book is to present the development of methods with which such problems were...
1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's El...
The common solutions of a finite number of polynomial equations in a finite number of variables constitute an algebraic variety. The degrees of freedom of a moving point on the variety is the dimension of the variety. A one-dimensional variety is a curve and a two-dimensional variety is a surface. A three-dimensional variety may be called asolid. Most points of a variety are simple points. Singularities are special points, or points of multiplicity greater than one. Points of multiplicity two are double points, points of multiplicity three are tripie points, and so on. A nodal point of a...
The common solutions of a finite number of polynomial equations in a finite number of variables constitute an algebraic variety. The degrees of freedo...
The finite simple groups are basic objects in algebra since many questions about general finite groups can be reduced to questions about the simple groups. Finite simple groups occur naturally in certain infinite families, but not so for all of them: the exceptions are called sporadic groups, a term used in the classic book of Burnside Bur] to refer to the five Mathieu groups. There are twenty six sporadic groups, not definitively organized by any simple theme. The largest of these is the monster, the simple group of Fischer and Griess, and twenty of the sporadic groups are involved in the...
The finite simple groups are basic objects in algebra since many questions about general finite groups can be reduced to questions about the simple gr...
An attractive book on the intersection of analysis and numerical analysis, deriving classical boundary integral equations arising from the potential theory and acoustics. This self-contained monograph can be used as a textbook by graduate/postgraduate students. It also contains a lot of carefully chosen exercises.
An attractive book on the intersection of analysis and numerical analysis, deriving classical boundary integral equations arising from the potentia...
Cyclotomic fields have always occupied a central place in number theory, and the so called "main conjecture" on cyclotomic fields is arguably the deepest and most beautiful theorem known about them. It is also the simplest example of a vast array of subsequent, unproven "main conjectures'' in modern arithmetic geometry involving the arithmetic behaviour of motives over p-adic Lie extensions of number fields. These main conjectures are concerned with what one might loosely call the exact formulae of number theory which conjecturally link the special values of zeta and L-functions to purely...
Cyclotomic fields have always occupied a central place in number theory, and the so called "main conjecture" on cyclotomic fields is arguably the d...
Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. The book also contains some examples of computations and applications.
Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieti...
The present book is an English translation of Algebre Locale - Multiplicites published by Springer-Verlag as no. 11 of the Lecture Notes series. The original text was based on a set of lectures, given at the College de France in 1957-1958, and written up by Pierre Gabriel. Its aim was to give a short account of Commutative Algebra, with emphasis on the following topics: a) Modules (as opposed to Rings, which were thought to be the only subject of Commutative Algebra, before the emergence of sheaf theory in the 1950s); b) H omological methods, a la Cartan-Eilenberg; c) Intersection...
The present book is an English translation of Algebre Locale - Multiplicites published by Springer-Verlag as no. 11 of the Lecture Notes series. The o...
