This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

Real and Complex Analysis

What do you do to start reading dimension theory? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this dimension theory.

Aspects of topology: On dimension theory

Logic, Semantics, Metamathematics. By A. Tarski. Translated by J. H. Woodger. Pp. 471. 60s. 1956. (Clarendon Press)

The development of techniques for autonomous navigation in real-world environments constitutes one of the major trends in the current research on robotics. An important problem in autonomous navigation is the need to cope with the large amount of uncertainty that is inherent of natural environments. Fuzzy logic has features that make it an adequate tool to address this problem. In this paper, we review some of the possible uses of fuzzy logic in the field of autonomous navigation. We focus on four issues: how to design robust behavior-producing modules; how to coordinate the activity of several such modules; how to use data from the sensors; and how to integrate high-level reasoning and low-level execution. For each issue, we review some of the proposals in the literature, and discuss the pros and cons of fuzzy logic solutions.

The uses of fuzzy logic in autonomous robot navigation

MV-algebras can be viewed either as the Lindenbaum algebras of Łukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] n . The maximal spectrum of Free n is canonically homeomorphic to [0, 1] n , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic l...

Invariant Measures in Free MV-Algebras

MV-algebras can be viewed either as the Lindenbaum algebras of Lukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free_n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0,1]^n. The maximal spectrum of Free_n is canonically homeomorphic to [0,1]^n, and the automorphisms of the algebra are in 1-1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this paper we prove that the only probability measure on [0,1]^n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure.

Invariant measures in free MV-algebras

Adding structure to MV-algebras

We give a self-contained geometric proof of the completeness theorem for the infinite-valued sentential calculus of Łukasiewicz.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0DF2C1B8F4F980ABB25EFEE1C1B92EC0/S0022481200018661a.pdf/div-class-title-a-geometric-proof-of-the-completeness-of-the-lukasiewicz-calculus-div.pdf

A geometric proof of the completeness of the Lukasiewicz calculus

The Minkowski question mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the Monkemeyer continued fraction algorithm with an appropriate tent map.

Giovanni Panti

Papers

Invariant Measures in Free MV-Algebras

Invariant measures in free MV-algebras

Adding structure to MV-algebras

A geometric proof of the completeness of the Lukasiewicz calculus

Multidimensional continued fractions and a Minkowski function