The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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

Computational modelling of impact damage in brittle materials

A fast incremental/iterative solution procedure that handles "snap-through"

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Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

Interfacing nonmatching FEM meshes: the zero moment rule

A general three-field variational principle is obtained for the motion of an acoustic fluid enclosed in a rigid or flexible container by the method of canonical decomposition applied to a modified form of the wave equation in the displacement potential The general principle is specialized to a mixed two-field principle that contains the fluid displacement potential and pressure as independent fields Semidiscrete finite-element equations of motion based on this principle are displayed

Treatment of coupled fluid-structure interaction problems by a mixed variational principle

Penalty Spring Stabilization of Singular Jacobians

: This report constitutes a reference manual for the Underwater Shock Analysis (USA) Code, a computer program for calculation of the transient response of a submerged structure to a spherical shock wave of arbitrary pressure profile and source location. The code considers the structure to be linear-elastic and treats the surrounding fluid as an infinite acoustic medium. A discrete-element (finite-element, finite-difference) computational model is used for the structure, while the computational model for the fluid is based upon the Doubly Asymptotic Approximation. (Author)

The Underwater Shock Analysis (USA) Code, A Reference Manual.

We present a procedure for matching a displacement field to a given strain field, or vice-versa, over an arbitrary domain, which can be a finite element. The fitting criterion used is minimization of a dislocation energy functional. The strain field, whether given or fitted, need not be compatible. The method has four immediate applications: (i) finite element stiffness formulation based on fitting assumed-natural-strain (ANS) fields to node displacements; (ii) pointwise recovery of an internal displacement field in ANS elements as required for consistent mass, body load or geometric stiffness computations; (iii) recovery of smoothed strains from node displacements for stress post-processing, and (iv) system identification and damage detection from experimental data. The article focuses on application (i) for the strain fitting (SF) problem and (ii) for the displacement fitting (DF) problem. The separation of mean and deviatoric strains is emphasized whenever it is found convenient to simplify calculations.

Carlos A. Felippa

Papers

Interfacing nonmatching FEM meshes: the zero moment rule

Treatment of coupled fluid-structure interaction problems by a mixed variational principle

Penalty Spring Stabilization of Singular Jacobians

The Underwater Shock Analysis (USA) Code, A Reference Manual.

Fitting strains and displacements by minimizing dislocation energy