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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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

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The surface chemistry of amorphous silica. Zhuravlev model

1: Magnetic Particles: Preparation, Properties and Applications: M. Ozaki. 2: Maghemite (gamma-Fe2O3): A Versatile Magnetic Colloidal Material C.J. Serna, M.P. Morales. 3: Dynamics of Adsorption and Oxidation of Organic Molecules on Illuminated Titanium Dioxide Particles Immersed in Water M.A. Blesa, R.J. Candal, S.A. Bilmes. 4: Colloidal Aggregation in Two-Dimensions A. Moncho-Jorda, F. Martinez-Lopez, M.A. Cabrerizo-Vilchez, R. Hidalgo Alvarez, M. Quesada-PMerez. 5: Kinetics of Particle and Protein Adsorption Z. Adamczyk.

Surface and Colloid Science

This Review discusses ionic electrets: their preparation, their mechanisms of formation, tools for their characterization, and their applications. An electret is a material that has a permanent, macroscopic electric field at its surface; this field can arise from a net orientation of polar groups in the material, or from a net, macroscopic electrostatic charge on the material. An ionic electret is a material that has a net electrostatic charge due to a difference in the number of cationic and anionic charges in the material. Any material that has ions at its surface, or accessible in its interior, has the potential to become an ionic electret. When such a material is brought into contact with some other material, ions can transfer between them. If the anions and cations have different propensities to transfer, the unequal transfer of these ions can result in a net transfer of charge between the two materials. This Review focuses on the experimental evidence and theoretical models for the formation of ionic electrets through this ion-transfer mechanism, and proposes--as a still-unproved hypothesis--that this ion-transfer mechanism may also explain the ubiquitous contact electrification ("static electricity") of materials, such as organic polymers, that do not explicitly have ions at their surface.

Electrostatic Charging Due to Separation of Ions at Interfaces: Contact Electrification of Ionic Electrets

Water purification using magnetic assistance: a review.

Synthesis of hydroxyapatite for biomedical applications

Water and electrolyte solutions were exposed for 5 min to a weak static magnetic field (B = 15 mT). Their conductivity and the amount of evaporated water were then measured as a function of time. Simultaneously, these quantities were determined for magnetically untreated samples, as reference systems. It was found that a magnetic field influences these two parameters and their changes depend on the thermodynamic functions of hydration of these ions. A roughly linear change in conductivity versus 'scaled' functions was obtained. On this basis it was concluded that the magnetic field causes changes in the hydration shells of the ions.

http://www.tma-ram.pl/pdf/publications/Article-Journal.of.Colloid.and.Interface.Science.pdf

Effects of a static magnetic field on water and electrolyte solutions.

The thin-layer wicking technique was used to determine the solid surface free energy components for thin-layer chromatography silica (Merck) by the Washburn equation. It has been found that wick-wetting is a reproducible process and the results of the wetting rates expressed as the relationship of the penetrated distances squared, x2 , are a linear function of time, t. However, in some systems an inflection appeared at a distance 3 5 x 5 5 cm, giving thus two straight-line sections with somewhat different slopes. nAlkanes, diiodomethane, water, and formamide were used as the penetrating liquids. Van Oss et al.'s approach to the surface free energy was applied. Thus, apolar, ysLw, and polar electron donor, ys, and electron acceptor, ys+, components were determined for both the laboratory-made glass plates with a deposited Si02 thin layer as well as commercial ones. Practically the same value of ysLw was obtained for both kinds of plates. However, the commercial plates showed higher values of the yscomponent. It is probably due to the presence of some adhesive additives. The polar ys+ component is small for both plates. It is believed that this technique and the approach to the surface free energy components are very useful for testing the wetting properties of solids by liquids.

Use of the Washburn equation for surface free energy determination

The contact angle is one of the most sensitive experimental values describing a junction between three phases, being influenced by the composition and properties of contacting media as well as the ...

https://www.icevirtuallibrary.com/doi/pdf/10.1680/jsuin.19.00007

Contact angles: history of over 200 years of open questions

Water was exposed for different times to weak static magnetic field (MF) generated from a stack of magnets (B = 15 mT) or from a single permanent magnet (B = 0.27 T) at flow conditions. The water conductivity and the amount of evaporated water were measured as a function of time following the application of MF. It was found that the MF decreases the water conductivity, which is inversely proportional to the flow rate, and increases the amount of evaporated water, even after the water's distillation. The effects are due to the hydrogen bond network strengthening and the perturbation of gas/liquid interface from the air nanobubbles in the water.

Lucyna Hołysz

Papers

Synthesis of hydroxyapatite for biomedical applications

Effects of a static magnetic field on water and electrolyte solutions.

Use of the Washburn equation for surface free energy determination

Contact angles: history of over 200 years of open questions

Effects of static magnetic field on water at kinetic condition