to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

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 Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate.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

Introduction to lie algebras and representation theory

Cohesion-tension transport of water is an energetically efficient way to carry large amounts of water from the roots up to the leaves. However, the cohesion-tension mechanism places the xylem water under negative hydrostatic pressure ( P x ), rendering it susceptible to cavitation. There are conflicts among the structural requirements for minimizing cavitation on the one hand vs maximizing efficiency of transport and construction on the other. Cavitation by freeze-thaw events is triggered by in situ air bubble formation and is much more likely to occur as conduit diameter increases, creating a direct conflict between conducting efficiency and sensitivity to freezing induced xylem failure. Temperate ring-porous trees and vines with wide diameter conduits tend to have a shorter growing season than conifers and diffuse-porous trees with narrow conduits. Cavitation by water stress occurs by air seeding at interconduit pit membranes. Pit membrane structure is at least partially uncoupled from conduit size, leading to a much less pronounced trade-off between conducting efficiency and cavitation by drought than by freezing. Although wider conduits are generally more susceptible to drought-induced cavitation within an organ, across organs or species this trend is very weak. Different trade-offs become apparent at the level of the pit membranes that interconnect neighbouring conduits. Increasing porosity of pit membranes should enhance conductance but also make conduits more susceptible to air seeding. Increasing the size or number of pit membranes would also enhance conductance, but may weaken the strength of the conduit wall against implosion. The need to avoid conduit collapse under negative pressure creates a significant trade-off between cavitation resistance and xylem construction cost, as revealed by relationships between conduit wall strength, wood density and cavitation pressure. Trade-offs involving cavitation resistance may explain the correlations between wood anatomy, cavitation resistance, and the physiological range of negative pressure experienced by species in their native habitats.

https://www.ualberta.ca/~hacke/pdf/Perspectives2001.pdf

Functional and ecological xylem anatomy

Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems.

https://www.mdpi.com/2073-8994/2/2/658/pdf

Lie Symmetries of Differential Equations: Classical Results and Recent Contributions

We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit constant solutions to which there correspond non-constant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.

When nonautonomous equations are equivalent to autonomopus ones

In this paper, we consider the unsteady equations that govern two- and three-dimensional flows of a perfect gas. We explicitly characterize various classes of exact solutions by introducing some invertible transformations suggested by the invariance with respect to Lie groups of point symmetries and using suitable transformations known in literature as substitution principles.

Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles

In this paper, we consider the equations governing an inviscid, thermally non-conducting fluid of infinite electrical conductivity in the presence of a magnetic field and subject to no extraneous force. By using conveniently the Lie point symmetries admitted, we map the governing system into an equivalent autonomous form. The transformed system can be directly inspected in order to find some simple solutions that, written in terms of the original variables, provide non-trivial exact solutions of the system at hand. Use is made of some finite transformations known in the literature as substitution principles, enabling us to build exact solutions containing some arbitrary functions. The link between the substitution principle and the recently discovered Bogoyavlenskij symmetries for equilibrium magnetohydrodynamics is also discussed. Some of the recovered solutions are considered to solve well-known physically relevant boundary value problems and the linear stability analysis is performed, thus generalizing well-established results.

http://iopscience.iop.org/article/10.1088/0305-4470/38/40/019/pdf

Exact solutions to the ideal magneto-gas-dynamics equations through Lie group analysis and substitution principles

We adopt an operatorial method based on the so-called creation, annihilation and number operators in the description of different systems in which two populations interact and move in a two-dimensional region. In particular, we discuss diffusion processes modeled by a quadratic Hamiltonian. This general procedure will be adopted, in particular, in the description of migration phenomena. With respect to our previous analogous results, we use here fermionic operators since they automatically implement an upper bound for the population densities.

http://www1.unipa.it/fabio.bagarello/ricerca/articolipdf/migrazione_revised2.pdf

Francesco Oliveri

Papers

Lie Symmetries of Differential Equations: Classical Results and Recent Contributions

When nonautonomous equations are equivalent to autonomopus ones

Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles

Exact solutions to the ideal magneto-gas-dynamics equations through Lie group analysis and substitution principles

A phenomenological operator description of interactions between populations with applications to migration