There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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).

A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

The result of this paper is that any fast-decaying function can be represented as an integral over the canonical Maslov operator, on a special Lagrangian manifold, acting on a specific function. This representation enables one to construct effective explicit formulas for asymptotic solutions of a vast class of linear hyperbolic systems with variable coefficients.

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Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations

It is shown that the recently discovered finite-zone, almost-periodic solutions may, on the one hand, serve as the foundation for the development of the multiphase WKB method in nonlinear equations (the method of Whitham) and, on the other hand, serve to define Lagrangian manifolds with complex germs which can be (second) quantized in the quasiclassical approximation.

Finite-zone, almost-periodic solutions in WKB approximations

Multi-phase solutions of the Benjamin-Ono equation and their averaging

Linear problems in mathematical physics where the adiabatic approximation is used in a wide sense are studied. From the idea that all these problems can be treated as problems with an operator-valued symbol, a general regular scheme of adiabatic approximation based on operator methods is proposed. This scheme is a generalization of the Born–Oppenheimer and Maslov methods, the Peierls substitution, etc. The approach proposed in this paper allows one to obtain “effective” reduced equations for a wide class of states inside terms (i.e., inside modes, subbands of dimensional quantization, etc.) with possible degeneration taken into account. Next, by application of asymptotic methods, in particular the semiclassical approximation method, to the reduced equation, the states corresponding to a distinguished term (effective Hamiltonian) can be classified. It is shown that the adiabatic effective Hamiltonian and the semiclassical Hamiltonian can be different, which results in the appearance of “nonstandard characteristics” while passing to classical mechanics. This approach is used to construct solutions of several problems in wave and quantum mechanics, particularly problems in molecular physics, solid-state physics, nanophysics and hydrodynamics.

Operator separation of variables for adiabatic problems in quantum and wave mechanics

A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr—Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk<n (incomplete dimension). The construction method associates the families of such tori with spectral series covering the region of “intermediate” quantum numbers. The construction includes, in particular, new quantization conditions of Bohr—Sommerfeld type in which other characteristics of the tori appear instead of the Maslov index. Applications and also generalizations of the theory to Lie groups will be presented in subsequent publications of the series.

S. Yu. Dobrokhotov

Papers

Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations

Finite-zone, almost-periodic solutions in WKB approximations

Multi-phase solutions of the Benjamin-Ono equation and their averaging

Operator separation of variables for adiabatic problems in quantum and wave mechanics

Semiclassical maslov asymptotics with complex phases. I. General approach