/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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).

Time is divided by geologists according to marked shifts in Earth's state. Recent global environmental changes suggest that Earth may have entered a new human-dominated geological epoch, the Anthropocene. Here we review the historical genesis of the idea and assess anthropogenic signatures in the geological record against the formal requirements for the recognition of a new epoch. The evidence suggests that of the various proposed dates two do appear to conform to the criteria to mark the beginning of the Anthropocene: 1610 and 1964. The formal establishment of an Anthropocene Epoch would mark a fundamental change in the relationship between humans and the Earth system.

Defining the Anthropocene

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

https://iopscience.iop.org/article/10.1088/0034-4885/35/1/306/pdf

Semiclassical approximations in wave mechanics

This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.

Idempotent functional analysis: An algebraic approach

CONTENTSPreface § 1. The connection between geometric quantization and deformations of Poisson brackets and the theory of pseudodifferential operators (PDO) § 2. The calculus of PDO's with symbols on general symplectic manifolds § 3. Spectral series of nearly integrable Hamiltonians References

https://iopscience.iop.org/article/10.1070/RM1984v039n06ABEH003183/pdf

Asymptotic and geometric quantization

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

We consider tuples {N
 jk
 }, j = 1, 2, ..., k = 1, ..., q
 j
 , of nonnegative integers such that $$
\sum\limits_{j = 1}^\infty {\sum\limits_{k = 1}^{q_j } {jN_{jk} } } \leqslant M.
$$
 Assuming that q
 j
 ∼ j
 d−1, 1 < d < 2, we study how the probabilities of deviations of the sums $$
\sum\nolimits_{j = j_1 }^{j_2 } {\sum\nolimits_{k = 1}^{q_j } {N_{jk} } } 
$$
 N
 jk
 from the corresponding integrals of the Bose-Einstein distribution depend on the choice of the interval [j
 1,j
 2].

On the Distribution of Integer Random Variables Related by a Certain Linear Inequality: I

In the present paper, we describe an approach to thermodynamics that does not involve Bogolyubov chains or Gibbs ensembles. We present isotherms, isochores, and isobars of various pure gases, as well as binodals, i.e., lines along which gas becomes liquid, and spinodals (endpoints of isotherms). We study supercritical phenomena for values of temperature and pressure above the critical ones. A lot of attention is paid to the region of negative pressures. The superfluid component for supercritical phenomena is described, as well as the thermodynamics of nanostructures and superfluidity in nanotubes.

V. P. Maslov

Papers

Idempotent functional analysis: An algebraic approach

Asymptotic and geometric quantization

Finite-zone, almost-periodic solutions in WKB approximations

On the Distribution of Integer Random Variables Related by a Certain Linear Inequality: I

Undistinguishing statistics of objectively distinguishable objects: Thermodynamics and superfluidity of classical gas