Table of contents

This review paper is intended to give a useful guide for those who want to apply the discrete wavelet transform in practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to the corresponding literature. The multiresolution analysis and fast wavelet transform have become a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for the achievement of a goal. Analysis of various functions with the help of wavelets allows one to reveal fractal structures, singularities etc. The wavelet transform of operator expressions helps solve some equations. In practical applications one often deals with the discretized functions, and the problem of stability of the wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves to a few examples only. The authors would be grateful for any comments which would move us closer to the goal proclaimed in the first phrase of the abstract.

A supersymmetric field-theoretical scheme is derived based on the Langevin equation, which enables memory and nonergodicity effects in a nonequilibrium stochastic system with quenched disorder to be described in an optimal manner. This scheme is applied to a disordered heteropolymer whose effective Hamiltonian is found to be simply the free energy as a function of the compositional order parameter. Instead of the Langevin equation, an effective equation of motion is used here to describe the way different monomers alternate as we move along a polymer chain. The isothermal and adiabatic susceptibilities, memory parameter, and irreversible response are determined as functions of the temperature and the intensity of quenched disorder.

The paper reviews the basic experimental facts and a number of theoretical models relevant to the understanding of the pseudogap state in high-temperature superconductors. The state is observed in the region of less-than-optimal current-carrier concentrations in the HTSC cuprate phase diagram and manifests itself as various anomalies in the electronic properties, presumably due to the antiferromagnetic short-range-order fluctuations that occur as the antiferromagnetic region of the phase diagram is approached. The interaction of current carriers with these fluctuations leads to an anisotropic transformation of the electron spectrum and causes the system to behave as a non-Fermi liquid in certain regions of the Fermi surface. Simple theoretical models for describing the basic properties of the pseudogap state, in particular renormalization-induced anomalies in the superconducting state, are discussed.

It is shown that at zero temperature the magnetic field μH ≫ T_K does not move the system from the strong coupling to the weak coupling regime. As a result, the average of the impurity spin approaches its saturation value as a power of the small parameter (2T_K/μH)². The study of the high-temperature expansion of the free energy shows that the Kondo problem contains at least two energy scales and that these scales are separated by the coupling constant. The Hamiltonian of the Kondo problem is not renormalizable.