Free and nonsteady-state oscillations of elastic fluid-filled systems situated on an elastic base

We develop a method of computing the nonsteady-state and free oscillations of a framed elastic structure situated on an elastic base and containing an ideal compressible fluid. The solution uses the method of integral transforms in conjunction with the method of orthogonal polynomials. In the transform space the problem reduces to systems of linear algebraic equations. The Fourier transform is applied to return to the original space. Examples of the computation are given.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 69–72.     

Isolation of singularities in the density and the kernel of a two-dimensional integral equation for complex boundary surfaces

We consider the solution of integral equations in problems of three-dimensional electromagnetic fields in piecewise-homogeneous media. An algorithm is proposed for analytical isolation of singularities in the kernel and the density of the integral equation of potential theory for the case of complex boundary surfaces.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 14–17, 1985     

Numerical analysis of near electromagnetic fields of a parabolic cylinder and a paraboloid of revolution

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 119–127.     

Asymptotic description of solutions of the fourth Painlevé equations on the stokes rays analogs

The proof of the existence of classical solutions is given for the Painlevé type nonlinear ordinary differential equations. The solutions have asymptotic formulas, which can be obtained by the isomonodromic deformation method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 101–109, 1989.     

Analytic calculation of scaling dimensions: Tricritical hard squares and critical hard hexagons

The finite-size corrections, central chargesc, and scaling dimensionsx of tricritical hard squares and critical hard hexagons are calculated analytically. This is achieved by solving the special functional equation or inversion identity satisfied by the commuting row transfer matrices of these lattice models at criticality. The results are expressed in terms of Rogers dilogarithms. For tricritical hard squares we obtainc=7/10,x=3/40, 1/5, 7/8, 6/5 and for hard hexagons we obtainc=4/5,x=2/15, 4/5, 17/15, 4/3, 9/5, in accord with the predictions of conformal and modular invariance.     

Stationary turbulent closure via the Hopf functional equation

Exact closed-form solutions are exhibited for the Hopf equation for stationary incompressible 3D Navier-Stokes flow, for the cases of homogeneous forced flow (including a solution with depleted nonlinearity) and inhomogeneous flow with arbitrary boundary conditions. This provides an exact method for computing two- and higher-point moments, given the mean flow.     

Unitarity condition in covariant quantum field theory with indefinite metric

V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 3, pp. 347–358, June, 1989.     

Quantum interference of electrons in disordered metals

Investigation of the galvanomagnetic properties of disordered metals in weak magnetic fields [r(H)?l, wherer(H) is the electron trajectory radius andl, the electron free path], proved to be one of the effective experimental methods of studying disordered metals. The phase difference between the interfering electron waves is affected by the presence of magnetic flux in the sample. One of the observable effects is the oscillatory magnetoresistanceK(H) of multiconnected samples predicted by Altshuleret al (1981). The period ofK(H) oscillations for the hollow cylinders, networks or chains with orifices cross-sections areasS isΔH=φ ₀/2S [whereφ ₀=hc/e]. The amplitude and the phase of the oscillations depend on the spin orbit interaction, the intensity of superconductive fluctuation etc. It should be noted that in small “mesoscopic” single loops the oscillations with the periodΔH?φ ₀/S were also observed recently (see also Altshuleret al 1987 included in this issue).     

Scattering of solitons for the Schrödinger equation coupled to a particle

We establish soliton-like asymptotics for finite energy solutions to the Schrödinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold converges to a sum of a travelling wave and an outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto the solitary manifold in the Hilbert phase space.