The Central Limit Problem for Random Vectors with Symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries. This work was done while at Stanford University.     

Relation between the Hydrodynamic and Elastic Parameters in Surface Wave Diffraction on a Floating Plate

The problem of surface wave diffraction on a floating elastic plate is considered. The relation between the parameters of the elastic vibrations of the plate and the transmitted and reflected wave amplitudes is investigated. It is shown that the maximum amplitudes of the plate stresses and deflections depend nonmonotonically on the incident wave frequency and are reached simultaneously with the maxima of both the transmission coefficient and the length of the wave penetrating into the plate. This makes it possible to use the transmission coefficient as a parameter for investigating the maximum and minimum amplitudes of the hydroelastic vibrations of the plate. As an example, this criterion is used to minimize the vibrations of a plate whose leading edge is elastically connected to the bottom.     

Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x ₀ and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated. Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003 This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support     

Pressure effect on cation exchange distribution of some fission products on a strongly acidic cation exchanger of sulfonic acid type in nitric acid

Distribution coefficients of fission products on a cation exchanger in 1M nitirc acid were measured as a function of the pressure by means of the column method. The distribution coefficients were found to decrease with the pressure, and this became more pronounced with increasing charge of the ions. The distribution coefficients of yttrium and RuNO³⁺ decreased with the pressure to a relatively greter extent than lighter lanthanides, and RuNO³⁺ appeared at a separate peak from europium in the elution chromatogram as the pressure was increased up to 900 kg/cm².     

Singularities and noncommutative Frobenius manifolds

We prove that the quaternionic miniversal deformations of an A _n singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory.     

Runge–Kutta convolution quadrature methods for well-posed equations with memory

Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity, of the exact solution. Numerical illustrations are provided.     

Zeroth-order phase transitions and Zipf law quantization

From the standpoint of thermodynamic averaging of fission microprocesses, we investigate the origin of radioactive release in an NPP after an accident or after resource depletion. The genesis of the NPP release is interpreted as a new thermodynamic phenomenon, a zeroth-order phase transition. This problem setting results in a problem in probabilistic number theory. We prove the corresponding theorem leading to quantization of the Zipf law for the frequency of a zeroth-order phase transition with different values of the jump of the Gibbs thermodynamic potential. We introduce the notion of hole dimension. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 118–142, January, 2007.