Attenuation of the interference slow wave in a cracked layer

The attenuation of the interference slow wave propagating in a cracked fluid-filled layer sandwiched between two elastic half-spaces is evaluated. It is proposed that the attenuation is induced by the inelasticity of the fluid. It is shown that the attenuation of the wave may be very small if the thickness of the layer is much less than the wavelength. This explains the possibility of observing the slow wave in real field seismic experiments. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 153–160. Translated by P. V. Krauklis.     

Weighted minimal contrast estimates and simulation problems

While complex systems are being designed, a number of statistical experiments are carried out. The present paper is devoted to the development of methods of correlated processes which increase the efficiency of statistical tests by designing dependent experiments and simultaneous processing of the obtained data. The deficiency of the usual approaches in this field consists in their specialization in complex systems modelled by functionals of the mathematical expectaition only. We consider here the technique of minimal contrast estimation which allows us to extend the methods of correlated processes to a large family of functionals. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 150–163, Perm, 1990.     

What information about a wave function is given by “measuring” it with the help of the tomographic method of recovering the Wigner distribution?

Mathematical problems connected with “measuring” a wave function by the tomographic method of recovering the Wigner distribution are considered. Inequalities showing what information is contained in a finite number of projections of the Wigner distribution are obtained. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 189–200.     

On a special function used in the description of electromagnetic surface waves

The tangential component of the electric field of a surface wave at any distance from the transmitting antenna lying in the interface plane of two homogeneous media can be represented in terms of a function of two complex variables Ŵ(q,ξ) for arbitrary parameters of the interface. In this paper, representations of the function Ŵ(q,ξ) in the form of series are given that allow one to quickly calculate the values of Ŵ(q,ξ) and to investigate the analytic properties of this function. The dependence of the field of the surface wave on time is determined using the inverse Laplace transform, where the path of integration is chosen in such a way that the integrand rapidly decreases at infinity, which drastically improves the computation speed compared with the method based on the Fourier transform.     

Diffraction on a semi-infinite screen of a plane nonstationary wave the amplitude of which increases linearly along the front set

An exact analytic solution of the two-variables nonstationary problem of diffraction on an ideal half-infinite screen is obtained by the Smirnov-Sobolev method. The source of the field is an incident plane acoustic wave with a δ-function profile. The wave amplitude is a linear function increasing along the front set. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 138–152.     

Importance and Sensitivity Analysis in Dynamic Reliability

In dynamic reliability, the evolution of a system is governed by a piecewise deterministic Markov process, which is characterized by different input data. Assuming such data to depend on some parameter p ∈ P, our aim is to compute the first-order derivative with respect to each p ∈ P of some functionals of the process, which may help to rank input data according to their relative importance, in view of sensitivity analysis. The functionals of interest are expected values of some function of the process, cumulated on some finite time interval [0,t], and their asymptotic values per unit time. Typical quantities of interest hence are cumulated (production) availability, or mean number of failures on some finite time interval and similar asymptotic quantities. The computation of the first-order derivative with respect to p ∈ P is made through a probabilistic counterpart of the adjoint state method, from the numerical analysis field. Examples are provided, showing the good efficiency of this method, especially in case of a large P.     

Test closures of sets in Banach spaces generated by linear functionals

Some criteria are proved for an element of a Banach space to belong to the test or Korovkin closure of a given set. The operations of closure in question are determined by linear functionals. The cases of arbitrary linear functionals and of positive functionals are considered. Some sufficient conditions are given for the test closure to coincide with the Korovkin closure. Bibliography: 33 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 114–145. Translated by S. Yu. Pilyugin.     

On some problems of nonparametric estimation

New classes of efficient estimators are proposed for functionals of the distribution density function or of the spectral density function. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 60–76, 1993. Translated by A. Sudakov.     

Distribution of functionals of bridges for diffusions with jumps

The paper deals with a method of calculation of distributions for functionals of bridges of a process which is a generalization of a diffusion with jumps. The approach to calculation of distributions for integral functionals of bridges is the same as for the diffusion itself. This approach is based on calculation of the Laplace transform of distributions of the integral functionals. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 34–47.     

Estimates of suitable weak solutions to the Navier-Stokes equations in critical Morrey spaces

We prove some estimates for suitable weak solutions to the nonstationary three-dimensional Navier-Stokes equations under the assumption that certain invariant functionals of the velocity field are bounded. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 199–210.     

Diffraction on a wedge-shaped domain with curved sides

The problem of diffraction of side waves overrunning a wedge-shaped body ω with curved sides is considered. The incidence wave is given by a ray asymptotic expansion. The scattered field is found as the sum of a geometric-optical field and a diffraction field. Bibliography: 2 tittles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 263–273. Translated by N. Ya. Kirpichnikova.     

On symmetric configurations in some problems on extremal decomposition

Some problems on extremal decomposition in families of systems of nonoverlapping domains of various types with free parameters are studied. The functionals considered contain a sum of the reduced modules of biangles. For the problems under consideration, the existence of extremal configurations with appropriate symmetry is investigated. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 160–172.     

Inversion of a theorem on action of analytic functions and multiplicative properties of some subclasses of the hardy space H∞

In this paper, function spaces V∩l _A ^p (w) are considered in the context of their multiplicative structure. The space V is determined by conditions on the values of a function in a disk (for example, C_A,Lip _Aα). We denote by l _A ^p (w) the space of power series such that their Taylor coefficients are p-summable with weight w. For an analytic function Φ acting in a space of this type, we prove the following alternative: either Φ″(z)≡0, or the space is a Banach algebra with respect to pointwise multiplication. For a wide class of weights w, we establish the continuity of the identity embeddingmult(V∩l _A ^p (w))↪multl _A ^p . An estimate for the l^p-multiplicative norm of random polynomials is found. This estimate can be considered as an extension of the known result by Salem-Zygmund. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 50–72. Translated by S. Shimorin.     

Investigation of the tangency of front sets of two transversal waves in transversely isotropic elastic media

A special case of a transversely isotropic elastic medium is considered. This medium is excited by a point force perpendicular to the anisotropy axis. The wave field in this medium is constructed and investigated. The front sets of SV and SH waves are in contact with one another at a point. In order to investigate a neighborhood of this point, we derive simpler expressions for the wave field and establish a special function describing the tangency of two front sets. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 206–216.     

Diffusion Processes That Correspond to the Hypergeometric Equation

Two integral functionals of special type are considered. For diffusion processes corresponding to the hypergeometric equation, such characteristics as the scale, speed density, Green function, and transition density function are investigated. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 79–91.     

T-differentiable functionals and ther critical points

We study critical points of functionalsF: D⊂X→ℝdefined on “nonlinear” setsD in topological vector spacesX. For such functionals, we suggest a notion ofT-derivative and study its connection with other relevant structures. The concept of weak critical point is introduced and the Coleman principle is justified forT-differentiable functionals. Institute of Cybernetics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 720–728, June, 1994.     

Construction of algorithms of reconstruction tomography

An algorithm for the recovery of the parameter field of elastic waves by the method diffraction tomography is suggested. A priori information is taken into account in probabilistic form. Under different assumptions on the statistical correlations of the parameter field, different modifications of the algorithm are considered. The notion of the information sensitivity of the measurement field with respect to a fixed linear parameter field is introduced. It is shown that the greater the energy of the useful signal that has passed through the medium with a nonhomogeneity, the higher the information sensitivity of a given system of tomographic functionals.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 225–235.This work was supported in part by the Russian Foundation for Basic Research (Grant No. 96-05-65904).     

Localization properties of highly singular generalized functions

We study the localization properties of generalized functions defined on a broad class of spaces of entire analytic test functions. This class, which includes all Gelfand-Shilov spaces S _α ^β (R ^k ) with β < 1, provides a convenient language for describing quantum fields with a highly singular infrared behavior. We show that the carrier cone notion, which replaces the support notion, can be correctly defined for the considered analytic functionals. In particular, we prove that each functional has a uniquely determined minimal carrier cone. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 179–194, May, 2007.     

Counterexample to Peano's theorem in infinite-dimensionalF′-spaces

LetE be a nonnormable Fréchet space, and letE′ be the space of all continuous linear functionals onE in the strong topology. A continuous mappingf:E′→E′ such that for anyt ₀∈ℝ,x ₀∈E′, the Cauchy problemx=f(x), x(t ₀ )=x ₀ has no solutions is constructed. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 128–137, July, 1997. Translated by V. N. Dubrovsky     

Norm-attaining functionals onC(Q, X)

Functionals (vector measures) defined on the spaceC(Q, X) of continuous abstract functions (whereQ is a compact Hausdorff space andX is a Banach space) and attaining their norm on the unit sphere are considered. A characterization of such functionals is given in terms of the Radon-Nikodym derivative of the vector measure with respect to the variation of the measure and in terms of analogs of the derivative. Applications to the characterization of finite-codimensional subspaces with the best approximation property are given. Similar results are obtained for the spaceB(Q, Σ, X) of uniform limits of simple functions. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 45–56, January, 1997. Translated by V. E. Nazaikinskii