The inverse problem of estimating the film structure from X-ray reflection data

A new model of an X-ray reflection experiment is proposed for studying the structure of a film on a substrate. The experimental data are processed to yield the density distribution across the film, and an appropriate approximation produces numerical values of the parameters { δ_i, β_i, σ_i, d ₁} for each of the n layers in a multilayer film model. So far there has been no method for determining the film structure from experimental data and the problem has been usually solved by trial and error. __________ Translated from Prikladnaya Matematika i Informatika, No. 21, pp. 19–29, 2005.     

Two questions on the exterior geometry of the plücker embeddings of the Grassmann manifolds

The Plücker model of the Grassmann manifold G _p,p+q ⁺ is considered. The structure of intersections of G _p,p+q ⁺ with tangent spaces of G _p,p+q ⁺ regarded as subspaces of the ambient exterior algebra is described. An explicit formula for the second fundamental form of G _2,4 ⁺ as of a hypersurface in the five-dimensional sphere is given. The level sets of the normal curvature functions for this hypersurface are studied. Bibliography: 3 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 5–12. Translated by N. Yu. Netsvetaev.     

On the universal ultrametric space

An ultrametric space in which any separable ultrametric space can be isometrically imbedded is constructed. We describe the method for isometric imbedding of any separable ultrametric space intol ₁,l ₂ andc ₀ based on the application of this universal space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1700–1706, December, 1994.     

Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice

The Ising model on a Bethe lattice of orderk≥2 is considered. For maximum or minimum translation-invariant Gibbs states of this model, the relations between the von Neumann algebras generated by these states for the Gelfand-Neimark-Segal representation are found. These algebras can be of types III_λ, λ∈(0, 1), and III₁. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 88–93, April, 2000.     

Approximation of functions of boundedp-variation by polynomials in terms of the faber-schauder system

In this paper the best polynomial approximation in terms of the system of Faber-Schauder functions in the spaceC _p [0, 1] is studied. The constant in the estimate of Jackson’s inequality for the best approximation in the metric ofC _p [0, 1] and the estimate of the modulus of continuity ω_1−1/p are refined. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 363–371, September, 1997. Translated by N. K. Kulman     

Sarason's transform in a Sobolev space

The lassical Sarason transform is the canonical isomorphism of the model space H²/gqH² (θ is an inner function generated by a single point mass) onto the standard L² space. In the paper the image of the space of smooth functions H ₂ ¹ ={f: f′ ε H²} under this transform is described in explicit terms. Pointwise estimates of the model operator in H ₂ ¹ /θH ₂ ¹ are obtained. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI. Vol. 206, 1993, pp. 33–39. Translated by I. Boricheva.     

Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics

We obtain asymptotic equalities for the least upper bounds of approximations of classes of Poisson integrals of periodic functions by a linear approximation method of special form in the metrics of the spaces C and L _p . Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 976–982, July, 2008.     

Yang-baxterization of the quantum dilogarithm

A new solution of the Yang-Baxter equation with spectral parameter is found. The resulting R-matrix R(x) is an operator inH⊗H, whereH=L ₂(ℝ). This R-matrix is required to justify the solution of the sine-Gordon model on the discrete space-time. Bibliography: 18 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 146–154. Translated by B. Bekker.     

Geometry of real grassmann manifolds. Part III

A canonical decomposition for an element of the tangent bundle of the Grassmann manifold G _p,n ⁺ in its Plücker model is constructed. This decomposition is used for introducing the notion of stationary angles between oriented planes. The relation with the stationary angles in the nonoriented case is established. An explicit formula for the exponential mapping is given, which allows calculating the diameter and the injectivity radius of the manifold G _p,n ⁺ . The problem of uniqueness of the canonical decomposition is reduced to a similar problem on the decomposition of bivectors that realizes their mass (the latter has earlier been solved by the author). The techniques developed are used for describing the structure of the closure of an arbitrary geodesic in the manifolds G _p,n ⁺ and G_p,n. (For the manifolds G_p,n, this result was announced by Wong without proof.) Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 108–129. Translated by N. Yu. Netsvetaev.     

Effective action and collective bose-excitations in the T-J-model

The two-dimensional (2D) t-T-model is investigated using the functional integration method. The effective action is obtained, and the spectrum of collective Bose-excitations in the superconductive phase of the system is calculated both in the phase-transition vicinity (Ginsburg-Landau region |T−T_c| ≪ T_c) and in the low-temperature region T ≪ T_c. Bibliography: 26 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 300–309. Translated by P. A. Sevastianov.     

On methods of deriving equations describing effective models of layered media

Methods of deriving equations describing effective models of layered periodic media are presented. Elastic and fluid media, as well as porous Biot media, may be among these media. First, effective models are derived by a rigorous method, and then some operations in the derivation are replaced by simpler ones providing correct results. As a consequence, a comparatively simple and justified method of deriving equations of an effective model is established. In particular, this method allows us to simplify to a degree and justify the derivation of an effective model for media containing Biot layers; this method also produces equations of an effective model of a porous layered medium intersected by fractures with slipping contacts. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998 pp. 219–243. Translated by L. A. Molotkov.     

L 2-estimates of solutions of boundary value problems for equations with mixed elliptic-parabolic structure

For the linear equation L u=0 in which L is a product of a 2b₁-parabolic operator and a 2b₂r-elliptic operator (b₁, b₂, and r are integers), we obtain L₂-estimates for solutions of model boundary value problems, namely, for the Cauchy problem and the half-space problem. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 4–27, 1992. Translated by N. A. Karazeeva.     

The solvability of the initial boundary-value problem for equations of motion of a viscous compressible barotropic liquid in the spacesW 2 l+1,l/2+1 (Q T )

A new method of estimating the solutions of the Navier-Stokes equations for a viscous compressible barotropic fluid in a bounded domain Ω⊂ℝ³ is suggested, which makes it possible to investigate the problem for the whole scale of anisotropic spaces W ₂ ^l+2,l/2+1 (Q_T), Q_T=Ω×(0,T), for arbitrary l>1/2. Bibliography: 10 titles. To dear Olga Alexandrovna Ladyzenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 177–186. Translated by V. A. Solonnikov.     

Bounds for exponential sums modulo <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

In this paper we consider exponential sums over subgroups G ⊂ ℤ _q ^* . Using Stepanov’s method, we obtain nontrivial bounds for exponential sums in the case where q is a square of a prime number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 81–94, 2005.     

Some properties of the basis invariants of the unitary group W(K5)

The basis invariants of degrees 6, 12, and 18 of the unitary groupW(K ₅) are constructed in explicit form by a new method.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 92–96.     

A Richardson-type iterative approach for identification of delamination boundaries

A direct problem of applied mathematical modelling is to determine the response of a system given the governing partial differential equations, the geometry of interest, the complete boundary and initial conditions, and material properties. When one or more of the conditions for the solution of the direct problem are unknown, an inverse problem can be formulated. One of the methods frequently used for the solution of inverse problems involves finding the values of the unknowns in a mathematical formulation such that the behavior calculated with the model matches the measured response to a degree evaluated in terms of the classical L ₂ norm. Considered in this sense, the inverse problem is equivalent to an ill-posed optimization problem for the estimation of parameters whose solution in the majority of cases is a real mathematical challenge. In this contribution, we report a novel approach that avoids the mathematical difficulties inspired by the ill-posed character of the model. Our method is devoted to the computation of inverse problems furnished by second-order elliptical systems of partial differential equations and falls in the same conceptual line with the method initiated by Kozlov et al. and further extended and algorithmized by Weikl et al. We construct and employ a weak version of the algorithm found by Weikl et al. Proofs for the convergence and regularity of this version are given for the case of a single layer. The computational realization of the algorithm (called briefly AICRA) is applied and numerical results are obtained. Comparison with experiments demonstrates a good significance and representativeness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 209–230, 2006.     

The riquier problem for a nonlinear equation unresolved with respect to the Lévy iterated laplacian

We present a method of solving for the nonlinear equationf(U(x),Δ _L ² U(x)) = Δ _L U(x) (Δ _L is an infinite-dimensional Laplacian) unresolved with respect to an iterated infinite-dimensional Laplacian and for the Riquier problem for this equation. Ukrainian NII MOD, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 423–427, March, 1999.     

On the optimization of direct methods for solving fredholm integral equations of the second kind with infinitely smooth kernels

We give a direct method, optimal inL ₂, for solving the Fredholm integral equation of the second kind with operators acting into the space of functions harmonic in a disk or into the space of functions that can be analytically extended to an infinite strip. The exact order of the error of this method is determined.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1695–1701, December, 1993.     

Quantitative Whitney-type theorems

The crucial point in the construction of an almost optimal decomposition for the couple (L_q, W _p ^k ) was the covering theorem in which the parameter α (α-capacity) was controlled. Here we give a detailed proof of this theorem for negative α. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 96–113. Translated by S. V. Kislyakov.     

The opening of a nonthrough longitudinal crack in a cylindrical shell taking account of elastoplastic deformation

In the context of an analog of the _k —model the problem of the stress-strain state of a shallow cylindrical shell with a nonthrough longitudinal crack is reduced to a system of singular integral equations that is solved by the small-parameter method. For a shell under an internal pressure we study the dependence of the opening of the crack on the geometric and physico-mechanical parameters of the shell.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 66–70.