Some conic bundless

A representation of the anticanonical K3 surface of a singular pencil of conics is described. This generalizes the well-known Shokurov theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 903–910, June, 1998.The author is greatly indebted to V. A. Iskovskikh, Yu. G. Prokhorov, and I. A. Chel'tsov for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00820 and by INTAS under grant No. 93-2805.00-00-00.     

On the separation of spectra in the analysis of Berezin kernels

The problem of restricting a highest weight representation of the group U(p, q) to the subgroup O(p, q) is considered. This restriction has an intricate spectrum that contains representations of different types. We construct a decomposition of this representation into reducible representations each of which has a single-type spectrum. Some integrals over classical groups are also calculated; these integrals generalize those of Hua. Partially supported by grant RFBR 98-01-00303 and the Russian program of support of leading scientific schools (grant RFBR 96-01-96249). Moscow Institute of Electronics and Mathematics. Translated from Funktsional'nyi analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 49–62, July–September, 2000. Translated by Yu. A. Neretin     

Jumps of displacements and stresses as seismic sources in Biot's medium

Wave fields excited in a homogeneous isotropic Biot medium by point sources described in terms of discontinuities of displacements and stresses are determined. The results are represented in the form of relations involving Fourier-Bessel or Mellin integrals and in the form of Stokes-type formulas. The interrelations between these representations are established. Among all possible point sources exciting Biot's medium, the elementary sources, in terms of which any complicated linear source can be described, are selected. The special case where the wave fields in the two phases of the Biot medium are independent of one another is considered, and the corresponding sources in the Biot medium are compared with the known sources in elastic and fluid media.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 164–196.This work was supported by the Russian Foundation for Basic Research under grant No. 96-05-65904.     

Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry

Stability criteria of boundary equilibria for dynamical systems in the three critical cases, (n, k)=(3, 0), (2, 1), and (1, 1), are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 572–578, April, 1998.The author wishes to thank V. I. Yudovich for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01791 and by the International Science Foundation under grant NRQ000.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

Isospectrality and galois projective geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 660–664, May, 1998. This research was partially supported by the Russian Foundation for Basic Research and by INTAS under joint grant No. 96-01-00043.     

Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G₂ and F₄. Supported by RFFR grant No. 96-01-01893, the program “Universities of Russia,” and by International Science Foundation and Government of Russia grant No. RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 663–684, November–December, 1996.     

Weight functions on groups and an amenability criterion for Beurling algebras

The paper is devoted to the study of weights on groups. A connection between weight functions and harmonic functions is established. A relationship between the weight theory on groups with the “Tychonoff property” and the theory of bounded cohomology is presented. It is proved that the Beurling algebraℓ1 (G, ω) constructed for the weightω is amenable if and only if the groupG is amenable and the weightω is equivalent to a multiplicative characterχ:G→ℝ₊. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 448–460, September, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00974 and by the INTAS Foundation under grant No. 94-3420.     

The compatibility condition for the embedding problem related top-extensions of groups

Embedding problems with a group and its Sylow p-subgroup over p-extensions of fields are considered. The Faddeev-Hasse compatibility condition for these problems is studied. It is proved that the compatibility conditions for the problems under consideration are equivalent if the kernel of the problem is a supersolvable group or a Sylow p-subgroup is an invariant subgroup. Bibliography: 3titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 100–105.The present paper was supported by the Russian Foundation for Basic Research, grant No. 96-01-00S54.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

The problem of continuation with least coanalytic deviation

The problem of continuing a function from the unit circle to the unit disk so that the continuation has the least deviation from the Sobolev subspace of analytic functions is considered. A mathematical model of this problem is constructed. It is proved that the problem is well-posed.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 45–57, July, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00621.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

Noncompact leaves of foliations of Morse forms

In this paper foliations determined by Morse forms on compact manifolds are considered. An inequality involving the number of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form is obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 862–865, June, 1998.The author wishes to thank Professor A. S. Mishchenko for his interest in this work and stimulating discussions.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00276.     

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained. Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G0008.     

The boundary behavior of components of polyharmonic functions

We consider the following representation of polyharmonic functions on the unit ballD ^m : 450-1 where the _j are harmonic onD ^m . We study the relation between uniform boundary properties ofƒ (its smoothness and growth while approaching the boundary) and the same properties of the terms in this representation. The theorems proved in this paper generalize some results obtained by Dolzhenko in the theory of polyanalytic functions.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 518–530, October, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01366.     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus

We consider dynamical systems with two degrees of freedom whose configuration space is a torus and which admit first integrals polynomial in velocity. We obtain constructive criteria for the existence of conditional linear and quadratic integrals on the two-dimensional torus. Moreover, we show that under some additional conditions the degree of an irreducible integral of the geodesic flow on the torus does not exceed 2.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 37–44, July, 1998.The author wishes to express his thanks to V. V. Kozlov for his interest and his help in this work.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00747.     

Integral invariants of the Hamilton equations

Conditions are found for the existence of integral invariants of Hamiltonian systems. For two-degrees-of-freedom systems these conditions are intimately related to the existence of nontrivial symmetry fields and multivalued integrals. Any integral invariant of a geodesic flow on an analytic surface of genus greater than 1 is shown to be a constant multiple of the Poincaré-Cartan invariant. Poincaré's conjecture that there are no additional integral invariants in the restricted three-body problem is proved. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 379–393, September, 1995. The work was financially supported by the Russian Foundation for Basic Research (grant No. 242 93-013-16244), International Science Foundation (grant No. MCY 000), and INTAS (grant No. 93-339).     

Holomorphic functions and embedded real surfaces

The paper is devoted to the study of necessary and sufficient topological conditions for an embedded real surface to lie in a strictly pseudoconvex domain on a complex surface. These results are used to construct Stein domains on algebraic manifolds and to describe envelopes of holomorphy of real surfaces in P ² and in some other complex surfaces.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 599–606, April, 1998.The author is grateful to A. Vitushkin for the statement of the problem and for constant encouragement.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01002 and by the International Science Foundation under grant No. a97-1850.