One-dimensional perturbations of singular unitary operators

Under some natural restrictions, we prove that any one-dimensional perturbation of a singular unitary operator on a Hilbert space is unitarily equivalent to a model operator on a space determined (in a certain way) by two functions from the Hardy space H². Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 118–122. Translated by V. V. Kapustin.     

Extension of functions preserving the modulus of continuity

The problem of the extension of a real-valued function from a subset of a metric space to the entire space is treated. An extension operator preserving the modulus of continuity of a function is proposed and its properties are studied. An application to the problem of the trace of a locally Lipschitz function on a compact subset of a metric space is given. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 236–245, February, 1997. Translated by N. K. Kulman     

Four counterexamples to the Fubini theorem

In this paper we study signed measures. Our main results are as follows: the Fubini theorem is not true in the general case; the Jordan parts of a transition measure are not necessarily transition measures; the operation of taking the Jordan parts does not necessarily commute with multiplying by the initial measure; the product of σ-bounded measures need not be a σ-bounded measure. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 124–127, July, 1997. Translated by S. S. Anisov     

Convergence of a sequence of weakly regular set functions

The present paper is devoted to generalizations of the Dieudonné theorem claiming that the convergence of sequences of regular Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all Borelian subsets of this space. The Dieudonné theorem is proved in the case for which the set functions are weakly regular, nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary σ-topological space, and take values in a uniform space. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 103–110, July, 1997. Translated by O. V. Sipacheva     

Space-time gaussian beams of electromagnetic waves

Space-time Gaussian beams in a nonhomogeneous electromagnetic medium concentrated near space rays are constructed. Bibliography: 4 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 115–121, 1990. Translated by A. P. Kachalov.     

Parametrization of the varieties of symmetrizers for operators in finite-dimensional vector spaces

We determine the space that parametrizes the cone of symmetrizers of a semisimple operator with real spectrum in a finite-dimensional vector space. It is proved that the set that determines the cone of symmetrizers is parametrized by the portion of the unit sphere of the vector space lying the first octant.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 1–3.     

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 214–227, April–June, 2000. Translated by V. Mackevičius     

Geometry of distributions of flags on Grassmann manifolds of a projective space. II

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 335–349, July–September, 2000. Translated by V. Mackevičius     

Reconstruction of 3-submanifolds of large codimension in euclidean spaces from their gauss image

We present necessary and sufficient conditions for a regular three-dimensional manifold in the Grassmannian manifoldG(m, m+3) to be the Gauss image of a regular 3-submanifold in (m+3)-dimensional Euclidean space form>4. Translated fromMatemalicheskie Zametki, Vol. 62, No. 5, pp. 694–699, November, 1997. Translated by S. K. Lando     

The Functor of Order-Preserving Functionals of Finite Degree

It is proved that the space of all order-preserving functionals with finite supports is a compact, and if the supports are one-point sets, then this space is the Stone-Chech compactification of a given Tychonoff space. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 313, 2004, pp. 135–138.     

Justification of a construction of the solution of the problem on a point source ofSH waves in a layered medium with vertical boundary

The solution is determined by the method of separation of variables. The method is justified by using a suitable space of generalized functions. Boundary conditions “are hidden” in this space. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 56–66. Original article This work was supported by the Russian Foundation of Fundamental Research (Grant No. 93-011-16148). Translated by T. N. Surkova.     

On a multidimensional version of the tomographic problem

The main result: If the dimension of a Euclidean space with measure is high enough, only the first observation from a long series of observations with random independent choice of projection gives essential information on the measure. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 295–301. Translated by V. Sudakov.     

On the spectra of finite-dimensional quadratic Bose operators

A complete study of the spectrum of a finite-dimensional Bose operator is carried out in the paper. The cases in which the spectrum is discrete or continuous are studied. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 835–854, June, 1997. Translated by A. M. Chebotarev     

Intrinsic normalizations of a hyperplane distribution on the Grassmann manifold. I

We consider intrinsic normalizations of a hyperplane distribution on the Grassmann manifold of ann-dimensional projective space. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 257–273, April–June, 1999. Translated by V. Mackevičius     

Conical algorithm for the global minimization of linearly constrained decomposable concave minimization problems

In this paper, we are concerned with the linearly constrained global minimization of the sum of a concave function defined on ap-dimensional space and a linear function defined on aq-dimensional space, whereq may be much larger thanp. It is shown that a conical algorithm can be applied in a space of dimensionp + 1 that involves only linear programming subproblems in a space of dimensionp +q + 1. Some computational results are given.This research was accomplished while the second author was a Fellow of the Alexander von Humboldt Foundation, University of Trier, Trier, Germany.     

A space decomposition method for parabolic equations

A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank–Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O(| log τ |) steps of iteration at each time level are needed, where τ is the time-step size. Applications to overlapping domain decomposition and to a two-level method are given for a second-order parabolic equation. The analysis shows that only a one-element overlap is needed. Discussions about iterative and noniterative methods for parabolic equations are presented. A method that combines the two approaches and utilizes some of the good properties of the two approaches is tested numerically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 27–46, 1998     

A Note on Bruhat Decomposition of GL(n) over Local Principal Ideal Rings

Let A be a local commutative principal ideal ring. We study the double coset space of GL _n (A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A ⁿ . We introduce some invariants of the double cosets. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depends on the ring in question. For n = 3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.     

Lattice of subalgebras of the ring of continuous functions and Hewitt spaces

The latticeA(X) of all possible subalgebras of the ring of all continuous ℝ-valued functions defined on an ℝ-separated spaceX is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line ℝ. The main achievement of the paper is the proof of the fact that any Hewitt spaceX is determined by the latticeA(X). An original technique of minimal and maximal subalgebras is applied. It is shown that the latticeA(X) is regular if and only ifX contains at most two points. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 687–693, November, 1997. Translated by A. I. Shtern     

A topological structure on a set of continous functions with various domains

We construct a topological space of continuous functions which generalizes the previously studied space of functions defined on closed intervals. For the new space, metrizability properties are studied. The results can be applied in the topological theory of ordinary differential equations. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 76–88, July, 1999.     

Integrals along trajectories of a dynamical system and the existence of homogeneous Lyapunov-Krasovsky functions

Linear equations with respect to a dynamical system on a metric space are considered. A new proof of the existence of homogeneous Lyapunov-Krasovsky functions for a homogeneous system in the Euclidean space is given. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 352, 2008, pp. 106–113.