The cauchy problem for a nonlinear integro-differential transport equation

In the present paper, we study the Cauchy problem for a nonlinear time-dependent kinetic neutrino transport equation. We prove the existence and uniqueness theorem for the solution of the Cauchy problem, establish uniform bounds int for the solution of this problem, and prove the existence and uniqueness of a stationary trajectory and the stabilization ast→∞ of the solution of the time-dependent problem for arbitrary initial data. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 677–686, May, 1997. Translated by A. M. Chebotarev     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

Criterion for uniform invertibility of families of strong approximations to one-dimensional singular integral operators with continuous coefficients

A criterion is obtained for the applicability of the approximation method based on strongly approximating operator families converging to a one-dimensional singular integral operator with coefficients continuous in the circle. Some special cases are considered. Translated fromMatematischeskie Zametki, Vol. 62, No. 3, pp. 430–439, September, 1997. Translated by A. M. Chebotarev     

Doubly stochastic operators on a finite-dimensional simplex

Generalizing the concept of a quadratic doubly stochastic operator, we introduce the concept of an arbitrary doubly stochastic operator. We give a necessary condition for double stochasticity. Moreover, we prove an ergodic theorem for doubly stochastic operators. Original Russian Text Copyright ? 2009 Shahidi F. A. __________ Tashkent. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 463–468, March–April, 2009.     

Summation rational operators of the Jackson type

Summation rational positive operatorsD _4n−n(x; f) of the Jackson type are constructed on the real axis. The corresponding approximations of continuous functionsf onℝwith coinciding finite limits lim_x→−∞ f(x) and lim_x→+∞ f(x) are estimated. Translated fromMatematischeskie Zametki, Vol. 61, No. 2, pp. 270–277, February, 1997. Translated by N. K. Kulman     

On a transformation operator

We prove the existence of a transformation operator that takes the solution of the equationy″=λ²ⁿ y to the solution of the equation

Spectral functions of self-adjoint and symmetric multiplication operators inL 2(X, μ)-spaces

Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution inL ²(X, μ)-space and a convolution-type operator inL ²(ℝ ⁿ )-spaces. Symmetric convolution and multiplication operators are considered inL ²(X, μ) andL ²(ℝ ⁿ )-spaces. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 803–810, June, 2000.     

On the existence of a complete set of solutions of the Lur’e equation

The algebraic Lur’e equations are considered in the general case of an arbitrary pair (A, B) and a nonsingular matrix Γ of quadratic form. The necessary and sufficient conditions for the existence of a complete set of solutions of such equations are obtained. These conditions are other than in the standard case of a definite matrix Γ. For the standard case, the constraints on the pair (A, B) are maximally relaxed. Then the results are extended to the case of a singular matrix Γ. A special representation of Hamiltonian matrices, which forms the basis for the proofs, is developed. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 29, Voronezh Conference-1, 2005.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

Invertibility and the fredholm property of difference operators

We obtain conditions for the invertibility and the Fredholm property of the difference operator (Dx)(n)=x(n) -U(n)x(n − 1),n ε ℤ, in the Banach space l _p (ℤ, X),p ε [1, ∞], of vector sequences, whereX is a Banach space andU is a bounded operator function. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 816–827, June, 2000.     

Systems of matrix linear differential equations of first order

Sufficient conditions for the solvability in quadratures of systems of matrix linear ordinary differential equations of first order with one-sided multiplication by variable matrix coefficients are given in this paper. These conditions are stated in terms of the theory of Lie algebras. We consider matrix equations of higher orders that are equivalent to such systems. An illustrative example is considered. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 63–75, July, 1999.     

The index of singular integral operators in reflexive Orlicz spaces

We consider the Banach algebra of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz spaceL _M ⁿ (Γ). We assume that Γ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra in terms of the symbol of this operator. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 383–396, September, 1998.     

Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators

For boundary value problems generated by a second-order differential equation with regular nonseparated boundary conditions, criteria for the eigenvalues to be multiple are given and the relative position of the eigenvalues is studied. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 369–381, March, 2000.     

On normal matrices with normal principal submatrices

Let A be a normal n×n matrix. This paper discusses in detail under what conditions and in what way A can be dilated to a normal matrix of order n+1 or n+2. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 63–94. Translated by Kh. D. Ikramov.     

Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (a_ij) ∈ ℂ ^n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ² − | a_ii|², i = 1, …, n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D⁻¹C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77.     

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     

Composition operators between area-type Nevanlinna classes

This paper is concerned with the composition operators between the area-type Nevanlinna classes. Some sufficient and necessary conditions are given in terms of the concept of Carleson measure and the standard techniques of Montel Theorem for the composition operator C _φ: N _a ^p → N _a ^q to be bounded or compact, where 1 < p ⩽ q. Moreover, the inducing maps which induce invertible or Fredholm composition operators on N _a ^p are characterized. __________ Translated from Journal of Wuhan University (Natural Science Edition), 2004, 50(1): 1–5. This work was supported by the National Natural Science Foundation of China under grant number 19771063     

Degenerate Nevanlinna-Pick problem

A general solution of the degenerate Nevanlinna-Pick problem is described in terms of fractional-linear transformations. A resolvent matrix of the problem is obtained in the form of a J-expanding matrix of full rank. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1334–1343, October, 2005.     

Some models in differential geometry

Some variants of the axiomatics of the algebras of “vector fields” in models of noncommutative differential geometry are considered. In the case of a commutative model (the de Rham complex) a matrix analogue of the Kadomtsev-Petviashvili hierarchy is constructed. The corresponding Sato system is presented. The method of deformations of D-modules is used. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 122–153. Translated by A. M. Nikitin.     

On rank-one corrections of complex symmetric matrices

Let a matrix A ∈ M_n(C) be a rank-one perturbation of a complex symmetric matrix, i.e., A = X + Y for some unknown matrices X and Y such that X = X^T and rank Y = 1. The problem of determining the matrices X and Y is solved. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 78–83.