On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials

We discuss at first in this paper the Gauge equivalence among several u‐linear Hamiltonian operators and present explicitly the associated Gauge transformation of Bäcklund type among them. We then establish the sufficient and necessary conditions for the linear superposition of the discussed u‐linear operators and matrix differential operators with constant coefficients of arbitrary order to be Hamiltonian, which interestingly shows that the resulting Hamiltonian operators survive only up to the third differential order. Finally, we explore a few illustrative examples of integrable hierarchies from Hamiltonian pairs embedded in the resulting Hamiltonian operators.     

Graphs of linear operators

In 2006 the author proposed an algorithm for constructing graphs of difference operators. In this paper, the following question is studied: to which linear operators $ \mathcal{A} In 2006 the author proposed an algorithm for constructing graphs of difference operators. In this paper, the following question is studied: to which linear operators does this algorithm apply? Graphs of difference operators are used to determine the complexity of a sequence in the sense of Arnold, so the algorithm makes it possible to determine the complexity of any sequence. Original Russian Text ? A.I. Garber, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 64–71.     

Approximation gewisser Operatoren durch Operatoren endlichen Ranges und eine Kollokationsmethode

Summary Certain bounded linear operators from Hilbert spaces H into function spaces C(X) are approximated by operators of finite rank with the aid of orthogonal projections. As examples for such operators we present special integral operators and linear partial differential operators. As application we consider a collocation method.     

On stationarity of stochastic retarded linear equations with unbounded drift operators

This article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

A new bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation

The complexity of computing the Galois group of a linear differential equation is of general interest. In a recent work, Feng gave the first degree bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation. This bound is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group (see Definition 2.7) and is sextuply exponential in the order of the differential equation. In this paper, we use Szántó’s algorithm of triangular representation for algebraic sets to analyze the complexity of computing the Galois group of a linear differential equation and we give a new bound which is triple exponential in the order of the given differential equation.     

On the purely algebraic data‐sparse approximation of the inverse and the triangular factors of sparse matrices

The approximation of the inverse and the factors of the LU decomposition of general sparse matrices by hierarchical matrices is investigated. In this first approach, we present and motivate a new matrix partitioning algorithm which is based on the matrix graph by proving logarithmic‐linear complexity of the approximant in the case of bounded condition numbers. In contrast to the usual partitioning, the new algorithm allows to treat general grids if the origin of the sparse matrix is the finite element discretization of differential operators. Numerical examples indicate that the restriction to bounded condition numbers has only technical reasons. Copyright © 2010 John Wiley & Sons, Ltd.     

On Iterations of the Green Integrals and their Applications to Elliptic Differential Complexes

Convergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operators P of order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equation Pu = f in the Sobolev space W^p,²(D) and, as a corollary, necessary and sufficient conditions for the vanishing of the first cohomology group of elliptic differential complexes. Also a criterion for the solvability of a P-Neumann problem for elliptic differential operators is proved.     

Brakhage's Implicit Iteration Method and the Information Complexity of Equations with Operators Having Closed Range

An a posteriori stopping rule connected with monitoring the norm of the second residual is introduced for Brakhage's implicit nonstationary iteration method, applied to ill-posed problems involving linear operators with closed range. It is also shown that for some classes of equations with such operators, the algorithm consisting in combination of Brakhage's method with some new discretization scheme is order optimal in the sense of information-based complexity.     

Equivalence of some coercive estimates for linear partial differential operators

The equivalence of certain coercive estimates for linear partial differential operators on ℝⁿ in a Banach space X is studied. Such estimates play an important role in the theory of differential equations and operators. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

On Conformal Invariants Built from Spinor Fields

By means of a conformal covariant differentiation process we construct generating systems for conformally invariant symmetric (r, s)–spinors in an arbitrary curved space–time. Extending this method to conformally invariant linear differential operators acting on symmetric spinor fields some classes of such operators are derived.     

A wavelet-based nested iteration–inexact conjugate gradient algorithm for adaptively solving elliptic PDEs

In Cohen et al. (Math Comput 70:27–75, 2001), a new paradigm for the adaptive solution of linear elliptic partial differential equations (PDEs) was proposed, based on wavelet discretizations. Starting from a well-conditioned representation of the linear operator equation in infinite wavelet coordinates, one performs perturbed gradient iterations involving approximate matrix–vector multiplications of finite portions of the operator. In a bootstrap-type fashion, increasingly smaller tolerances guarantee convergence of the adaptive method. In addition, coarsening performed on the iterates allow one to prove asymptotically optimal complexity results when compared to the wavelet best N-term approximation. In the present paper, we study adaptive wavelet schemes for symmetric operators employing inexact conjugate gradient routines. Inspired by fast schemes on uniform grids, we incorporate coarsening and the adaptive application of the elliptic operator into a nested iteration algorithm. Our numerical results demonstrate that the runtime of the algorithm is linear in the number of unknowns and substantial savings in memory can be achieved in two and three space dimensions.     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.     

An analog of the rolle theorem for differential operators andL-spline interpolation

An analog of the Rolle theorem is established for linear differential operators with continuous periodic coefficients. By using this result, exact values of the deviations of interpolationalL-splines are obtained on certain classes of functions given by a linear differential operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1122–1128, August, 1993.     

Holomorphic Operators Generating Dense Images

The existence of infinite dimensional closed linear spaces of holomorphic functions f on a domain G in the complex plane such that Tf has dense images on certain subsets of G, where T is a continuous linear operator, is analyzed. Necessary and sufficient conditions for T to have the latter property are provided and applied to obtain a number of concrete examples: infinite order differential operators, composition operators and multiplication operators, among others. This work was supported in part by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and by MEC DGES Grants MTM2006-13997-C02-01 and MTM2004-21420-E.     

A class of linear positive operators in weighted spaces

In this paper, we introduce a class of linear positive operators based on q-integers. For these operators we give some convergence properties in weighted spaces of continuous functions and present an application to differential equation related to q-derivatives. Furthermore, we give a Stancu-type remainder.     

On the exact WKB analysis of higher order simple-pole type operators

Higher order simple-pole type operators, that is, higher order linear ordinary differential operators with a large parameter η whose coefficients have simple poles at the origin, are discussed from the viewpoint of the exact WKB analysis. Making use of the technique of microdifferential operators, we clarify the singularity structure of the Borel transform of their WKB solutions.     

Linear differential operators with unbounded operator coefficients and semigroups of bounded operators

Associated with a family of evolution operators in a complex Banach space is a linear unbounded operator, which is studied with the aid of a semigroup of difference operators and a difference operator in a sequence space. Some formulas for the spectra of the linear operators in question (in particular, for abstract hyperbolic differential operators) and the spectrum mapping theorem for the semigroup of difference operators are obtained. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 811–820, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00032 and by the International Science Foundation under grant No. NZA000 and grant No. NZA300.