An inner product and orthonormalization inl p -spaces and solution of infinite systems of infinite linear equations

Let the rows of an infinite square matrixM be elements ofl ^p -space (p>1) andX be an infinite column vector of unknowns andC an infinite column vector of real numbers. To our knowledge the solvability ofMX=C has nowhere been satisfactorily studied in the literature. This is also true of Riesz’classical work [2]. A reason for this is that not until recently [1] an appropriate inner product and the corresponding orthonormalization forp≠2 has been introduced. In this paper, based on [1], it is shown thatMX=C has a solution which is an element ofl ^q if and only if upon our process of orthonormalization of the rows ofM the system yields an equivalent systemAX=K where the rows ofA form an orthonormal sequence (in our sense) of elements ofl ^p andK becomes an element ofl ^q withp ⁻¹+q ⁻¹=1. A solution is then given byX=(A ^(q) (AA ^(q) )⁻¹)K whereA ^(q) is ourq-transpose ofA. This solution is the solution of the minimall ^q -norm. Otherwise, the obvious dual concept of the best approximating solution inl ^q -norm is introduced and obtained. 1980 Mathematics Subject Classification: Primary 46C10, Secondary 15A06     

Necessary solvability conditions of systems of linear extremal equations

Systems of linear equations of the form A?X = B?X and of the form A?X = A?Y over the structure based on linearly ordered commutative group (G, ?, ≤) where the role of ⊕ plays the maximum are treated. Necessary solvability conditions are derived using known results concerning eigenvectors of matrices in such structures. In the special case of idempotent, increasing matrices A and B a condition is given which is necessary and sufficient for the existence of a non-trivial solution.     

On solvability recognition for interval linear systems of equations

The paper considers the problem of recognizing solvability (nonemptiness of the solution set) for interval systems of linear algebraic equations. We introduce a quantitative measure of the membership of a point in the solution set, the so-called “recognizing functional” of the solution set. As the result, the decision on solvability of the interval linear systems reduces to global maximization of the recognizing functional. Additionally, the specific value of this maximum and its argument provide us with important quantitative information of the solvability supply or its deficiency, which can used for the correction of the interval system in a desired sense.     

A short note on solvability of systems of interval linear equations

In this note we formulate necessary and sufficient conditions for strong solvability and feasibility of systems of linear interval equations in terms of absolute value inequalities.     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁ +A₂X₂B₂ +A₃X₃B₃ =C,(ii) A₁XB₁ =C₁ A₂XB₂ =C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A ₁ X ₁ B ₁+A ₂ X ₂ B ₂+A ₃ X ₃ B ₃+A ₄ X ₄ B ₄=C, (iv) A ₁ XB ₁=C ₁ A ₂ XB ₂=C ₂ A ₃ XB ₃=C ₃ A ₄ XB ₄=C ₄, (v) AXB+CXD=E are also considered.     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁+A₂X₂B₂+A₃X₃B₃=C,(ii) A₁XB₁=C₁A₂XB₂=C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A₁X₁B₁+A₂X₂B₂+A₃X₃B₃+A₄X₄B₄=C, (iv) A₁XB₁=C₁A₂XB₂=C₂A₃XB₃=C₃A₄XB₄=C₄, (v) AXB+CXD=E are also considered.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

Lyapunov coupled equations for continuous-time infinite Markov jump linear systems

This paper deals with Lyapunov equations for continuous-time Markov jump linear systems (MJLS). Out of the bent which wends most of the literature on MJLS, we focus here on the case in which the Markov chain has a countably infinite state space. It is shown that the infinite MJLS is stochastically stabilizable (SS) if and only if the associated countably infinite coupled Lyapunov equations have a unique norm bounded strictly positive solution. It is worth mentioning here that this result do not hold for mean square stabilizability (MSS), since SS and MSS are no longer equivalent in our set up (see, e.g., [J. Baczynski, Optimal control for continuous time LQ-problems with infinite Markov jump parameters, Ph.D. Thesis, Federal University of Rio de Janeiro, UFRJ/COPPE, 2000]). To some extent, a decomplexification technique and tools from operator theory in Banach space and, in particular, from semigroup theory are the very technical underpinning of the paper.     

Checking solvability of systems of interval linear equations and inequalities via mixed integer programming

This paper deals with the problems of checking strong solvability and feasibility of linear interval equations, checking weak solvability of linear interval equations and inequalities, and finding control solutions of linear interval equations. These problems are known to be NP$\mathit{NP}$-hard. We use some recently developed characterizations in combination with classical arguments to show that these problems can be equivalently stated as optimization tasks and provide the corresponding linear mixed 0–1 programming formulations.     

Some new conditions for the solvability of the Cauchy problem for systems of linear functional-differential equations

We establish efficient conditions sufficient for the unique solvability of certain classes of Cauchy problems for systems of linear functional-differential equations. The conditions obtained are optimal in a certain sense.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 867–884, July, 2004.     

On solvability of systems of polynomial equations

We study the computational complexity of the solvability problem of systems of polynomial equations over finite algebras. We prove a new dichotomy theorem that extends most of the dichotomy results which have been obtained over different families of finite algebras so far. As a corollary, for example, we get that if \mathbbA{\mathbb{A}} is a finite algebra of finite signature and omits the Hobby-McKenzie type 1, then the problem is solvable in polynomial time whenever \mathbbA{\mathbb{A}} is a reduct of a generalized affine algebra, and NP-complete otherwise.     

Nonnegative solvability of linear equations in certain ordered rings

In the integers and in certain densely ordered rings that are not fields, projections of the solution set of finitely many homogeneous weak linear inequalities may be defined by finitely many congruence inequalities, where a congruence inequality combines a weak inequality with a system of congruences. These results extend well-known facts about systems of weak linear inequalities over ordered fields and imply corresponding analogues of Farkas' Lemma on nonnegative solvability of systems of linear equations.

     

Pathology of infinite systems of linear first order differential equations with constant coefficients

Summary The properties of solutions of finite systems are analyzed and it is shown by examples how these properties may be distorted and ultimately lost in passing from finite to infinite systems. To Enrico Bompiani on his Scientific Jubilee This research was supported in part by United States Army through its Office of Ordnance Research under Grant DA-ORD-12. — Most of the results have been presented to various organisations such as Section A of the American Association for the Advancement of Science and mathematics colloquia at the universities of Leningrad, Moscow, and Toronto.     

Compactness in the fine topology

For reasonable spaces (including topological manifolds) X, Y, we characterize compact subsets of the space of continuous maps from X to Y, topologized with the fine (Whitney) C⁰-topology. In the case of smooth manifolds, we characterize also compact subsets of the space of C^r maps in the Whitney C^r topology.     

On the solvability and numerical methods for solution of linear integro-algebraic equations

We study solvability conditions for a system of Volterra equations with some identically degenerate or rectangular matrix at the main term. Connection is discussed of the solvability conditions and applicability of numerical methods for solving these systems. In particular, the conditions of the convergence of the least squares method with the error functional defined in Sobolev spaces are presented.     

Criteria for the unique solvability of a linear two-point boundary value problem for systems of integro-differential equations

For a linear boundary value problem for a Fredholm integro-differential equation, we obtain necessary and sufficient conditions for the unique solvability in terms of a matrix Q _ν ^m (h) formed on the basis of the matrices of boundary conditions, the differential part, the integral term, and a partition with increment h > 0 of the interval on which the problem is defined.