A review on some contributions to perturbation theory, singular limits and well-posedness

In the beginning of the 1990s we devoted a sequence of papers to perturbation theory, singular limits and well-posedness problems. In particular, the strong well-posedness of the initial-boundary value problem for the compressible Euler equations was demonstrate for the first time. Our method also allowed singular limit results in the strong norm, even under assumptions weaker than the current ones in the literature (where the strong norm is not reached). It is worth noting that, until now, the above method and results have not been substantially improved. Hence an introduction to it still looks timely. Actually, in a forthcoming paper, by returning to this method, we improve (in a very substantial way) some important results recently appeared in the literature.     

The singular values of and

We obtain several inequalities relating the singular values of A+B and those of A+iB when A and B are hermitian, and when one or both of them are positive semidefinite.     

On the existence of bubble-type solutions of nonlinear singular problems

Considered in this paper is a class of singular boundary value problem, arising in hydrodynamics and nonlinear field theory, when centrally bubble-type solutions are sought: \((p(t)u0)0 = c(t)p(t)f(u); u0(0) = 0; u(+1) = L > 0\) in the half-line \([0;+1)\), where \(p(0) = 0\). We are interested in strictly increasing solutions of this problem in \([0;1)\) having just one zero in \((0;+1) \)and finite limit at zero, which has great importance in applications or pure and applied mathematics. Su±cient conditions of the existence of such solutions are obtained by applying the critical point theory and by using shooting argument [9,10] to better analysis the properties of certain solutions associated with the singular di®erential equation. To the authors' knowledge, for the first time, the above problem is dealt with when f satis¯es non-Lipschitz condition. Recent results in the literature are generalized and signi¯cantly improved.     

Low rank update of singular values

The notion of a low rank update arises in many important applications. This paper deals with the inverse problem of updating a rectangular matrix by additive low rank matrices so as to reposition the associated singular values. The setting is analogous to the classical pole assignment problem where eigenvalues of a square matrix are relocated. Precise and easy-to-check necessary and sufficient conditions under which the problem is solvable are completely characterized, generalizing some traditional Weyl inequalities for singular values. The constructive proof makes it possible to compute such a solution numerically. A pseudo algorithm is outlined.

     

Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices

We study the perturbation theory for the eigenvalue problem of a formal matrix product A ₁ ^{s 1} ··· A _p ^{s p}, where all A _k are square and s _k {–1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eigenvalues of such formal matrix products. As an application we then extend the structured perturbation theory for the eigenvalue problem of Hamiltonian matrices to Hamiltonian/skew-Hamiltonian pencils.     

Lower bound for the spectrum and the presence of pure point spectrum of a singular discrete Hamiltonian system

In this paper, criteria for limit-point (n) case of a singular discrete Hamiltonian system are established. Furthermore, the lower bound of the essential spectrum is obtained and the present of pure point spectrum is discussed for such system by using the spectral theory of self-adjoint operators in a Hilbert space.     

A note on Krylov subspace methods for singular systems

Recently, Calvetti et al. have published an interesting paper [Linear Algebra Appl. 316 (2000) 157–169] concerning the least-squares solution of a singular system by using the so-called range restricted GMRES (RRGMRES) method. However, one of the main results (cf. [loc. cit., Theorem 3.3]) seems to be incomplete. As a complement of paper [loc. cit.], in this note we first make an example to show the incompleteness of that theorem, then we give a modified result.     

Asymptotic behavior of the singular values of a generalization of the operator fractional integration

In this paper we find the first term in the asymptotics of singular values of the generalized fractional integration operator.     

Twisted traces of singular values

Let p be a prime for which the congruence group Γ₀(p)^* is of genus zero, and j _p ^* be the corresponding Hauptmodul. We investigate the twisted traces of singular values of j _p ^* and construct infinite products related to them. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-331-C00006).     

Association schemes based on maximal totally isotropic subspaces in singular pseudo-symplectic spaces

This paper provides some new families of symmetric association schemes based on maximal totally isotropic subspaces in (singular) pseudo-symplectic spaces. All intersection numbers of these schemes are computed.     

Riesz-Nágy singular functions revisited

In 1952 F. Riesz and Sz.-Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-Nágy family of functions and we relate it to a system for real number representation which we call (τ,τ−1)-expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.     

Solution of a class of singular boundary value problems

In this work a class of singular ordinary differential equations is considered. These problems arise from many engineering and physics applications such as electro-hydrodynamics and some thermal explosions. Adomian decomposition method is applied to solve these singular boundary value problems. The approximate solution is calculated in the form of series with easily computable components. The method is tested for its efficiency by considering four examples and results are compared with previous known results. Techniques that can be applied to obtain higher accuracy of the present method has also been discussed.     

Monotone method for singular BVP in the presence of upper and lower solutions

In this paper we examine existence of monotone approximations of solutions of singular boundary value problem -(p(x)y^′(x))^′=q(x)f(x,y,py^′) for 0<x?b and lim_x→0⁺p(x)y^′(x)=0,α₁y(b)+β₁p(b)y^′(b)=γ₁. Under quite general conditions on f(x,y,py^′) we show that solution of the singular two point boundary value problem is unique. Here is allowed to have integrable singularity at x=0 and we do not assume .     

Perturbation analysis of the heat transfer in porous media with small thermal conductivity

An approximate analytical solution for the one-dimensional problem of heat transfer between an inert gas and a porous semi-infinite medium is presented. Perturbation methods based on Laplace transforms have been applied using the solid thermal conductivity as small parameter. The leading order approximation is the solution of Nusselt (or Schumann) problem. Such solution is corrected by means of an outer approximation. The boundary condition at the origin has been taking into account using an inner approximation for a boundary layer. The gas temperature presents a discontinuous front (due to the incompatibility between initial and boundary conditions) which propagates at constant velocity. The solid temperature at the front has been smoothed out using an internal layer asymptotic approximation. The good accuracy of the resulting asymptotic expansion shows its usefulness in several engineering problems such as heat transfer in porous media, in exhausted chemical reactions, mass transfer in packed beds, or in the analysis of capillary electrochromatography techniques.     

Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.