We rigorously derive energy estimates for the second order vector wave equation with gauge condition for the electric field with non-constant electric permittivity function. This equation is used in the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system. Our numerical experiments illustrate efficiency of the modified hybrid scheme in two and three space dimensions when the method is applied for generation of backscattering data in the reconstruction of the electric permittivity function.

Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system

We obtain a simple closed form solution for gradient elasticity involving a mode III crack of semi infinite length, with tractionless surfaces in an infinite plate. This solution is the gradient elastic equivalent of the small scale yielding solution of linear elasticity, e.e. the asymptotic solution

The asymptotic solution of gradient elasticity for mode III

A robust Kalman filter is proposed for the discrete-time system with norm-bounded parametric uncertainties. The uncertainties are described by the energy bound constraint, i.e. the sum quadratic constraint (SQC). It is shown that the SQC can be converted into an indefinite quadratic cost function to be minimised in the Krein space, and it is found that the Krein space Kalman filter is a solution of the minimisation problem. After introducing a Krein space state-space model, which includes the uncertainty, one can easily write a robust version of the Krein space Kalman filter by modifying the measurement matrix and the variance of measurement noises in the original Krein space Kalman filter. Since the resulting robust Kalman filter has the same recursive structure as a conventional Kalman filter, a robust filtering scheme can be readily designed using the proposed method. A numerical example demonstrates that the proposed filter achieves robustness against parameter variation and improvement in performance when compared with a conventional Kalman filter and an existing robust Kalman filter, respectively.

Robust Kalman filtering via Krein space estimation

Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1

Chebyshev polynomial solutions of second-order linear partial differential equations

Inner Product Structures

In this very short chapter we present some problems connected with the inner product structures. This collection of problems is by no means complete and our purpose is to direct the attention of interested readers to problems that are mentioned by different authors. We give references to where the problem is first stated as well as references to where the latest results (to the best knowledge of the present author) can be found.

A Collection of Problems

The notion of indefinite inner product space was first considered by the famous physicist P. A. M. Dirac in 1942 and independently by the famous Soviet-Russian mathematician L. S. Pontrjagin 1944, who was motivated by some questions of S.L. Sobolev related to a problem in mechanics (published, much later, in 1960). The subject has attracted the interest of many mathematicians. For a detailed account of the subject we refer to the excellent survey articles by Krein and Iohvidov (1956, 1959) and to the book by Bognar (1974); for applications the readers may wish to consult the book by Daleckiĭ and Krein (1970) and the excellent papers by Synge (1954) and Bracci (1975).

Indefinite Inner Product Structures

The theory of complete inner product spaces with reproducing kernel functions has received a growing interest partly because of its own beauty and partly because of numerous applications in the theory of functions of one or several variables and in the theory of partial differential equations. It seems worthy of notice that, perhaps, the space in which the existence of a reproducing kernel was first recognized was the space of harmonic functions on a complex domain and which are elements of L2. This fact was stated by S. Zaremba in 1908. The theory of reproducing kernel functions was decisively influenced by the works of S. Bergman in which the so-called ‘Bergman kernel’ was introduced and studied intensively. The abstract theory of reproducing kernel functions was greatly influenced by the works of Aronszajn (1943, 1950). For more information about the applications of the theory of reproducing kernels we refer to Bergman (1950), Bergman and Schiffer (1951), Epstein (1965) as well as to the references quoted in our bibliography.

https://link.springer.com/content/pdf/10.1007%2F978-94-009-3713-0_11.pdf

Reproducing Kernels and Inner Product Spaces. Applications

Let H be a complex inner product space and let L(H) be the set of all bounded linear operators on H. In this chapter we present some basic facts about the set L(H) as well as about the set of unbounded operators on H. We also present some classes of operators whose structure is better understood, among which we mention the class of hermitian, unitary and normal operators, selfadjoint and essentially selfadjoint operators, and the class of unbounded normal operators. Several examples illustrate the exposed results. Also, since there has recently been a growing interest in new classes of unbounded operators, we also present some classes of unbounded operators related to bounded normal operators, namely the classes of quasinorma1 operators, subnormal operators, hyponormal operators, operators of class (N), W–N operators, θ operators, etc. For more detail on the classes of bounded operators we refer to the author’s book (1981).

Vasile Ion Istrăţescu

Papers

Inner Product Structures

A Collection of Problems

Indefinite Inner Product Structures

Reproducing Kernels and Inner Product Spaces. Applications

Bounded and Unbounded Linear Operators