We consider the symmetric Darlington synthesis of a p × p rational symmetric Schur function S with the constraint that the extension is of size 2p × 2p and we investigate what happens when we impose that S and its extension have real coefficients. In this case, under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the an upper bound for th...

Let Mm×n be the set of all complex matrices of size m × n with Mn = Mn×n. For A ∈ Mm×n, we denote the conjugate transpose of A by A ∗ and call A strictly contractive if I − A∗A is positive definite, where I is the identity matrix of appropriate size. If A ∈ Mn is Hermitian positive (semi)definite, then we write A(≥) > 0. Also, we identify A > (≥)B with A−B > (≥)0, called the Löwner partial orde...

Let $(\mathbb{D}^2,\mathcal{F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb{D}^2$ defined by linear vector field \[ z \,\frac{\partial}{\partial z}+ \lambda \,w w}, \] where $\lambda\in\mathbb{C}^*$. Such has non-degenerate linearized singularity at $0$. $T$ harmonic current directed $\mathcal{F}$ which does not give mass to any of two separatrices $(z=0)$ and $(w=0)$...

We study the wave equation in exterior of a bounded domain K with dissipative boundary condition $$\partial _{\nu } u - \gamma (x) \partial _t = 0$$ on $$\Gamma $$ and $$\gamma > 0.$$ The solutions are described by contraction semi-group $$V(t) e^{tG}, \, t \ge eigenvalues $$\lambda _k$$ G $$\text{ Re }\,\lambda _k < yield asymptotically disappearing $$u(t, x) e^{\lambda t} f(x)$$ having expone...

This paper is a survey devoted to the transformations $$\begin{aligned} C&\mapsto \frac{1}{(2\pi i)^2}\int _{\Gamma _1}\int _2}f(\lambda ,\mu )\,R_{1,\,\lambda }\,C\, R_{2,\,\mu }\,{\mathrm{d}}\mu \,{\mathrm{d}}\lambda ,\\ \frac{1}{2\pi i}\int }g(\lambda )R_{1,\,\lambda R_{2,\,\lambda }\,{\mathrm{d}}\lambda , \end{aligned}$$ where $$R_{1,\,(\cdot )}$$ and $$R_{2,\,(\cdot are pseudo-resolvents a...

A new iterative method for solving the time-dependent multifrequency radiative transfer equations is described. The method is applicable to semi-implicit time discretizations that generate a linear steady-state multifrequency transport problem with pseudo-scattering within each time step. The standard “lambda” iteration method is shown to often converge slowly for such problems, and the new gre...