State space formulas for stable rational matrix solutions of a Leech problem

Given stable rational matrix functions $G$ and $K$, a procedure is presented to compute a stable rational matrix solution $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$. The solution is given in the form of a state space realization, where the matrices involved in this realization are computed from state space realizations of the data functions $G$ and $K$.Comment: 25 page

State space formulas for a suboptimal rational Leech problem I: Maximum entropy solution

For the strictly positive case (the suboptimal case) the maximum entropy solution $X$ to the Leech problem $G(z)X(z)=K(z)$ and $\|X\|_\infty=\sup_{|z|\leq 1}\|X(z)\|\leq 1$, with $G$ and $K$ stable rational matrix functions, is proved to be a stable rational matrix function. An explicit state space realization for $X$ is given, and $\|X\|_\infty$ turns out to be strictly less than one. The matrices involved in this realization are computed from the matrices appearing in a state space realization of the data functions $G$ and $K$. A formula for the entropy of $X$ is also given.Comment: 19 page

All solutions to the relaxed commutant lifting problem

A new description is given of all solutions to the relaxed commutant lifting problem. The method of proof is also different from earlier ones, and uses only an operator-valued version of a classical lemma on harmonic majorants.Comment: 15 page

State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions

For the strictly positive case (the suboptimal case), given stable rational matrix functions $G$ and $K$, the set of all $H^\infty$ solutions $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions $G$ and $K$. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.Comment: 28 page

Exploring the temporally resolved electron density evolution in EUV induced plasmas

We measured for the first time the electron density in an Extreme Ultra-Violet induced plasma. This is achieved in a low-pressure argon plasma by using a method called microwave cavity resonance spectroscopy. The measured electron density just after the EUV pulse is $2.6\cdot10^{16}$ m$^{-3}$. This is in good agreement with a theoretical prediction from photo ionization, which yields a density of $4.5\cdot10^{16}$ m$^{-3}$. After the EUV pulse the density slightly increase due to electron impact ionization. The plasma (i.e. electron density) decays in tens of microseconds.Comment: 3 pages, 4 figure

Why do little kids ask to hear the same story over and over?

One way people learn new words is through reading books and stories. Little kids love hearing their favorite stories over and over and are also very good at learning new words. We wondered if reading the same stories could be helping preschool kids learn new words. Our research tested if it was better to read the same stories over and over or to read a few different stories. Here we tell you about three studies that show preschool kids learn more words from the same stories over and over. Our research suggests that it’s easier to learn new words from stories when you have heard the story before and know what is going to happen

New economic geography, empirics, and regional policy

There are doubts about the effectiveness of regional policy. Well known are the fruitless attempts of Italy to bridge the gap between the Mezzogiorno and the North, of Germany to bridge the gap between the Neue Länder and the West, and of the European Commission to reduce regional disparities in general. We validate one explanation: agglomeration advantages lock business activity in relatively prosperous core regions, even though wages – and thus production costs – tend to be higher there. We set off from the ‘New Economic Geography’, a set of general equilibrium models that focus on location choice. Theory, descriptive statistics, and econometric analysis support the conclusion that the European economic geography is characterized by a network of local and stable core periphery systems. This implies that disparities between core regions and their peripheries at a (sub) provincial level of regional aggregation are with us to stay, as regional policy targeted on peripheries tends to be insufficient to counter centripetal market forces. Moreover, even if such policy has an impact, it may be adverse, as core regions may benefit disproportionately in the long run. A focus of regional policy on local agglomerations, which have a realistic chance to hold on to economic activity, is therefore desirable.