On matrices for which norm bounds are attained

Let $\|A\|_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$ columns by the H\"older $\ell_p$ and $\ell_q$ norms on $R^n$ and $R^m$ (or $C^n$ and $C^m$), respectively. It is easy to find an upper bound for the ratio $\|A\|_{r,s}/\|A\|_{p,q}$. In this paper we study the classes of matrices for which the upper bound is attained. We shall show that for fixed $A$, attainment of the bound depends only on the signs of $r-p$ and $s-q$. Various criteria depending on these signs are obtained. For the special case $p=q=2$, the set of all matrices for which the bound is attained is generated by means of singular value decompositions

A First-Principles Implementation of Scale Invariance Using Best Matching

We present a first-principles implementation of spatial scale invariance as a local gauge symmetry in geometry dynamics using the method of best matching . In addition to the 3-metric, the proposed scale invariant theory also contains a 3-vector potential $A_k$ as a dynamical variable. Although some of the mathematics is similar to Weyl's ingenious but physically questionable theory, the equations of motion of this new theory are second order in time-derivatives. Thereby we avoid the problems associated with fourth order time derivatives that plague Weyl's original theory. It is tempting to try to interpret the vector potential $A_k$ as the electromagnetic field. We exhibit four independent reasons for not giving into this temptation. A more likely possibility is that it can play the role of "dark matter". Indeed, as noted in scale invariance seems to play a role in the MOND phenomenology. Spatial boundary conditions are derived from the free-endpoint variation method and a preliminary analysis of the constraints and their propagation in the Hamiltonian formulation is presented.Comment: 11 page

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S−1/2B*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H∞ for positive real transfer functions of the form D+S−1/2B*(author−A)−1,B

University Innovation and the Professor’s Privilige

Publisher PD

R&D-Persistency, Metropolitan Externalities and Productivity

Firms display persistent differences as regards both internal and external characteristics, and these differences correspond to asymmetries in the performance of firms with regard to productivity level and growth as well as innovativeness. This paper focuses on one internal characteristic and one external factor by distinguishing between firms with persistent R&D efforts and other firms and firms located in a metropolitan region versus firms with other locations. Applying Swedish data on individual firms and their location, the paper shows that firms that follow a strategy with persistent R&D efforts have a distinctly higher level of productivity across all types of location. In addition, the productivity level of firms with persistent R&D is augmented in a significant way when such firms have a metropolitan location and, in particular, a location in a metropolitan city

Analysis of a model for the dynamics of prions II

A new mathematical model for the dynamics of prion proliferation involving an ordinary differential equation coupled with a partial integro-differential equation is analyzed, continuing earlier work. We show the well-posedness of this problem in a natural phase space, i.e. there is a unique global semiflow in the phase space associated to the problem. A theorem of threshold type is derived for this model which is typical for mathematical epidemics. If a certain combination of kinetic parameters is below or at the threshold, there is a unique steady state, the disease-free equilibrium, which is globally asymptotically stable; above the threshold it is unstable, and there is another unique steady state, the disease equilibrium, which inherits that property