Stability analysis in continuous and discrete time, using the Cayley transform

For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman distance. This is extended to a class of Desch–Schappacher perturbations

Growth of semigroups in discrete and continuous time

We show that the growth rates of solutions of the abstract differential equations x˙(t)=Ax(t), x˙(t)=A −1 x(t) and the difference equation xd(n+1)=(A+I)(A−I)−1 xd(n) are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup (e A−1t) t≥0 is O(√4t) and for ((A+I)(A−I)−1)n it is O(√4n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of ((A+I)(A−I) −1 )n is O(1), i.e., the operator is power bounded

Maths fights flooding

Due to climate changes that are expected in the coming years, the characteristics of the rainfall will change. This can potentially cause flooding or have negative influences on agriculture and nature. In this research, we study the effects of this change in rainfall and investigate what can be done to reduce the undesirable consequences of these changes. \u

Math Fights Flooding

Due to climate changes that are expected in the coming years, the characteristics of the rainfall will change. This can potentially cause flooding or have negative influences on agriculture and nature. In this research, we study the effects of this change in rainfall and investigate what can be done to reduce the undesirable consequences of these changes