Internalization of external economic effects on urban sprawl affected areas. An example of the Krakow Metropolitan Area, The study is a discussion on economic externalities, with particular emphasis on technological effects. Attention is also paid to the problem of internalization of economic external effects caused by movement (transport) in areas affected by the urban sprawl process. The research was conducted for all communes of the Krakow Metropolitan Area (KOM), as a result of which the value of: directly incurred financial losses and the value of lost time generated by the necessity of commuting and return in the space of KOM were presented.
Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.