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Recently land-use change has been the main concern for worldwide environment change and is being used by city and regional planners to design sustainable cities. Nakuru in the central Rift Valley of Kenya has undergone rapid urban growth in last decade. This paper focused on urban growth using multi-sensor satellite imageries and explored the potential benefits of combining data from optical sensors (Landsat, Worldview-2) with Radar sensor data from Advanced Land Observing Satellite (ALOS) Phased Array type L-band Synthetic Aperture Radar (PALSAR) data for urban land-use mapping. Landsat has sufficient spectral bands allowing for better delineation of urban green and impervious surface, Worldview-2 has a higher spatial resolution and facilitates urban growth mapping while PALSAR has higher temporal resolution compared to other operational sensors and has the capability of penetrating clouds irrespective of weather conditions and time of day, a condition prevalent in Nakuru, because it lies in a tropical area. Several classical and modern classifiers namely maximum likelihood (ML) and support vector machine (SVM) were applied for image classification and their performance assessed. The land-use data of the years 1986, 2000 and 2010 were compiled and analyzed using post classification comparison (PCC). The value of combining multi-temporal Landsat imagery and PALSAR was explored and achieved in this research. Our research illustrated that SVM algorithm yielded better results compared to ML. The integration of Landsat and ALOS PALSAR gave good results compared to when ALOS PAL- SAR was classified alone. 19.70 km2 of land changed to urban land-use from non-urban land-use between the years 2000 to 2010 indicating rapid urban growth has taken place. Land-use information is useful for the comprehensive land-use planning and an integrated management of resources to ensure sustainability of land and to achieve social Eq- uity, economic efficiency and environmental sustainability.
The investment portfolio with stochastic returns can be represented as a maximum flow generalized network with stochastic multipliers. Modern portfolio theory (MPT)  provides a myopic short horizon solution to this network by adding a parametric variance constraint to the maximize flow objective function. MPT does not allow the number of securities in solution portfolios to be specified. Integer constraints to control portfolio size in MPT results in a nonlinear mixed integer problem and is not practical for large universes. Digital portfolio theory (DPT)  finds a non-myopic long-term solution to the nonparametric variance constrained portfolio network. This paper discusses the long horizon nature of DPT and adds zero-one (0-1) variables to control portfolio size. We find optimal size constrained allocations from a universe of US sector indexes. The feasible size of optimal portfolios depends on risk. Large optimal portfolios are infeasible for low risk investors. High risk investors can increase portfolio size and diversification with little effect on return.