The paper describes a parsimonious approach for generating continuous daily stream-¯ow time-series from observed daily rainfall data in a catchment. The key characteristic in the method is a duration curve. It is used to convert the daily rainfall information from source rain gauges into a continuous daily hydrograph at the destination river site. For each source rain gauge a time-series of rainfall related`current precipitation index' is generated and its duration curve is established. The current precipitation index re¯ects the current catchment wetness and is de®ned as a continuous function of precipitation, which accumulates on rainy days and exponentially decays during the periods of no rainfall. The process of rainfall-to-runo conversion is based on the assumption that daily current precipitation index values at rainfall site(s) in a catchment and the destination site's daily¯ows correspond to similar probabilities on their respective duration curves. The method is tested in several small catchments in South Africa. The method is designed primarily for application at ungauged sites in data-poor regions where the use of more complex and information consuming techniques of data generation may not be justi®ed.
For several common parent laws, the number of vertices of a sample convex hull follows a kind of law of large numbers. We exhibit an example of a parent law which contradicts a general conjecture about this matter.
We consider a large class of fast growing sequences of numbers Un like the nth superfactorial Q k=n k=1 k!, the nth hyperfactorial Q k=n k=1 k k and similar ones. We show that their mantissas are distributed following Benford's law in the sense of the natural density. We prove that this is also verified by Vn = Q k=n k=1 U k , by Q k=n k=1 V k and is passed down to all the sequences obtained by iterating this design process. We also consider the superprimorial numbers and the products of logarithms of integers.
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