Modeling Bed Shear Stress Distribution in Rectangular Channels Using the Entropic Parameter The evaluation of bed shear stress distribution is fundamental to predicting the transport of sediments and pollutants in rivers and to designing successful stable open channels. Such distribution cannot be determined easily as it depends on the velocity field, the shape of the cross section, and the bed roughness conditions. In recent years, information theory has been proven to be reliable for estimating shear stress along the wetted perimeter of open channels. The entropy models require the knowledge of the shear stress maximum and mean values to calculate the Lagrange multipliers, which are necessary to the resolution of the shear stress probability distribution function. This paper proposes a new formulation which stems from the maximization of the Tsallis entropy and simplifies the calculation of the Lagrange coefficients in order to estimate the bed shear stress distribution in openchannel flows. This formulation introduces a relationship between the dimensionless mean shear stress and the entropic parameter which is based on the ratio between the observed mean and maximum velocity of an openchannel cross section. The validity of the derived expression was tested on a large set of literature laboratory measurements in rectangular cross sections having different bed and sidewall roughness conditions as well as various water discharges and flow depths. A detailed error analysis showed good agreement with the experimental data, which allowed linking the smallscale dynamic processes to the largescale kinematic ones.

(Generalized) Maximum Cumulative Direct, Residual, and Paired xcexa6 Entropy Approach A distribution that maximizes an entropy can be found by applying two different principles. On the one hand, Jaynes (1957a,b) formulated the maximum entropy principle (MaxEnt) as the search for a distribution maximizing a given entropy under some given constraints. On the other hand, Kapur (1994) and Kesavan and Kapur (1989) introduced the generalized maximum entropy principle (GMaxEnt) as the derivation of an entropy for which a given distribution has the maximum entropy property under some given constraints. In this paper, both principles were considered for cumulative entropies. Such entropies depend either on the distribution function (direct), on the survival function (residual) or on both (paired). We incorporate cumulative direct, residual, and paired entropies in one approach called cumulative    xcexa6    entropies. Maximizing this entropy without any constraints produces an extremely Ushaped (bipolar) distribution. Maximizing the cumulative entropy under the constraints of fixed mean and variance tries to transform a distribution in the direction of a bipolar distribution, as far as it is allowed by the constraints. A bipolar distribution represents socalled contradictory information, which is in contrast to minimum or no information. In the literature, to date, only a few maximum entropy distributions for cumulative entropies have been derived. In this paper, we extended the results to well known flexible distributions (like the generalized logistic distribution) and derived some special distributions (like the skewed logistic, the skewed Tukey    xcexbb    and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey    xcexbb    distribution and the Fechner family of skewed distributions. Finally, cumulative entropies were estimated such that the data was drawn from a maximum entropy distribution. This estimator will be applied to the daily Su0026P500 returns and time durations between mine explosions.