The method of Webster has been at the center of discussion about the problem of apportionment. Bias is often used as an evaluation criterion for fair apportioment. Hence it is significant to evaluate the bias of the Webster method. First we define the bias of apportionment methods based on information theory. Then, fixing the number of states, the bias of the Webster method is expressed as a function of the house size.

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The unit commitment problem is to determine the schedule of power generating units and the generating level of each unit. The decisions involve which units to commit at each time period and at what level to generate power to meet the electricity demand. We consider the column generation algorithm to solve the problem. Previous method used the approach in which each column corresponds to the start-stop schedule and output level. We present a new solution algorithm based on the column generation. It is shown that the new approach is effective to solve the problem.

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Using Numerov method, the recursive transfer method (RTM) was developed as a numerical method for solving scattering problems caused by localized waves. However, the application scope of RTM was limited because of the restriction on Numerov method. Therefore, the new discretization scheme was proposed to extend the scope of RTM. Analyzing one-dimensional scattering problem under the proposed discretization scheme, the RTM error was evaluated and the possibility to achieve high accuracy was shown.

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We recently proposed a method for generating true orbits of one-dimensional piecewise linear fractional maps (Physica D, 268 (2014), 100-105). By applying the method to the continued fraction transformation and a modified Bernoulli map, we show that we can obtain true orbits which display the same properties as typical orbits of the two maps. We also conduct verification of Brillhart's cubic irrational that is known to display an unusual occurrence of large partial quotients in its continued fraction expansion, and we report that we can obtain results which support the safety of our true orbit generation method.

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