This paper treats the apportionment problem where seats in a legislature are distributed based proportionally on the population of electoral districts. The purpose of this paper is to establish that the relaxed divisor method is proportional which has been recently proposed by the author in this journal. Another purpose is to compare the five historical methods derived by Huntington's pairwise comparison tests to all the relaxed divisor methods where the five historical methods are those of Adams, Dean, Hill, Webster and Jefferson.

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This is a survey of the recent researches on the global complexity bounds of the gradient methods and the Newton-type methods for the nonlinear programming problem. The classical benchmarks of these iterative methods, such as global convergence and rates of convergence, do not represent a total computational time. The global complexity bound is the worst number of iterations to find an appropriate solution, and hence it is proportional to the worst computational time. In recent years the global complexity bounds of the gradient methods and the Newton-type methods have been vigorously investigated. In the survey we report some of these results.

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The aim of this paper is to propose a linear programming (LP) model to answer what roads should be improved to make a road network robust against the disruption of roads at the time of disaster. To this end, we first give a rigorous definition of an alternative route, which requires that an alternative route should share few roads with a usual route, and the travel time along an alternative route should be almost the same as that along a usual route. Then, a road network having many alternative routes would be robust against the disruption of roads, and the LP model is designed for ensuring many alternative routes for various origin and destination (OD) pairs. The Tokai earthquake would cause great damage to the Tokai area including Aichi prefecture. And so, we applied the LP model, for example, to problems with OD pairs of (i) Aichi and Osaka prefectures and (ii) major cities in Aichi and Shizuoka prefectures. In the case of (i) with the data of 1995, we obtained an alternative route similar to Shinmeishin highway, which opened in 2005 and is actually an alternate to Meishin highway. Hence, it can be said that this result indicates the effectiveness of the LP model. In the case of (ii) with the data of 2003, we obtained alternative routes along the coast. This result seems to be strongly affected by the structure of the road network in Shizuoka prefecture. It is socially recognized that Shintomei highway, which opened in 2012, drastically improved the structure. Therefore, we recalculated the result with the data of Shintomei highway. Then, we additionally obtained a suitable alternative route between Hamamatsu and Shizuoka. This result would theoretically support the social recognition for Shintomei highway.

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After the latest world-wide financial crisis, Residential Mortgage-Backed Securities (RMBS) become more and more important in financial instruments for long-term investment. Since RMBS have longer maturities and more frequent payoffs than the ordinary bonds, it is desired to evaluate their values quickly in business transactions, and especially in risk managements where valuations are repeated under various kinds of economic conditions. In this article, we derive semi-analytical pricing formulas of RMBS where prepayment rates have a term structure and a dependence on the interest rates. As an stochastic interest rate model, we adopt CIR++ model or Quadratic Gaussian++ model because their future interest rates become positive and these models can perfectly fit the initial term structure of the market rates. Moreover, we consider the time-dependence of the interest rate sensitivities of prepayment rates. The technical key points are the forward-neutral pricing method and the Feynman-Kac theorem. The obtained pricing formulas have three different expressions, and the interest-rate sensitivity of prepayment rates determines which one is used among the three expressions. All formulas can be calculated so quickly that they would be useful in practice.

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