In this paper, we present a geometrical model which describes the spatial and temporal distribution of traffic within a circular city whose transportation network consists of a radial-ring system of roads. We extend the earlier ideas of two-dimentional traffic flow models by giving explicit consideration of the time variation of the traffic flow. The model is focused upon the morning peak period of commuter traffic during which the greatest demands on the existing road networks are observed. The results are useful in explorations of the effects of staggered working hours on the distribution of traffic over the urban transport infrastructure.

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This paper proposes a problem of adding relation to an organization structure which is a complete binary tree of height H. When one edge between a node with a depth M and its descendant with a depth N is added, an optimal pair of depth (M,N)^* is obtained by maximizing the total shortening path length which is the sum of shortening lengths of shortest paths between every pair of all nodes. (M,N)^* = (0,2) for H = 2,3,4,5 and (M,N)^* = (0,4) for H = 6, 7, … are shown.

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In this paper we derive a formula for indefinite integration of analytic functions over (-1,s) where -1 < s < 1, with possible singularity at the end points s = ±1 of the integrand, by means of the sinc approximation followed by the double exponential transformation. It can be regarded as a double exponential analogue of the formula by S. Haber. The error of our formula behaves approximately as exp (-c_1N/log c_2N) where N is the number of function evaluations of the integrand. This error term shows a much faster convergence when N becomes large than that of the formula by Haber. Several numerical examples also indicate high efficiency of our formula.

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In this paper, using the electric circuit method, we consider a connection method of complete graphs by vertices or edges where the number of trees becomes the maximum. A necessary condition for the maximum number of trees is that both vertex and edge connection graphs are primitive connection graphs, and then the primitive connection graph is equivalent to its corresponding adjacency graph in regard to the number of trees. Therefore, we can use general graphs with the maximum number of trees to find the above-mentioned connection method of complete graphs.

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By the different modification of J_<2a+2n-k>(x) Σ^∞_<i=0>Γ(2b-2a+i)/(i!Γ(2b-2a)). J_<-2a-2n+2i>(x), we obtain an identity A(a, b, n, k, l) ≡ B(a, b, n, k, l), where A(a, b, n, k, l) and B(a, b, n, k, l) are given in the paper. Rewriting A(a, b, n, k, l) and B(a, b, n, k, l) with Pochhammer's symbol, setting l = n, substituting 2n to k and using the relation (0)_0 = 1, (0)_i = 0 (i ≥ 1), we have discovered the following summation formula of the finite generalized hypergeometric series, [numerical formula], When n (≥ 0) is an integer. Similarly, setting l = n, substituting 2n + 1 to k, we have also obtained the following formula [numerical formula].

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In general, zeros of a Lagrange interpolation polynomial can be calculated only numerically, but if the interpolated function is given as the form 1/(z-c) and the sampling points are equally distributed on an ellipse, then the zeros can be represented explicitly and they are also equally distributed on an ellipse of common foci. We will show this fact and prove a theorem that presents a sufficient condition for this method to be generalized.

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The objective of this paper is to show the theoretical relationship between spatial distribution of urban workers and commuting or business-trip distances, using the Vaughan's joint distribution of home and workplace locations. First, we describe macroscopic relationship of the locations between home and workplace, which is based on Vaughan's quadvariate normal distribution whose marginal distributions are according to Sherratt's bivariate normal distribution, by introducing intensive parameters, i.e., jobs-housing balance and spatial correlations. Then we derive the analytical expression of commuting distance or business-trip distance by using those parameters. We find out the optimal workplace distribution that minimizes the weighted sum of average commute travel distance and average business-trip distance. Commuting distance has a minimum value with respect to jobs-housing balance, when spatial correlation is constant. The decentralization of jobs is reasonable policy when the spatial correlation is sufficiently large and the weight of business-trip is low. Thus, jobs-housing balance and spatial correlation of home and workplaces play important roles in identifying travel distance, and also in determining optimal spatial distribution of urban workers.

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