The study on distribution of objects in space could be divided into two parts, that of form and process. Although the form of distribution could be thought including distribution pattern and spacial dispersion, neither the difference between them is recognized clearly, nor much attention has been paid to the study about the spatial dispersion.
This paper mainly aims to built a index measuring how much densely or dispersively do objects distribute in space. The objects dealt with here are treated as points, lines, surfaces and solids, being called as point-objects, line-objects, surface-objects and solid-objects respectively (fig. 1), and meanwhile are treated as the ones spread in one, two and three-dimensional space.
Scatteration measure of objects in one or two-dimensional space often used up to now is standard deviation or something like it. For example socalled ‘standard distance’ defined by Bachi in 1957 is calculated as follow.
S_d=(1/NΣNi=1d_ci²)^1/2 (*)
where, d_ci, represents the distance between object i(i=1, 2, …, N) and their mean center, C.
It has been introduced that the standard distance could be used to distinguish the difference of scatteration between different groups of objects.
In fact, that is not correct. If we consider the square lattice points as shown in fig. 2 in the paper, the value of standard distance of the square lattice points should remain same despite of their size, since that kind of lattice points spread uniformly in a plain. But the value, S_d, of the square lattice points calculated by the way mentioned in formula (*) is not so but becomes bigger as the number of objects studied, N, increases (see formula 4 in the paper).
Based on some discussing made in appendix about so-called ‘uniform point set’, the paper suggests a new measure of the scatteration of point-objects as follow:
P_{cb, m}=GΣNi=1ΣNj=1d_ij^b/N^(2m+b)/m (**)
where, P_{cb, m}, named as ‘scatteration measure’, represents the scatteration of point-objects in m-dimensional space (m=1, 2, 3). N is the number of the point-objects concerned. d_ij is distance between point i and j, while b could be 1, 2, …, and G be chosen properly case by case.
When objects, on the other hand, are considered as line-objects, surface-objects and solid-objects, the formula (**) could also be used to calculate their scatteration measure. In these cases, N represents the number of points of uniform point set included within those kinds of objects.
In addition, how could the formula (**) be used to calculate the scatteration measure if objects (points, lines and surfaces) spread on a limited curve or a limited curved surface is also discussed.
The value of scatteration measure built up in the way shown in (**) will not affected by the size of the objects concerned, but depends on the condition of how dispersively the objects distribute in space only. And it is not necessary for calculation of the scatteration measure to have a accurate boundary surrounding the objects concerned.
Besides, the scatteration measure has a obvious geometric meaning as shown in fig. 4 in the paper (m=2). Supposing that there are k point-objects with same number of points, and that their scatteration measure are P_c1, P_c2, …, P_ck (P_ci>P_ci+1, i=1, 2, …, k-1)

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From the view point of ‘migrations in one's life history’, entire migration careers should be considered. A migration career means a sequence of moves an individual has experienced throughout one's life. Conditions of migration careers would give important clues to deeper understanding of migration behavior. Unfortunately, many of the previous longitudinal data of migration histories were too fragmented to clarify entire patterns of individual migration careers. Therefore, an analysis of migration histories, using more suitable materials, is necessary. In this paper, based on my field survey, I investigated American mobility, through description and classification of migration careers. Statistical methods for the procedures were proposed.
Field survey and subjects:
This field survey was conducted in two small university towns, one on the West Coast and the other in the East South Central States-Santa Barbara, CA, and Lexington, KY (Fig. 1). Materials on the migration history were collected through interviews with 126 subjects (48 in Santa Barbara and 78 in Lexington). The subjects had the common characteristic of being ‘White Anglo-Saxon Protestants and belonging to an upper middle or middle class’.
Statistical procedures:
1) Migration-career graph-A migration career can be drawn as a graph which has the horizontal axis representing age and the vertical axis representing the number of moves. 8 examples were shown in Fig. 4.
2) Transit vector-I proposed a transit vector [θ₁, θ₂, θ₃, θ₄] to abstract geometrical features from a migration-career graph, Change of mobility through one's life course can be summarized as elements in a vector. θ₁ indicates mobility (moves/years) in an age-bracket of 0≤<18. In the same way, θ₂ is for 18≤<40, θ₃ for 40≤<60, and θ₄ for 60≤<90. Each θ is assigned to an integral value from 0 to 9. For example, 0 indicates 0 moves/the whole period, and 9 more than 1 move/a year. If a subject is younger than 40, his (or her) transit vector is expressed as [θ₁, θ_{2, -, -}]
3) Classification-index-The combination of θ₁ and θ_t was a simpler classification-index than the transit vector itself. θ₁ represents an initial status of a migration-career graph. θ_t means mobility through one's life (total moves/one's entire life (years)), and it closely relates to θ₂ (R=0.79). For most of the subjects, θ₂ has the highest value among the four vector elements. Due to these reasons, the index works in graph-grouping very well (Fig. 7).
Findings:
a) The subjects have high mobility. Numbers of moves experienced in one's life time were distributed in the range of 2-35, and the mean was 12.12. There were two-groups, a lower-mobility group of about 10 moves and a higher-mobility group of about 17 moves (Fig. 2). The ratio of these two groups was around 7:3. These groups could not be characterized by region nor gender. No significant differences were found among generations (Table 2).
b) As a result of transit-vector reduction, the following typical pattern of migration careers was detected: [θ₁<θ₂>θ₃≥θ₄=0 or ≥1] (Fig. 5 and Table 3).
c) Younger generations have higher mobility than older ones have. The rise of mobility during the age-bracket of 18≤<40 causes the change (Fig. 6).
d) The percentage of intra-urban migration has been increasing. Most of all the moves were carried out within a fairly small sphere.

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This paper seeks to review the gentrification studies in the Western countries, and obtain implications for Japanese cities.
The article begins with a review of the literature concerned with social effects of gentrification. In the capitalist countries, at the outset, many people expect that gentrification is caused by private ecnomy. However the effects of gentrification are not so expected for the several reasons. One reason is that most of inmovers to the gentrified neighborhoods are not part of a ‘back to the city movement’, but ‘staying in the city’. The other is that gentrifiers revitalizing abandoned areas are limited in number. On the contrary, it causes the displacement of the socioeconomically weak. Most of them are low income, elderly and minority. They feel the sting of the displacement caused by urban revitalization. Furthermore it results in producing a lot of homeless people.
The third chapter treats with five theoretical issues, which are institutions, stage models, rent-gap theory, the new middle class, and marginal gentrifiers. According to the institutionalist approach, central and local government, estate agents, and building societies, are the inducers of gentrification. The stage model explains the gentrification process positively by inmovers' attitude to accept risks in the deteriorated areas. Rent-gap theory explains gentrification structurally by the movement of capital, back to the inner city. The new middle class is on the rise due to industrial restructuring.
They prefer to live near the city center, so they cause gentrification. They prefer not only historical architecture, but also modern amenities. Inmovers to the gentrified neighborhoods, are not only the new middle class. There is also the formation of reproducing marginal gentrifiers. Marginal gentrifiers come to live in the inner city because of alternative life-styles. Many researchers agree that no approach cannot explain the phenomenon alone, and some of them seek to integrate several approaches.
In section four I argue the applicability of researches on social effects and theoretical approaches in Western countries for Japanese cases. First I show two bases for the occurrence of gentrification in Japanese cities. One is the recent trend of upgrading living spaces. Most Japanese houses are built of wood, so they become obsolete without maintenance. It is easy to scrap obsolete houses and renovate new ones. Recently there are many cases of rehabilitating modern Western-style buildings and reforming the living layout of condominiums. The other is the restructuring of the inner city. In the 1970s most central cities lost affluent people. Many heavy industries scattered from metropolitan regions to nonmetropolitan areas. Although the inner city area in the central cities lost population, it also provides opportunities to increase population again. Actually recurrence of population appeared in some of those large cities during the late 1980s. I argue that three primary factors may cause gentrification in Japanese cities. The first one is industrial restructuring. Industrial restructuring produces new professionals. They may be potential gentrifiers. The second one is suburbanization. Expanding urban regions make a long commute to the office in the central city. So many people prefer to live not so far from the office. The third one is the supply of condominiums. Many people invested in them during the late 1980s, because of lower interest rates. Most of them located in the inner city and induce inmovers.
Second, I discuss future directions for research on gentrification in Japanese cities. There may be three main issues. The first one is resettlement in the inner city. Municipal officials of most large central cities are working to prevent the population from decreasing.

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In the previous literature, the urban system is a result of corporate hierarchy. In reality, however, the urban system is also a determinant of allocation of branch offices. In this paper, the optimal allocation of branch offices is analyzed within a given urban system.
In preparation for the analysis, an urban system is defined in section II. There are two major models of urban systems, that is, Christaller's and Pred's urban systems. However, these systems do not conflict, because Christaller's system is comprised in Pred's. Although Pred's urban system is more general, it is not capable of “identifying”the existing urban system as a particular system. Therefore, in this paper, it is supposed that urban system is Christaller's system, which is identifiable. It is necessary to suppose such a condition, in order to get the optimal locations of branch offices from the optimal number of branch offices.
In section III, optimal number n^* and locations of branch offices are analyzed. Suppose that the manufacturing firm has a nationwide market area of goods, a head office in the highest-ranked central place, and n(≥0) branch offices in the country. Each branch office is of the same scale. The territories of the branch offices necessarily correspond with the complementary areas of central places with a certain rank, and never intersect. And branch offices have the budgets, have no rank, and cannot contact without intervention of the head office.
The roles of branch office are as follows, that is, ‘communication to head’ and ‘communication to market.’ T_R and T_M, respectively represent the communication costs corresponding to those two roles, and C represents operational cost of offices. Then, T_R, T_M and C are defined as the functions of n. And it is supposed that the behavioral principle of the firm is cost minimizing. In the case that C is a linearly increasing function of n, there are two cases about the value of n^*, [see figure 1];
•when the three functions are all increasing, n^*=0 (named the centralized type)
•when some decreasing functions are contained, n^*>0.
Next, this model is linked to central place theory. In the central place rank, i=0, 1, …, m, i=0 represents the highest-ranked central place, the larger i becomes, the lower the central place. Because branch offices can only be located at central place, there is a relation, that is, n=kⁱ-1 (k means Christaller's k, k>1), between n and i. By this formula, we can decide the optimal i, that is, the arrangement of branch offices [see figure 2]. Although the value of i corresponding to n^* may not be an integer, it is an optimal i that a firm usually decides in advance. The value of n^* is decided on optimali later.
Central place rank i indicates totally, not only the number of central places, but also their scale or arrangement. The largest merit of the approach, in which the optimal arrangement of branch offices is solved as the equation of i, is that i gives much information.

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