A mathematical modeling approach to the formation of urban and rural areas: Convergence of global solutions of the mixed problem for the master equation in sociodynamics

Urban and rural areas are formed by human migration from thinly populated areas to densely populated areas. It is known in sociodynamics that human migration is described by a nonlinear integro-partial differential equation whose unknown function denotes the population density. This equation is called the master equation. The master equation has its origin in statistical physics, and is regarded as one of the most fundamental equations in natural sciences, as its name suggests. We describe the formation of urban and rural areas by making use of global solutions of the mixed problem for this equation. In this paper we prove sufficient conditions for the mixed problem to have a unique global solution that converges to a two-tier step function as the time variable tends to infinity. This step function is a stationary solution of the master equation, and the higher (lower, respectively) step represents a stationary urban (rural, respectively) area. This result mathematically describes the formation of urban and rural areas in the real world.     

An Entropy-Based Tool to Help the Interpretation of Common-Factor Spaces in Factor Analysis

This paper proposes a method for deriving interpretable common factors based on canonical correlation analysis applied to the vectors of common factors and manifest variables in the factor analysis model. First, an entropy-based method for measuring factor contributions is reviewed. Second, the entropy-based contribution measure of the common-factor vector is decomposed into those of canonical common factors, and it is also shown that the importance order of factors is that of their canonical correlation coefficients. Third, the method is applied to derive interpretable common factors. Numerical examples are provided to demonstrate the usefulness of the present approach.     

Statistical solution to the capacity problem in direct-sequence code-division multiple-access communication systems

Tohru Kohda Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 812-8581, Japan Minoru Tabata Department of Mathematical Sciences, Osaka Prefecture University, Osaka 599-8531, Japan Email: eshima{at}med.oita-u.ac.jp Received on September 21, 2005; Accepted on May 2, 2006 Capacity in the direct-sequence code-division multiple-access(DS/CDMA) communication system was considered according to thecode acquisition performance with the conventional serial-searchmethod, because code acquisition needs a difficult operation.Since capacity in DS/CDMA systems is defined by the maximumnumber of users that can simultaneously transmit their signalswith the same carrier frequencies, assuring a larger capacityis very important in DS/CDMA systems with respect to the economyof frequency source. This paper reconsiders the capacity problemthrough a statistical approach to code acquisition. First, aDS/CDMA system model is reviewed. Second, properties of thecounting method for code acquisition are discussed. It is provedthat the method can acquire the target user's code under anynumber of interference users that simultaneously transmit theirsignals, and that the method can guarantee a considerable precisionin code acquisition. Third, an observation time necessary forcode acquisition is given from a statistical discussion, andthe original counting method is modified into a simpler one.It is concluded that the capacity of the DS/CDMA communicationsystem can be set by the ‘bit error-rate-based capacityor signal-to-noise ratio-based capacity’, i.e. ‘post-acquisition-basedcriterion’, rather than the ‘acquisition-based capacity’.     

A mathematical-model approach to human population explosions caused by migration

If the human population density becomes extremely high in a small area, then we say that a population explosion occurs in the area. Geographical movements of human population can form a regional overconcentration of population. If such an overconcentration becomes excessive, then it often forms a population explosion. In this paper, by taking a mathematical-model approach to human population explosions caused by migration, we obtain a sufficient condition for population to explode. It is known in sociodynamics that geographical population movements are described by a nonlinear integro-partial differential equation whose unknown function denotes the population density. This equation is called the master equation, and has its origin in statistical physics. We express a population explosion as a blow-up solution to the initial-value problem for this equation. We will study a population explosion as an interdisciplinary subject among human population dynamics, statistical physics, and the theory of nonlinear functional equations. The principal result is as follows: if a human population migrates within a sufficiently small domain, if the gradient of initial population density is sufficiently large, if the population gravitates strongly toward densely populated areas, and if a cost incurred in moving is sufficiently small, then a population explosion occurs.     

Statistical approach to the code acquisition problem in direct-sequence spread-spectrum communication systems

^** Email: eshima{at}med.oita-u.ac.jp In direct-sequence spread-spectrum (DS/SS) communication, users'original signals are modulated into higher frequencies withthe users' codes. DS/SS communication has the attractive propertythat multiple users' signals can be simultaneously transmitted;however, communication cannot be performed without synchronizationof users' spread-spectrum (SS) signal. Synchronization is typicallyperformed in two steps, i.e. code acquisition and tracking.This paper gives a statistical solution to the question as tohow code acquisition can be performed effectively and precisely.First, properties of matched-filter outputs of SS signal arediscussed. Second, a theoretical method of code acquisitionis proposed according to statistical decision theory. The methoduses all matched-filter outputs for code acquisition. Third,matched-filter outputs are dichotomized with a threshold valueand the dichotomous outputs are used for code acquisition. Asimple and effective method for code acquisition is proposed.Numerical simulations are also given to illustrate the effectivenessof the proposed method. Finally, a further discussion and conclusionto this study are provided.     

An extension of Krugman’s core–periphery model to the case of a continuous domain: Existence and uniqueness of solutions of a system of nonlinear integral equations in spatial economics

We consider a model that is an extension of Krugman’s core–periphery model to the case of a bounded closed domain included in a Euclidean space. We can describe the relation of the density of workers, the density of nominal wages, and the density of real wages by the system of nonlinear integral equations of the model. If we obtain a solution of the system under the condition that the density of workers is given, then the solution is called a short-run equilibrium. In this paper we prove that this model has a short-run equilibrium, and we obtain a sufficient condition for its uniqueness. Moreover we obtain upper and lower estimates for short-run equilibria, and we construct a useful iteration scheme to numerically obtain short-run equilibria.     

The Kramers–Moyal expansion of the master equation that describes human migration in a bounded domain

It is known in quantitative sociodynamics that human migration in a bounded domain can be described by a nonlinear integro-partial differential equation, which is called the master equation. This equation has its origin in statistical physics. At a physical level of rigor we can formally expand the nonlinear integral operator contained in the master equation into an infinite series whose terms are nonlinear partial differential operators. The infinite series thus obtained is called the Kramers–Moyal expansion. The purpose of this paper is to give a mathematical justification of this formal expansion.