scholarly journals Emergence of Industrial Hierarchy in One-dimensional Space: Bifurcation Analysis of a Multi-industry Urban Location Model with Communication Externality

2010 ◽  
Vol 27 (0) ◽  
pp. 285-295
Author(s):  
Yuki TAKAYAMA ◽  
Takashi AKAMATS
Keyword(s):  
Bifurcation Analysis ◽  
Dimensional Space ◽  
Location Model ◽  
One Dimensional ◽  
Urban Location

Parametric surfaces

1949 ◽  
Vol 45 (1) ◽  
pp. 14-23 ◽  
Author(s):  
A. S. Besicovitch
Keyword(s):  
Dimensional Space ◽  
Parametric Surfaces ◽  
Rectifiable Curve ◽  
One Dimensional ◽  
Dimensional Measure

In 1914 Carathéodory defined m–dimensional measure in n–dimensional space. He considered one-dimensional measure as a generalization of length and he proved that the length of a rectifiable curve coincides with its one-dimensional measure.

scholarly journals A new method for solving a class of heat conduction equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1205-1210
Author(s):  
Yi Tian ◽  
Zai-Zai Yan ◽  
Zhi-Min Hong
Keyword(s):  
Monte Carlo ◽  
Heat Conduction ◽  
Dimensional Space ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Governing Equations ◽  
One Dimensional ◽  
Sparse System ◽  
Linear Algebraic Equations ◽  
Crank Nicolson

A numerical method for solving a class of heat conduction equations with variable coefficients in one dimensional space is demonstrated. This method combines the Crank-Nicolson and Monte Carlo methods. Using Crank-Nicolson method, the governing equations are discretized into a large sparse system of linear algebraic equations, which are solved by Monte Carlo method. To illustrate the usefulness of this technique, we apply it to two problems. Numerical results show the performance of the present work.

scholarly journals Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials

2020 ◽  
Vol 10 (24) ◽  
pp. 9123
Author(s):  
Yan Zeng ◽  
Hong Zheng ◽  
Chunguang Li
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Finite Volume Method ◽  
Difference Method ◽  
Legendre Polynomials ◽  
Dimensional Space ◽  
Numerical Manifold Method ◽  
One Dimensional ◽  
Numerical Manifold ◽  
Volume Method

Traditional methods such as the finite difference method, the finite element method, and the finite volume method are all based on continuous interpolation. In general, if discontinuity occurred, the calculation result would show low accuracy and poor stability. In this paper, the numerical manifold method is used to capture numerical discontinuities, in a one-dimensional space. It is verified that the high-degree Legendre polynomials can be selected as the local approximation without leading to linear dependency, a notorious “nail” issue in Numerical Manifold Method. A series of numerical tests are carried out to evaluate the performance of the proposed method, suggesting that the accuracy by the numerical manifold method is higher than that by the later finite difference method and finite volume method using the same number of unknowns.

scholarly journals Hopf bifurcation and the existence and stability of closed orbits in three-sector models of optimal endogenous growth

2019 ◽  
Vol 23 (4) ◽  
Author(s):  
Kazuo Nishimura ◽  
Tadashi Shigoka
Keyword(s):  
Hopf Bifurcation ◽  
Endogenous Growth ◽  
Bifurcation Analysis ◽  
Stable Manifold ◽  
Closed Orbit ◽  
One Dimensional ◽  
Convex Structure ◽  
Equilibrium Dynamics ◽  
Closed Orbits ◽  
Existence And Stability

Abstract The present paper constructs a family of three-sector models of optimal endogenous growth, and conducts exact bifurcation analysis. In so doing, original six-dimensional equilibrium dynamics is decomposed into five-dimensional stationary autonomous dynamics and one-dimensional endogenously growing component. It is shown that the stationary dynamics thus decomposed undergoes supercritical Hopf bifurcation. It is inferred from the convex structure of our model that the dimension of a stable manifold of each closed orbit thus bifurcated in this five-dimensional dynamics should be two.

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