AbstractA grain surplus and the grain trade are closely related to the urbanization of developing countries. However, existing literature has not given adequate attention to this issue. Our paper develops a theoretical model to explain the mechanisms whereby changes in the grain surplus constrain the level of urbanization in a closed economy context and the grain trade affects the level of urbanization by acting on the grain surplus in an open economy context. A test of the theoretical model applied to some developing countries in Asia during the period 1993-2010 shows that international trade is generally negatively correlated with level of urbanization. However, cereal and non-cereal trade vary in terms of their relation to urbanization: whereas the former is positively correlated with level of urbanization, the latter is negatively correlated with it. Since the net import of cereals relaxes the constraint imposed on the level of urbanization by the domestic grain surplus, it can have a marked positive effect on the course of urbanization. Our research findings show that provided grain production or grain security is guaranteed, developing countries may adopt a policy of importing an appropriate amount of grain to increase their level of urbanization. 相似文献
Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (Stat Comput 27(4):1065–1082, 2017. https://doi.org/10.1007/s11222-016-9671-0), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.
