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The original v2-based sectional method assumes that the selected property quantity of particles is uniformly distributed in each section, which makes particle size distribution (PSD) fluctuate dramatically in the entire size range. The number concentration in each section as well as the zeroth moment of PSD also cannot be correctly predicted in case there are not enough sections used in calculation. In order to provide a more appropriate representation of PSD, different approximate models are used to close the conservation equations. The paper aims to discuss these issues.

The uniform distribution of the selected property quantity of particles in each section is not necessarily satisfied. Instead, the distribution is approximated using an expression with an approximation factor. Different models are investigated on recovering the initial size distribution and predicting the time evolution of size distribution as well as the first three moments so that the advantages and disadvantages of each model can be compared.

The approximate model with an approximation factor of 0.8 is capable of predicting the time evolution of the zeroth moment accurately no matter how many sections are used in simulations. The original v2-based model is recommended to calculate the first and second moments as long as the section number is larger than 50, otherwise, the model with an approximation factor of 0.15 would be a preferred choice.

Different approximate models can be used to improve the accuracy of the results supposing we know which moment is of great importance in calculation.

The purpose of this paper is to develop new types of direct expansion method of moments (DEMM) by using the n/3th moments for simulating nanoparticle Brownian coagulation in the free molecule regime. The feasibilities of new proposed DEMMs with n/3th moments are investigated to describe the evolution of aerosol size distribution, and some of the models will be applied to further simulation of physical processes.

The accuracy and efficiency of some kinds of methods of moments are mainly compared including the quadrature method of moments (QMOM), Taylor-expansion method of moments (TEMOM), the log-normal preserving method of moments proposed by Lee (LMM) and the derived DEMM in this paper. QMOM with 12 quadrature approximation points is taken as a reference to evaluate other methods.

The newly derived models, namely DEMM(4/3,4) and DEMM(2,6), as well as the previous DEMM(2,4), are considered to be qualified models due to their high accuracy and efficiency. They are confirmed to be valid and alternative models to describe the evolution of aerosol size distribution for particle dynamical process involving the n/3th moments.

The n/3th moments, which have clear physical interpretations when n stands for first several integers, are first introduced in the DEMM method for simulating nanoparticle Brownian coagulation in the free molecule regime.

There is a recent growing interest to find a lasting intervention to rural poverty (RP) in developing countries based on farmer entrepreneurship and innovation. The purpose of this paper, therefore, is to examine the relation between entrepreneurship and RP alleviation in two resource-constrained provinces of China. This paper assesses the influence of three capabilities of farm entrepreneurs – educational, economic and socio-cultural – on farmer entrepreneurship growth and how these, in turn, impact alleviation of RP.

Household survey data comprising 363 respondents were taken from four deprived communities in two provinces of China. The paper employed structural equation modeling (SEM), using AMOS 21.0 alongside SPSS 20.0 to test the relations between the constructs.

The results show that a statistically significant and positive relation exists between entrepreneurship and RP alleviation in China. The findings of the study further reveal that qualitative growth of entrepreneurship has a stronger positive influence on RP alleviation than on quantitative growth, and socio-cultural capabilities of respondents significantly and positively affect entrepreneurial growth of farmers, rather than education and economic capabilities.

The use of data from four communities in two provinces tends to limit the ability to generalize the findings of the study. Furthermore, the survey did not collect information on non-farm entrepreneurs, making it impossible to compare the findings from farm entrepreneurs with non-farm entrepreneurs.

The findings have practical implications for policy makers in rural China toward addressing targeted RP. This paper, therefore, suggests that entrepreneurship should be pursued vigorously among farmers in rural areas of China to help solve poverty. The paper also presents a useful lesson for various stakeholders in poverty alleviation programs in other developing countries.

This paper contributes to the academic literature on the entrepreneurship–RP alleviation nexus by combining the theory of capability and SEM in the analysis of an emerging economy such as China.