Over-expansion flow can generate asymmetric shock wave interactions, which lead to significant lateral forces on a nozzle. However, there is still a lack of a suitable theory to explain the phenomenon of asymmetry. The current work carefully investigates the configurations of shock wave interactions in a planar nozzle, and proposes a theoretical method to analyse the asymmetry of over-expansion flows. First, various possible flow patterns of over-expansion flows are discussed, including regular and Mach reflections. Second, the free interaction theory and the minimum entropy production principle are used to analyse the boundary layer flow and main shock wave interactions, establish the relationship between the separation shock strength and separation position, and predict asymmetric configurations. Finally, experiments are conducted to validate the theoretical method, and similar experiments from other studies are discussed to demonstrate the effectiveness of the proposed method. Results demonstrate that the direction of asymmetric over-expansion flow is random, and the separated flow strives to adopt a pattern with minimal total pressure loss. Asymmetric interaction is a mechanism through which the flow can achieve a more efficient thermodynamic balance by minimising entropy production.

In recent years, there has been an increasing interest in detecting disease-related rare variants in sequencing studies. Numerous studies have shown that common variants can only explain a small proportion of the phenotypic variance for complex diseases. More and more evidence suggests that some of this missing heritability can be explained by rare variants. Considering the importance of rare variants, researchers have proposed a considerable number of methods for identifying the rare variants associated with complex diseases. Extensive research has been carried out on testing the association between rare variants and dichotomous, continuous or ordinal traits. So far, however, there has been little discussion about the case in which both genotypes and phenotypes are ordinal variables. This paper introduces a method based on the γ-statistic, called OV-RV, for examining disease-related rare variants when both genotypes and phenotypes are ordinal. At present, little is known about the asymptotic distribution of the γ-statistic when conducting association analyses for rare variants. One advantage of OV-RV is that it provides a robust estimation of the distribution of the γ-statistic by employing the permutation approach proposed by Fisher. We also perform extensive simulations to investigate the numerical performance of OV-RV under various model settings. The simulation results reveal that OV-RV is valid and efficient; namely, it controls the type I error approximately at the pre-specified significance level and achieves greater power at the same significance level. We also apply OV-RV for rare variant association studies of diastolic blood pressure.

With a large number of quantitative trait loci being identified in genome-wide association studies, researchers have become more interested in detecting interactions among genes or single nucleotide polymorphisms (SNPs). In this research, we carried out a two-stage model selection procedure to detect interacting gene pairs or SNP pairs associated with four important traits of outbred mice, including glucose, high-density lipoprotein cholesterol, diastolic blood pressure and triglyceride. In the first stage, a variance heterogeneity test was used to screen for candidate SNPs. In the second stage, the Lasso method and single pair analysis were used to select two-way interactions. Moreover, the shared Gene Ontology information about the selected interacting gene pairs was considered to study the interactions auxiliarily. Based on this method, we not only replicated the identification of important SNPs associated with each trait of outbred mice, but also found some SNP pairs and gene pairs with significant interaction effects on each trait. Simulation studies were also conducted to evaluate the performance of the two-stage method in different situations.

Quantitative trait is always controlled by multiple latent genetic loci, and genetic markers have been used to map quantitative trait loci (QTLs) auxiliarily. The method of multiple interval mapping (MIM) provides an appropriate way for mapping QTL using genetic makers. However, the computation in the MIM seems infeasible for a large number of marker intervals. Nowadays, the Dantzig selector (DS) method proves to be a more efficient method to estimate model effects in a linear model when the number of parameters is (much) larger than the sample size, which has not been applied to genetic mapping for QTL. In this paper, we developed a two-step method for mapping QTL based on the MIM, and we also illustrate the feasibility of adopting the DS to estimate marker or QTL effects. Simulation results showed that the proposed method performed satisfactorily well by comparisons with the existing MIM method, and the analysis to real data set also tested the practicability and efficiency of the DS method in genetic mapping.