In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use th...In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use the previous model after data transformation. However, an appropriate transformation may not exist or may be difficult to find. Also this approach can raise interpretation issues. An interesting alternative is to consider a skew-normal mixture model in standard IM, and the resulting method is here denoted as skew-normal IM. This flexible model that includes the usual symmetric normal distribution as a special case is important, allowing continuous variation from normality to non-normality. In this paper we briefly introduce the main peculiarities of the skew-normal distribution. The maximum likelihood estimates of parameters of the skew-normal distribution are obtained by the expectation-maximization (EM) algorithm. The proposed model is illustrated with real data from an intercross experiment that shows a significant departure from the normality assumption. The performance of the skew-normal IM is assessed via stochastic simulation. The results indicate that the skew-normal IM has higher power for QTL detection and better precision of QTL location as compared to standard IM and nonparametric IM.展开更多
In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder con...In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder continuity for solutions to the com-plex Monge-Ampère equation with the right-hand side in Lp for any given p>1.Our proof uses various PDE techniques but not the pluripotential theory.展开更多
In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument,using a recent ...In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument,using a recent global regularity of optimal transportation in convex domains by the authors.展开更多
In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary.This expansion formula shows the singularity profi...In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary.This expansion formula shows the singularity profile of solutions at the boundary.We deal with both linear and nonlinear elliptic equations,including fully nonlinear elliptic equations and equations of Monge-Ampère type.展开更多
The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).Necessary and sufficient conditions have been found by Firey(1967)and Berg(1969),by using the...The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).Necessary and sufficient conditions have been found by Firey(1967)and Berg(1969),by using the Green function of the Laplacian on the sphere.Expressing the Christoffel problem as the Laplace equation on the entire space R^(n+1),we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equation.Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem.We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem,for the case p≥2.展开更多
基金Project supported in part by Foundation for Science and Technology(FCT) (No.SFRD/BD/5987/2001)the Operational ProgramScience,Technology,and Innovation of the FCT,co-financed by theEuropean Regional Development Fund (ERDF)
文摘In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use the previous model after data transformation. However, an appropriate transformation may not exist or may be difficult to find. Also this approach can raise interpretation issues. An interesting alternative is to consider a skew-normal mixture model in standard IM, and the resulting method is here denoted as skew-normal IM. This flexible model that includes the usual symmetric normal distribution as a special case is important, allowing continuous variation from normality to non-normality. In this paper we briefly introduce the main peculiarities of the skew-normal distribution. The maximum likelihood estimates of parameters of the skew-normal distribution are obtained by the expectation-maximization (EM) algorithm. The proposed model is illustrated with real data from an intercross experiment that shows a significant departure from the normality assumption. The performance of the skew-normal IM is assessed via stochastic simulation. The results indicate that the skew-normal IM has higher power for QTL detection and better precision of QTL location as compared to standard IM and nonparametric IM.
文摘In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder continuity for solutions to the com-plex Monge-Ampère equation with the right-hand side in Lp for any given p>1.Our proof uses various PDE techniques but not the pluripotential theory.
基金supported by Australian Research Council (Grant Nos. FL130100118 and DP170100929)
文摘In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument,using a recent global regularity of optimal transportation in convex domains by the authors.
基金supported by Australian Research Council(Grant No.FL130100118)National Natural Science Foundation of China(Grant Nos.11771237 and 11871432)。
文摘In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary.This expansion formula shows the singularity profile of solutions at the boundary.We deal with both linear and nonlinear elliptic equations,including fully nonlinear elliptic equations and equations of Monge-Ampère type.
基金Dedicated to the Memory of Professor S.S. Chern.Supported by the Natural Science Foundation of China and the Australian Research Council. Acknowledgement This work was finished in December 2004 when the author was visiting at Nankai University as a Yangtze River fellow. He is very grateful to Professor S,S. Chern for his constant encouragement and many conversations which benefited him enormously, He would also like to thank his colleagues at Nankai Institute of Mathematics for their support.
基金supported by the One-Thousand-Young-Talents Program of Chinasupported by National Natural Science Foundation of China(Grant No.11871345)supported by Australian Research Council(Grant Nos.DP170100929 and DP200101084)。
文摘The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).Necessary and sufficient conditions have been found by Firey(1967)and Berg(1969),by using the Green function of the Laplacian on the sphere.Expressing the Christoffel problem as the Laplace equation on the entire space R^(n+1),we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equation.Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem.We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem,for the case p≥2.
