The Schrodinger equation -△u+λ2u=|u|2q-2u has a unique positive radial solution Uλ, which decays exponentially at infinity. Hence it is reasonable that the Schrolinger system -△u1+u1=|u1|2q-1u1-εb(x)|u2...The Schrodinger equation -△u+λ2u=|u|2q-2u has a unique positive radial solution Uλ, which decays exponentially at infinity. Hence it is reasonable that the Schrolinger system -△u1+u1=|u1|2q-1u1-εb(x)|u2|1|u1|q-1u1,-△u2+u2=|u2|2q-2u2-εb(x)|u1|1|u2|q-1u2 has multiple-bump solutions which behave like Uλ in the neighborhood of some points. For u=(u1,u2)∈H1(R3)×H1(R3), a nonlinear functional Iε(u)=I1(u1)+I2(u2)-ε/q∫R3b(x)|u1|q|u2|qdx,is defined,where I1(u1)=1/2||u1||2-1/2q∫R3|u1|2qdx and I2(u2)=1/2||u2||2ω-1/2q∫R3|u2|2qdx. It is proved that the solutions of the system are the critical points of I,. Let Z be the smooth solution manifold of the unperturbed problem and TzZ is the tangent space. The critical point of I is rewritten as the form of z + w, where w ∈ (TzZ)⊥. Using some properties of Iε, it is proved that there exists a critical point of I, close to the form which is a multi-bump solution.展开更多
Hot deformation behavior and microstructure evolution of hot isostatically pressed FGH96 P/M superalloy were studied using isothermal compression tests. The tests were performed on a Gleeble-1500 simulator in a temper...Hot deformation behavior and microstructure evolution of hot isostatically pressed FGH96 P/M superalloy were studied using isothermal compression tests. The tests were performed on a Gleeble-1500 simulator in a temperature range of 1000-1150 °C and strain rate of 0.001-1.0 s-1, respectively. By regression analysis of the stress—strain data, the constitutive equation for FGH96 superalloy was developed in the form of hyperbolic sine function with hot activation energy of 693.21 kJ/mol. By investigating the deformation microstructure, it is found that partial and full dynamical recrystallization occurs in specimens deformed below and above 1100 °C, respectively, and dynamical recrystallization (DRX) happens more readily with decreasing strain rate and increasing deformation temperature. Finally, equations representing the kinetics of DRX and grain size evolution were established.展开更多
The varying-coefficient model is flexible and powerful for modeling the dynamic changes of regression coefficients. We study the problem of variable selection and estimation in this model in the sparse, high- dimensio...The varying-coefficient model is flexible and powerful for modeling the dynamic changes of regression coefficients. We study the problem of variable selection and estimation in this model in the sparse, high- dimensional case. We develop a concave group selection approach for this problem using basis function expansion and study its theoretical and empirical properties. We also apply the group Lasso for variable selection and estimation in this model and study its properties. Under appropriate conditions, we show that the group least absolute shrinkage and selection operator (Lasso) selects a model whose dimension is comparable to the underlying mode], regardless of the large number of unimportant variables. In order to improve the selection results, we show that the group minimax concave penalty (MCP) has the oracle selection property in the sense that it correctly selects important variables with probability converging to one under suitable conditions. By comparison, the group Lasso does not have the oracle selection property. In the simulation parts, we apply the group Lasso and the group MCP. At the same time, the two approaches are evaluated using simulation and demonstrated on a data example.展开更多
基金The National Natural Science Foundation of China(No.11171063)the Natural Science Foundation of Jiangsu Province(No.BK2010404)
文摘The Schrodinger equation -△u+λ2u=|u|2q-2u has a unique positive radial solution Uλ, which decays exponentially at infinity. Hence it is reasonable that the Schrolinger system -△u1+u1=|u1|2q-1u1-εb(x)|u2|1|u1|q-1u1,-△u2+u2=|u2|2q-2u2-εb(x)|u1|1|u2|q-1u2 has multiple-bump solutions which behave like Uλ in the neighborhood of some points. For u=(u1,u2)∈H1(R3)×H1(R3), a nonlinear functional Iε(u)=I1(u1)+I2(u2)-ε/q∫R3b(x)|u1|q|u2|qdx,is defined,where I1(u1)=1/2||u1||2-1/2q∫R3|u1|2qdx and I2(u2)=1/2||u2||2ω-1/2q∫R3|u2|2qdx. It is proved that the solutions of the system are the critical points of I,. Let Z be the smooth solution manifold of the unperturbed problem and TzZ is the tangent space. The critical point of I is rewritten as the form of z + w, where w ∈ (TzZ)⊥. Using some properties of Iε, it is proved that there exists a critical point of I, close to the form which is a multi-bump solution.
文摘Hot deformation behavior and microstructure evolution of hot isostatically pressed FGH96 P/M superalloy were studied using isothermal compression tests. The tests were performed on a Gleeble-1500 simulator in a temperature range of 1000-1150 °C and strain rate of 0.001-1.0 s-1, respectively. By regression analysis of the stress—strain data, the constitutive equation for FGH96 superalloy was developed in the form of hyperbolic sine function with hot activation energy of 693.21 kJ/mol. By investigating the deformation microstructure, it is found that partial and full dynamical recrystallization occurs in specimens deformed below and above 1100 °C, respectively, and dynamical recrystallization (DRX) happens more readily with decreasing strain rate and increasing deformation temperature. Finally, equations representing the kinetics of DRX and grain size evolution were established.
基金supported by National Natural Science Foundation of China(GrantNos.71271128 and 11101442)the State Key Program of National Natural Science Foundation of China(GrantNo.71331006)+2 种基金National Center for Mathematics and Interdisciplinary Sciences(NCMIS)Shanghai Leading Academic Discipline Project A,in Ranking Top of Shanghai University of Finance and Economics(IRTSHUFE)Scientific Research Innovation Fund for PhD Studies(Grant No.CXJJ-2011-434)
文摘The varying-coefficient model is flexible and powerful for modeling the dynamic changes of regression coefficients. We study the problem of variable selection and estimation in this model in the sparse, high- dimensional case. We develop a concave group selection approach for this problem using basis function expansion and study its theoretical and empirical properties. We also apply the group Lasso for variable selection and estimation in this model and study its properties. Under appropriate conditions, we show that the group least absolute shrinkage and selection operator (Lasso) selects a model whose dimension is comparable to the underlying mode], regardless of the large number of unimportant variables. In order to improve the selection results, we show that the group minimax concave penalty (MCP) has the oracle selection property in the sense that it correctly selects important variables with probability converging to one under suitable conditions. By comparison, the group Lasso does not have the oracle selection property. In the simulation parts, we apply the group Lasso and the group MCP. At the same time, the two approaches are evaluated using simulation and demonstrated on a data example.