This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p...This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.展开更多
A vector potential of a magnetic field in Lagrangian is defined as the necessary partial solution of a inhomogeneous differential equation. The ＂gradient transformation＂ is an addition of arbitrary general solution ...A vector potential of a magnetic field in Lagrangian is defined as the necessary partial solution of a inhomogeneous differential equation. The ＂gradient transformation＂ is an addition of arbitrary general solution of the corresponding homogeneous equation that does not change the Lagrange equations. When dynamics is described by momenta and coordinates, this transformation is not the vector potential modification, which does not change expressions for other physical quantities, but a canonical transformation of momentum, which changes expressions for all fimctions of momentum, not changing the Poisson brackets, and, hence, the integrals of motion. The generating function of this transformation must reverse sign under the time-charge reversal. In quantum mechanics the unitary transformation corresponds to this canonical transformation. It also does not change the commutation relations. The phase of this unitary operator also must reverse sign under the time-charge reversal. Examples of necessary vector potentials for some magnetic fields are presented.展开更多
We utilize Park＇s maximal element theorem in H-space to prove the existence theorems of solutions of the complementarity problems for multivalued non-monotone operators in Banach spaces.
Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional q...Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional quasigeostrophic model in this paper. The difference between the linear and nonlinear fastest growing perturbations is demonstrated. Moreover, local nonlinear fastest growing perturbations are also found numerically. This is one of the essential differences between linear and nonlinear theories, since in former case there is no local fastest growing perturbation. The results show that the nonlinear local fastest growing perturbations play a more important role in the study of the first kind of predictability than the nonlinear global fastest growing perturbation.展开更多
Let B R^n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It ...Let B R^n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It is well-known that there exists a λ*> 0 such that the biharmonic equation has a solution for λ∈ (0, λ*) and has a unique weak solution u*with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies u*≤ r^(-4/ (p-1)) - 1 on the unit ball, which actually solve a part of the open problem left in [D`avila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193] .展开更多
文摘TB96 2003053892两种反射法确定光学常数的非单值性的研究=Investiga-tion of ambiguities in determining the oPtieal eonstants fortwo refleetion methods[刊,中1/姜良广(鞍山师范学院物理系.辽宁。
文摘This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.
文摘A vector potential of a magnetic field in Lagrangian is defined as the necessary partial solution of a inhomogeneous differential equation. The ＂gradient transformation＂ is an addition of arbitrary general solution of the corresponding homogeneous equation that does not change the Lagrange equations. When dynamics is described by momenta and coordinates, this transformation is not the vector potential modification, which does not change expressions for other physical quantities, but a canonical transformation of momentum, which changes expressions for all fimctions of momentum, not changing the Poisson brackets, and, hence, the integrals of motion. The generating function of this transformation must reverse sign under the time-charge reversal. In quantum mechanics the unitary transformation corresponds to this canonical transformation. It also does not change the commutation relations. The phase of this unitary operator also must reverse sign under the time-charge reversal. Examples of necessary vector potentials for some magnetic fields are presented.
基金the Foundation of Jiangsu Education Committee (04KJD110170)the Foundation of Univer-sity of Science and Technology of Suzhou.
文摘We utilize Park＇s maximal element theorem in H-space to prove the existence theorems of solutions of the complementarity problems for multivalued non-monotone operators in Banach spaces.
基金the National Key Basic Research Project, "Research on the FormationMechanism and Prediction Theory of Severe Synoptic Disasters in China" (Grand No. G1998040910), the National Natural Science Foundation of China (Grand Nos. 49775262 and 49823002) and t
文摘Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional quasigeostrophic model in this paper. The difference between the linear and nonlinear fastest growing perturbations is demonstrated. Moreover, local nonlinear fastest growing perturbations are also found numerically. This is one of the essential differences between linear and nonlinear theories, since in former case there is no local fastest growing perturbation. The results show that the nonlinear local fastest growing perturbations play a more important role in the study of the first kind of predictability than the nonlinear global fastest growing perturbation.
基金supported by the National Natural Science Foundation of China(Nos.11201119,11471099)the International Cultivation of Henan Advanced Talents and the Research Foundation of Henan University(No.yqpy20140043)
文摘Let B R^n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It is well-known that there exists a λ*> 0 such that the biharmonic equation has a solution for λ∈ (0, λ*) and has a unique weak solution u*with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies u*≤ r^(-4/ (p-1)) - 1 on the unit ball, which actually solve a part of the open problem left in [D`avila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193] .
