Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 ...Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.展开更多
This article is concerned with the oscillation of the forced second order differ- ential equation with mixed nonlinearities（a（t）（x＇（t））γ）＇＋po（t）xγ（go（t））＋n∑t=1pi（t）｜x（gi（t））｜αi sgn x（gi...This article is concerned with the oscillation of the forced second order differ- ential equation with mixed nonlinearities（a（t）（x＇（t））γ）＇＋po（t）xγ（go（t））＋n∑t=1pi（t）｜x（gi（t））｜αi sgn x（gi（t））=e（t）,where γ is a quotient of odd positive integers αi〉0,i=1,2 ……n,a,e,and pi ∈ C（[0,∞）R）,a（t）〉0,gi：R→R are positive continuous functions on R with lim gi （t）=∞,i=0,1,……,n. Our results generalize and improve the results in a recent article by Sun and Wong[29].展开更多
This paper discussses the random singular integral of random process with second order moment, establishes the concepts of the random singular integral and proves that it's a linear bounded operator of space Ha(L)...This paper discussses the random singular integral of random process with second order moment, establishes the concepts of the random singular integral and proves that it's a linear bounded operator of space Ha(L)(m, s). Then Plemelj formula and some other properties for random singular integral are proved.展开更多
A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous f...A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].展开更多
The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t） ＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x） ＋ W（t, x）, K（t, x） is neit...The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t） ＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x） ＋ W（t, x）, K（t, x） is neither a quadratic form in x nor periodic in t and W（t, x） is superquadratic in x.展开更多
By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral dela...By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral delay differential equation of second order.展开更多
The author establishes in this paper the following results: (1) In a quasiconstant curvature manifold M a parallel tensor of type is constant multiple of the metric tensor. (2) On a quasi_constant curvature manifold ...The author establishes in this paper the following results: (1) In a quasiconstant curvature manifold M a parallel tensor of type is constant multiple of the metric tensor. (2) On a quasi_constant curvature manifold there is no nonzero parallel 2_form. Unless the Ricci principal curvature corresponding to the generator of M is equal to zero.展开更多
This paper investigates the cluster consensus problem for second-order multi-agent systems by applying the pinning control method to a small collection of the agents. Consensus is attained independently for different ...This paper investigates the cluster consensus problem for second-order multi-agent systems by applying the pinning control method to a small collection of the agents. Consensus is attained independently for different agent clusters according to the community structure generated by the group partition of the underlying graph and sufficient conditions for both cluster and general consensus are obtained by using results from algebraic graph theory and the LaSalle Invariance Principle. Finally, some simple simulations are presented to illustrate the technique.展开更多
A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorith...A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.展开更多
This article investigates the consensus problem of the second-order multi-agent systems with an active leader and coupling time delay in direct graph. One decentralized state control rule is constructed for each agent...This article investigates the consensus problem of the second-order multi-agent systems with an active leader and coupling time delay in direct graph. One decentralized state control rule is constructed for each agent to track the active leader and it is proved that the proposed control scheme enables the consensus to be obtained when the adjacency topology is fixed/switched. Simulation results show effectiveness of the proposed theoretical analysis.展开更多
This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The...This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.展开更多
This paper investigates the consensus tracking problems of second-order multi-agent systems with a virtual leader via event-triggered control. A novel distributed event-triggered transmission scheme is proposed, which...This paper investigates the consensus tracking problems of second-order multi-agent systems with a virtual leader via event-triggered control. A novel distributed event-triggered transmission scheme is proposed, which is intermittently examined at constant sampling instants. Only partial neighbor information and local measurements are required for event detection. Then the corresponding event-triggered consensus tracking protocol is presented to guarantee second-order multi-agent systems to achieve consensus tracking. Numerical simulations are given to illustrate the effectiveness of the proposed strategy.展开更多
In this paper, the effects of both rotation and magnetic field of the peristaltic transport of a second-order fluid through a porous medium in a channel are studied analytically and computed numerically. The material ...In this paper, the effects of both rotation and magnetic field of the peristaltic transport of a second-order fluid through a porous medium in a channel are studied analytically and computed numerically. The material is represented by the constitutive equations for a second-order fluid. Closed-form solutions under the consideration of long wavelength and low Reynolds number is presented. The analytical expressions for the pressure gradient, pressure rise, friction force, stream function, shear stress, and velocity are obtained in the physical domain. The effects of the non-dimensional wave amplitude, porosity, magnetic field, rotation, and the dimensionless time-mean flow in the wave frame are analyzed theoretically and computed numerically. Numerical results are given and illustrated graphically in each case considered. Comparison was made with the results obtained in the presence and absence of rotation, magnetic field, and porosity. The results indicate that the effects of the non-dimensional wave amplitude, porosity, magnetic field, rotation, and the dimensionless time-mean flow are very pronounced in the phenomena.展开更多
In this paper we investigated the effect of the first order responses of mooring line on the second order mooring line damping.In the study of the slow oscillating of a moored floating structure by perturbation method...In this paper we investigated the effect of the first order responses of mooring line on the second order mooring line damping.In the study of the slow oscillating of a moored floating structure by perturbation method in frequency domain,the second order equations of the mooring line are divided into non homogeneous and homogeneous equations.The solutions are related to the first order responses of mooring line and second order floating structure oscillation respectively.In order to find the effect of the first order responses of mooring line,the second order mooring line tension and damping were determined by solving the non homogeneous equation and homogeneous equation.From the results,we found,although the second order mooring line tension obtained from the non homogenous equation is quite small compared with the total second order mooring line tension,the damping contributed from both of them are in the same order in quantity.So,in predicting the second order mooring line damping,the effect of the solution related to the non homogeneous equation can not be omitted.展开更多
This paper deals with the consensus problem of multi-agent systems with second-order dynamics. The objective is to design algorithms such that all agents will have same positions and velocities. First, a reference mod...This paper deals with the consensus problem of multi-agent systems with second-order dynamics. The objective is to design algorithms such that all agents will have same positions and velocities. First, a reference model based consensus algorithm is proposed. It is proved that the consensus can be achieved if the communication graph has a spanning tree. Different from most of the consensus algorithms proposed in the literature, the parameters of the control laws are different among agents. Therefore, each agent can design its control law independently. Secondly, it gives a consensus algorithm for the case that the velocities of the agents are not available. Thirdly, the effectiveness of the input delay and the communication delay is considered. It shows that consensus can be achieved if the input delay of every agent is smaller than a bound related to parameters in its control law. Finally, some numerical examples are given to illustrate the proposed results.展开更多
The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by em- ploying two-photon interference in Feynman＇s path integral theory. It is concluded that whether the sec...The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by em- ploying two-photon interference in Feynman＇s path integral theory. It is concluded that whether the second-order temporal interference pattern can or cannot be retrieved via two-photon coincidence counting rate is dependent on the resolution time of the detection system and the frequency difference between these two lasers. Two identical and tunable single-mode continuous-wave diode lasers are employed to verify the predictions. These studies are helpful to understand the physics of two-photon interference with photons of different spectra.展开更多
The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory ...The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.展开更多
Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable...Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.展开更多
We investigate the Hyers-Ulam stability(HUS)of certain second-order linear constant coefficient dynamic equations on time scales,building on recent results for first-order constant coefficient time-scale equations.In ...We investigate the Hyers-Ulam stability(HUS)of certain second-order linear constant coefficient dynamic equations on time scales,building on recent results for first-order constant coefficient time-scale equations.In particular,for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale,we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two roots.Conditions for instability are also given.展开更多
基金Supported by the Natural Science Foundation of Hunan Province（06JJ50008） Supported by the Natural Science Foundation of Guangdong Province（7004569）
文摘Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.
文摘This article is concerned with the oscillation of the forced second order differ- ential equation with mixed nonlinearities（a（t）（x＇（t））γ）＇＋po（t）xγ（go（t））＋n∑t=1pi（t）｜x（gi（t））｜αi sgn x（gi（t））=e（t）,where γ is a quotient of odd positive integers αi〉0,i=1,2 ……n,a,e,and pi ∈ C（[0,∞）R）,a（t）〉0,gi：R→R are positive continuous functions on R with lim gi （t）=∞,i=0,1,……,n. Our results generalize and improve the results in a recent article by Sun and Wong[29].
基金The project is supported by the NSF of China (10271098)Education Foundation of Fujian (JB02083) Science & Technical Development Foundation of Fuzhou University (2003xy-11).
文摘This paper discussses the random singular integral of random process with second order moment, establishes the concepts of the random singular integral and proves that it's a linear bounded operator of space Ha(L)(m, s). Then Plemelj formula and some other properties for random singular integral are proved.
文摘A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].
基金Supported by National Natural Science Foundation of China (10771173)
文摘The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t） ＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x） ＋ W（t, x）, K（t, x） is neither a quadratic form in x nor periodic in t and W（t, x） is superquadratic in x.
文摘By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral delay differential equation of second order.
文摘The author establishes in this paper the following results: (1) In a quasiconstant curvature manifold M a parallel tensor of type is constant multiple of the metric tensor. (2) On a quasi_constant curvature manifold there is no nonzero parallel 2_form. Unless the Ricci principal curvature corresponding to the generator of M is equal to zero.
基金Project supported by the National Natural Science Foundation of China (Grant No. 70571059)
文摘This paper investigates the cluster consensus problem for second-order multi-agent systems by applying the pinning control method to a small collection of the agents. Consensus is attained independently for different agent clusters according to the community structure generated by the group partition of the underlying graph and sufficient conditions for both cluster and general consensus are obtained by using results from algebraic graph theory and the LaSalle Invariance Principle. Finally, some simple simulations are presented to illustrate the technique.
基金the National Science Foundation(60574075, 60674108)
文摘A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.
基金supported by the National Natural Science Foundation of China(11301492)the Ph.D.Programs Foundation of Ministry of Education of China(20130145120005)the TianYuan Special Funds of the National Natural Science Foundation of China(11226134)
文摘This article investigates the consensus problem of the second-order multi-agent systems with an active leader and coupling time delay in direct graph. One decentralized state control rule is constructed for each agent to track the active leader and it is proved that the proposed control scheme enables the consensus to be obtained when the adjacency topology is fixed/switched. Simulation results show effectiveness of the proposed theoretical analysis.
基金Project supported by the National Science Fund for Distinguished Young Scholars （Grant No 40425015）, the Cooperative Project of Chinese Academy Sciences and the China National 0ffshore oil Corporation （＂Behaviours of internal waves and their roles on the marine structures＂） and the National Natural Science Foundation of China （Grant No10461005）.
文摘This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61203147,61374047,and 61403168)
文摘This paper investigates the consensus tracking problems of second-order multi-agent systems with a virtual leader via event-triggered control. A novel distributed event-triggered transmission scheme is proposed, which is intermittently examined at constant sampling instants. Only partial neighbor information and local measurements are required for event detection. Then the corresponding event-triggered consensus tracking protocol is presented to guarantee second-order multi-agent systems to achieve consensus tracking. Numerical simulations are given to illustrate the effectiveness of the proposed strategy.
文摘In this paper, the effects of both rotation and magnetic field of the peristaltic transport of a second-order fluid through a porous medium in a channel are studied analytically and computed numerically. The material is represented by the constitutive equations for a second-order fluid. Closed-form solutions under the consideration of long wavelength and low Reynolds number is presented. The analytical expressions for the pressure gradient, pressure rise, friction force, stream function, shear stress, and velocity are obtained in the physical domain. The effects of the non-dimensional wave amplitude, porosity, magnetic field, rotation, and the dimensionless time-mean flow in the wave frame are analyzed theoretically and computed numerically. Numerical results are given and illustrated graphically in each case considered. Comparison was made with the results obtained in the presence and absence of rotation, magnetic field, and porosity. The results indicate that the effects of the non-dimensional wave amplitude, porosity, magnetic field, rotation, and the dimensionless time-mean flow are very pronounced in the phenomena.
文摘In this paper we investigated the effect of the first order responses of mooring line on the second order mooring line damping.In the study of the slow oscillating of a moored floating structure by perturbation method in frequency domain,the second order equations of the mooring line are divided into non homogeneous and homogeneous equations.The solutions are related to the first order responses of mooring line and second order floating structure oscillation respectively.In order to find the effect of the first order responses of mooring line,the second order mooring line tension and damping were determined by solving the non homogeneous equation and homogeneous equation.From the results,we found,although the second order mooring line tension obtained from the non homogenous equation is quite small compared with the total second order mooring line tension,the damping contributed from both of them are in the same order in quantity.So,in predicting the second order mooring line damping,the effect of the solution related to the non homogeneous equation can not be omitted.
基金supported by the National Natural Science Foundation of China (Grant No. 60904022)
文摘This paper deals with the consensus problem of multi-agent systems with second-order dynamics. The objective is to design algorithms such that all agents will have same positions and velocities. First, a reference model based consensus algorithm is proposed. It is proved that the consensus can be achieved if the communication graph has a spanning tree. Different from most of the consensus algorithms proposed in the literature, the parameters of the control laws are different among agents. Therefore, each agent can design its control law independently. Secondly, it gives a consensus algorithm for the case that the velocities of the agents are not available. Thirdly, the effectiveness of the input delay and the communication delay is considered. It shows that consensus can be achieved if the input delay of every agent is smaller than a bound related to parameters in its control law. Finally, some numerical examples are given to illustrate the proposed results.
基金Project supported by the National Natural Science Foundation of China(Grant No.11404255)the Doctor Foundation of Education Ministry of China(Grant No.20130201120013)
文摘The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by em- ploying two-photon interference in Feynman＇s path integral theory. It is concluded that whether the second-order temporal interference pattern can or cannot be retrieved via two-photon coincidence counting rate is dependent on the resolution time of the detection system and the frequency difference between these two lasers. Two identical and tunable single-mode continuous-wave diode lasers are employed to verify the predictions. These studies are helpful to understand the physics of two-photon interference with photons of different spectra.
基金Supported by the National Natural Science Foundation of China(11101096 )Guangdong Natural Science Foundation (S2012010010376, S201204006711)
文摘The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.
文摘Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.
基金supported by JSPS KAKENHI Grant Number JP20K03668。
文摘We investigate the Hyers-Ulam stability(HUS)of certain second-order linear constant coefficient dynamic equations on time scales,building on recent results for first-order constant coefficient time-scale equations.In particular,for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale,we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two roots.Conditions for instability are also given.
