The stiction of a thin plate induced by the capillary force has attracted much attention in the broad range of applications. A novel method is presented to calculate the capillary adhesion problem of the plate through...The stiction of a thin plate induced by the capillary force has attracted much attention in the broad range of applications. A novel method is presented to calculate the capillary adhesion problem of the plate through analytical method. The expressions of the surface energy, the strain energy and the total potential energy of the plate-substrate system have been analyzed and delineated. By means of continuum mechanics and the principle of minimum potential energy, the governing equation of the plate with an arbitrary shape and the corresponding transversality boundary condition due to the moving bound have been derived. Then the critical adhesion radius of the circular plate has been solved according to the supplementary transversality condition. Thus the deflections of the plates are analytically calculated with different critical adhesion radii. The results may be beneficial to the engineering application and the micro/nanomeasurement.展开更多
Elasto-capillarity phenomena are prevalent in various industrial fields such as mechanical engineering,material science,aerospace,soft robotics,and biomedicine.In this study,two typical peeling processes of slender be...Elasto-capillarity phenomena are prevalent in various industrial fields such as mechanical engineering,material science,aerospace,soft robotics,and biomedicine.In this study,two typical peeling processes of slender beams driven by the parallel magnetic field are investigated based on experimental and theoretical analysis.The first is the adhesion of two parallel beams,and the second is the self-folding of a long beam.In these two cases,the energy variation method on the elastica is used,and then,the governing equations and transversality boundary conditions are derived.It is shown that the analytical solutions are in excellent agreement with the experimental data.The effects of magnetic induction intensity,distance,and surface tension on the deflection curve and peeling length of the elastica are fully discussed.The results are instrumental in accurately regulating elasto-capillarity in structures and provide insights for the engineering design of programmable microstructures on surfaces,microsensors,and bionic robots.展开更多
In this study, we developed a general method to analytically tackle a kind of movable boundary problem from the viewpoint of energy variation. Having grouped the adhesion of a micro-beam, droplet and carbon nanotube ...In this study, we developed a general method to analytically tackle a kind of movable boundary problem from the viewpoint of energy variation. Having grouped the adhesion of a micro-beam, droplet and carbon nanotube （CNT） ring on a substrate into one framework, we used the developed line of reasoning to investigate the adhesion behaviors of these systems. Based upon the derived governing equations and transversality conditions, explicit solutions involving the critical parameters and morphologies for the three systems are successfully obtained, and then the parameter analogies and common characteristics of them are thor- oughly investigated. The presented method has been verified via the concept of energy release rate in fracture mechanics. Our analyses provide a new approach for exploring the mechanism of different systems with similarities as well as for understanding the unity of nature. The analysis results may be beneficial for the design of nano-structured materi- als, and hold potential for enhancing their mechanical, chemical, optical and electronic properties.展开更多
The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is nece...The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s equations.The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics.Here,the fractional Pfaff-Birkhoff variational problem is presented and studied.The definitions of fractional derivatives,the formulae for integration by parts and some other preliminaries are firstly given.Secondly,the fractional Pfaff-Birkhoff principle and the fractional Birkhoff′s equations in terms of RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.展开更多
Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic i...Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].展开更多
In this paper,the (?)-equivariant (s, t)-equivalence relation and (?)-equivariant infinitesimally (r, s)-stability of (?)-equivariant bifurcation problem are defined. The criterion for (?)-equivariant infinitesimally ...In this paper,the (?)-equivariant (s, t)-equivalence relation and (?)-equivariant infinitesimally (r, s)-stability of (?)-equivariant bifurcation problem are defined. The criterion for (?)-equivariant infinitesimally (r, s)-stability is proven when (?) is a compact finite Lie group .Transversality condition is used to characterize the stability.展开更多
A geometrical method using the exponential dichotomy and the invariant manifold thoery is given to set up the criteria for the existence of transversal and tangential heterodinic orbits under the most general degenera...A geometrical method using the exponential dichotomy and the invariant manifold thoery is given to set up the criteria for the existence of transversal and tangential heterodinic orbits under the most general degenerate cases. Conclusions given here extend and contain the relevant known results.展开更多
The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variabl...The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variable size population and two control strategies where creating awareness and treatment are considered as controls. We intend to find the optimal combination of these two control strategies that will minimize the cost of the two control measures and as a result the number of infectious individuals will decrease. We establish the existence for the optimal controls and Pontryagin’s maximum principle is used to characterize the optimal controls. The numerical simulation suggests that optimal control technique is much more effective to minimize the infected individuals and the corresponding cost of the two controls. It is also monitored that in the case of high contact rate, controls have to work for longer period of time to get the desired result. Numerical simulation reveals that the spread of Nipah virus can be controlled effectively if we apply control strategy at early stage.展开更多
Let (■_n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift spaceΣ_A defined by Fibonacci incident matrix A.We consider the measureμ_λof the probab...Let (■_n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift spaceΣ_A defined by Fibonacci incident matrix A.We consider the measureμ_λof the probability distribution of the random seriesΣ_(n=0)~∞■_nλ~n(0<λ<1).It is proved thatμ_λis singular ifλ∈(0,(5^(1/2)-1)/2) and thatμ_λis absolutely continuous for almost allλ∈((5^(1/2)-1)/2,0.739).展开更多
The implementation of optimal control strategies involving preventive measures and antiviral treatment can significantly reduce the number of clinical cases of influenza. In this paper, a model for the transmission dy...The implementation of optimal control strategies involving preventive measures and antiviral treatment can significantly reduce the number of clinical cases of influenza. In this paper, a model for the transmission dynamics of influenza is formulated and two control strategies involving preventive measures (awareness campaign, washing hand, using hand sanitizer, wearing mask) and treatment are considered and used to minimize the total number of infected individuals and associated cost of using these two controls. The resulting optimality system is solved numerically. Hamiltonian is formulated to investigate the existence of the optimal control, in the optimal control model. Pontryagin’s Maximum Principle is applied to describe the control variables and the objective function is designed to reduce both the infection and the cost of interventions. From the numerical simulation, it is observed that in the case of high contact rate (β = 3), both the controls work for a longer period of time to reduce the disease burden. The optimal control analysis and numerical simulations reveal that the interventions reduce the number of exposed and infected individuals.展开更多
In this study,we considered the wetting phenomenon on a general substrate from a new viewpoint of continuum mechanics.The analyses first show how the Wenzel and the Cassie models deviate the practical results in some ...In this study,we considered the wetting phenomenon on a general substrate from a new viewpoint of continuum mechanics.The analyses first show how the Wenzel and the Cassie models deviate the practical results in some special substrates,and then elucidate the mechanism of the triple contact line(TCL) moving.Based upon variational theory of the total free functional dealing with the movable boundary condition,we show that the macroscopic contact angle(MCA) expression is the corresponding transversality condition.It manifests that the MCA depends only on the chemical and geometric property at the TCL,and is not affected by the gravity of the droplet and the contact area beneath the liquid.Our continuum model also shows the exploration of the pinning effect on a sharp wedge or the interface between two different phases.This investigation will help designing super-hydrophobic materials for novel micro-fluidic devices.展开更多
By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of applicat...By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of application are also given.展开更多
Using the exponential dichotomies,the transversality theory and the generalized Melnikov method,we consider the conditions for the persistence and the transversality of the singular orbit,with high degeneracy,situated...Using the exponential dichotomies,the transversality theory and the generalized Melnikov method,we consider the conditions for the persistence and the transversality of the singular orbit,with high degeneracy,situated on the heteroclinic or homoclinic manifold under perturbation.The results obtained extend,include and improve the corresponding ones given in certain papers well known in this area.展开更多
The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations wit...The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.展开更多
The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in R2,2 to R2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used b...The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in R2,2 to R2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used by Dai and Terng to construct monopoles with continuous scattering data, and then the equation can be linearized by the scattering data, allowing one to use the inverse scattering method to solve the Cauchy problem with rapidly decaying small initial data. In this paper, we use the terminology of holomorphic bundle and transversality of certain maps, parametrized by initial data, to give more initial data, with which we can use scattering method to solve the Cauchy problem of the monopole equation up to gauge transformation.展开更多
A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeo...A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeomorphisms.展开更多
基金supported by Scientific Research Foundation of China University of Petroleum(Y081513)National Natural Science Foundation of China(10802099)Doctoral Fund of Ministry of Education of China(200804251520)
文摘The stiction of a thin plate induced by the capillary force has attracted much attention in the broad range of applications. A novel method is presented to calculate the capillary adhesion problem of the plate through analytical method. The expressions of the surface energy, the strain energy and the total potential energy of the plate-substrate system have been analyzed and delineated. By means of continuum mechanics and the principle of minimum potential energy, the governing equation of the plate with an arbitrary shape and the corresponding transversality boundary condition due to the moving bound have been derived. Then the critical adhesion radius of the circular plate has been solved according to the supplementary transversality condition. Thus the deflections of the plates are analytically calculated with different critical adhesion radii. The results may be beneficial to the engineering application and the micro/nanomeasurement.
基金supported by the National Natural Science Foundation of China(12372027 and 12211530028)the Natural Science Foundation of Shandong Province(ZR202011050038)Special Funds for the Basic Scientific Research Expenses of Central Government Universities(2472022X03006A).
文摘Elasto-capillarity phenomena are prevalent in various industrial fields such as mechanical engineering,material science,aerospace,soft robotics,and biomedicine.In this study,two typical peeling processes of slender beams driven by the parallel magnetic field are investigated based on experimental and theoretical analysis.The first is the adhesion of two parallel beams,and the second is the self-folding of a long beam.In these two cases,the energy variation method on the elastica is used,and then,the governing equations and transversality boundary conditions are derived.It is shown that the analytical solutions are in excellent agreement with the experimental data.The effects of magnetic induction intensity,distance,and surface tension on the deflection curve and peeling length of the elastica are fully discussed.The results are instrumental in accurately regulating elasto-capillarity in structures and provide insights for the engineering design of programmable microstructures on surfaces,microsensors,and bionic robots.
基金supported by the National Natural Science Foundation of China (11272357 and 11102140)Doctoral Fund of Ministry of Education of China (200804251520 and 20110141120024)Natural Science Foundation of Shandong Province (ZR2009AQ006)
文摘In this study, we developed a general method to analytically tackle a kind of movable boundary problem from the viewpoint of energy variation. Having grouped the adhesion of a micro-beam, droplet and carbon nanotube （CNT） ring on a substrate into one framework, we used the developed line of reasoning to investigate the adhesion behaviors of these systems. Based upon the derived governing equations and transversality conditions, explicit solutions involving the critical parameters and morphologies for the three systems are successfully obtained, and then the parameter analogies and common characteristics of them are thor- oughly investigated. The presented method has been verified via the concept of energy release rate in fracture mechanics. Our analyses provide a new approach for exploring the mechanism of different systems with similarities as well as for understanding the unity of nature. The analysis results may be beneficial for the design of nano-structured materi- als, and hold potential for enhancing their mechanical, chemical, optical and electronic properties.
基金Supported by the National Natural Science Foundation of China(10972151,11272227)the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province(CXZZ11_0949)the Innovation Program for Postgraduate of Suzhou University of Science and Technology(SKCX11S_050)
文摘The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s equations.The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics.Here,the fractional Pfaff-Birkhoff variational problem is presented and studied.The definitions of fractional derivatives,the formulae for integration by parts and some other preliminaries are firstly given.Secondly,the fractional Pfaff-Birkhoff principle and the fractional Birkhoff′s equations in terms of RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.
基金Supported by the National Natural Science Foundation of China Shanghai Natural Science Foundation.
文摘Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].
基金Supported by the National Nature Science Foundation of China (10261002)
文摘In this paper,the (?)-equivariant (s, t)-equivalence relation and (?)-equivariant infinitesimally (r, s)-stability of (?)-equivariant bifurcation problem are defined. The criterion for (?)-equivariant infinitesimally (r, s)-stability is proven when (?) is a compact finite Lie group .Transversality condition is used to characterize the stability.
基金Project supported by the National Natural Science Foundation of China.
文摘A geometrical method using the exponential dichotomy and the invariant manifold thoery is given to set up the criteria for the existence of transversal and tangential heterodinic orbits under the most general degenerate cases. Conclusions given here extend and contain the relevant known results.
文摘The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variable size population and two control strategies where creating awareness and treatment are considered as controls. We intend to find the optimal combination of these two control strategies that will minimize the cost of the two control measures and as a result the number of infectious individuals will decrease. We establish the existence for the optimal controls and Pontryagin’s maximum principle is used to characterize the optimal controls. The numerical simulation suggests that optimal control technique is much more effective to minimize the infected individuals and the corresponding cost of the two controls. It is also monitored that in the case of high contact rate, controls have to work for longer period of time to get the desired result. Numerical simulation reveals that the spread of Nipah virus can be controlled effectively if we apply control strategy at early stage.
文摘Let (■_n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift spaceΣ_A defined by Fibonacci incident matrix A.We consider the measureμ_λof the probability distribution of the random seriesΣ_(n=0)~∞■_nλ~n(0<λ<1).It is proved thatμ_λis singular ifλ∈(0,(5^(1/2)-1)/2) and thatμ_λis absolutely continuous for almost allλ∈((5^(1/2)-1)/2,0.739).
文摘The implementation of optimal control strategies involving preventive measures and antiviral treatment can significantly reduce the number of clinical cases of influenza. In this paper, a model for the transmission dynamics of influenza is formulated and two control strategies involving preventive measures (awareness campaign, washing hand, using hand sanitizer, wearing mask) and treatment are considered and used to minimize the total number of infected individuals and associated cost of using these two controls. The resulting optimality system is solved numerically. Hamiltonian is formulated to investigate the existence of the optimal control, in the optimal control model. Pontryagin’s Maximum Principle is applied to describe the control variables and the objective function is designed to reduce both the infection and the cost of interventions. From the numerical simulation, it is observed that in the case of high contact rate (β = 3), both the controls work for a longer period of time to reduce the disease burden. The optimal control analysis and numerical simulations reveal that the interventions reduce the number of exposed and infected individuals.
基金supported by the National Natural Science Foundation of China(Grant Nos.10802099,11272357 and 11102140)the Doctoral Fund of Ministry of Education of China(Grant No.20110141120024)+2 种基金the Natural Science Foundation of Shandong Province(Grant No.ZR2009AQ006)the Opening Project of State Key Laboratory of Explosion Science and Technology(Beijing Institute of Technology)(Grant No. KFJJ12-11M)the support from the Brain Korea 21 Program at Seoul National University
文摘In this study,we considered the wetting phenomenon on a general substrate from a new viewpoint of continuum mechanics.The analyses first show how the Wenzel and the Cassie models deviate the practical results in some special substrates,and then elucidate the mechanism of the triple contact line(TCL) moving.Based upon variational theory of the total free functional dealing with the movable boundary condition,we show that the macroscopic contact angle(MCA) expression is the corresponding transversality condition.It manifests that the MCA depends only on the chemical and geometric property at the TCL,and is not affected by the gravity of the droplet and the contact area beneath the liquid.Our continuum model also shows the exploration of the pinning effect on a sharp wedge or the interface between two different phases.This investigation will help designing super-hydrophobic materials for novel micro-fluidic devices.
文摘By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of application are also given.
基金Project supported by the National Natural Science Foundation of ChinaShanghai Natural Science Foundation
文摘Using the exponential dichotomies,the transversality theory and the generalized Melnikov method,we consider the conditions for the persistence and the transversality of the singular orbit,with high degeneracy,situated on the heteroclinic or homoclinic manifold under perturbation.The results obtained extend,include and improve the corresponding ones given in certain papers well known in this area.
基金National Natural Science Foundations of China(Nos.11572212,11272227,10972151)the Innovation Program for Scientific Research of Nanjing University of Science and Technology,Chinathe Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(No.KYLX15_0405)
文摘The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.
基金Supported by NSFC Grant No.10671171NSF (Jiangsu Province,China) Grant No.BK2007073
文摘The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in R2,2 to R2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used by Dai and Terng to construct monopoles with continuous scattering data, and then the equation can be linearized by the scattering data, allowing one to use the inverse scattering method to solve the Cauchy problem with rapidly decaying small initial data. In this paper, we use the terminology of holomorphic bundle and transversality of certain maps, parametrized by initial data, to give more initial data, with which we can use scattering method to solve the Cauchy problem of the monopole equation up to gauge transformation.
文摘A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeomorphisms.
