This paper proposes second-order consensus protocols with time-delays and gives the measure of the robustness of the protocols to the time-delay existing in the network of agents with second-order dynamics. By employi...This paper proposes second-order consensus protocols with time-delays and gives the measure of the robustness of the protocols to the time-delay existing in the network of agents with second-order dynamics. By employing a frequency domain method, it is proven that the information states and their time derivatives of all the agents in the network achieve consensus asymptotically, respectively, for appropriate communication timedelay if the topology of weighted network is connected. Particularly, a tight upper bound on the communication time-delay that can be tolerated in the dynamic network is found. The consensus protocols are distributed in the sense that each agent only needs information from its neighboring agents, which reduces the complexity of connections between neighboring agents significantly. Numerical simulation results are provided to demonstrate the effectiveness and the sharpness of the theoretical results for second-order consensus in networks in the presence of communication time-delays.展开更多
The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the po...The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations (jumps) resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.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 purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),w...The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.展开更多
A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different ...A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex nonsmooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function. The R(x)-orthonormal bases in the linear space of boundary functions are constructed, of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x). When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix A :=[aij ],aij= 0 if i + j is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of A can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions half lower. Hence, we can sequentially solve the natural frequencies in closed-form owing to the specialty of A. We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.展开更多
Nonlinear wave loads can induce low-frequency and high-frequency resonance motions of a moored platform in deep water. For the analysis of the nonlinear response of an offshore platform under the action of irregular w...Nonlinear wave loads can induce low-frequency and high-frequency resonance motions of a moored platform in deep water. For the analysis of the nonlinear response of an offshore platform under the action of irregular waves, the most widely used method in practice is the Cummins method, in which the second-order exciting forces in the time domain are computed by a two-term Volterra series model based on incident waves, first-order body motion response, and quadratic transfer functions(QTFs). QTFs are bichromatic waves acting on a body and are computed in the frequency domain in advance. For moving bodies, QTFs are related to the first-order body response, which is to be determined in the simulation process of body motion response but is unknown in the computation procedure of QTFs. In solving this problem, Teng and Cong(2017) proposed a method to divide the QTFs into different components,which are unrelated to the body response. With the application of the new QTF components, a modified Cummins method can be developed for the simulation of the nonlinear response of a moored floating platform. This paper presents a review of the theory.展开更多
The selection of wave force models will significantly impact the structural responses of floating wind turbines.In this study,comparisons of wave force model effects on the structural responses and fatigue loads of a ...The selection of wave force models will significantly impact the structural responses of floating wind turbines.In this study,comparisons of wave force model effects on the structural responses and fatigue loads of a semi-submersible floating wind turbine(SFWT)were conducted.Simulations were performed by employing the Morison equation(ME)with linear or second-order wave kinematics and potential flow theory(PFT)with first-or second-order wave forces.A comparison of regular waves,irregular waves,and coupled wind/waves analyses with the experimental data showed that many of the simulation results and experimental data are relatively consistent.However,notable discrepancies are found in the response amplitude operators for platform heave,tower base bending moment,and tension in mooring lines.PFT models give more satisfactory results of heave but more significant discrepan-cies in tower base bending moment than the ME models.In irregular wave analyses,low-frequency resonances were captured by PFT models with second-order difference-frequency terms,and high-frequency resonances were captured by the ME models or PFT models with second-order sum-frequency terms.These force models capture the response frequencies but do not reasonably predict the response amplitudes.The coupled wind/waves analyses showed more satisfactory results than the wave-only analyses.However,an important detail to note is that this satisfactory result is based on the overprediction of wind-induced responses.展开更多
We describe an implementation of the cluster-in-molecule (CIM) resolution of the identity (RI) approximation second-order Moller-Plesset perturbation theory (CIM-RI-MP2), with the purpose of extending RI-MP2 cal...We describe an implementation of the cluster-in-molecule (CIM) resolution of the identity (RI) approximation second-order Moller-Plesset perturbation theory (CIM-RI-MP2), with the purpose of extending RI-MP2 calculations to very large systems. For typical conformers of several large polypeptides, we calculated their conformational energy differences with the CIM-RI-MP2 and the generalized energy-based fragmentation MP2 (GEBF-MP2) methods, and compared these results with the density functional theory (DFT) results obtained with several popular functionals. Our calculations show that the conformational energy differences obtained with CIM-RI-MP2 and GEBF-MP2 are very close to each other. In comparison with the GEBF-MP2 and CIM-RI-MP2 relative energies, we found that the DFT functionals (CAM-B3LYP-D3, LC-ωPBE-D3, M05-2X, M06-2X and coB97XD) can give quite accurate conformational energy differences for structurally similar conformers, but provide less-accurate results for structurally very different conformers.展开更多
The ground state of osmium monoxide(OsO) has long been controversial. In this paper, the low-lying Λ–S and ? electronic states of OsO have been comprehensively studied by the high-precision multi-reference calculati...The ground state of osmium monoxide(OsO) has long been controversial. In this paper, the low-lying Λ–S and ? electronic states of OsO have been comprehensively studied by the high-precision multi-reference calculations. The ground state of OsO is unexpectedly the closed-shell1Σ+state with a double bond instead of the previously reported3Φ or5Σ+state;after including the spin–orbit coupling effects, the ground state becomes3Π2. With the help of the theoretical spectroscopic constants and transition dipole moments, the emission spectra in the region of 405 nm–875 nm are assigned.Our results will facilitate the future studies of absorption and emission spectra of OsO.展开更多
Density flmctional theory(DFT)was employed to calculate the geometrical structures,UV-Vis absorption spectra and second-order nonlinear optical(NLO)properties of a family of iridium(Ⅲ)complexes,which possess of diffe...Density flmctional theory(DFT)was employed to calculate the geometrical structures,UV-Vis absorption spectra and second-order nonlinear optical(NLO)properties of a family of iridium(Ⅲ)complexes,which possess of different cyclometallated ligands(C∧N)and ancillary ligands[pyridme-2-carboxylate(pic)].It was found that the mo-dification of the LUMO energy levels was achieved by changing pic ligands and the energy gaps between the HOMO and LUMO were notably increased or decreased.In addition,the degree of conjugation was significantly changed with the substituent groups varied,which led to that the first hyperpolarizabilityβcould be effectively modulated.Through the analysis of time-dependent DFT(TD-DFT)results,we predicted that these studied complexes withπ→π^*charge transfer was beneficial to the large second-order NLO properties.Therefore,we hope that these studied iridium(III)complexes can be considered as versatile second-order NLO materials.展开更多
In this paper,we consider an asymptotically linear second-order ordinary differential system with Dirchlet boundary value conditions. Under some conditions,we show the multiplicity of solutions to the system by the Mo...In this paper,we consider an asymptotically linear second-order ordinary differential system with Dirchlet boundary value conditions. Under some conditions,we show the multiplicity of solutions to the system by the Morse theory and an index theory.展开更多
A new full time-domain nonlinear coupled method has been established and then applied to predict the responses of a Truss Spar in irregular wave. For the coupled analysis, a second-order time-domain approach is develo...A new full time-domain nonlinear coupled method has been established and then applied to predict the responses of a Truss Spar in irregular wave. For the coupled analysis, a second-order time-domain approach is developed to calculate the wave forces, and a finite element model based on rod theory is established in three dimensions in a global coordinate system. In nu- merical implementation, the higher-order boundary element method (HOBEM) is employed to solve the velocity potential, and the 4th-order Adams-Bashforth-Moultn scheme is used to update the second-order wave surface. In deriving convergent solu- tions, the hull displacements and mooring tensions are kept consistent at the fairlead and the motion equations of platform and mooring-lines/risers are solved simultaneously using Newmark-fl integration scheme including Newton-Raphson iteration. Both the coupled quasi-static analysis and the coupled dynamic analysis are performed. The numerical simulation results are also compared with the model test results, and they coincide very well as a whole. The slow-drift responses can be clearly ob- served in the time histories of displacements and mooring tensions. Some important characteristics of the coupled responses are concluded.展开更多
The mechanism of a cycloaddition reaction between singlet methylidenesilene and ethylene has been investigated with MP2/6-31G^* and B3LYP/6-31G^* methods, including geometry optimization and vibrational analysis for...The mechanism of a cycloaddition reaction between singlet methylidenesilene and ethylene has been investigated with MP2/6-31G^* and B3LYP/6-31G^* methods, including geometry optimization and vibrational analysis for the involved stationary points on the potential energy surface. Energies of the involved conformers were calculated by CCSD(T)//MP2/6-31G* and CCSD(T)//B3LYP/6-31 G* methods, respectively. The results show that the dominant reaction pathway of the cycloaddition reaction is that a complex intermediate is firstly formed between the two reactants through a barrier-free exothermic reaction of 13.3 kJ/mol, and the complex is then isomefized to a four-membered ring product P2,1 via a transition state TS2.1 with a barrier of 32.0 kJ/mol.展开更多
The mechanism of cycloaddition reaction between singlet alkylidene carbene and ethylene has been investigated with second-order Moller-Plesset perturbation theory (MP2). By using 6–31G* basis, geometry optimization, ...The mechanism of cycloaddition reaction between singlet alkylidene carbene and ethylene has been investigated with second-order Moller-Plesset perturbation theory (MP2). By using 6–31G* basis, geometry optimization, vibrational analysis and energetics have been calculated for the involved stationary points on the potential energy surface. The results show that the title reaction has two major competition channels. An energy-rich intermediate (INT) is firstly formed between alkylidene carbene and ethylene through a barrier-free exothermic reaction of 63.62 kJ/mol, and the intermediate then isomerizes to a three-membered ring product (Pl) and a four-membered ring product (P2) via transition state TS1 and TS2, in which energy barriers are 47.00 and 51.02 kl/mol. respectively. PI is the main product.展开更多
We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang(2012)for the unique complete Kahler-Einstein metric of Cheng and Yau(1980),Kobayashi(1984),Tian and Yau(1987)and Bando(1990)on quasi-...We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang(2012)for the unique complete Kahler-Einstein metric of Cheng and Yau(1980),Kobayashi(1984),Tian and Yau(1987)and Bando(1990)on quasi-projective manifolds.The main tools are the solution formula for second-order ordinary differential equations(ODEs)with constant coefficients and spectral theory for the Laplacian operator on a closed manifold.展开更多
基金supported by the National Natural Science Foundation of China (6057408860274014)
文摘This paper proposes second-order consensus protocols with time-delays and gives the measure of the robustness of the protocols to the time-delay existing in the network of agents with second-order dynamics. By employing a frequency domain method, it is proven that the information states and their time derivatives of all the agents in the network achieve consensus asymptotically, respectively, for appropriate communication timedelay if the topology of weighted network is connected. Particularly, a tight upper bound on the communication time-delay that can be tolerated in the dynamic network is found. The consensus protocols are distributed in the sense that each agent only needs information from its neighboring agents, which reduces the complexity of connections between neighboring agents significantly. Numerical simulation results are provided to demonstrate the effectiveness and the sharpness of the theoretical results for second-order consensus in networks in the presence of communication time-delays.
基金supported by the National Board for Higher Mathematics,Mumbai,India under Grant No.2/48(5)/2013/NBHM(R.P.)/RD-II/688 dt 16.01.2014
文摘The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations (jumps) resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.
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This research is supported by the Shandong Provincial Natural Science Foundation of China(ZR2017MA043).
文摘The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.
文摘A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex nonsmooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function. The R(x)-orthonormal bases in the linear space of boundary functions are constructed, of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x). When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix A :=[aij ],aij= 0 if i + j is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of A can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions half lower. Hence, we can sequentially solve the natural frequencies in closed-form owing to the specialty of A. We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.
基金the National Key R&D Program of China (Grant No.2016YFE0200100)the National Natural Science Foundation of China (Grant Nos.51490672 and 51479026).
文摘Nonlinear wave loads can induce low-frequency and high-frequency resonance motions of a moored platform in deep water. For the analysis of the nonlinear response of an offshore platform under the action of irregular waves, the most widely used method in practice is the Cummins method, in which the second-order exciting forces in the time domain are computed by a two-term Volterra series model based on incident waves, first-order body motion response, and quadratic transfer functions(QTFs). QTFs are bichromatic waves acting on a body and are computed in the frequency domain in advance. For moving bodies, QTFs are related to the first-order body response, which is to be determined in the simulation process of body motion response but is unknown in the computation procedure of QTFs. In solving this problem, Teng and Cong(2017) proposed a method to divide the QTFs into different components,which are unrelated to the body response. With the application of the new QTF components, a modified Cummins method can be developed for the simulation of the nonlinear response of a moored floating platform. This paper presents a review of the theory.
基金funded by the National Natural Science Foundation of China(No.51809135)the Shandong Provincial Natural Science Foundation(No.ZR2018BEE 047)+1 种基金the National Natural Science Foundation of China–Shandong Joint Fund(No.U2006229)the SKL of HESS(No.HESS-1808).
文摘The selection of wave force models will significantly impact the structural responses of floating wind turbines.In this study,comparisons of wave force model effects on the structural responses and fatigue loads of a semi-submersible floating wind turbine(SFWT)were conducted.Simulations were performed by employing the Morison equation(ME)with linear or second-order wave kinematics and potential flow theory(PFT)with first-or second-order wave forces.A comparison of regular waves,irregular waves,and coupled wind/waves analyses with the experimental data showed that many of the simulation results and experimental data are relatively consistent.However,notable discrepancies are found in the response amplitude operators for platform heave,tower base bending moment,and tension in mooring lines.PFT models give more satisfactory results of heave but more significant discrepan-cies in tower base bending moment than the ME models.In irregular wave analyses,low-frequency resonances were captured by PFT models with second-order difference-frequency terms,and high-frequency resonances were captured by the ME models or PFT models with second-order sum-frequency terms.These force models capture the response frequencies but do not reasonably predict the response amplitudes.The coupled wind/waves analyses showed more satisfactory results than the wave-only analyses.However,an important detail to note is that this satisfactory result is based on the overprediction of wind-induced responses.
基金supported by the National Natural Science Foundation of China(21073086,21333004)the National Basic Research Program of China(2011CB808501)
文摘We describe an implementation of the cluster-in-molecule (CIM) resolution of the identity (RI) approximation second-order Moller-Plesset perturbation theory (CIM-RI-MP2), with the purpose of extending RI-MP2 calculations to very large systems. For typical conformers of several large polypeptides, we calculated their conformational energy differences with the CIM-RI-MP2 and the generalized energy-based fragmentation MP2 (GEBF-MP2) methods, and compared these results with the density functional theory (DFT) results obtained with several popular functionals. Our calculations show that the conformational energy differences obtained with CIM-RI-MP2 and GEBF-MP2 are very close to each other. In comparison with the GEBF-MP2 and CIM-RI-MP2 relative energies, we found that the DFT functionals (CAM-B3LYP-D3, LC-ωPBE-D3, M05-2X, M06-2X and coB97XD) can give quite accurate conformational energy differences for structurally similar conformers, but provide less-accurate results for structurally very different conformers.
基金Project supported by the National Natural Science Foundation of China (Grant No. 22073072)the Double First-Class University Construction Project of Northwest University。
文摘The ground state of osmium monoxide(OsO) has long been controversial. In this paper, the low-lying Λ–S and ? electronic states of OsO have been comprehensively studied by the high-precision multi-reference calculations. The ground state of OsO is unexpectedly the closed-shell1Σ+state with a double bond instead of the previously reported3Φ or5Σ+state;after including the spin–orbit coupling effects, the ground state becomes3Π2. With the help of the theoretical spectroscopic constants and transition dipole moments, the emission spectra in the region of 405 nm–875 nm are assigned.Our results will facilitate the future studies of absorption and emission spectra of OsO.
基金the“12th Five-year”Science and Technology Research Project of the Education Departaent of Jilin Province,China(No.[2016]494)and the National Natural Science Foundation of China(No.21173035).
文摘Density flmctional theory(DFT)was employed to calculate the geometrical structures,UV-Vis absorption spectra and second-order nonlinear optical(NLO)properties of a family of iridium(Ⅲ)complexes,which possess of different cyclometallated ligands(C∧N)and ancillary ligands[pyridme-2-carboxylate(pic)].It was found that the mo-dification of the LUMO energy levels was achieved by changing pic ligands and the energy gaps between the HOMO and LUMO were notably increased or decreased.In addition,the degree of conjugation was significantly changed with the substituent groups varied,which led to that the first hyperpolarizabilityβcould be effectively modulated.Through the analysis of time-dependent DFT(TD-DFT)results,we predicted that these studied complexes withπ→π^*charge transfer was beneficial to the large second-order NLO properties.Therefore,we hope that these studied iridium(III)complexes can be considered as versatile second-order NLO materials.
文摘In this paper,we consider an asymptotically linear second-order ordinary differential system with Dirchlet boundary value conditions. Under some conditions,we show the multiplicity of solutions to the system by the Morse theory and an index theory.
基金supported by the National High Technology Research and Development Program of China(Grant No.2006AA09A109-3)the Important National Science&Technology Specific Projects(Grant No.2008ZX05026-06-02)
文摘A new full time-domain nonlinear coupled method has been established and then applied to predict the responses of a Truss Spar in irregular wave. For the coupled analysis, a second-order time-domain approach is developed to calculate the wave forces, and a finite element model based on rod theory is established in three dimensions in a global coordinate system. In nu- merical implementation, the higher-order boundary element method (HOBEM) is employed to solve the velocity potential, and the 4th-order Adams-Bashforth-Moultn scheme is used to update the second-order wave surface. In deriving convergent solu- tions, the hull displacements and mooring tensions are kept consistent at the fairlead and the motion equations of platform and mooring-lines/risers are solved simultaneously using Newmark-fl integration scheme including Newton-Raphson iteration. Both the coupled quasi-static analysis and the coupled dynamic analysis are performed. The numerical simulation results are also compared with the model test results, and they coincide very well as a whole. The slow-drift responses can be clearly ob- served in the time histories of displacements and mooring tensions. Some important characteristics of the coupled responses are concluded.
基金Project supported by the Natural Science Foundation of Shandong Province of China (No. Y2002B07).
文摘The mechanism of a cycloaddition reaction between singlet methylidenesilene and ethylene has been investigated with MP2/6-31G^* and B3LYP/6-31G^* methods, including geometry optimization and vibrational analysis for the involved stationary points on the potential energy surface. Energies of the involved conformers were calculated by CCSD(T)//MP2/6-31G* and CCSD(T)//B3LYP/6-31 G* methods, respectively. The results show that the dominant reaction pathway of the cycloaddition reaction is that a complex intermediate is firstly formed between the two reactants through a barrier-free exothermic reaction of 13.3 kJ/mol, and the complex is then isomefized to a four-membered ring product P2,1 via a transition state TS2.1 with a barrier of 32.0 kJ/mol.
基金the Natural Science Foundation of Shandong Province of China (No. Y2002B07).
文摘The mechanism of cycloaddition reaction between singlet alkylidene carbene and ethylene has been investigated with second-order Moller-Plesset perturbation theory (MP2). By using 6–31G* basis, geometry optimization, vibrational analysis and energetics have been calculated for the involved stationary points on the potential energy surface. The results show that the title reaction has two major competition channels. An energy-rich intermediate (INT) is firstly formed between alkylidene carbene and ethylene through a barrier-free exothermic reaction of 63.62 kJ/mol, and the intermediate then isomerizes to a three-membered ring product (Pl) and a four-membered ring product (P2) via transition state TS1 and TS2, in which energy barriers are 47.00 and 51.02 kl/mol. respectively. PI is the main product.
基金supported by National Natural Science Foundation of China(Grant Nos.11331001 and 11871265)the Hwa Ying Foundation for its financial support and thanks Professor Jian Song for his invitation。
文摘We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang(2012)for the unique complete Kahler-Einstein metric of Cheng and Yau(1980),Kobayashi(1984),Tian and Yau(1987)and Bando(1990)on quasi-projective manifolds.The main tools are the solution formula for second-order ordinary differential equations(ODEs)with constant coefficients and spectral theory for the Laplacian operator on a closed manifold.