In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 ...In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 and any 0【ε【14,E k(x+H)-E k(x)≤H(log x) -Aholds for x 12-14k+ε≤H≤x, here the implies constant depends at most on A and ε.展开更多
The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutio...The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutionsof the Z-K equation are obtained.The methods used to solve the Z-K equation can be employed in further work toestablish new solutions for other nonlinear partial differential equations.展开更多
We firstly propose two kinds of new multi-component BKP (mcBKP) hierarchy based on the eigenfunction symmetry reduction and nonstandard reduction, respectively. The first one contains two types of BKP equation with ...We firstly propose two kinds of new multi-component BKP (mcBKP) hierarchy based on the eigenfunction symmetry reduction and nonstandard reduction, respectively. The first one contains two types of BKP equation with self-consistent sources whose Lax representations are presented. The two mcBKP hierarchies both admit reductions to the k-constrained BKP hierarchy and to integrable (1+1)-dimensional hierarchy with self-consistent sources, which include two types of SK equation with self-consistent sources and of hi-directional SK equations with self-consistent展开更多
The thermodynamic properties of metamizol monohydrate in pure solvents(methanol,ethanol,n-propanol and isopropanol) and two binary mixed solvent systems including(methanol+ethanol) and(methanol+isopropanol) were measu...The thermodynamic properties of metamizol monohydrate in pure solvents(methanol,ethanol,n-propanol and isopropanol) and two binary mixed solvent systems including(methanol+ethanol) and(methanol+isopropanol) were measured from 283.15 K to 313.15 K by gravimetric method under atmospheric pressure thought as 0.1 MPa.The modi fied Apelblat equation,the CNIBS/R-K equation,the Hybrid model and the NRTL model were used to correlate the solubility of metamizol monohydrate,respectively.The results show that the solubility of metamizol monohydrate in all the tested solvents increases with the rising temperature which means that it has temperature dependence.What's more,the effects of solvent components of the binary solvent mixtures on solubility were discussed,it illustrates that the increasing of the molar fraction of methanol gives the system a greater dissolving power.Furthermore,according to the NRTL model,the enthalpy,the Gibbs energy and the entropy of the mixing process were also obtained and discussed.展开更多
We lind that the Fokker-Planck equation in complex variables can be conveniently solved in the context of bipartite entangled state representation and its relationship with SU(2) Lie algebraic generators' new reali...We lind that the Fokker-Planck equation in complex variables can be conveniently solved in the context of bipartite entangled state representation and its relationship with SU(2) Lie algebraic generators' new realization {(1/4)[(Q1 - Q2)^2 + (P1+ P2)^2], (1/4)[(Q1 +Q2)^2+ (P1 - P2)^2], and -(i/2)(Q1P2 + Q2P1)}, the quadratic combination of canonical operators.展开更多
The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful However, wh...The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful However, what are the su^cient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reitect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems. By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker-Planck equation of the chemical reaction systems. The obtained Fokker-Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies.展开更多
In this work we present a new method to solve the Perona Malik equation for the image denoising. The method is based on a modified fixed point algorithm which is fast and stable. We discretize the equation using a fin...In this work we present a new method to solve the Perona Malik equation for the image denoising. The method is based on a modified fixed point algorithm which is fast and stable. We discretize the equation using a finite volume method by integrating the equation using a fuzzy measure on the control volume. To make our algorithm move faster in time, we have used an optimized domain decomposition which generalize the wave relaxation method. Several test of noised images illustrate this approach and show the efficiency of the proposed new method.展开更多
To observe the effects of closed reduction and percutaneous K wires fixation of displacd supracondylar humerus fracture in children MethodsRetrospective review of fourteen patients who s...To observe the effects of closed reduction and percutaneous K wires fixation of displacd supracondylar humerus fracture in children MethodsRetrospective review of fourteen patients who sustained displaced supracondylar fracture of distal humerus treated by closed reduction and percutaneous K wires fixation Results. All patients’ K wires were removed at 4 weeks post operation Their elbow function regained at 8 weeks The average period of followed up was 10 month (varies from 6 to 18 month), all fractures healed very well without any permanent complications Two transient nerves palsy,ulnar and radial nerve each, recovered completely at 12 weeks and 16 weeks post operation respectively Conclusion. Closed reduction and percutaneous K wires fixation is a safe and efficient treatment for displaced humerus surpracondylar fracture in children展开更多
A two-body equation of the kaon-proton system with the lowest order relativistic corrections is derived and solved. The scattering lengths and the energy of an unstable bound state are calculated.
Data analysis and automatic processing is often interpreted as knowledge acquisition. In many cases it is necessary to somehow classify data or find regularities in them. Results obtained in the search of regularities...Data analysis and automatic processing is often interpreted as knowledge acquisition. In many cases it is necessary to somehow classify data or find regularities in them. Results obtained in the search of regularities in intelligent data analyzing applications are mostly represented with the help of IF-THEN rules. With the help of these rules the following tasks are solved: prediction, classification, pattern recognition and others. Using different approaches---clustering algorithms, neural network methods, fuzzy rule processing methods--we can extract rules that in an understandable language characterize the data. This allows interpreting the data, finding relationships in the data and extracting new rules that characterize them. Knowledge acquisition in this paper is defined as the process of extracting knowledge from numerical data in the form of rules. Extraction of rules in this context is based on clustering methods K-means and fuzzy C-means. With the assistance of K-means, clustering algorithm rules are derived from trained neural networks. Fuzzy C-means is used in fuzzy rule based design method. Rule extraction methodology is demonstrated in the Fisher's Iris flower data set samples. The effectiveness of the extracted rules is evaluated. Clustering and rule extraction methodology can be widely used in evaluating and analyzing various economic and financial processes.展开更多
In this paper,we investigate the{P,Q,k+1}-reflexive and anti-reflexive solutions to the system of matrix equations AX=C,XB=D and AXB=E.We present the necessary and sufficient conditions for the system men-tioned above...In this paper,we investigate the{P,Q,k+1}-reflexive and anti-reflexive solutions to the system of matrix equations AX=C,XB=D and AXB=E.We present the necessary and sufficient conditions for the system men-tioned above to have the{P,Q,k+1}-reflexive and anti-reflexive solutions.We also obtain the expressions of such solutions to the system by the singular value decomposition.Moreover,we consider the least squares{P,Q,k+1}-reflexive and anti-reflexive solutions to the system.Finally,we give an algorithm to illustrate the results of this paper.展开更多
The maximum entropy distribution, which consists of various recognized theoretical distributions, is a better curve to estimate the design thickness of sea ice. Method of moment and empirical curve fitting method are ...The maximum entropy distribution, which consists of various recognized theoretical distributions, is a better curve to estimate the design thickness of sea ice. Method of moment and empirical curve fitting method are common-used parameter estimation methods for maximum entropy distribution. In this study, we propose to use the particle swarm optimization method as a new parameter estimation method for the maximum entropy distribution, which has the advantage to avoid deviation introduced by simplifications made in other methods. We conducted a case study to fit the hindcasted thickness of the sea ice in the Liaodong Bay of Bohai Sea using these three parameter-estimation methods for the maximum entropy distribution. All methods implemented in this study pass the K-S tests at 0.05 significant level. In terms of the average sum of deviation squares, the empirical curve fitting method provides the best fit for the original data, while the method of moment provides the worst. Among all three methods, the particle swarm optimization method predicts the largest thickness of the sea ice for a same return period. As a result, we recommend using the particle swarm optimization method for the maximum entropy distribution for offshore structures mainly influenced by the sea ice in winter, but using the empirical curve fitting method to reduce the cost in the design of temporary and economic buildings.展开更多
In 5 G Ultra-dense Network(UDN), resource allocation is an efficient method to manage inter-small-cell interference. In this paper, a two-stage resource allocation scheme is proposed to supervise interference and reso...In 5 G Ultra-dense Network(UDN), resource allocation is an efficient method to manage inter-small-cell interference. In this paper, a two-stage resource allocation scheme is proposed to supervise interference and resource allocation while establishing a realistic scenario of three-tier heterogeneous network architecture. The scheme consists of two stages: in stage I, a two-level sub-channel allocation algorithm and a power control method based on the logarithmic function are applied to allocate resource for Macrocell and Picocells, guaranteeing the minimum system capacity by considering the power limitation and interference coordination; in stage II, an interference management approach based on K-means clustering is introduced to divide Femtocells into different clusters. Then, a prior sub-channel allocation algorithm is employed for Femtocells in diverse clusters to mitigate the interference and promote system performance. Simulation results show that the proposed scheme contributes to the enhancement of system throughput and spectrum efficiency while ensuring the system energy efficiency.展开更多
The surface wave generated by flow around a ship hull moving near free surface of water is simulated numerically in this study. The three-dimensional implicit finite volume method (FVM) is applied to solve Reynolds ...The surface wave generated by flow around a ship hull moving near free surface of water is simulated numerically in this study. The three-dimensional implicit finite volume method (FVM) is applied to solve Reynolds averaged Navier-Stokes (RANS) equation. The realizable k-e turbulence model has been implemented to capture turbulent flow around the ship hull in the free surface zone. The volume of fluid (VOF) method coupled with the Stokes wave theory has been used to determine the free surface effect of water. By using is a six degrees of freedom model, the ship hull's movement is numerically solved with the Stokes wave together. Under the action of Stokes waves on the sea, the interface between the air and water waves at the same regular pattem and so does the pressure and the vertical velocity. The ship hull moves in the same way as the wave. The amplitude of the ship hull's heave is less than the wave height because of the viscosity damping. This method could provide an important reference for the study of ships' movement, wave and hydrodynamics.展开更多
A new method, named relocation, was proposed to reduce the impact of sensor errors systematically, especially whenavailable data of sensors are abundant. The procedure includes evaluating the reliability of every sens...A new method, named relocation, was proposed to reduce the impact of sensor errors systematically, especially whenavailable data of sensors are abundant. The procedure includes evaluating the reliability of every sensors datum, processing the initiallocation by the credible data, and selecting a set of equations with optimal noise tolerance according to the relative relationshipbetween the initial location and sensors location, then calculating the final location by k-mean voting. The results obtained in thisresearch include comparing traditional location method with the presented method in both simulation and field experiment. In thefield experiment, the location error of relocation method reduced 41.8% compared with traditional location method. The resultssuggested that relocation method can improve the fault-tolerant performance significantly.展开更多
Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the d...Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained. From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively. Especiaily, the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation. The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variationai identity. Finally, we decompose the BK hierarchy of evolution equations into x-constrained flows and tn-eonstrained flows whose adjoint representations and the Lax pairs are given.展开更多
Let G be a graph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y 1,y 2} Y such that dist (y 1,y 2)=2. In this paper, we wi...Let G be a graph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y 1,y 2} Y such that dist (y 1,y 2)=2. In this paper, we will use the technique of the vertex insertion on l connected ( l=k or k+1,k≥2 ) claw free graphs to provide a unified proof for G to be hamiltonian or 1 hamiltonian, the sufficient conditions are expressed by the inequality concerning ∑ki=0N(Y i) and n(Y) for each essential set Y={y 0,y 1,...,y k} of G , where Y i={y i,y i-1 ,...,y i-(b-1) }Y for i∈{0,1,...,k} (the subscriptions of y j ’s will be taken modulo k+1 ), b ( 0【b【k+1 ) is an integer, and n(Y)={v∈V(G): dist (v,Y)≤2 }.展开更多
文摘In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 and any 0【ε【14,E k(x+H)-E k(x)≤H(log x) -Aholds for x 12-14k+ε≤H≤x, here the implies constant depends at most on A and ε.
基金Supported by the National Natural Science Foundation of China under Grant No.10974160
文摘The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutionsof the Z-K equation are obtained.The methods used to solve the Z-K equation can be employed in further work toestablish new solutions for other nonlinear partial differential equations.
基金supported by National Basic Research Program of China (973 Program) under Grant No.2007CB814800National Natural Science Foundation of China under Grant No.10601028the Natural Science Foundation of Fujian Province under Grant No.2008J0199
文摘We firstly propose two kinds of new multi-component BKP (mcBKP) hierarchy based on the eigenfunction symmetry reduction and nonstandard reduction, respectively. The first one contains two types of BKP equation with self-consistent sources whose Lax representations are presented. The two mcBKP hierarchies both admit reductions to the k-constrained BKP hierarchy and to integrable (1+1)-dimensional hierarchy with self-consistent sources, which include two types of SK equation with self-consistent sources and of hi-directional SK equations with self-consistent
基金Supported by National Natural Science Foundation of China(21206109)China Ministry of Science and Major National Scientific Instrument Development Project(21527812)
文摘The thermodynamic properties of metamizol monohydrate in pure solvents(methanol,ethanol,n-propanol and isopropanol) and two binary mixed solvent systems including(methanol+ethanol) and(methanol+isopropanol) were measured from 283.15 K to 313.15 K by gravimetric method under atmospheric pressure thought as 0.1 MPa.The modi fied Apelblat equation,the CNIBS/R-K equation,the Hybrid model and the NRTL model were used to correlate the solubility of metamizol monohydrate,respectively.The results show that the solubility of metamizol monohydrate in all the tested solvents increases with the rising temperature which means that it has temperature dependence.What's more,the effects of solvent components of the binary solvent mixtures on solubility were discussed,it illustrates that the increasing of the molar fraction of methanol gives the system a greater dissolving power.Furthermore,according to the NRTL model,the enthalpy,the Gibbs energy and the entropy of the mixing process were also obtained and discussed.
基金Supported by National Natural Science Foundation of China under Grant No.10874174PHD Guiding Foundation of Chinese Education Ministry
文摘We lind that the Fokker-Planck equation in complex variables can be conveniently solved in the context of bipartite entangled state representation and its relationship with SU(2) Lie algebraic generators' new realization {(1/4)[(Q1 - Q2)^2 + (P1+ P2)^2], (1/4)[(Q1 +Q2)^2+ (P1 - P2)^2], and -(i/2)(Q1P2 + Q2P1)}, the quadratic combination of canonical operators.
基金Supported in part by the National Basic Research Program of China(973 Program)under Grants No.2007CB935903the National Nature Science Foundation of China under Grant No.11074259
文摘The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful However, what are the su^cient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reitect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems. By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker-Planck equation of the chemical reaction systems. The obtained Fokker-Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies.
文摘In this work we present a new method to solve the Perona Malik equation for the image denoising. The method is based on a modified fixed point algorithm which is fast and stable. We discretize the equation using a finite volume method by integrating the equation using a fuzzy measure on the control volume. To make our algorithm move faster in time, we have used an optimized domain decomposition which generalize the wave relaxation method. Several test of noised images illustrate this approach and show the efficiency of the proposed new method.
文摘To observe the effects of closed reduction and percutaneous K wires fixation of displacd supracondylar humerus fracture in children MethodsRetrospective review of fourteen patients who sustained displaced supracondylar fracture of distal humerus treated by closed reduction and percutaneous K wires fixation Results. All patients’ K wires were removed at 4 weeks post operation Their elbow function regained at 8 weeks The average period of followed up was 10 month (varies from 6 to 18 month), all fractures healed very well without any permanent complications Two transient nerves palsy,ulnar and radial nerve each, recovered completely at 12 weeks and 16 weeks post operation respectively Conclusion. Closed reduction and percutaneous K wires fixation is a safe and efficient treatment for displaced humerus surpracondylar fracture in children
基金the National Natural Science Foundation of China under,the High Performance Computing Center of China (Beijing) and partly undertaken on IBM RS/6000 SP at CCSE of Peking University,北京大学校科研和校改项目
文摘A two-body equation of the kaon-proton system with the lowest order relativistic corrections is derived and solved. The scattering lengths and the energy of an unstable bound state are calculated.
文摘Data analysis and automatic processing is often interpreted as knowledge acquisition. In many cases it is necessary to somehow classify data or find regularities in them. Results obtained in the search of regularities in intelligent data analyzing applications are mostly represented with the help of IF-THEN rules. With the help of these rules the following tasks are solved: prediction, classification, pattern recognition and others. Using different approaches---clustering algorithms, neural network methods, fuzzy rule processing methods--we can extract rules that in an understandable language characterize the data. This allows interpreting the data, finding relationships in the data and extracting new rules that characterize them. Knowledge acquisition in this paper is defined as the process of extracting knowledge from numerical data in the form of rules. Extraction of rules in this context is based on clustering methods K-means and fuzzy C-means. With the assistance of K-means, clustering algorithm rules are derived from trained neural networks. Fuzzy C-means is used in fuzzy rule based design method. Rule extraction methodology is demonstrated in the Fisher's Iris flower data set samples. The effectiveness of the extracted rules is evaluated. Clustering and rule extraction methodology can be widely used in evaluating and analyzing various economic and financial processes.
基金supported by the National Natural Science Foundation of China(11571220)
文摘In this paper,we investigate the{P,Q,k+1}-reflexive and anti-reflexive solutions to the system of matrix equations AX=C,XB=D and AXB=E.We present the necessary and sufficient conditions for the system men-tioned above to have the{P,Q,k+1}-reflexive and anti-reflexive solutions.We also obtain the expressions of such solutions to the system by the singular value decomposition.Moreover,we consider the least squares{P,Q,k+1}-reflexive and anti-reflexive solutions to the system.Finally,we give an algorithm to illustrate the results of this paper.
基金supported by the National Natural Science Foundation of China (Nos. 51279186, 51479183, 51509227)the Shandong Province Natural Science Foundation, China (No. ZR2014EEQ030)the Fundamental Research Funds for the Central Universities (No. 201413003)
文摘The maximum entropy distribution, which consists of various recognized theoretical distributions, is a better curve to estimate the design thickness of sea ice. Method of moment and empirical curve fitting method are common-used parameter estimation methods for maximum entropy distribution. In this study, we propose to use the particle swarm optimization method as a new parameter estimation method for the maximum entropy distribution, which has the advantage to avoid deviation introduced by simplifications made in other methods. We conducted a case study to fit the hindcasted thickness of the sea ice in the Liaodong Bay of Bohai Sea using these three parameter-estimation methods for the maximum entropy distribution. All methods implemented in this study pass the K-S tests at 0.05 significant level. In terms of the average sum of deviation squares, the empirical curve fitting method provides the best fit for the original data, while the method of moment provides the worst. Among all three methods, the particle swarm optimization method predicts the largest thickness of the sea ice for a same return period. As a result, we recommend using the particle swarm optimization method for the maximum entropy distribution for offshore structures mainly influenced by the sea ice in winter, but using the empirical curve fitting method to reduce the cost in the design of temporary and economic buildings.
基金partially supported by the Major Project of National Science and Technology of China under Grants No. 2016ZX03002010003 and No. 2015ZX03001033-002
文摘In 5 G Ultra-dense Network(UDN), resource allocation is an efficient method to manage inter-small-cell interference. In this paper, a two-stage resource allocation scheme is proposed to supervise interference and resource allocation while establishing a realistic scenario of three-tier heterogeneous network architecture. The scheme consists of two stages: in stage I, a two-level sub-channel allocation algorithm and a power control method based on the logarithmic function are applied to allocate resource for Macrocell and Picocells, guaranteeing the minimum system capacity by considering the power limitation and interference coordination; in stage II, an interference management approach based on K-means clustering is introduced to divide Femtocells into different clusters. Then, a prior sub-channel allocation algorithm is employed for Femtocells in diverse clusters to mitigate the interference and promote system performance. Simulation results show that the proposed scheme contributes to the enhancement of system throughput and spectrum efficiency while ensuring the system energy efficiency.
基金Foundation item: Supported by National Natural Science Foundation of China (51409031), Fundamental Research Funds for the Central Universities (3132015203) and China Postdoctoral Science Foundation (2014M561216).
文摘The surface wave generated by flow around a ship hull moving near free surface of water is simulated numerically in this study. The three-dimensional implicit finite volume method (FVM) is applied to solve Reynolds averaged Navier-Stokes (RANS) equation. The realizable k-e turbulence model has been implemented to capture turbulent flow around the ship hull in the free surface zone. The volume of fluid (VOF) method coupled with the Stokes wave theory has been used to determine the free surface effect of water. By using is a six degrees of freedom model, the ship hull's movement is numerically solved with the Stokes wave together. Under the action of Stokes waves on the sea, the interface between the air and water waves at the same regular pattem and so does the pressure and the vertical velocity. The ship hull moves in the same way as the wave. The amplitude of the ship hull's heave is less than the wave height because of the viscosity damping. This method could provide an important reference for the study of ships' movement, wave and hydrodynamics.
基金Projects(11472311,41272304,51504288)supported by the National Natural Science Foundation of China
文摘A new method, named relocation, was proposed to reduce the impact of sensor errors systematically, especially whenavailable data of sensors are abundant. The procedure includes evaluating the reliability of every sensors datum, processing the initiallocation by the credible data, and selecting a set of equations with optimal noise tolerance according to the relative relationshipbetween the initial location and sensors location, then calculating the final location by k-mean voting. The results obtained in thisresearch include comparing traditional location method with the presented method in both simulation and field experiment. In thefield experiment, the location error of relocation method reduced 41.8% compared with traditional location method. The resultssuggested that relocation method can improve the fault-tolerant performance significantly.
基金Supported by the National Science Foundation of China under Grant No.10971031the Natural Science Foundation of Shandong Province under Grant No.ZR2009AL021
文摘Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained. From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively. Especiaily, the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation. The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variationai identity. Finally, we decompose the BK hierarchy of evolution equations into x-constrained flows and tn-eonstrained flows whose adjoint representations and the Lax pairs are given.
文摘Let G be a graph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y 1,y 2} Y such that dist (y 1,y 2)=2. In this paper, we will use the technique of the vertex insertion on l connected ( l=k or k+1,k≥2 ) claw free graphs to provide a unified proof for G to be hamiltonian or 1 hamiltonian, the sufficient conditions are expressed by the inequality concerning ∑ki=0N(Y i) and n(Y) for each essential set Y={y 0,y 1,...,y k} of G , where Y i={y i,y i-1 ,...,y i-(b-1) }Y for i∈{0,1,...,k} (the subscriptions of y j ’s will be taken modulo k+1 ), b ( 0【b【k+1 ) is an integer, and n(Y)={v∈V(G): dist (v,Y)≤2 }.