This paper considers the graph sequence graph （f^n ）={x×f^n ： x ∈X} of the iterates f^n of a continuous map f ： X →X on a compact metric space （X, d）. The main aim is to introduce serval limit sets by usi...This paper considers the graph sequence graph （f^n ）={x×f^n ： x ∈X} of the iterates f^n of a continuous map f ： X →X on a compact metric space （X, d）. The main aim is to introduce serval limit sets by using different recurrent time sets, and discuss the relation between dynamics of the underlying compact system （X, f） and the structure of these limit sets, from a topological or fractal viewpoint.展开更多
ICM （Independent Continuous Mapping） method can solve topological optimization problems with the minimized weight as the objective and subjected to displacement constraints. To get a clearer topological configuratio...ICM （Independent Continuous Mapping） method can solve topological optimization problems with the minimized weight as the objective and subjected to displacement constraints. To get a clearer topological configuration, by introducing the discrete condition of topological variables and integrating with the original objective, an optimal model with multi-objectives is formulated to make the topological variables approach 0 or 1 as near as possible, and the model reduces the effect of deleting rate on the result. The image-filtering method is employed to eliminate the checkerboard patterns and mesh dependence that occurred in the topology optimization of a continuum structure. The computational efficiency is enhanced through selecting quasi-active displacement constraints and a design region. Numerical examples indicate that this algorithm is robust and practicable, though the number of iterations is slightly increased with respect to the original algorithm.展开更多
A topology optimization method based on the solid isotropic material with penalization interpolation scheme is utilized for designing gradient coils for use in magnetic resonance microscopy.Unlike the popular stream f...A topology optimization method based on the solid isotropic material with penalization interpolation scheme is utilized for designing gradient coils for use in magnetic resonance microscopy.Unlike the popular stream function method,the proposed method has design variables that are the distribution of conductive material.A voltage-driven transverse gradient coil is proposed to be used as micro-scale magnetic resonance imaging(MRI)gradient coils,thus avoiding introducing a coil-winding pattern and simplifying the coil configuration.The proposed method avoids post-processing errors that occur when the continuous current density is approximated by discrete wires in the stream function approach.The feasibility and accuracy of the method are verified through designing the z-gradient and y-gradient coils on a cylindrical surface.Numerical design results show that the proposed method can provide a new coil layout in a compact design space.展开更多
This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫...This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫∫_(R2N)|x|^(α1p)|y|^(α2p)|u(x) − u(y)|^(p)/|x − y|^(N+ps) dxdy)L_(p)^(s)u = |x| ^(β)f(u) in Ω,u = 0 in R^(N) \ Ω,where L_(p)^(s) is the generalized fractional p-Laplacian operator,N≥1,s∈(0,1),α_(1),α_(2),β∈R,Ω■R^(N) is a bounded domain with Lipschitz boundary,and M:R0^(+)→R0^(+),f:Ω→R are continuous functions.Firstly,we introduce a variational framework for the above problem.Then,the existence of least energy solutions is obtained by using variational methods,provided that the nonlinear term f has(θ_(p-1))-sublinear growth at infinity.Moreover,the existence of infinitely many solutions is obtained by using Krasnoselskii’s genus theory.Finally,we obtain the existence and multiplicity of solutions if f has(θ_(p-1))-superlinear growth at infinity.The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.展开更多
A new hybrid MMA-MGCMMA （HMM） algorithm for solving topology optimization problems is presented. This algorithm combines the method of moving asymptotes （MMA） algorithm and the modified globally convergent version...A new hybrid MMA-MGCMMA （HMM） algorithm for solving topology optimization problems is presented. This algorithm combines the method of moving asymptotes （MMA） algorithm and the modified globally convergent version of the method of moving asymptotes （MGCMMA） algorithm in the optimization process. This algorithm preserves the advantages of both MMA and MGCMMA. The optimizer is switched from MMA to MGCMMA automatically, depending on the numerical oscillation value existing in the calculation. This algorithm can improve calculation efficiency and accelerate convergence compared with simplex MMA or MGCMMA algorithms, which is proven with an example.展开更多
Non-dominated sorting genetic algorithm II（NSGA-II）with multiple constraints handling is employed for multi-objective optimization of the topological structure of telescope skin,in which a bit-matrix is used as the ...Non-dominated sorting genetic algorithm II（NSGA-II）with multiple constraints handling is employed for multi-objective optimization of the topological structure of telescope skin,in which a bit-matrix is used as the representation of a chromosome,and genetic algorithm（GA）operators are introduced based on the matrix.Objectives including mass,in-plane performance,and out-of-plane load-bearing ability of the individuals are obtained by fnite element analysis（FEA）using ANSYS,and the matrix-based optimization algorithm is realized in MATLAB by handling multiple constraints such as structural connectivity and in-plane strain requirements.Feasible confgurations of the support structure are achieved.The results confrm that the matrix-based NSGA-II with multiple constraints handling provides an effective method for two-dimensional multi-objective topology optimization.展开更多
This paper explores the realization of robotic motion planning, especially Findpath problem, which is a basic motion planning problem that arises in the development of robotics. Findpath means: Give the initial and de...This paper explores the realization of robotic motion planning, especially Findpath problem, which is a basic motion planning problem that arises in the development of robotics. Findpath means: Give the initial and desired final configurations of a robotic arm in 3-dimensionnl space, and give descriptions of the obstacles in the space, determine whether there is a continuous collision-free motion of the robotic arm from one configure- tion to the other and find such a motion if it exists. There are several branches of approach in motion planning area, but in reality the important things are feasibility, efficiency and accuracy of the method. In this paper ac- cording to the concepts of Configuration Space (C-Space) and Rotation Mapping Graph (RMG) discussed in [1], a topological method named Dimension Reduction Method (DRM) for investigating the connectivity of the RMG (or the topologic structure of the RMG )is presented by using topologic technique. Based on this ap- proach the Findpath problem is thus transformed to that of finding a connected way in a finite Characteristic Network (CN). The method has shown great potentiality in practice. Here a simulation system is designed to embody DRM and it is in sight that DRM can he adopted in the first overall planning of real robot sys- tem in the near future.展开更多
Integrated circuits(ICs)are the foundation of information technology development.The optimal design scheme of an analog IC is determined by iteratively running the simulation software and comparing the performance met...Integrated circuits(ICs)are the foundation of information technology development.The optimal design scheme of an analog IC is determined by iteratively running the simulation software and comparing the performance metrics.However,the simulation software of an analog IC is time-consuming,which leads to the low design efficiency.Due to the nonideal factors in analog ICs,the nonlinear relationship between design parameters and performance metrics cannot be well described by the deduced approximation equations.Inspired by the image and semantic recognition,a universal high-efficiency modeling method for analog ICs based on convolutional neural network(CNN)was proposed in the current work,named as CNN-IC.The sparse topology mapping method was proposed to map the design parameters into a sparse matrix,which includes the spatial and transistor characteristics of analog IC.The CNN model with three convolutional kernels was constructed to extract“transistor-circuit module-integrate circuit”features level by level,which can replace the simulation software to effectively improve the training efficiency and accuracy.Two typical analog ICs were selected to verify the effectiveness of the CNN-IC model.The results show that the accuracy of the CNN-IC model could reach over 99%and that its convergence rate was the fastest compared with the machine learning models in the state of the art.展开更多
To address the challenges for vibration suppression in the precision sensors of underwater vehicles,phononic crystals have attracted significant attention due to the superior capabilities in elastic wave manipulation ...To address the challenges for vibration suppression in the precision sensors of underwater vehicles,phononic crystals have attracted significant attention due to the superior capabilities in elastic wave manipulation and vibration suppression.Unlike conventional damping materials,the phononic crystals can effectively suppress the wave propagation in the specific frequency ranges due to the unique periodic microstructures.展开更多
This paper concerns the existence of affine-periodic solutions for perturbed affine-periodic systems.This kind of affine-periodic solutions has the form of x(t+T)≡Qx(t) with some nonsingular matrix Q,which may be qua...This paper concerns the existence of affine-periodic solutions for perturbed affine-periodic systems.This kind of affine-periodic solutions has the form of x(t+T)≡Qx(t) with some nonsingular matrix Q,which may be quasi-periodic when Q is an orthogonal matrix. It can be even unbounded but x(t)/|x(t)| is quasi-periodic,like a helical line. for example x(t)=e^(at)(cos ωt, sin ωt), when Q is not an orthogonal matrix. The averaging method of higher order for finding affine-periodic solutions is given by topological degree.展开更多
In this paper,an algorithm for solving the multi-target correlation and co-location problem of aerial-ground heterogeneous system is investigated.Aiming at the multi-target correlation problem,the fusion algorithm of ...In this paper,an algorithm for solving the multi-target correlation and co-location problem of aerial-ground heterogeneous system is investigated.Aiming at the multi-target correlation problem,the fusion algorithm of visual axis correlation method and improved topological similarity correlation method are adopted in view of large parallax and inconsistent scale between the aerial and ground perspectives.First,the visual axis was preprocessed by the threshold method,so that the sparse targets were initially associated.Then,the improved topological similarity method was used to further associate dense targets with the relative position characteristics between targets.The shortcoming of dense target similarity with small di®erence was optimized by the improved topological similarity method.For the problem of colocation,combined with the multi-target correlation algorithm in this paper,the triangulation positioning model was used to complete the co-location of multiple targets.In the experimental part,simulation experiments and°ight experiments were designed to verify the e®ectiveness of the algorithm.Experimental results show that the proposed algorithm can e®ectively achieve multi-target correlation positioning,and that the positioning accuracy is obviously better than other positioning methods.展开更多
An efficient path planning algorithm based on topologic method is presented in this paper.The colli- sion free path planning for three-joint robotic arm consists of three parts:partition of C-space,construc- tion of C...An efficient path planning algorithm based on topologic method is presented in this paper.The colli- sion free path planning for three-joint robotic arm consists of three parts:partition of C-space,construc- tion of CN and search for a path in CN.We mainly solved the problems of partitioning the C-space and judging the connectivity between connected blocks,etc.For the motion planning of a robotic arm with a gripper,we developed the concepts of global planning and local planning,and discussed the basic fac- tors for constructing the planning system.In the paper,some evaluation and analysis of the complexity and reliability of the algorithm are given,together with some ideas to improve the efficiency and increase the reliability.At last,some experimental results are presented to show the efficiency and accuracy of the nigorithm.展开更多
In this paper,we study the existence of"weak solution"for a class of p(x)-Kirchhoff type problem involving the p(x)-Laplacian-like operator depending on two real parameters with Neumann boundary condition.Us...In this paper,we study the existence of"weak solution"for a class of p(x)-Kirchhoff type problem involving the p(x)-Laplacian-like operator depending on two real parameters with Neumann boundary condition.Using a topological degree for a class of demicontinuous operator of generalized(S_(+))type and the theory of the variable exponent Sobolev space,we establish the existence of"weak solution"of this problem.展开更多
By using topological method, we study a class of boundary value problem for a system of nonlinear ordinary differential equations. Under suitable conditions, we prove the existence of positive solution of the problem.
Let(Σ,g)be a compact Riemann surface with smooth boundary■Σ,Δ_(g) be the Laplace-Beltrami operator,and h be a positive smooth function.Using a min-max scheme introduced by Djadli and Malchiodi(2008)and Djadli(2008...Let(Σ,g)be a compact Riemann surface with smooth boundary■Σ,Δ_(g) be the Laplace-Beltrami operator,and h be a positive smooth function.Using a min-max scheme introduced by Djadli and Malchiodi(2008)and Djadli(2008),we prove that ifΣis non-contractible,then for anyρ∈(8kπ,8(k+1)π)with k∈N^(*),the mean field equation{Δgu=ρhe^(u)/∫∑he^(u)dv_(g)in∑,u=0 on■∑has a solution.This generalizes earlier existence results of Ding et al.(Ann Inst H PoincaréAnal Non Linéaire,1999)and Chen and Lin(2003)in the Euclidean domain.Also we consider the corresponding Neumann boundary value problem.If h is a positive smooth function,then for anyρ∈(4kπ,4(k+1)π)with k∈N^(*),the mean field equation{Δgu=ρhe^(u)/∫_(∑)he^(u)dv_(g)-1/|∑|in∑,■u/■v=0 on■∑has a solution,where v denotes the unit normal outward vector on ■Σ.Note that in this case we do not require the surface to be non-contractible.展开更多
Applying topological method,this paper investigates the existence of positive solu-tions to singular m-point boundary value problems of a coupled system of differential equations.
文摘This paper considers the graph sequence graph （f^n ）={x×f^n ： x ∈X} of the iterates f^n of a continuous map f ： X →X on a compact metric space （X, d）. The main aim is to introduce serval limit sets by using different recurrent time sets, and discuss the relation between dynamics of the underlying compact system （X, f） and the structure of these limit sets, from a topological or fractal viewpoint.
基金supported by the National Natural Science Foundation of China(10472003)Beijing Natural Science(3002002)+1 种基金Beijing Educational Committee Foundations(KM200410005019)Suspensofled by American MSC Company.
文摘ICM （Independent Continuous Mapping） method can solve topological optimization problems with the minimized weight as the objective and subjected to displacement constraints. To get a clearer topological configuration, by introducing the discrete condition of topological variables and integrating with the original objective, an optimal model with multi-objectives is formulated to make the topological variables approach 0 or 1 as near as possible, and the model reduces the effect of deleting rate on the result. The image-filtering method is employed to eliminate the checkerboard patterns and mesh dependence that occurred in the topology optimization of a continuum structure. The computational efficiency is enhanced through selecting quasi-active displacement constraints and a design region. Numerical examples indicate that this algorithm is robust and practicable, though the number of iterations is slightly increased with respect to the original algorithm.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.51675506 and 51275504)the German Research Foundation(DFG)(Grant Nos.#ZA 422/5-1 and#ZA 422/6-1)
文摘A topology optimization method based on the solid isotropic material with penalization interpolation scheme is utilized for designing gradient coils for use in magnetic resonance microscopy.Unlike the popular stream function method,the proposed method has design variables that are the distribution of conductive material.A voltage-driven transverse gradient coil is proposed to be used as micro-scale magnetic resonance imaging(MRI)gradient coils,thus avoiding introducing a coil-winding pattern and simplifying the coil configuration.The proposed method avoids post-processing errors that occur when the continuous current density is approximated by discrete wires in the stream function approach.The feasibility and accuracy of the method are verified through designing the z-gradient and y-gradient coils on a cylindrical surface.Numerical design results show that the proposed method can provide a new coil layout in a compact design space.
基金supported by National Natural Science Foundation of China(11601515)Fundamental Research Funds for the Central Universities(3122017080)+3 种基金the second author acknowledges the support of the Slovenian Research Agency grants P1-0292,J1-8131,N1-0064,N1-0083,N1-0114the third author was supported by National Natural Science Foundation of China(11871199and 12171152)Shandong Provincial Natural Science Foundation,PR China(ZR2020MA006)Cultivation Project of Young and Innovative Talents in Universities of Shandong Province。
文摘This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫∫_(R2N)|x|^(α1p)|y|^(α2p)|u(x) − u(y)|^(p)/|x − y|^(N+ps) dxdy)L_(p)^(s)u = |x| ^(β)f(u) in Ω,u = 0 in R^(N) \ Ω,where L_(p)^(s) is the generalized fractional p-Laplacian operator,N≥1,s∈(0,1),α_(1),α_(2),β∈R,Ω■R^(N) is a bounded domain with Lipschitz boundary,and M:R0^(+)→R0^(+),f:Ω→R are continuous functions.Firstly,we introduce a variational framework for the above problem.Then,the existence of least energy solutions is obtained by using variational methods,provided that the nonlinear term f has(θ_(p-1))-sublinear growth at infinity.Moreover,the existence of infinitely many solutions is obtained by using Krasnoselskii’s genus theory.Finally,we obtain the existence and multiplicity of solutions if f has(θ_(p-1))-superlinear growth at infinity.The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.
基金This project is supported by National Basic Research Program of China(973Program, No.2003CB716207) and National Hi-tech Research and DevelopmentProgram of China(863 Program, No.2003AA001031).
文摘A new hybrid MMA-MGCMMA （HMM） algorithm for solving topology optimization problems is presented. This algorithm combines the method of moving asymptotes （MMA） algorithm and the modified globally convergent version of the method of moving asymptotes （MGCMMA） algorithm in the optimization process. This algorithm preserves the advantages of both MMA and MGCMMA. The optimizer is switched from MMA to MGCMMA automatically, depending on the numerical oscillation value existing in the calculation. This algorithm can improve calculation efficiency and accelerate convergence compared with simplex MMA or MGCMMA algorithms, which is proven with an example.
基金supported by the National Natural Science Foundation of China(Nos.50905085 and 91116020)the National Science Foundation for Post-doctoral Scientists of China(No.2012M511263)
文摘Non-dominated sorting genetic algorithm II（NSGA-II）with multiple constraints handling is employed for multi-objective optimization of the topological structure of telescope skin,in which a bit-matrix is used as the representation of a chromosome,and genetic algorithm（GA）operators are introduced based on the matrix.Objectives including mass,in-plane performance,and out-of-plane load-bearing ability of the individuals are obtained by fnite element analysis（FEA）using ANSYS,and the matrix-based optimization algorithm is realized in MATLAB by handling multiple constraints such as structural connectivity and in-plane strain requirements.Feasible confgurations of the support structure are achieved.The results confrm that the matrix-based NSGA-II with multiple constraints handling provides an effective method for two-dimensional multi-objective topology optimization.
文摘This paper explores the realization of robotic motion planning, especially Findpath problem, which is a basic motion planning problem that arises in the development of robotics. Findpath means: Give the initial and desired final configurations of a robotic arm in 3-dimensionnl space, and give descriptions of the obstacles in the space, determine whether there is a continuous collision-free motion of the robotic arm from one configure- tion to the other and find such a motion if it exists. There are several branches of approach in motion planning area, but in reality the important things are feasibility, efficiency and accuracy of the method. In this paper ac- cording to the concepts of Configuration Space (C-Space) and Rotation Mapping Graph (RMG) discussed in [1], a topological method named Dimension Reduction Method (DRM) for investigating the connectivity of the RMG (or the topologic structure of the RMG )is presented by using topologic technique. Based on this ap- proach the Findpath problem is thus transformed to that of finding a connected way in a finite Characteristic Network (CN). The method has shown great potentiality in practice. Here a simulation system is designed to embody DRM and it is in sight that DRM can he adopted in the first overall planning of real robot sys- tem in the near future.
基金supported by the Xidian University Specially Funded Project for Interdisciplinary Exploration(No:TZJH2024049)the National Natural Science Foundation of China of China(No:62090043,61934006)the National Key Research and Development Program of China(No:2022YFB4401300).
文摘Integrated circuits(ICs)are the foundation of information technology development.The optimal design scheme of an analog IC is determined by iteratively running the simulation software and comparing the performance metrics.However,the simulation software of an analog IC is time-consuming,which leads to the low design efficiency.Due to the nonideal factors in analog ICs,the nonlinear relationship between design parameters and performance metrics cannot be well described by the deduced approximation equations.Inspired by the image and semantic recognition,a universal high-efficiency modeling method for analog ICs based on convolutional neural network(CNN)was proposed in the current work,named as CNN-IC.The sparse topology mapping method was proposed to map the design parameters into a sparse matrix,which includes the spatial and transistor characteristics of analog IC.The CNN model with three convolutional kernels was constructed to extract“transistor-circuit module-integrate circuit”features level by level,which can replace the simulation software to effectively improve the training efficiency and accuracy.Two typical analog ICs were selected to verify the effectiveness of the CNN-IC model.The results show that the accuracy of the CNN-IC model could reach over 99%and that its convergence rate was the fastest compared with the machine learning models in the state of the art.
基金supported by the National Key Research and Development Programme of China(Grant No.2023YFB3406302)the National Natural Science Foundation of China(Grant No.52175120)Key Research,Development Plan of Shaanxi(Grant No.2024GH-ZDXM-29)。
文摘To address the challenges for vibration suppression in the precision sensors of underwater vehicles,phononic crystals have attracted significant attention due to the superior capabilities in elastic wave manipulation and vibration suppression.Unlike conventional damping materials,the phononic crystals can effectively suppress the wave propagation in the specific frequency ranges due to the unique periodic microstructures.
基金supported by National Basic Research Program of China (Grant No. 2013CB834100)National Natural Science Foundation of China (Grant Nos. 11571065,11171132 and 11201173)
文摘This paper concerns the existence of affine-periodic solutions for perturbed affine-periodic systems.This kind of affine-periodic solutions has the form of x(t+T)≡Qx(t) with some nonsingular matrix Q,which may be quasi-periodic when Q is an orthogonal matrix. It can be even unbounded but x(t)/|x(t)| is quasi-periodic,like a helical line. for example x(t)=e^(at)(cos ωt, sin ωt), when Q is not an orthogonal matrix. The averaging method of higher order for finding affine-periodic solutions is given by topological degree.
基金supported in part by the National Natural Science Foundation of China under Grant Nos.61876187 and 61806217.
文摘In this paper,an algorithm for solving the multi-target correlation and co-location problem of aerial-ground heterogeneous system is investigated.Aiming at the multi-target correlation problem,the fusion algorithm of visual axis correlation method and improved topological similarity correlation method are adopted in view of large parallax and inconsistent scale between the aerial and ground perspectives.First,the visual axis was preprocessed by the threshold method,so that the sparse targets were initially associated.Then,the improved topological similarity method was used to further associate dense targets with the relative position characteristics between targets.The shortcoming of dense target similarity with small di®erence was optimized by the improved topological similarity method.For the problem of colocation,combined with the multi-target correlation algorithm in this paper,the triangulation positioning model was used to complete the co-location of multiple targets.In the experimental part,simulation experiments and°ight experiments were designed to verify the e®ectiveness of the algorithm.Experimental results show that the proposed algorithm can e®ectively achieve multi-target correlation positioning,and that the positioning accuracy is obviously better than other positioning methods.
文摘An efficient path planning algorithm based on topologic method is presented in this paper.The colli- sion free path planning for three-joint robotic arm consists of three parts:partition of C-space,construc- tion of CN and search for a path in CN.We mainly solved the problems of partitioning the C-space and judging the connectivity between connected blocks,etc.For the motion planning of a robotic arm with a gripper,we developed the concepts of global planning and local planning,and discussed the basic fac- tors for constructing the planning system.In the paper,some evaluation and analysis of the complexity and reliability of the algorithm are given,together with some ideas to improve the efficiency and increase the reliability.At last,some experimental results are presented to show the efficiency and accuracy of the nigorithm.
文摘In this paper,we study the existence of"weak solution"for a class of p(x)-Kirchhoff type problem involving the p(x)-Laplacian-like operator depending on two real parameters with Neumann boundary condition.Using a topological degree for a class of demicontinuous operator of generalized(S_(+))type and the theory of the variable exponent Sobolev space,we establish the existence of"weak solution"of this problem.
基金The Project Supported by the National Natural Science Foundation of China (10371066).
文摘By using topological method, we study a class of boundary value problem for a system of nonlinear ordinary differential equations. Under suitable conditions, we prove the existence of positive solution of the problem.
基金supported by National Natural Science Foundation of China (Grant No.11721101)the National Key Research and Development Project (Grant No.SQ2020YFA070080)+1 种基金supported by Hubei Provincial Natural Science Foundation of China (Grant No.2021CFB400)National Natural Science Foundation of China (Grant No.11971358)。
文摘Let(Σ,g)be a compact Riemann surface with smooth boundary■Σ,Δ_(g) be the Laplace-Beltrami operator,and h be a positive smooth function.Using a min-max scheme introduced by Djadli and Malchiodi(2008)and Djadli(2008),we prove that ifΣis non-contractible,then for anyρ∈(8kπ,8(k+1)π)with k∈N^(*),the mean field equation{Δgu=ρhe^(u)/∫∑he^(u)dv_(g)in∑,u=0 on■∑has a solution.This generalizes earlier existence results of Ding et al.(Ann Inst H PoincaréAnal Non Linéaire,1999)and Chen and Lin(2003)in the Euclidean domain.Also we consider the corresponding Neumann boundary value problem.If h is a positive smooth function,then for anyρ∈(4kπ,4(k+1)π)with k∈N^(*),the mean field equation{Δgu=ρhe^(u)/∫_(∑)he^(u)dv_(g)-1/|∑|in∑,■u/■v=0 on■∑has a solution,where v denotes the unit normal outward vector on ■Σ.Note that in this case we do not require the surface to be non-contractible.
基金Supported by the National Natural Science Foundation of China (10671167).
文摘Applying topological method,this paper investigates the existence of positive solu-tions to singular m-point boundary value problems of a coupled system of differential equations.
