Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been anal...Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.展开更多
In the present paper, the singular perturbations for the higher-order scalar nonlinear boundary value problem epsilon(2)y(n)=f(t,epsilon y,y',...,y((n-2)),epsilon y((n-1)), t is an element of[0,1] H1(y(0,epsilon),...In the present paper, the singular perturbations for the higher-order scalar nonlinear boundary value problem epsilon(2)y(n)=f(t,epsilon y,y',...,y((n-2)),epsilon y((n-1)), t is an element of[0,1] H1(y(0,epsilon),...,y((n-3))(0,epsilon),epsilon y((n-2))(0,epsilon),epsilon y((n-1))(0,epsilon),epsilon)=0, H2(y(0,epsilon),y((n-1))(0,epsilon),y(1,epsilon)...,y((n-1))(1,epsilon),epsilon=0 are studied, where epsilon > 0 is a small parameter, n greater than or equal to 2. Under some mild assumptions, we prove the existence and local uniqueness of the perturbed solution and give out the uniformly valid asymptotic expansions up to its nth-order derivative function by employing the Banach/Picard fixed-point theorem. Then the existing results are extended and improved.展开更多
In this paper, we study a class singular perturbed elliptic equation boundary value problem with a super surface of turning point in n-dimensional space by using the method of multiple scales and the comparison theore...In this paper, we study a class singular perturbed elliptic equation boundary value problem with a super surface of turning point in n-dimensional space by using the method of multiple scales and the comparison theorem. The uniformly valid asymptotic approxmations of solutions for the boundary value problem is constructed.展开更多
Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and ...Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and Conclusion The criteria for singular points, closed orbits and hyperbolic equilibrium points of a second order autonomous Birkhoff system are given. Moreover the stability of equilibria, stable manifolds and unstable manifolds are obtained.展开更多
In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: -u′′(t) = λf(t, u(t)) for all t ∈ (0, 1) subjecting to u(0) = ...In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: -u′′(t) = λf(t, u(t)) for all t ∈ (0, 1) subjecting to u(0) = 0 and αu(η) = u(1), where η ∈ (0, 1), α ∈ [0, 1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ* ∈ (0, +∞], such that the above problem has at least two positive solutions for any λ ∈ (0, λ*) under certain conditions on the nonlinear term f.展开更多
In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the met...In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations(ODEs) are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.展开更多
By applying fixed point theorem, the existence of positive solution is considered for superlinear semipositone singular m-point boundary value problem -(Lφ)(x)=(p(x)φ′(x))′+q(x)φ(x) and ξi ∈ (0,...By applying fixed point theorem, the existence of positive solution is considered for superlinear semipositone singular m-point boundary value problem -(Lφ)(x)=(p(x)φ′(x))′+q(x)φ(x) and ξi ∈ (0,1)with 0〈ξ1〈ξ2……〈ξm-2〈1,αi ∈ R^+,f ∈C[(0,1)×R^+,R^+],f(x,φ) may be singular at x=0 and x=1,g(x):(0,1)→R is Lebesgue measurable, g may tend to negative infinity and have finitely many singularities.展开更多
Singular point(SP)extraction is a key component in automatic fingerprint identification system(AFIS).A new method was proposed for fingerprint singular points extraction,based on orientation tensor field and Laurent s...Singular point(SP)extraction is a key component in automatic fingerprint identification system(AFIS).A new method was proposed for fingerprint singular points extraction,based on orientation tensor field and Laurent series.First,fingerprint orientation flow field was obtained,using the gradient of fingerprint image.With these gradients,fingerprint orientation tensor field was calculated.Then,candidate SPs were detected by the cross-correlation energy in multi-scale Gaussian space.The energy was calculated between fingerprint orientation tensor field and Laurent polynomial model.As a global descriptor,the Laurent polynomial coefficients were allowed for rotational invariance.Furthermore,a support vector machine(SVM)classifier was trained to remove spurious SPs,using cross-correlation coefficient as a feature vector.Finally,experiments were performed on Singular Point Detection Competition 2010(SPD2010)database.Compared to the winner algorithm of SPD2010 which has best accuracy of 31.90%,the accuracy of proposed algorithm is 45.34%.The results show that the proposed method outperforms the state-of-the-art detection algorithms by large margin,and the detection is invariant to rotational transformations.展开更多
We consider the bifurcation of singular points near a double fold point in Z2 -symmetric nonlinear equations with two parameters,where the linearization has a two dimensional null space spanned by a symmetric null vec...We consider the bifurcation of singular points near a double fold point in Z2 -symmetric nonlinear equations with two parameters,where the linearization has a two dimensional null space spanned by a symmetric null vector and an ami-symmetric null vector. In particular, we show the existence of a turning point path and a pitchfork point path passing ihrough the double fold point and they are the only singular points nearby. Their nondegeneracy is confirmed. A supporting numerical example is also provided. The main tools for our analysis as well as the compulation are some extended systems.展开更多
In this paper, we prove that the chord method and the modified chord method are also convergent to the solution x<sup>?</sup> of F(x)=0 if the dimension of the null space of F’(x<sup>?</sup>...In this paper, we prove that the chord method and the modified chord method are also convergent to the solution x<sup>?</sup> of F(x)=0 if the dimension of the null space of F’(x<sup>?</sup>) is】1.展开更多
Under some certain assumptions, the physical model of the air combustion system was simplified to a laminar flame system. The mathematical model of the laminar flame system, which was built according to thermodynamics...Under some certain assumptions, the physical model of the air combustion system was simplified to a laminar flame system. The mathematical model of the laminar flame system, which was built according to thermodynamics theory and the corresponding conservative laws, was studied. With the aid of qualitative theory and method of ordinary differential equations, the location of singular points on the Rayleigh curves is determined, the qualitative structure and the stability of the singular points of the laminar flame system, which are located in the areas of deflagration and detonation, are given for different parameter values and uses of combustion. The phase portraits of the laminar flame system in the reaction-stagnation enthalpy and combustion velocity-stagnation enthalpy planes are shown in the corresponding figures.展开更多
The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for th...The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.展开更多
In this paper, we study the topological structure of the singular points of the third order phase locked loop equations with the character of detected phase being g(?) =(1+k)sin?/1+kcos?.
Given an irreducible plane algebraic curve of degree d 〉 3, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular p...Given an irreducible plane algebraic curve of degree d 〉 3, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular points are ordinary. The numerical procedures rely on computing numerical solutions of polynomial systems by homotopy continuation method and a reliable method that calculates multiple roots of the univariate polynomials accurately using standard machine precision. It is completely different from the traditional symbolic computation and provides singular points and their related properties of some plane algebraic curves that the symbolic software Maple cannot work out. Without using multiprecision arithmetic, extensive numerical experiments show that our numerical procedures are accurate, efficient and robust, even if the coefficients of plane algebraic curves are inexact.展开更多
We consider in this paper the boundary value problems of nonlinear systems the form εY″=F(t,Y,Y′,ε), -1<t<1, Y(-1,ε)=A(ε), Y(1,ε)=B(ε). Supoosing some or all of the components of F , that is, ...We consider in this paper the boundary value problems of nonlinear systems the form εY″=F(t,Y,Y′,ε), -1<t<1, Y(-1,ε)=A(ε), Y(1,ε)=B(ε). Supoosing some or all of the components of F , that is, f i satisfy 2 f y′ 2 i t =0 =0, we say that F possesses a generalized turning point at t =0. Our goal is to give sufficient conditions for the existence of solution of the problems and to study the asymptotic behavior of the solution when F possesses a generalized turning point at t =0. We mainly discuss regular singular crossings.展开更多
In thispaper,theexistence oftravelling frontsolution fora classofcom petition-diffu- sion system w ith high-order singular point w it = diw ixx - w αii fi(w ),x ∈R,t> 0,i= 1,2 (Ⅰ) is studied,w here di,αi>...In thispaper,theexistence oftravelling frontsolution fora classofcom petition-diffu- sion system w ith high-order singular point w it = diw ixx - w αii fi(w ),x ∈R,t> 0,i= 1,2 (Ⅰ) is studied,w here di,αi> 0 (i= 1,2) and w = (w 1(x,t),w 2(x,t)).Under the certain assum ptions on f,itis show ed thatifαi< 1 for som e i,then (Ⅰ) has no travelling frontsolution,ifαi≥1 for i= 1,2,then there isa c0,c:c0≥c> 0,w herecis called the m inim alwavespeed of(Ⅰ),such thatifc≥c0 orc= c,then (Ⅰ) has a travelling frontsolution,ifc< c,then (Ⅰ) hasno travel- ling frontsolution by using the shooting m ethod in com bination w ith a com pactness argum ent.展开更多
In this work, we investigate one class of Volterra type integral equation, in model case, when kernels have first order fixed singularity and logarithmic singularity. In detail study the case, when n = 3. In depend of...In this work, we investigate one class of Volterra type integral equation, in model case, when kernels have first order fixed singularity and logarithmic singularity. In detail study the case, when n = 3. In depend of the signs parameters solution to this integral equation can contain three arbitrary constants, two arbitrary constants, one constant and may have unique solution. In the case when general solution of integral equation contains arbitrary constants, we stand and investigate different boundary value problems, when conditions are given in singular point. Besides for considered integral equation, the solution found cane represented in generalized power series. Some results obtained in the general model case.展开更多
The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with...The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.展开更多
In this paper, the singularity and its effect on learning dynamics in the complex-valued neural network are elucidated. It has learned that the linear combination structure in the updating rule of the complex-valued n...In this paper, the singularity and its effect on learning dynamics in the complex-valued neural network are elucidated. It has learned that the linear combination structure in the updating rule of the complex-valued neural network increases the speed of moving away from the singular points, and the complex-valued neural network cannot be easily influenced by the singular points, whereas the learning of the usual real-valued neural network can be attracted in the neighborhood of singular points, which causes a standstill in learning. Simulation results on the learning dynamics of the three-layered real-valued and complex-valued neural networks in the neighborhood of singularities support the analytical results.展开更多
文摘Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.
文摘In the present paper, the singular perturbations for the higher-order scalar nonlinear boundary value problem epsilon(2)y(n)=f(t,epsilon y,y',...,y((n-2)),epsilon y((n-1)), t is an element of[0,1] H1(y(0,epsilon),...,y((n-3))(0,epsilon),epsilon y((n-2))(0,epsilon),epsilon y((n-1))(0,epsilon),epsilon)=0, H2(y(0,epsilon),y((n-1))(0,epsilon),y(1,epsilon)...,y((n-1))(1,epsilon),epsilon=0 are studied, where epsilon > 0 is a small parameter, n greater than or equal to 2. Under some mild assumptions, we prove the existence and local uniqueness of the perturbed solution and give out the uniformly valid asymptotic expansions up to its nth-order derivative function by employing the Banach/Picard fixed-point theorem. Then the existing results are extended and improved.
文摘In this paper, we study a class singular perturbed elliptic equation boundary value problem with a super surface of turning point in n-dimensional space by using the method of multiple scales and the comparison theorem. The uniformly valid asymptotic approxmations of solutions for the boundary value problem is constructed.
文摘Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and Conclusion The criteria for singular points, closed orbits and hyperbolic equilibrium points of a second order autonomous Birkhoff system are given. Moreover the stability of equilibria, stable manifolds and unstable manifolds are obtained.
基金supported by the National Natural Science Foundation of China (11071149, 10771128)the NSF of Shanxi Province (2006011002, 2010011001-1)
文摘In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: -u′′(t) = λf(t, u(t)) for all t ∈ (0, 1) subjecting to u(0) = 0 and αu(η) = u(1), where η ∈ (0, 1), α ∈ [0, 1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ* ∈ (0, +∞], such that the above problem has at least two positive solutions for any λ ∈ (0, λ*) under certain conditions on the nonlinear term f.
基金supported by the National Natural Science Foundation of China (11132004 and 51078145)the Natural Science Foundation of Guangdong Province (9251064101000016)
文摘In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations(ODEs) are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.
基金Foundation item: Supported by the National Natural Science Foundation of China(10671167) Supported by the Research Foundation of Liaocheng University(31805)
文摘By applying fixed point theorem, the existence of positive solution is considered for superlinear semipositone singular m-point boundary value problem -(Lφ)(x)=(p(x)φ′(x))′+q(x)φ(x) and ξi ∈ (0,1)with 0〈ξ1〈ξ2……〈ξm-2〈1,αi ∈ R^+,f ∈C[(0,1)×R^+,R^+],f(x,φ) may be singular at x=0 and x=1,g(x):(0,1)→R is Lebesgue measurable, g may tend to negative infinity and have finitely many singularities.
基金Project(11JJ3080)supported by Natural Science Foundation of Hunan Province,ChinaProject(11CY012)supported by Cultivation in Hunan Colleges and Universities,ChinaProject(ET51007)supported by Youth Talent in Hunan University,China
文摘Singular point(SP)extraction is a key component in automatic fingerprint identification system(AFIS).A new method was proposed for fingerprint singular points extraction,based on orientation tensor field and Laurent series.First,fingerprint orientation flow field was obtained,using the gradient of fingerprint image.With these gradients,fingerprint orientation tensor field was calculated.Then,candidate SPs were detected by the cross-correlation energy in multi-scale Gaussian space.The energy was calculated between fingerprint orientation tensor field and Laurent polynomial model.As a global descriptor,the Laurent polynomial coefficients were allowed for rotational invariance.Furthermore,a support vector machine(SVM)classifier was trained to remove spurious SPs,using cross-correlation coefficient as a feature vector.Finally,experiments were performed on Singular Point Detection Competition 2010(SPD2010)database.Compared to the winner algorithm of SPD2010 which has best accuracy of 31.90%,the accuracy of proposed algorithm is 45.34%.The results show that the proposed method outperforms the state-of-the-art detection algorithms by large margin,and the detection is invariant to rotational transformations.
文摘We consider the bifurcation of singular points near a double fold point in Z2 -symmetric nonlinear equations with two parameters,where the linearization has a two dimensional null space spanned by a symmetric null vector and an ami-symmetric null vector. In particular, we show the existence of a turning point path and a pitchfork point path passing ihrough the double fold point and they are the only singular points nearby. Their nondegeneracy is confirmed. A supporting numerical example is also provided. The main tools for our analysis as well as the compulation are some extended systems.
文摘In this paper, we prove that the chord method and the modified chord method are also convergent to the solution x<sup>?</sup> of F(x)=0 if the dimension of the null space of F’(x<sup>?</sup>) is】1.
基金theNaturalScienceFoundationofBeijingMunicipalGovernment (No .1 0 42 0 0 7)andtheScientificResearchFoundationfortheReturnedOverseasChineseScholars,StateEducationMinistry (No .Lxkyjj2 0 0 41 6)
文摘Under some certain assumptions, the physical model of the air combustion system was simplified to a laminar flame system. The mathematical model of the laminar flame system, which was built according to thermodynamics theory and the corresponding conservative laws, was studied. With the aid of qualitative theory and method of ordinary differential equations, the location of singular points on the Rayleigh curves is determined, the qualitative structure and the stability of the singular points of the laminar flame system, which are located in the areas of deflagration and detonation, are given for different parameter values and uses of combustion. The phase portraits of the laminar flame system in the reaction-stagnation enthalpy and combustion velocity-stagnation enthalpy planes are shown in the corresponding figures.
基金Supported by the Key Program of National Natural Science Foundation of China (60634020), Doctoral Program Foundation of Ministry of Education of China (20050533028, 20070533132), Natural Science Foundation of Hunan Province (06J35145), and Program for New Century Excellent Talents in University (NCET-07-0867)
文摘The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.
文摘In this paper, we study the topological structure of the singular points of the third order phase locked loop equations with the character of detected phase being g(?) =(1+k)sin?/1+kcos?.
基金The NSF (61033012,10801023,11171052,10771028) of China
文摘Given an irreducible plane algebraic curve of degree d 〉 3, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular points are ordinary. The numerical procedures rely on computing numerical solutions of polynomial systems by homotopy continuation method and a reliable method that calculates multiple roots of the univariate polynomials accurately using standard machine precision. It is completely different from the traditional symbolic computation and provides singular points and their related properties of some plane algebraic curves that the symbolic software Maple cannot work out. Without using multiprecision arithmetic, extensive numerical experiments show that our numerical procedures are accurate, efficient and robust, even if the coefficients of plane algebraic curves are inexact.
文摘We consider in this paper the boundary value problems of nonlinear systems the form εY″=F(t,Y,Y′,ε), -1<t<1, Y(-1,ε)=A(ε), Y(1,ε)=B(ε). Supoosing some or all of the components of F , that is, f i satisfy 2 f y′ 2 i t =0 =0, we say that F possesses a generalized turning point at t =0. Our goal is to give sufficient conditions for the existence of solution of the problems and to study the asymptotic behavior of the solution when F possesses a generalized turning point at t =0. We mainly discuss regular singular crossings.
文摘In thispaper,theexistence oftravelling frontsolution fora classofcom petition-diffu- sion system w ith high-order singular point w it = diw ixx - w αii fi(w ),x ∈R,t> 0,i= 1,2 (Ⅰ) is studied,w here di,αi> 0 (i= 1,2) and w = (w 1(x,t),w 2(x,t)).Under the certain assum ptions on f,itis show ed thatifαi< 1 for som e i,then (Ⅰ) has no travelling frontsolution,ifαi≥1 for i= 1,2,then there isa c0,c:c0≥c> 0,w herecis called the m inim alwavespeed of(Ⅰ),such thatifc≥c0 orc= c,then (Ⅰ) has a travelling frontsolution,ifc< c,then (Ⅰ) hasno travel- ling frontsolution by using the shooting m ethod in com bination w ith a com pactness argum ent.
文摘In this work, we investigate one class of Volterra type integral equation, in model case, when kernels have first order fixed singularity and logarithmic singularity. In detail study the case, when n = 3. In depend of the signs parameters solution to this integral equation can contain three arbitrary constants, two arbitrary constants, one constant and may have unique solution. In the case when general solution of integral equation contains arbitrary constants, we stand and investigate different boundary value problems, when conditions are given in singular point. Besides for considered integral equation, the solution found cane represented in generalized power series. Some results obtained in the general model case.
基金Project supported by the National Natural Science Foundation of China(No.42207182)the Research Grants Council of the Hong Kong Special Administrative Region Government of China(Nos.HKU 17207518 and R5037-18)。
文摘The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.
文摘In this paper, the singularity and its effect on learning dynamics in the complex-valued neural network are elucidated. It has learned that the linear combination structure in the updating rule of the complex-valued neural network increases the speed of moving away from the singular points, and the complex-valued neural network cannot be easily influenced by the singular points, whereas the learning of the usual real-valued neural network can be attracted in the neighborhood of singular points, which causes a standstill in learning. Simulation results on the learning dynamics of the three-layered real-valued and complex-valued neural networks in the neighborhood of singularities support the analytical results.