AIM:To build a functional generalized estimating equation(GEE)model to detect glaucomatous visual field progression and compare the performance of the proposed method with that of commonly employed algorithms.METHODS:...AIM:To build a functional generalized estimating equation(GEE)model to detect glaucomatous visual field progression and compare the performance of the proposed method with that of commonly employed algorithms.METHODS:Totally 716 eyes of 716 patients with primary open angle glaucoma(POAG)with at least 5 reliable 24-2 test results and 2y of follow-up were selected.The functional GEE model was used to detect perimetric progression in the training dataset(501 eyes).In the testing dataset(215 eyes),progression was evaluated the functional GEE model,mean deviation(MD)and visual field index(VFI)rates of change,Advanced Glaucoma Intervention Study(AGIS)and Collaborative Initial Glaucoma Treatment Study(CIGTS)scores,and pointwise linear regression(PLR).RESULTS:The proposed method showed the highest proportion of eyes detected as progression(54.4%),followed by the VFI rate(34.4%),PLR(23.3%),and MD rate(21.4%).The CIGTS and AGIS scores had a lower proportion of eyes detected as progression(7.9%and 5.1%,respectively).The time to detection of progression was significantly shorter for the proposed method than that of other algorithms(adjusted P≤0.019).The VFI rate displayed moderate pairwise agreement with the proposed method(k=0.47).CONCLUSION:The functional GEE model shows the highest proportion of eyes detected as perimetric progression and the shortest time to detect perimetric progression in patients with POAG.展开更多
We introduce a new method to study the asymptotic behavior of solutions on the basis of the continuation theory for k-set contractions.We apply this technique to show the existence of nontrivial decaying solutions to ...We introduce a new method to study the asymptotic behavior of solutions on the basis of the continuation theory for k-set contractions.We apply this technique to show the existence of nontrivial decaying solutions to the sup-linear generalized Emden-Fowler equation and the existence of asymptotically linear solutions to the sub-linear one.展开更多
In this paper,we investigate the generalized quasilinear Schrödinger equation:-div(g2(u)▽u)+g(u)g'(u)|▽u|2+u=P(εx)|u|αp-2u,x∈R^(N),where N>3,g:R→R+is a C1 even function,g(0)=1,g'(s)≥0 for all s...In this paper,we investigate the generalized quasilinear Schrödinger equation:-div(g2(u)▽u)+g(u)g'(u)|▽u|2+u=P(εx)|u|αp-2u,x∈R^(N),where N>3,g:R→R+is a C1 even function,g(0)=1,g'(s)≥0 for all s≥0,g(s)=β|s|α-1+O(|s|γ-1)as s→∞for some constantsα∈[1,2],β>0,γ<αand(α-1)g(s)≥g'(s)s for all s≥0,ε>0 is a positive parameter,and p∈(2,2^(*)).We will study the impact of the nonlinearity’s coefficient P(x)on the quantity of positive solutions.展开更多
A hyperelliptic curve digital signature algorithm （HECDSA） can be viewed as the hyperelliptic curve analogue of the standard digital signature algorithm （DSA）. This article discusses divisor evaluations, the basic...A hyperelliptic curve digital signature algorithm （HECDSA） can be viewed as the hyperelliptic curve analogue of the standard digital signature algorithm （DSA）. This article discusses divisor evaluations, the basic HECDSA, variants, two HECDSA equations and a 4-tuple HECDSA scheme, and puts forward a generalized equation for HECDSA. From this generalized equation, seven general HECDSA types are derived based on the efficiency requirements. Meanwhile, the securities of these general HECDSA types are analyzed in detail.展开更多
Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:...Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:U→Y is a nonsmooth function and F:X■Y is a set-valued mapping with closed graph.We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method.Specifically,under the point-based approximation of f on U and metrically regular property of f+F,we present quadratic rate of convergence of this method.Furthermore,superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f+F is metrically regular.An example of nonsmooth functions that have p-point-based approximation is given.Moreover,a numerical experiment is given which illustrates the theoretical result.展开更多
Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain...Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain some results.展开更多
In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzent...In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.展开更多
Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer alg...Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraicsystem, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, wecan not only successfully recover the previously known travelling wave solutions found by existing various tanh methodsand other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shapedsolitons, bell-shaped solitons, singular solitons, and periodic solutions.展开更多
In this article, generalized Birkhoff equations are put forward by adding supplementary terms to the Birkhoff equations. A conformal invariance of the Birkhoff equations can be used to study the generalized Birkhoff E...In this article, generalized Birkhoff equations are put forward by adding supplementary terms to the Birkhoff equations. A conformal invariance of the Birkhoff equations can be used to study the generalized Birkhoff Equations, and two examples are presented to illustrate the application of the results.展开更多
Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new...Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new explicit solutions of the generalized Zakharov-Kuznetsov equation.展开更多
By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic functio...By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.展开更多
Using a complete discrimination system for polynomials, new exact traveling wave solutions for generalized Ginzburg-Landau equation are obtained. The method has general meaning for many similar problems.
The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamil- ton principle for nonholonomic systems on time scales is established, and the corresponding generalized ...The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamil- ton principle for nonholonomic systems on time scales is established, and the corresponding generalized Chaplygin equa- tions are deduced. The reduced Chaplygin equations are also presented. Two special cases of the generalized Chaplygin equations on time scales, where the time scales are equal to the set of real numbers and the integer set, are discussed. Finally, several examples are given to illustrate the application of the results.展开更多
This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete mu...This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.展开更多
In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the...In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.展开更多
This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a n...This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.展开更多
It is obtained the existence of the weak solution for a degenerate generalized Burgers equation under the restriction u0 ∈ L∞. The main method is to add viscosity perturbation and obtain some estimates in L1 norm. M...It is obtained the existence of the weak solution for a degenerate generalized Burgers equation under the restriction u0 ∈ L∞. The main method is to add viscosity perturbation and obtain some estimates in L1 norm. Meanwhile it is obtained the solution is exponential decay when the initial data has compact support.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
In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under d...In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.展开更多
基金Supported by the Korea Health Technology R&D Project through the Korea Health Industry Development Institute(KHIDI),funded by the Ministry of Health&Welfare,Republic of Korea(No.HR20C0026)the National Research Foundation of Korea(NRF)(No.RS-2023-00247504)the Patient-Centered Clinical Research Coordinating Center,funded by the Ministry of Health&Welfare,Republic of Korea(No.HC19C0276).
文摘AIM:To build a functional generalized estimating equation(GEE)model to detect glaucomatous visual field progression and compare the performance of the proposed method with that of commonly employed algorithms.METHODS:Totally 716 eyes of 716 patients with primary open angle glaucoma(POAG)with at least 5 reliable 24-2 test results and 2y of follow-up were selected.The functional GEE model was used to detect perimetric progression in the training dataset(501 eyes).In the testing dataset(215 eyes),progression was evaluated the functional GEE model,mean deviation(MD)and visual field index(VFI)rates of change,Advanced Glaucoma Intervention Study(AGIS)and Collaborative Initial Glaucoma Treatment Study(CIGTS)scores,and pointwise linear regression(PLR).RESULTS:The proposed method showed the highest proportion of eyes detected as progression(54.4%),followed by the VFI rate(34.4%),PLR(23.3%),and MD rate(21.4%).The CIGTS and AGIS scores had a lower proportion of eyes detected as progression(7.9%and 5.1%,respectively).The time to detection of progression was significantly shorter for the proposed method than that of other algorithms(adjusted P≤0.019).The VFI rate displayed moderate pairwise agreement with the proposed method(k=0.47).CONCLUSION:The functional GEE model shows the highest proportion of eyes detected as perimetric progression and the shortest time to detect perimetric progression in patients with POAG.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12001397,12171039)the Science&Technology Development Fund of Tianjin Education Commission for Higher Education(Grant No.2022KJ204).
文摘We introduce a new method to study the asymptotic behavior of solutions on the basis of the continuation theory for k-set contractions.We apply this technique to show the existence of nontrivial decaying solutions to the sup-linear generalized Emden-Fowler equation and the existence of asymptotically linear solutions to the sub-linear one.
基金supported by the NSFC(12161007)the Guangxi Natural Science(2023GXNSFAA026190)+1 种基金supported by the National Natural Science Foundation of China(12301145,12261107)the Yunnan Fundamental Research Projects(202301AU070144,202401AU070123)。
文摘In this paper,we investigate the generalized quasilinear Schrödinger equation:-div(g2(u)▽u)+g(u)g'(u)|▽u|2+u=P(εx)|u|αp-2u,x∈R^(N),where N>3,g:R→R+is a C1 even function,g(0)=1,g'(s)≥0 for all s≥0,g(s)=β|s|α-1+O(|s|γ-1)as s→∞for some constantsα∈[1,2],β>0,γ<αand(α-1)g(s)≥g'(s)s for all s≥0,ε>0 is a positive parameter,and p∈(2,2^(*)).We will study the impact of the nonlinearity’s coefficient P(x)on the quantity of positive solutions.
基金supported by the National Natural Science Foundation of China (60763009)the Science and Technology Key Project of the Ministry of Education of China (207089)Zhejiang Natural Science Foundation of Outstanding Youth Team Project (R1090138)
文摘A hyperelliptic curve digital signature algorithm （HECDSA） can be viewed as the hyperelliptic curve analogue of the standard digital signature algorithm （DSA）. This article discusses divisor evaluations, the basic HECDSA, variants, two HECDSA equations and a 4-tuple HECDSA scheme, and puts forward a generalized equation for HECDSA. From this generalized equation, seven general HECDSA types are derived based on the efficiency requirements. Meanwhile, the securities of these general HECDSA types are analyzed in detail.
基金supported by CAS-President International Fellowship Initiative (PIFI), Chinese Academy of Sciences, Beijing, Chinasupported by National Natural Science Foundation of China (Grants Nos. 11688101 and 11331012)
文摘Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:U→Y is a nonsmooth function and F:X■Y is a set-valued mapping with closed graph.We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method.Specifically,under the point-based approximation of f on U and metrically regular property of f+F,we present quadratic rate of convergence of this method.Furthermore,superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f+F is metrically regular.An example of nonsmooth functions that have p-point-based approximation is given.Moreover,a numerical experiment is given which illustrates the theoretical result.
文摘Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized complex algebraic differential equations and obtain some results.
基金Project supported by the National Natural Science Foundation of China （Grant No 10461006）, the High Education Science Research Program （Grant No NJ02035） of Inner Mongolia Autonomous Region, Natural Science Foundation of Inner Mongolia Autonomous Region （Grant No 2004080201103） and the Youth Research Program of Inner Mongolia Normal University （Grant No QN005023）.
文摘In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.
基金The project supported by National Natural Science Foundation of China under Grant No.10072013the National Key Basic Research Development Program under Grant No.G1998030600
文摘Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraicsystem, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, wecan not only successfully recover the previously known travelling wave solutions found by existing various tanh methodsand other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shapedsolitons, bell-shaped solitons, singular solitons, and periodic solutions.
基金the National Natural Science Foundation of China(10572021 and 10772025)the Doctoral Program Foundation of Institution of Higher Education,China(20040007022)
文摘In this article, generalized Birkhoff equations are put forward by adding supplementary terms to the Birkhoff equations. A conformal invariance of the Birkhoff equations can be used to study the generalized Birkhoff Equations, and two examples are presented to illustrate the application of the results.
基金The project supported by Natural Science Foundation of Shandong Province of China under Grant 2004 zx 16The authors would like to thank professor Bai Cheng-Lin and the referees for their valuable advices.
文摘Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new explicit solutions of the generalized Zakharov-Kuznetsov equation.
基金Project supported by the State Key Program for Basic Research of China (Grant No 2004CB418304)the National Natural Science Foundation of China (Grant No 40405010)
文摘By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.
文摘Using a complete discrimination system for polynomials, new exact traveling wave solutions for generalized Ginzburg-Landau equation are obtained. The method has general meaning for many similar problems.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11572212 and 11272227)the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(Grant No.KYLX16-0414)
文摘The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamil- ton principle for nonholonomic systems on time scales is established, and the corresponding generalized Chaplygin equa- tions are deduced. The reduced Chaplygin equations are also presented. Two special cases of the generalized Chaplygin equations on time scales, where the time scales are equal to the set of real numbers and the integer set, are discussed. Finally, several examples are given to illustrate the application of the results.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10572119, 10772147 and 10632030)the Doctoral Program Foundation of Education Ministry of China (Grant No 20070699028)+1 种基金the National Natural Science Foundation of Shaanxi Province of China (Grant No 2006A07)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment
文摘This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.
基金Supported by the National Natural Science Foundation of China under Grant No. 10974160the Scientific Research Foundation of the Education Department of Sichuan Province of China under Grant No. 10ZA004
文摘In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.
基金supported by the National Natural Science Foundation of China (Grant Nos 10572021 and 10772025)
文摘This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.
文摘It is obtained the existence of the weak solution for a degenerate generalized Burgers equation under the restriction u0 ∈ L∞. The main method is to add viscosity perturbation and obtain some estimates in L1 norm. Meanwhile it is obtained the solution is exponential decay when the initial data has compact support.
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 the NNSF of China(60464001) Guangxi Science Foundation(0575092).
文摘In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.
