If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary con...If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary constant to this field.Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial.As for the opposite direction,we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field.These automorphisms are determined by linear rational functions,i.e.,Möbius transformations.Intrinsic properties of rational parametrizations,in combination with the particular shape of such automorphisms,lead to a number of necessary conditions on the existence of general solutions in this solution class.Furthermore,the desired linear rational function can be determined by solving a comparatively simple differential system over the ODE’s field of definition.These results hold for arbitrary differential fields of characteristic zero.展开更多
Beal conjecture is a famous world mathematical problem and was proposed by American banker Beal, so to solve it is more difficult than Fermat’s last theorem. This paper uses relationship between the mathematical form...Beal conjecture is a famous world mathematical problem and was proposed by American banker Beal, so to solve it is more difficult than Fermat’s last theorem. This paper uses relationship between the mathematical formula and corresponding graph, and by characteristics of graph, combined with the algebraic transformation and congruence theory of number theory;it is proved that the equation can only be formed under having a common factor and Beal conjecture is correct.展开更多
Indefinite equation is an unsolved problem in number theory. Through explo-ration, the author has been able to use a simple elementary algebraic method to solve the solutions of all three variable indefinite equations...Indefinite equation is an unsolved problem in number theory. Through explo-ration, the author has been able to use a simple elementary algebraic method to solve the solutions of all three variable indefinite equations. In this paper, we will introduce and prove the solutions of Pythagorean equation, Fermat’s the-orem, Bill equation and so on.展开更多
In this paper, we study the intersection multiplicity of algebraic curves at a point both in R^2 and in real projective plane P^2. We introduce the fold point of curves and provide conditions for the relations between...In this paper, we study the intersection multiplicity of algebraic curves at a point both in R^2 and in real projective plane P^2. We introduce the fold point of curves and provide conditions for the relations between the intersection multiplicity of curves at a point and the folds of the point.展开更多
Generalized Partial Computation (GPC) is a program transformation method utilizing partial information about input data, properties of auxiliary functions and the logical structure of a source program. GPC uses both a...Generalized Partial Computation (GPC) is a program transformation method utilizing partial information about input data, properties of auxiliary functions and the logical structure of a source program. GPC uses both an inference engine such as a theorem prover and a classical partial evaluator to optimize programs. Therefore, GPC is more powerful than classical partial evaluators but harder to implement and control. We have implemented an experimental GPC system called WSDFU (Waseda Simplify Distribute Fold Unfold). This paper discusses the power of the program transformation system, its theorem prover and future works.展开更多
The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integ...The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Biicklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new （2 ＋ 1）-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.展开更多
文摘If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary constant to this field.Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial.As for the opposite direction,we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field.These automorphisms are determined by linear rational functions,i.e.,Möbius transformations.Intrinsic properties of rational parametrizations,in combination with the particular shape of such automorphisms,lead to a number of necessary conditions on the existence of general solutions in this solution class.Furthermore,the desired linear rational function can be determined by solving a comparatively simple differential system over the ODE’s field of definition.These results hold for arbitrary differential fields of characteristic zero.
文摘Beal conjecture is a famous world mathematical problem and was proposed by American banker Beal, so to solve it is more difficult than Fermat’s last theorem. This paper uses relationship between the mathematical formula and corresponding graph, and by characteristics of graph, combined with the algebraic transformation and congruence theory of number theory;it is proved that the equation can only be formed under having a common factor and Beal conjecture is correct.
文摘Indefinite equation is an unsolved problem in number theory. Through explo-ration, the author has been able to use a simple elementary algebraic method to solve the solutions of all three variable indefinite equations. In this paper, we will introduce and prove the solutions of Pythagorean equation, Fermat’s the-orem, Bill equation and so on.
基金Teaching reform research project of Shandong vocational education(2017228)
文摘In this paper, we study the intersection multiplicity of algebraic curves at a point both in R^2 and in real projective plane P^2. We introduce the fold point of curves and provide conditions for the relations between the intersection multiplicity of curves at a point and the folds of the point.
文摘Generalized Partial Computation (GPC) is a program transformation method utilizing partial information about input data, properties of auxiliary functions and the logical structure of a source program. GPC uses both an inference engine such as a theorem prover and a classical partial evaluator to optimize programs. Therefore, GPC is more powerful than classical partial evaluators but harder to implement and control. We have implemented an experimental GPC system called WSDFU (Waseda Simplify Distribute Fold Unfold). This paper discusses the power of the program transformation system, its theorem prover and future works.
文摘The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Biicklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new （2 ＋ 1）-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.
