本文以一元Birkhoff插值研究结果为基础,对二元Birkhoff插值泛函组的适定性问题进行了研究。通过提出弱Gröbner基的概念及其发现其性质,提出了一种利用两条不同次数代数曲线相交的点,构造出二元Birkhoff插值问题适定泛函组的新方法...本文以一元Birkhoff插值研究结果为基础,对二元Birkhoff插值泛函组的适定性问题进行了研究。通过提出弱Gröbner基的概念及其发现其性质,提出了一种利用两条不同次数代数曲线相交的点,构造出二元Birkhoff插值问题适定泛函组的新方法,从而将该方法所得到的结果推广到一般情形。并得到了构造平面代数曲线二元Birkhoff插值适定泛函组的一般性方法和实用性较强的理论,最后给出了具体实验算例,对所得研究结论给予了验证。This paper takes the research results of univariate Birkhoff interpolation as its foundation to study the stability problem of two-dimensional Birkhoff interpolation generalized function sets. By introducing the concept and properties of weak Gröbner bases, a new method is obtained which utilizes the intersection of any two arbitrary algebraic curves to construct the two-dimensional Birkhoff well-posed interpolation functional systems. This method extends the research direction’s findings to general cases, providing a general approach and practical theory for constructing planar algebraic curve well-posed interpolation functional systems. Finally, specific experimental examples are provided to validate the research conclusions obtained.展开更多
It is known that monotone recurrence relations can induce a class of twist homeomorphisms on the high-dimensional cylinder,which is an extension of the class of monotone twist maps on the annulus or two-dimensional cy...It is known that monotone recurrence relations can induce a class of twist homeomorphisms on the high-dimensional cylinder,which is an extension of the class of monotone twist maps on the annulus or two-dimensional cylinder.By constructing a bounded solution of the monotone recurrence relation,the main conclusion in this paper is acquired:The induced homeomorphism has Birkhoff orbits provided there is a compact forward-invariant set.Therefore,it generalizes Angenent's results in low-dimensional cases.展开更多
The method of integrating factors is used to study the conservation laws of the Herglotz type Birkhoffian systems in this paper.Firstly,the definition of the integrating factors of the Herglotz type Birkhoffian system...The method of integrating factors is used to study the conservation laws of the Herglotz type Birkhoffian systems in this paper.Firstly,the definition of the integrating factors of the Herglotz type Birkhoffian systems is given.Secondly,the relationship between the integrating factors and conservation laws is studied,and the conservation theorems of Herglotz type Birkhoff's equations and their inverse theorems are established.Thirdly,two types of generalized Killing equations for calculating integrating factors are given.Finally,as an example,a linear damped oscillator is taken.This example can be transformed into a Herglotz type Birkhoffian system.The resulting conservation theorems are used to find the conserved quantities for this example.展开更多
以二元Birkhoff插值研究为基础,进一步研究了三维欧氏空间中马鞍面上的Birkhoff插值。首先给出了马鞍面上的多元Birkhoff插值相关定义,对插值条件组的拓扑结构进行了较为深入的研究,然后给出了构造多元函数插值适定泛函组的添加马鞍面法...以二元Birkhoff插值研究为基础,进一步研究了三维欧氏空间中马鞍面上的Birkhoff插值。首先给出了马鞍面上的多元Birkhoff插值相关定义,对插值条件组的拓扑结构进行了较为深入的研究,然后给出了构造多元函数插值适定泛函组的添加马鞍面法,最后给出具体实例进行验证。Based on the research of two-dimensional Birkhoff interpolation, this study further investigates Birkhoff interpolation on a saddle surface in three-dimensional Euclidean space. First, relevant definitions of multivariate Birkhoff interpolation on saddle surfaces are provided. An in-depth study of the topological structure of the interpolation condition set is conducted. Then, the method of adding saddle surface techniques to construct a suitable functional set for multi-variable function interpolation is introduced. Finally, specific examples are provided for verification.展开更多
文章以二元Birkhoff插值研究结果为基础,对三元Birkhoff插值泛函组的适定性问题进行了研究。并提出了空间代数曲线上和代数曲面上的Birkhoff插值适定泛函组的基本概念,研究了空间代数曲线上和代数曲面上的Birkhoff插值适定泛函组的某些...文章以二元Birkhoff插值研究结果为基础,对三元Birkhoff插值泛函组的适定性问题进行了研究。并提出了空间代数曲线上和代数曲面上的Birkhoff插值适定泛函组的基本概念,研究了空间代数曲线上和代数曲面上的Birkhoff插值适定泛函组的某些基本理论和拓扑结构,得到了构造空间代数曲线上Birkhoff插值适定泛函组的添加曲线交点法。方法是以迭加方式完成的,因此便于在计算机上实现其构造过程。最后给出了具体实验算例。Based on the results of the two-dimensional Birkhoff interpolation, the study investigates the well-posedness of the three-dimensional Birkhoff interpolation functional systems. The fundamental concepts of well-posed Birkhoff interpolation functional systems on space algebraic curves and algebraic surfaces are proposed. The research delves into some basic theories and topological structures of well-posed Birkhoff interpolation functional systems on space algebraic curves and surfaces. The study presents the method of constructing well-posed Birkhoff interpolation functional systems on space algebraic curves through the addition of intersection points of curves. This method is performed in an iterative manner, making it feasible to implement the construction process on a computer. Finally, specific experimental examples are provided.展开更多
主要研究了二元Birkhoff插值泛函组适定性问题。给出平面代数曲线上的二元Birkhoff插值适定结点组的定义并证明了相关性质定理,在过去已得到的构造适定二元切触插值泛函组的基础上给出了构造二元Birkhoff插值适定泛函组的一种新的构造...主要研究了二元Birkhoff插值泛函组适定性问题。给出平面代数曲线上的二元Birkhoff插值适定结点组的定义并证明了相关性质定理,在过去已得到的构造适定二元切触插值泛函组的基础上给出了构造二元Birkhoff插值适定泛函组的一种新的构造方法——添加曲线交点法。该方法是通过迭加过程来实现的。因此便于在计算机上实现其构造过程。最后给出了具体实验算例。We mainly studied the problem of fitness of binary Birkhoff interpolation functional groups. We defined the fitness node group of binary Birkhoff interpolation on planar algebraic curves and proved the relevant property theorems. Based on the construction of fitness binary tangent interpolation functional groups that have been obtained in the past, we proposed a new construction method for constructing fitness binary Birkhoff interpolation functional groups—the method of adding curve intersection points. This method is implemented through the superposition process, making it easy to implement its construction process on computers. Finally, specific experimental examples were provided.展开更多
文摘本文以一元Birkhoff插值研究结果为基础,对二元Birkhoff插值泛函组的适定性问题进行了研究。通过提出弱Gröbner基的概念及其发现其性质,提出了一种利用两条不同次数代数曲线相交的点,构造出二元Birkhoff插值问题适定泛函组的新方法,从而将该方法所得到的结果推广到一般情形。并得到了构造平面代数曲线二元Birkhoff插值适定泛函组的一般性方法和实用性较强的理论,最后给出了具体实验算例,对所得研究结论给予了验证。This paper takes the research results of univariate Birkhoff interpolation as its foundation to study the stability problem of two-dimensional Birkhoff interpolation generalized function sets. By introducing the concept and properties of weak Gröbner bases, a new method is obtained which utilizes the intersection of any two arbitrary algebraic curves to construct the two-dimensional Birkhoff well-posed interpolation functional systems. This method extends the research direction’s findings to general cases, providing a general approach and practical theory for constructing planar algebraic curve well-posed interpolation functional systems. Finally, specific experimental examples are provided to validate the research conclusions obtained.
基金Supported by the National Natural Science Foundation of China(12201446)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(22KJB110005)the Shuangchuang Program of Jiangsu Province(JSSCBS20220898)。
文摘It is known that monotone recurrence relations can induce a class of twist homeomorphisms on the high-dimensional cylinder,which is an extension of the class of monotone twist maps on the annulus or two-dimensional cylinder.By constructing a bounded solution of the monotone recurrence relation,the main conclusion in this paper is acquired:The induced homeomorphism has Birkhoff orbits provided there is a compact forward-invariant set.Therefore,it generalizes Angenent's results in low-dimensional cases.
基金Supported by the National Natural Science Foundation of China(12272248)。
文摘The method of integrating factors is used to study the conservation laws of the Herglotz type Birkhoffian systems in this paper.Firstly,the definition of the integrating factors of the Herglotz type Birkhoffian systems is given.Secondly,the relationship between the integrating factors and conservation laws is studied,and the conservation theorems of Herglotz type Birkhoff's equations and their inverse theorems are established.Thirdly,two types of generalized Killing equations for calculating integrating factors are given.Finally,as an example,a linear damped oscillator is taken.This example can be transformed into a Herglotz type Birkhoffian system.The resulting conservation theorems are used to find the conserved quantities for this example.
文摘以二元Birkhoff插值研究为基础,进一步研究了三维欧氏空间中马鞍面上的Birkhoff插值。首先给出了马鞍面上的多元Birkhoff插值相关定义,对插值条件组的拓扑结构进行了较为深入的研究,然后给出了构造多元函数插值适定泛函组的添加马鞍面法,最后给出具体实例进行验证。Based on the research of two-dimensional Birkhoff interpolation, this study further investigates Birkhoff interpolation on a saddle surface in three-dimensional Euclidean space. First, relevant definitions of multivariate Birkhoff interpolation on saddle surfaces are provided. An in-depth study of the topological structure of the interpolation condition set is conducted. Then, the method of adding saddle surface techniques to construct a suitable functional set for multi-variable function interpolation is introduced. Finally, specific examples are provided for verification.
文摘文章以二元Birkhoff插值研究结果为基础,对三元Birkhoff插值泛函组的适定性问题进行了研究。并提出了空间代数曲线上和代数曲面上的Birkhoff插值适定泛函组的基本概念,研究了空间代数曲线上和代数曲面上的Birkhoff插值适定泛函组的某些基本理论和拓扑结构,得到了构造空间代数曲线上Birkhoff插值适定泛函组的添加曲线交点法。方法是以迭加方式完成的,因此便于在计算机上实现其构造过程。最后给出了具体实验算例。Based on the results of the two-dimensional Birkhoff interpolation, the study investigates the well-posedness of the three-dimensional Birkhoff interpolation functional systems. The fundamental concepts of well-posed Birkhoff interpolation functional systems on space algebraic curves and algebraic surfaces are proposed. The research delves into some basic theories and topological structures of well-posed Birkhoff interpolation functional systems on space algebraic curves and surfaces. The study presents the method of constructing well-posed Birkhoff interpolation functional systems on space algebraic curves through the addition of intersection points of curves. This method is performed in an iterative manner, making it feasible to implement the construction process on a computer. Finally, specific experimental examples are provided.
文摘主要研究了二元Birkhoff插值泛函组适定性问题。给出平面代数曲线上的二元Birkhoff插值适定结点组的定义并证明了相关性质定理,在过去已得到的构造适定二元切触插值泛函组的基础上给出了构造二元Birkhoff插值适定泛函组的一种新的构造方法——添加曲线交点法。该方法是通过迭加过程来实现的。因此便于在计算机上实现其构造过程。最后给出了具体实验算例。We mainly studied the problem of fitness of binary Birkhoff interpolation functional groups. We defined the fitness node group of binary Birkhoff interpolation on planar algebraic curves and proved the relevant property theorems. Based on the construction of fitness binary tangent interpolation functional groups that have been obtained in the past, we proposed a new construction method for constructing fitness binary Birkhoff interpolation functional groups—the method of adding curve intersection points. This method is implemented through the superposition process, making it easy to implement its construction process on computers. Finally, specific experimental examples were provided.
