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
For an in-depth study on the integration problem of the constrained mechanical systems the method of integration for the Birkhoffian system with constraints is discussed and the method of variation of parameters for s...For an in-depth study on the integration problem of the constrained mechanical systems the method of integration for the Birkhoffian system with constraints is discussed and the method of variation of parameters for solving the dynamical equations of the constrained Birkhoffian system is provided.First the differential equations of motion for the constrained Birkhoffian system as well as for the corresponding free Birkhoffian system are established.Secondly a system of auxiliary equations is constructed and the general solution of the equations is found.Finally by varying the parameters and utilizing the properties of the generalized canonical transformation of the Birkhoffian system the solution of the problem can be obtained.The proposed method reveals the inherent relationship between the solution of a free Birkhoffian system and that of a constrained Birkhoffian system. The research results are of universal significance which can be further used in a variety of constrained mechanical systems such as non-conservative systems and nonholonomic systems etc.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.
基金The National Natural Science Foundation of China(No.10972151,11272227)
文摘For an in-depth study on the integration problem of the constrained mechanical systems the method of integration for the Birkhoffian system with constraints is discussed and the method of variation of parameters for solving the dynamical equations of the constrained Birkhoffian system is provided.First the differential equations of motion for the constrained Birkhoffian system as well as for the corresponding free Birkhoffian system are established.Secondly a system of auxiliary equations is constructed and the general solution of the equations is found.Finally by varying the parameters and utilizing the properties of the generalized canonical transformation of the Birkhoffian system the solution of the problem can be obtained.The proposed method reveals the inherent relationship between the solution of a free Birkhoffian system and that of a constrained Birkhoffian system. The research results are of universal significance which can be further used in a variety of constrained mechanical systems such as non-conservative systems and nonholonomic systems etc.
