The Legendre-Stirling numbers were discovered by Everitt, Littlejohn and Wellman in 2002 in a study of the spectral theory of powers of the classical second-order Legendre differential operator. In 2008, Andrews and L...The Legendre-Stirling numbers were discovered by Everitt, Littlejohn and Wellman in 2002 in a study of the spectral theory of powers of the classical second-order Legendre differential operator. In 2008, Andrews and Littlejohn gave a combinatorial interpretation of these numbers in terms of set partitions. In 2012, Mongelli noticed that both the Jacobi-Stirling and the Legendre-Stirling numbers are in fact specializations of certain elementary and complete symmetric functions and used this observation to give a combinatorial interpretation for the generalized Legendre-Stirling numbers. In this paper we provide a second combinatorial interpretation for the generalized Legendre-Stirling numbers which more directly generalizes the definition of Andrews and Littlejohn and give a combinatorial bijection between our interpretation and the Mongelli interpretation. We then utilize our interpretation to prove a number of new identities for the generalized Legendre-Stirling numbers.展开更多
We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in...We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.展开更多
Polysurfacic tori or kideas are three-dimensional objects formed by rotating a regular polygon around a central axis. These toric shapes are referred to as “polysurfacic” because their characteristics, such as the n...Polysurfacic tori or kideas are three-dimensional objects formed by rotating a regular polygon around a central axis. These toric shapes are referred to as “polysurfacic” because their characteristics, such as the number of sides or surfaces separated by edges, can vary in a non-trivial manner depending on the degree of twisting during the revolution. We use the term “Kideas” to specifically denote these polysurfacic tori, and we represent the number of sides (referred to as “facets”) of the original polygon followed by a point, while the number of facets from which the torus is twisted during its revolution is indicated. We then explore the use of concave regular polygons to generate Kideas. We finally give acceleration for the algorithm for calculating the set of prime numbers.展开更多
The purpose of this paper is to introduce to you, the Western people, nowadays a “widely unknown” Japanese thermodynamicist by the name of Motoyosi Sugita and his study on the thermodynamics of transient phenomena a...The purpose of this paper is to introduce to you, the Western people, nowadays a “widely unknown” Japanese thermodynamicist by the name of Motoyosi Sugita and his study on the thermodynamics of transient phenomena and his theory of life. This is because although he was one of the top theoretical physicists in Japan before, during and after WWII and after WWII he promoted the establishment of the biophysical society of Japan as one of the founding members, he himself and his studies themselves have seemed to be totally forgotten nowadays in spite that his study was absolutely important for the study of life. Therefore, in this paper I would like to present what kind of person he was and what he studied in physics as a review on the physics work of Motoyosi Sugita for the first time. I will follow his past studies to introduce his ideas in theoretical physics as well as in biophysics as follows: He proposed the bright ideas such as the quasi-static change in the broad sense, the virtual heat, and the field of chemical potential etc. in order to establish his own theory of thermodynamics of transient phenomena, as the generalization of the Onsager-Prigogine’s theory of the irreversible processes. By the concept of the field of chemical potential that acquired the nonlinear transport, he was seemingly successful to exceed and go beyond the scope of Onsager and Prigogine. Once he established his thermodynamics, he explored the existence of the 4th law of thermodynamics for the foundation of theory of life. He applied it to broad categories of transient phenomena including life and life being such as the theory of metabolism. He regarded the 4th law of thermodynamics as the maximum principle in transient phenomena. He tried to prove it all life long. Since I have recently found that his maximum principle can be included in more general maximum principle, which was known as the Pontryagin’s maximum principle in the theory of optimal control, I would like to explain such theories produced by Motoyosi Sugita as detailed as possible. And also I have put short history of Motoyosi Sugita’s personal life in order for you to know him well. I hope that this article helps you to know this wonderful man and understand what he did in the past, which was totally forgotten in the world and even in Japan.展开更多
Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure...Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.展开更多
文摘The Legendre-Stirling numbers were discovered by Everitt, Littlejohn and Wellman in 2002 in a study of the spectral theory of powers of the classical second-order Legendre differential operator. In 2008, Andrews and Littlejohn gave a combinatorial interpretation of these numbers in terms of set partitions. In 2012, Mongelli noticed that both the Jacobi-Stirling and the Legendre-Stirling numbers are in fact specializations of certain elementary and complete symmetric functions and used this observation to give a combinatorial interpretation for the generalized Legendre-Stirling numbers. In this paper we provide a second combinatorial interpretation for the generalized Legendre-Stirling numbers which more directly generalizes the definition of Andrews and Littlejohn and give a combinatorial bijection between our interpretation and the Mongelli interpretation. We then utilize our interpretation to prove a number of new identities for the generalized Legendre-Stirling numbers.
文摘We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.
文摘Polysurfacic tori or kideas are three-dimensional objects formed by rotating a regular polygon around a central axis. These toric shapes are referred to as “polysurfacic” because their characteristics, such as the number of sides or surfaces separated by edges, can vary in a non-trivial manner depending on the degree of twisting during the revolution. We use the term “Kideas” to specifically denote these polysurfacic tori, and we represent the number of sides (referred to as “facets”) of the original polygon followed by a point, while the number of facets from which the torus is twisted during its revolution is indicated. We then explore the use of concave regular polygons to generate Kideas. We finally give acceleration for the algorithm for calculating the set of prime numbers.
文摘The purpose of this paper is to introduce to you, the Western people, nowadays a “widely unknown” Japanese thermodynamicist by the name of Motoyosi Sugita and his study on the thermodynamics of transient phenomena and his theory of life. This is because although he was one of the top theoretical physicists in Japan before, during and after WWII and after WWII he promoted the establishment of the biophysical society of Japan as one of the founding members, he himself and his studies themselves have seemed to be totally forgotten nowadays in spite that his study was absolutely important for the study of life. Therefore, in this paper I would like to present what kind of person he was and what he studied in physics as a review on the physics work of Motoyosi Sugita for the first time. I will follow his past studies to introduce his ideas in theoretical physics as well as in biophysics as follows: He proposed the bright ideas such as the quasi-static change in the broad sense, the virtual heat, and the field of chemical potential etc. in order to establish his own theory of thermodynamics of transient phenomena, as the generalization of the Onsager-Prigogine’s theory of the irreversible processes. By the concept of the field of chemical potential that acquired the nonlinear transport, he was seemingly successful to exceed and go beyond the scope of Onsager and Prigogine. Once he established his thermodynamics, he explored the existence of the 4th law of thermodynamics for the foundation of theory of life. He applied it to broad categories of transient phenomena including life and life being such as the theory of metabolism. He regarded the 4th law of thermodynamics as the maximum principle in transient phenomena. He tried to prove it all life long. Since I have recently found that his maximum principle can be included in more general maximum principle, which was known as the Pontryagin’s maximum principle in the theory of optimal control, I would like to explain such theories produced by Motoyosi Sugita as detailed as possible. And also I have put short history of Motoyosi Sugita’s personal life in order for you to know him well. I hope that this article helps you to know this wonderful man and understand what he did in the past, which was totally forgotten in the world and even in Japan.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10771028 60533060)+1 种基金the programof New Century Excellent Fellowship of NECCfunded by a DoD fund (Grant No.DAAD19-03-1-0375)
文摘Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.
