n-Colour self-inverse compositions

MacMahon’s definition of self-inverse composition is extended ton-colour self-inverse composition. This introduces four new sequences which satisfy the same recurrence relation with different initial conditions like the famous Fibonacci and Lucas sequences. For these new sequences explicit formulas, recurrence relations, generating functions and a summation formula are obtained. Two new binomial identities with combinatorial meaning are also given.     

由几何反演导出的分形几何

本文介绍了由几何反演而导出的自反分形概念.构造了一个对称的,分叉自反分形曲线和一个自反分形尘集.推广了分形密切的概念而且提出了一个新的概念——分形包络.最后给出了“肥皂”和“鸡蛋”的两个自反分形的实例.     

关于自反的n-colour有序分拆的一个关系式

给出了2v＋1的自反n-colour有序分拆数与v的n-colour有序分拆数之间的一个关系式,并由此关系式得到了一个组合恒等式.此外,还给出了正整数2v＋1的自反n-colour有序分拆数与正整数2v的＂奇-偶＂无序分拆数、2v分拆成奇数的有序分拆数等几类分拆数之间的一些分拆恒等式.     

与自反的n-color有序分拆相关的一些恒等式

首先,给出了偶数2v的自反的n-color有序分拆与v+1,v-1的n-color有序分拆之间的一个组合双射,并利用相应的计数公式得到了一个组合恒等式.其次,给出了正整数自反的n-color有序分拆数与Fibonacci数、Lucas数之间的一个关系式,并利用此关系式给出了偶数与奇数的自反的n-color有序分拆之间的一个组合双射.最后,给出了一些涉及正整数v的自反的n-color有序分拆数与其它有约束条件的有序分拆数之间的分拆恒等式.     

Transport in a disordered one-dimensional system: A fractal view

We reexamine the calculation of the transmission coefficient of a random array ofN isotopic defects in an otherwise perfect, harmonic, one-dimensional crystal lattice. The thermal conductivity of this model system has been studied under steady state conditions in which there is a kinetic temperature difference across, and an associated energy flux through, the array of defects. An exact expression for the transmission coefficient is obtained in terms of the magnitude of anNth-order determinant. Rubin reduced the evaluation of the determinant to the evaluation of a sequence ofN–1 nonlinear transformations drawn from a set of transformations parametrized by the nearest-neighbor spacing of the isotopic defects. These transformations are self-inverse and provide an example of what Mandelbrot has termed aself-inverse fractal. The variety of limiting distributions of values obtained under these transformations will be illustrated.     

关于正整数不含分部量2的有序分拆的几个组合双射

利用正整数有序分拆的共轭分拆,分别给出了偶数2k、奇数2k+1和正整数n的不含分部量2的自反的有序分拆数的递推关系式的组合双射证明.此外,还给出了NAGI关于正整数n不含分部量2的有序分拆数的一个恒等式的不同组合双射.     

二项式型自反多项式

In this paper, we define the self-inverse sequences related to sequences of polynomials of binomial type, and give some interesting results of these sequences. Moreover, we study the self-inverse sequences related to the Laguerre polynomials.     

Fractal geometry derived from geometric inversion

In this paper,the conception of fractal geometry derived from geometric inversion isintroduced.A ramified self-inverse fractal with symmetry and a self-inverse fractal dust setare constructed.The authors extend the conception of the fractal osculation and propose anew notion—fractal envelope.Finally,two examples of self-inverse fractal(soap and egg)are given.