探索以(Fn-k,Fn,Fn)为边长的Fibonacci三角形

Fibonacci三角形是边长为Fibonacci数、面积为整数的三角形.存在以(F<,n-k>,F<,n>.F<,n>)为边长的Fibonacci三角形的情形可以被划分为三类(k时,不存在边长为(F<,n-k>,F<,n>.F<,n>)的Fibonacci三角形.     

The Lucas p-triangle

In this note, we study the Lucas p-numbers and introduce the Lucas p-triangle, which generalize the Lucas triangle is defined by Feinberg. We derive an expansion for the Lucas p-numbers by using some properties of our triangle.     

The Lucas p-matrix

In this note, we study the Fibonacci and Lucas p-numbers. We introduce the Lucas p-matrix and companion matrices for the sums of the Fibonacci and Lucas p-numbers to derive some interesting identities of the Fibonacci and Lucas p-numbers.     

The lotto numbers L(n,3,p,2)

An (n,k,p,t)‐lotto design is an n‐set N and a set of k‐subsets of N (called blocks) such that for each p‐subset P of N, there is a block for which . The lotto number L(n,k,p,t) is the smallest number of blocks in an (n,k,p,t)‐lotto design. The numbers C(n,k,t) = L(n,k,t,t) are called covering numbers. It is easy to show that, for n ≥ k(p ? 1), For k = 3, we prove that equality holds if one of the following holds:

• (i) n is large, in particular
• (ii)
• (iii) 2 ≤ p ≤ 6.

© 2006 Wiley Periodicals, Inc. J Combin Designs 14: 333–350, 2006     

低汉明重量序列的分布

数n的汉明重量是指n的二进制字符串表达式中数字1的个数,用Ham(n)来表示.低汉明重量序列在密码系统和编码理论中有非常广泛的应用.本文建立了低汉明重量数的序列表达式,并且利用指数和的上界以及Erd?s-Turán不等式证明低汉明重量序列的均匀分布性质,从而确保密码算法的随机性和运算效率.     

涉及广义Fibonacci-Lucas数的幂的一些级数的近似值

计算出涉及广义Fibonacci-Lucas数的幂的一些级数的近似值。     

Continued fractions,the group GL(2, ℤ), and Pisot numbers

The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.     

关于一些级数取整值问题的注记

利用Pell方程的理论,讨论了与广义Lucas数有关的一些级数的取整值问题.在一定条件下,解决了一些级数的取整值问题.     

Yet another way of calculating moments of the Kesten's distribution and its consequences for Catalan numbers and Catalan triangles

We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers.     

ζ(k)的部分和五阶和式的计算

u1,u2…是独立、同分布于(0,1)区间上均匀分布的随机变量.本文证明了1-u1u2…uk的n-1阶矩(n≥1)是以调和数的部分和ζn(r)=∑j=1n 1/jr,r≥1为变元的指数型完全Bell多项式,因此Riemann-Zeta函数ζ(k),k≥2能够被展开成第一类无符号Stirling数s(n,k)的级数,从而计算出与ζn(r)有关的全部6个五阶和式.它们都是ζ(5)与ζ(2)ζ(3)的有理组合.     

线性递归序列的矩阵表示

先给出k阶线性递归序列的矩阵表示,然后用矩阵方法得到2阶线性递归序列的通项及一些恒等式.     

The geometry of SO(n)\SO0(n, 1)

Consider the Lie group SO₀(n, 1) with the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left and right. This paper studies the geometry of the quotient space of the homogeneous submersion SO₀(n, 1) → SO(n)\SO₀(n, 1). It is a cohomogeneity one manifold, which can be expressed as a warped product. Its group of isometries, geodesics, and sectional curvatures are calculated.     

Mean Ramsey–Turán numbers

A ρ‐mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most ρ. For a graph H and for ρ ≥ 1, the mean Ramsey–Turán number RT(n, H,ρ ? mean) is the maximum number of edges a ρ‐mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. It is conjectured that where is the maximum number of edges a k edge‐colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. We prove the conjecture holds for . We also prove that . This result is tight for graphs H whose clique number equals their chromatic number. In particular, we get that if H is a 3‐chromatic graph having a triangle then . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 126–134, 2006     

Geometric Characterizations for Subgroups of PU（1, n; C）

In this paper, we discuss non-elementary subgroups and discontinuous subgroups of PU(1,n; C), and give their geometric characterizations.     

PU(1 ,n ;C)元素的一些性质(英文)

得到了PU( 1 ,n ;C)元素的一个类型判别 ,并讨论了斜驶元素和抛物元素的某些性质 .     

The (p,q) property in families of d-intervals and d-trees

Given integers $p\ge q>1$, a family of sets satisfies the $(p,q)$ property if among any $p$ members of it some $q$ intersect. We prove that for any fixed integer constants $p\ge q>1$, a family of $d$-intervals satisfying the $(p,q)$ property can be pierced by $O\left(d^{\frac{q}{q-1}}\right)$ points, with constants depending only on $p$ and $q$. This extends results of Tardos, Kaiser and Alon for the case $q=2$, and of Kaiser and Rabinovich for the case $p=q=\lceil lo{g}_2(d+2)\rceil$. We further show that similar bounds hold in families of subgraphs of a tree or a graph of bounded tree-width, each consisting of at most $d$ connected components, extending results of Alon for the case $q=2$. Finally, we prove an upper bound of $O\left(d^{\frac{1}{p-1}}\right)$ on the fractional piercing number in families of $d$-intervals satisfying the $(p,p)$ property, and show that this bound is asymptotically sharp.