The Andrews-Stanley partition function and Al-Salam-Chihara polynomials

For any partition λ let ω(λ) denote the four parameter weight

ω(λ)=a^{∑_i≥1⌈λ_2i−1/2⌉}b^{∑_i≥1⌊λ_2i−1/2⌋}c^{∑_i≥1⌈λ_2i/2⌉}d^{∑_i≥1⌊λ_2i/2⌋},     

Pairs of noncrossing free Dyck paths and noncrossing partitions

Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length 2n and noncrossing partitions of [2n+1] with n+1 blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.     

Labeled partitions with colored permutations

In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations. We use this structure to derive the generating function of the indices of colored permutations. We further give a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations. Based on labeled partitions, we provide an involution that implies the generating function formula due to Gessel and Simon for signed q-counting of the major indices. This involution can be extended to signed permutations. This gives a combinatorial interpretation of a formula of Adin, Gessel and Roichman.     

Arbitrarily vertex decomposable suns with few rays

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,…,n_k) of positive integers with n₁+?+n_k=n, there exists a partition (V₁,…,V_k) of the vertex set of G such that V_i induces a connected subgraph of order n_i, for all i=1,…,k. A sun with r rays is a unicyclic graph obtained by adding r hanging edges to r distinct vertices of a cycle. We characterize all arbitrarily vertex decomposable suns with at most three rays. We also provide a list of all on-line arbitrarily vertex decomposable suns with any number of rays.     

Multi-variable subordination distributions for free additive convolution

Let k be a positive integer and let Dc(k) denote the space of joint distributions for k-tuples of selfadjoint elements in C^∗-probability space. The paper studies the concept of “subordination distribution of μ?ν with respect to ν” for μ,ν∈Dc(k), where ? is the operation of free additive convolution on Dc(k). The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space.Multi-variable subordination turns out to have nice relations to a process of evolution towards ?-infinite divisibility on Dc(k) that was recently found by Belinschi and Nica (arXiv: 0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion.     

节点应力连续的四边形单元

节点应力连续的四边形单元Q4-CNS是一种基于单位分解理论的混合的有限元无网格法.Q4-CNS可以视作FE-LSPIM QUAD4的发展.Q4-CNS形函数的导数在节点处是连续的,因此可以自然的得到节点应力,而不需要使用节点应力磨平算法.数值实验表明,与传统四边形单元(QUAD4)相比,Q4-CNS具有更好的计算精度和更高的收敛速度.在扭曲网格下,Q4-CNS也能取得满意的数值精度.然而,QUAD4的数值精度则会随着网格的扭曲明显的变差.基于Kirchhoff-Love假设的非协调板单元计算中,不仅要求形函数在单元的交界面上要保持C0连续性,而且要求形函数在节点处具有C1连续性,所以在任意的四边形单元上构造满足插值条件的非协调板单元形函数较为困难.Q4-CNS形函数的导数在节点处是连续的,所以Q4-CNS在求解基于Kirchhoff-Love假设的板单元问题中具有潜在的应用价值.     

基于数据最优分区间相似度算法及应用

通过定义了一种基于数据最优分区间相似度算法,利用学习样本得单位相似度向量,并得各维数据的最优分区间.利用最优分区间得预测样本与学习样本的单位相似度向量,从而得预测样本的预测值.通过实例表明,算法所预测的结果相对误差可达百分位,并且本算法能应用到其它数据处理中,具有较广泛的通用性.     

On Ramanujan’s modular identities

For Ramanujan’s modular identities connected with his well-known partition congruences for the moduli 5 or 7, we had given, in an earlier paper, natural and uniform proofs through the medium of modular forms. Analogous (modular) identities corresponding to the (more difficult) case of the modulus 11 are provided here, with the consequent partition congruences; the relationship with relevant results of N J Fine is also sketched.     

Generators

In this note we indicate the importance of the notion of “generator” for the classification problem in ergodic theory, and we give a simple new construction of a finite generator for an ergodic transformation of finite entropy This revised version was published online in June 2006 with corrections to the Cover Date.     

Novel topological index for research on structure-property relationships of complex organic compounds

Based on bonding parameters such as Yang's Electronegative Force Gauge Y(i), electronic number of valence layer Z(i), number of combined hydrogen atoms h(i), number of bonding electron b(i), and quantum number such as the highest main quantum number of valence layer n(i), a novel atomic valence delta(i) (Y) is defined and a novel topological index (1)chi(Y) is derived from the atomic valence. The atomic valence is defined as delta(i) (Y) = (Z(i) - h(i))b(i)/n(i) (2)Y(i), while the topological index is expressed as (1)chi(Y) summation operator (i,j=1) (m) (delta(i) (Y)delta(j) (Y))(-1/2). Subsequently, the index (1)chi(Y) is utilized to study the structure-property relationships of complex organic compounds. The results of correlativity showed that the index is highly and extensively correlated with such properties as solubility of phenyl chlorides, gas chromatographic retention index of alkoxyl silanes, and toxicity of heterocyclic nitrogen-containing compounds. Moreover, predicted values are quite consistent with experimental ones when the index is employed to predict the partition coefficient (log P) of fatty alcohols, phenyl chlorides, and barbitals. Compared to the topological indices reported in the literature, the universality and reliability of (1)chi(Y) to the properties of complex organic compounds have been distinctively improved, and its calculating process is simple and convenient.