A Telescoping method for Double Summations

We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r$ and rational functions $R_1(n,i,j),R_2(n,i,j)$ such that $L F = \Delta_i (R_1 F) + \Delta_j (R_2 F)$. Based on simple divisibility considerations, we show that the denominators of $R_1$ and $R_2$ must possess certain factors which can be computed from $F(n, i,j)$. Using these factors as estimates, we may find the numerators of $R_1$ and $R_2$ by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, Carlitz's identities, the Ap\'ery-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkov\v{s}ek-Wilf-Zeilberger identity.Comment: 22 pages. to appear in J. Computational and Applied Mathematic

Applicability of the qq-Analogue of Zeilberger's Algorithm

The applicability or terminating condition for the ordinary case of Zeilberger's algorithm was recently obtained by Abramov. For the $q$-analogue, the question of whether a bivariate $q$-hypergeometric term has a $qZ$-pair remains open. Le has found a solution to this problem when the given bivariate $q$-hypergeometric term is a rational function in certain powers of $q$. We solve the problem for the general case by giving a characterization of bivariate $q$-hypergeometric terms for which the $q$-analogue of Zeilberger's algorithm terminates. Moreover, we give an algorithm to determine whether a bivariate $q$-hypergeometric term has a $qZ$-pair.Comment: 15 page

Phytoestrogen Biological Actions on Mammalian Reproductive System and Cancer Growth

Phytoestrogens are a family of diverse polyphenolic compounds derived from nature plant that structurally or functionally mimic circulating estrogen in the mammalian reproductive system. They induce estrogenic and anti-estrogenic effects in the brain-pituitary-gonad axis (a principal endocrine system involving in reproductive regulation) and peripheral reproductive organs. The dichotomy of phytoestrogen-mediated actions elucidates that they play the biological activities via complex mechanisms and belong to various chemical classes. In comparison with their unobvious physiological functions in normal reproductive tissues, there are increasing investigations showing that phytoestrogen induces significant inhibitory effects on the growth of breast and ovarian cancers through different signaling pathways. This review summarized the results of the previous studies regarding principal signaling transductions for mediating the growth of the ovarian and breast cancers. Phytoestrogen potentially modulates the signaling molecules via: (1) blocking the nuclear and membrane estrogen receptors (ER), (2) interfering with the growth factor receptor, (3) inhibiting the G protein-coupled receptor in ER-deficient cells, (4) activating apoptosis and nullifying anti-apoptotic signals

Spin Current and Current-Induced Spin Transfer Torque in Ferromagnet-Quantum Dot-Ferromagnet Coupled Systems

Based on Keldysh's nonequilibrium Green function method, the spin-dependent transport properties in a ferromagnet-quantum dot (QD)-ferromagnet coupled system are investigated. It is shown the spin current shows quite different characteristics from its electrical counterpart, and by changing the relative orientation of both magnetizations, it can change its magnitude even sign. The current-induced spin transfer torque (CISTT) is uncovered to be greatly enhanced when the bias voltage meets with the discrete levels of the QD at resonant positions. The relationship between the CISTT, the electrical current and the spin current is also addressed.Comment: 21 pages, 8 figure

Role of Inflammatory Factors in Regulation of Osteogenesis in Tissue-Engineered Bone

It was traditionally considered that the inhibition of inflammatory reaction is necessary for osteogenesis, but the latest issue argued inflammation is unavoidable in the process of bone trauma, and physiological inflammatory reaction is essential to achieve bone formation. Tissue-engineered bone graft is not only associated with osteoblast-related cells; the inflammatory reaction is the initial physiological process, mainly with neutrophil infiltration, which secretes MCP-1, IL-8, and other chemokines and further promotes dendritic cells, lymphocytes, and mononuclear macrophages to move in. The activation pathways of macrophages have a direct effect on the outcome of the inflammatory reaction and the healing, which are divided into the classical approach (M1) and the alternative approach (M2). The M1 pathway secretes IL-1 beta, IL-6, TNF-α, and other pro-inflammatory factors, while the M2 pathway secretes arginase, IL-1Ra, IL-4, and other anti-inflammatory cytokines, also with bone-healing-related growth factors, which promote homing of bone mesenchymal stem cells (bMSCs)