Koebe 1/4-Theorem and Inequalities in N=2 Super-QCD

The critical curve ${\cal C}$ on which ${\rm Im}\,\hat\tau =0$, $\hat\tau=a_D/a$, determines hyperbolic domains whose Poincar\'e metric is constructed in terms of $a_D$ and $a$. We describe ${\cal C}$ in a parametric form related to a Schwarzian equation and prove new relations for $N=2$ Super $SU(2)$ Yang-Mills. In particular, using the Koebe 1/4-theorem and Schwarz's lemma, we obtain inequalities involving $u$, $a_D$ and $a$, which seem related to the Renormalization Group. Furthermore, we obtain a closed form for the prepotential as function of $a$. Finally, we show that $\partial_{\hat\tau} \langle {\rm tr}\,\phi^2\rangle_{\hat \tau}={1\over 8\pi i b_1}\langle \phi\rangle_{\hat\tau}^2$, where $b_1$ is the one-loop coefficient of the beta function.Comment: 11 pages, LaTex file, Expanded version: new results, technical details explained, misprints corrected and references adde

Fusion Rules of the Lowest Weight Representations of osp_q(1|2) at Roots of Unity: Polynomial Realization and Degeneration at Roots of Unity

The degeneracy of the lowest weight representations of the quantum superalgebra $osp_q(1|2)$ and their tensor products at exceptional values of %when deformation parameter $q$ takes exceptional values is studied. The main features of the structures of the finite dimensional lowest weight representations and their fusion rules are illustrated using realization of group generators as finite-difference operators acting in the space of the polynomials. The complete fusion rules for the decompositions of the tensor products at roots of unity are presented. The appearance of indecomposable representations in the fusions is described using Clebsh-Gordan coefficients derived for general values of $q$ and at roots of unity.Comment: 28 pages, 3 figures; completed (section 5, section 6); version published in JP

IGFBP3 mRNA expression in benign and malignant breast tumors

INTRODUCTION: Most previous studies have focused on evaluating the association between circulating insulin-like growth factor binding protein 3 (IGFBP-3) levels and breast cancer risk. Emerging evidence over the past few years suggests that IGFBP-3 may act directly on mammary epithelial cells. METHODS: To understand the role of IGFBP-3 in breast tumorigenesis, we investigated IGFBP3 mRNA expression levels in benign and malignant breast tumors and their adjacent normal tissues using real-time quantitative PCR. RESULTS: Cancer tissues had significantly lower IGFBP3 expression than benign tumor tissues (p < 0.001). IGFBP3 expressions in both tumor and adjacent tissues were higher in patients who had proliferative benign tumors than in those who had non-proliferative benign tumors. Among patients with benign breast disease, IGFBP3 expression in the tumor was significantly higher than that in their adjacent normal tissue. There were no apparent associations of IGFBP3 expression in cancer tissues with either overall survival or disease-free survival in a cohort of 521 patients with breast cancer. CONCLUSION: Our findings suggest that the expression level of IGFBP3 in breast tissues may be involved in breast tumorigenesis