The fluctuations of the giant cluster for percolation on random split trees

A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant cluster as $n \rightarrow \infty$ are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work generalizes the results for the random $m$-ary recursive trees in Berzunza (2015). Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.Comment: 43 page

Automorphisms of surfaces of general type with q>=2 acting trivially in cohomology

A compact complex manifold X is said to be rationally cohomologically rigidified if its automorphism group Aut(X) acts faithfully on the cohomology ring H*(X,Q). In this note, we prove that, surfaces of general type with irregularity q>2 are rationally cohomologically rigidified, and so are minimal surfaces S with q=2 unless K^2=8X. This answers a question of Fabrizio Catanese in part. As examples we give a complete classification of surfaces isogenous to a product with q=2 that are not rationally cohomologically rigidified. These surfaces turn out however to be rigidified.Comment: 18 pages; a remark and a closely relevant reference are adde

Protective effect of Rhizoma drynariae extract on osteoporosis in ovariectomized rat model

Purpose: To investigate the therapeutic effect of Rhizoma Drynariae extract (RDE) on ovariectomyinduced osteoporosis in rats.Methods: Female Sprague-Dawley rats were randomly assigned to a sham-operated group (control) and five ovariectomy (OVX) subgroups: OVX with vehicle (OVX), OVX with 17ß-estradiol (E2, 25 μg/kg/day), and OVX with RDE doses (40, 80, and 160 mg/kg/day). Daily oral administration of E2 or RDE started 4 weeks after OVX and lasted for 16 weeks. The bone mineral density (BMD) of the L4 vertebrae and right femurs was estimated. The length of each femur was measured with a  micrometer gauge, and the center of the diaphysis determined. Three   representatives L4 vertebrae were selected to evaluate the trabecular  microarchitecture. Serum alkaline phosphatase (ALP), urinary calcium (U-Ca),urinary phosphorus (U-P), urinary creatinine (Cr) and osteocalcin (OC) levels were measured.Results: The study showed that high-dose of RDE significantly inhibited the bone mineral density (BMD) reduction of L4 vertebrae (0.20 ± 0.02 g/cm3, p < 0.05) and femurs (0.18 ± 0.02 g/cm3, p < 0.05) caused by OVX and prevented the  deterioration of trabecular microarchitecture (p < 0.05), which were accompanied by a significant decrease in skeletal remodeling (p < 0.05) as evidenced by the lowerlevels of bone turnover markers. High-dose of RDE improved morphometric parameters, namely, Tb-N (3.8 ± 0.2 mm, p < 0.05), Tb-Th (0.083 ± 0.011 mm, p < 0.05) and Tb-Sp (0.19 ± 0.01 mm, p < 0.05) in L4 vertebrae significantly. The present study indicates that the administration of RDE at higher doses over a  16-week period can prevent OVX-induced osteoporosis in rats without hyperplastic effects on the uterus.Conclusion: Thus, RDE is a potential natural alternative for postmenopausal  osteoporosis treatment in elderly women.Keywords: Rhizoma Drynariae, Postmenopausal osteoporosis, Ovariectomy, Bone mineral density, Morphometri