Infinite Transitivity on Affine Varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

Cortical bone loss at the tibia in postmenopausal women with osteoporosis is associated with incident non-vertebral fractures: Results of a randomized controlled ancillary study of HORIZON

BACKGROUND In postmenopausal women, yearly intravenous zoledronate (ZOL) compared to placebo (PLB) significantly increased bone mineral density (BMD) at lumbar spine (LS), femoral neck (FN), and total hip (TH) and decreased fracture risk. The effects of ZOL on BMD at the tibial epiphysis (T-EPI) and diaphysis (T-DIA) are unknown. METHODS A randomized controlled ancillary study of the HORIZON trial was conducted at the Department of Osteoporosis of the University Hospital of Berne, Switzerland. Women with ≥1 follow-up DXA measurement who had received ≥1 dose of either ZOL (n=55) or PLB (n=55) were included. BMD was measured at LS, FN, TH, T-EPI, and T-DIA at baseline, 6, 12, 24, and 36 months. Morphometric vertebral fractures were assessed. Incident clinical fractures were recorded as adverse events. RESULTS Baseline characteristics were comparable with those in HORIZON and between groups. After 36 months, BMD was significantly higher in women treated with ZOL vs. PLB at LS, FN, TH, and T-EPI (+7.6%, +3.7%, +5.6%, and +5.5%, respectively, p<0.01 for all) but not T-DIA (+1.1%). The number of patients with ≥1 incident non-vertebral or morphometric fracture did not differ between groups (9 ZOL/11 PLB). Mean changes in BMD did not differ between groups with and without incident fracture, except that women with an incident non-vertebral fracture had significantly higher bone loss at predominantly cortical T-DIA (p=0.005). CONCLUSION ZOL was significantly superior to PLB at T-EPI but not at T-DIA. Women with an incident non-vertebral fracture experienced bone loss at T-DIA.

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

This article provides importance sampling algorithms for computing the probabilities of various types ruin of spectrally negative Lévy risk processes, which are ruin over the infinite time horizon, ruin within a finite time horizon and ruin past a finite time horizon. For the special case of the compound Poisson process perturbed by diffusion, algorithms for computing probabilities of ruins by creeping (i.e. induced by the diffusion term) and by jumping (i.e. by a claim amount) are provided. It is shown that these algorithms have either bounded relative error or logarithmic efficiency, as t,x→∞t,x→∞, where t>0t>0 is the time horizon and x>0x>0 is the starting point of the risk process, with y=t/xy=t/x held constant and assumed either below or above a certain constant.

The Effect of 6 versus 9 Years of Zoledronic Acid Treatment in Osteoporosis: A Randomized Second Extension to the HORIZON-Pivotal Fracture Trial (PFT).

While bisphosphonates reduce fracture risk over 3 to 5 years, the optimal duration of treatment is uncertain. In a randomized extension study (E1) of the Health Outcomes and Reduced Incidence with Zoledronic Acid Once Yearly - Pivotal Fracture Trial (HORIZON-PFT), zoledronic acid (ZOL) 5 mg annually for 6 years showed maintenance of bone mineral density (BMD), decrease in morphometric vertebral fractures, and a modest reduction in bone turnover markers (BTMs) compared with discontinuation after 3 years. To investigate the longer-term efficacy and safety of ZOL, a second extension (E2) was conducted to 9 years in which women on ZOL for 6 years in E1 were randomized to either ZOL (Z9) or placebo (Z6P3) for 3 additional years. In this multicenter, randomized, double-blind study, 190 women were randomized to Z9 (n=95) and Z6P3 (n=95). The primary endpoint was change in total hip BMD at year 9 vs. year 6 in Z9 compared with Z6P3. Other secondary endpoints included fractures, BTMs, and safety. From year 6 to 9, the mean change in total hip BMD was -0.54% in Z9 vs. -1.31% in Z6P3 (difference 0.78%; 95% confidence interval [CI]: -0.37%, 1.93%; p=0.183). BTMs showed small, non-significant increases in those who discontinued after 6 years compared with those who continued for 9 years. The number of fractures was low and did not significantly differ by treatment. While generally safe, there was a small increase in cardiac arrhythmias (combined serious and non-serious) in the Z9 group but no significant imbalance in other safety parameters. The results suggest almost all patients who have received six annual ZOL infusions can stop medication for up to 3 years with apparent maintenance of benefits. This article is protected by copyright. All rights reserved.