Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.

Effect of osteoporosis on bone mineral density and fracture repair in a rat femoral fracture model

Osteoporosis (OP) is one of the most prevalent bone diseases worldwide with bone fracture the major clinical consequence. The effect of OP on fracture repair is disputed and although it might be expected for fracture repair to be delayed in osteoporotic individuals, a definitive answer to this question still eludes us. The aim of this study was to clarify the effect of osteoporosis in a rodent fracture model. OP was induced in 3-month-old rats (n = 53) by ovariectomy (OVX) followed by an externally fixated, mid-diaphyseal femoral osteotomy at 6 months (OVX group). A further 40 animals underwent a fracture at 6 months (control group). Animals were sacrificed at 1, 2, 4, 6, and 8 weeks postfracture with outcome measures of histology, biomechanical strength testing, pQCT, relative BMD, and motion detection. OVX animals had significantly lower BMD, slower fracture repair (histologically), reduced stiffness in the fractured femora (8 weeks) and strength in the contralateral femora (6 and 8 weeks), increased body weight, and decreased motion. This study has demonstrated that OVX is associated with decrease in BMD (particularly in trabecular bone) and a reduction in the mechanical properties of intact bone and healing fractures. The histological, biomechanical, and radiological measures of union suggest that OVX delayed fracture healing. (C) 2007 Orthopaedic Research Society. Published by Wiley Periodicals.