In this paper it is proved that the set of primitive elements of a nonabelian free group has density zero, i.e. the ratio of primitive elements in increasingly large balls is arbitrarily small. Two notions of density (natural and exponential density) are defined and some of their properties are studied. A class of subsets of the free group (graphical sets) is defined restricting the occurrence ...

We show a hardness-preserving construction of a PRF from any length doubling PRG which improves upon known constructions whenever we can put a non-trivial upper bound q on the number of queries to the PRF. Our construction requires only O(log q) invocations to the underlying PRG with each query. In comparison, the number of invocations by the best previous hardness-preserving construction (GGM ...

A graph is edge-primitive if its automorphism group acts primitively on the edge set of graph. Edge-primitive graphs form an important subclass symmetric graphs. In this paper, order as a product two distinct primes are completely determined. This depends non-abelian simple groups with subgroup index pq being classified, where p>q odd primes.

Peripheral primitive neuroectodermal tumor (pPNET) is a highly aggressive small round cell tumor belonging to PNET/Ewing sarcoma family. Ovarian tumors composed of primitive neuroectodermal elements are extremely rare. Herein we reported two cases of peripheral primitive neuroectodermal tumors of ovary in two patients with different clinical presentations. Definite diagnoses were made bas...

We show that most of the genus-zero subgroups braid group $\mathbb{B}_3$ (which are roughly monodromy groups trigonal curves on Hirzebruch surfaces) irrelevant as far Alexander invariant is concerned: there a very restricted class \enquote{primitive} such these and their intersections determine all invariants. Then, we classify primitive in special subclass. This result implies known classifica...

objectives: cerebral palsy (cp) is a static encephalopathy with abnormal reflexes that cause motor developmental delay. in addition to conventional methods, complementary medicine like homeopathy has been used in treatment of neuro-developmental disorders. this study has been done to determine the effect of adding homeopathic treatment to rehabilitation on abnormal reflexes of children with spa...