THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE SPHERE

In this paper, we determine the lower central and derived series for the braid groups of the sphere. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is important in its own right. The braid groups of the 2-sphere S(2) were studied by Fadell, Van Buskirk and Gillette during the 1960s, and are of particular interest due to the fact that they have torsion elements (which were characterised by Murasugi). We first prove that for all n epsilon N, the lower central series of the n-string braid group B(n)(S(2)) is constant from the commutator subgroup onwards. We obtain a presentation of Gamma(2)(Bn(S(2))), from which we observe that Gamma(2)(B(4)(S(2))) is a semi-direct product of the quaternion group Q(8) of order 8 by a free group F(2) of rank 2. As for the derived series of Bn(S(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group Bn(S(2)) being finite and soluble for n <= 3, the critical case is n = 4 for which the derived subgroup is the above semi-direct product Q(8) (sic) F(2). By proving a general result concerning the structure of the derived subgroup of a semi-direct product, we are able to determine completely the derived series of B(4)(S(2)) which from (B(4)(S(2)))(4) onwards coincides with that of the free group of rank 2, as well as its successive derived series quotients.

James Parkinson (1755-1824): A pioneer of child care

James Parkinson (1755-1824) of Parkinson's disease, is well recognized as a pioneer of clinical neurology; and is even more famous as a founder of modem palaeontology. We have reviewed from primary sources his extensive contributions to clinical child care and his pioneering advocacy for child welfare, protection and safety. His writings, outreach and advocacy for children's health characterizes him as one whose influence was an important springboard from which evolved the modern specialty of paediatrics. Parkinson was one of the first to write on child-rearing practices and in this context antedated Benjamin Spock by 150 years. Parkinson was a pioneer of child safety and the prevention of childhood trauma. He wrote of the resuscitation of near-drowned children and of first aid for injured children. This critical analysis reviews his pioneering description of child abuse and the development of post-abuse hydrocephalus. He wrote the datum description (in English) of the pathophysiology and pathology of appendicitis in children, of fatal rabies in children and highlighted the risk of death even when the biting dog was not clinically rabid. His advocacy for social reform for children's welfare was courageous and pioneering. James Parkinson, hitherto unacknowledged, was a significant founder of the evolving discipline of paediatrics and child health.

James W. Gerard: His image of imperial Germany, 1913-1918

The subject of this study was a typical, if in some respects well qualified, U.S. ambassadorial appointee for his time, the early twentieth century: an attorney, judge, and politician who served competently in his one diplomatic assignment, in Berlin, before returning to private life.—Ed.

Genus and Braid Index Associated to Sequences of Renormalizable Lorenz Maps

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant (K(f)(-), = K(f)(+)) = (X, Y) * (S, W) in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from the ones corresponding to the factors of the *-product. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant (X, Y) * (S,W)*(n), concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization.

A linguagem do espaço em Henry James: a fase madura

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)