COUNTABLE GROUPS OF ISOMETRIES ON BANACH SPACES

A group G is representable in a Banach space X if G is isomorphic to the group of isometrics on X in some equivalent norm. We prove that a countable group G is representable in a separable real Banach space X in several general cases, including when G similar or equal to {-1,1} x H, H finite and dim X >= vertical bar H vertical bar or when G contains a normal subgroup with two elements and X is of the form c(0)(Y) or l(p)(Y), 1 <= p < +infinity. This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space X and a countable bounded group G of isomorphisms on X containing -Id, there exists an equivalent norm on X for which G is equal to the group of isometrics on X. We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometrics. It follows that every real Banach space of dimension at least 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.

Risk factors for dental erosion in a group of 12- and 16-year-old Brazilian schoolchildren

Background. Dental erosion is a multifactorial disease and is associated with dietary habits in infancy and adolescence. Aim. To investigate possible associations among dental erosion and diet, medical history and lifestyle habits in Brazilian schoolchildren. Design. The sample consisted of a random single centre cluster of 414 adolescents (12- and 16-years old) of both genders from private and public schools in Bauru (Brazil). The O`Brien [Children`s Dental Health in the United Kingdom, 1993 (1994) HMSO, London] index was used for dental erosion assessment. Data on medical history, rate and frequency of food and drinks consumption, and lifestyle habits were collected by a self-reported questionnaire. Odds ratios with 95% confidence intervals were used to assess the univariate relationships between variables. Analysis of questionnaire items was performed by multiple logistic regression analysis. The statistical significance level was set at 5%. Results. The erosion present group comprised 83 subjects and the erosion absent group 331. There were no statistically significant correlations among dental erosion and the consumption of food and drinks, medical history, or lifestyle habits. Conclusion. The results indicate that there was no correlation between dental erosion and the risk factors analysed among adolescents in Bauru/Brazil and further investigations are necessary to clarify the multifactorial etiology of this condition.

Influence from the sulfonate group (RSO(3)(-)) on the conversion process and emission efficiency of poly(p-phenylene vinylene)

The discovery of an alternative route to convert poly(xylyliden tetrahydrothiophenium chloride) (PTHT) into poly(p-phenylene vinylene) (PPV) using dodecylbenzenesulfonate (DBS) has allowed the formation of ultrathin films with unprecedented control of architecture and emission properties. In this work, we show that this route may be performed with several sufonated compounds where RSO(3)(-) replaces the counter-ion (Cl(-)) of PTHT, some of which are even more efficient than DBS. Spin-coating films were produced from PTHT and azo-dye molecules, an azo-polymer and organic salts as counter-ions of PTHT. The effects of the thermal annealing step of PTHT/RSO(3)(-) films at 110 and 230 degrees C were monitored by measuring the absorption and emission spectra. The results indicate that the exchange of the counterion Cl(-) of PTHT by a linear long chain with RSO(3)(-) group is a general procedure to obtain PPV polymer at lower conversion temperature (ca. 110 degrees C) with significant increase in the emission efficiency, regardless of the chemical position and the number of sulfonate groups. With the enhanced emission caused by Congo Red and Tinopal as counter-ions, it is demonstrated that the new synthetic route is entirely generic, which may allow accurate control of conversion and emission properties. (C) 2010 Elsevier B.V. All rights reserved.

A group topology on the free abelian group of cardinality c that makes its square countably compact

Under p = c, we prove that it is possible to endow the free abelian group of cardinality c with a group topology that makes its square countably compact. This answers a question posed by Madariaga-Garcia and Tomita and by Tkachenko. We also prove that there exists a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group) whose square is countably compact. This answers a question posed by Grant.