Maximum curves and isolated points of entire functions

Given M(r; f) =maxjzj=r (jf(z)j) , curves belonging to the set of points M = fz : jf(z)j = M(jzj; f)g were de�ned by Hardy to be maximum curves. Clunie asked the question as to whether the set M could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions f1(z) and f2(z), if the maximum curve of f1(z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f1(z)f2(z). By means of these results an entire function of in�nite order is constructed for which the set M has an in�nite number of isolated points. A polynomial is also constructed with an isolated point.

Marginal adaptation and performance of bioactive dental restorative materials in deciduous and young permanent teeth

Objective: The aim of this study was to investigate the adaptation of different types of restorations towards deciduous and young permanent teeth. Materials and Methods: Class V cavities were prepared in deciduous and young permanent teeth and filled with different materials (a conventional glass-ionomer, a resin-modified glass-ionomer, a poly-acid-modified composite resin and a conventional composite resin). Specimens were aged in artificial saliva for 1, 6, 12 and 18 months, then examined by SEM. Results: The composite resin and the polyacid-modified composite had better marginal adaptation than the glass-ionomers,though microcracks developed in the enamel of the tooth. The glass-ionomers showed inferior marginal quality and durability, but no microcracking of the enamel. The margins of the resin-modified glass-ionomer were slightly superior to the conventional glass-ionomer. Conditioning improved the adaptation of the composite resin, but the type of tooth made little or no difference to the performance of the restorative material. All materials were associated with the formation of crystals in the gaps between the filling and the tooth; the quantity and shape of these crystals varied with the material. Conclusions: Resin-based materials are generally better at forming sound, durable margins in deciduous and young permanent teeth than cements, but are associated with microcracks in the enamel. All fluoride-releasing materials give rise to crystalline deposits.