A version of Kac’s lemma on first return times for suspension flows

In this paper, we study the mean return times to a given set for suspension flows. In the discrete time setting, this corresponds to the classical version of Kac’s lemma [11] that the mean of the first return time to a set with respect to the normalized probability measure is one. In the case of suspension flows we provide formulas to compute the mean return time. Positive measure sets on cross sections are also considered. In particular, this varies linearly with continuous reparametrizations of the flow and takes into account the mean escaping time from the original set. Relation with entropy and returns to positive measure sets on cross sections is also considered.

The Relation of the Neutral Zone to the Crest of the Residual Alveolar Ridge in Completely Edentulous Subjects – A Clinico-Radiological Study

ABSTRACT A Clinico-Radiological Study was conducted to establish a set relation between crest of residual alveolar ridge and neutral zone so that a uniform pattern could be followed for teeth setting. Thirty healthy edentulous patients with class – I ridge relation and who had not undergone any major oral surgical procedure was included in this study. Patients were divided into two groups, 1, who were edentulous from six months to two year and group 2, who were edentulous from two or more years. Fifteen patients were in each group. Results showed that in group 1 patients, neutral zone lied lingual to or on crest of alveolar ridge and in group 2 patients, neutral zone shifted to buccal side. Therefore, by studying the pattern of shift of neutral zone to buccal side we should come out with a reasonable plan for placing our teeth looking at duration of edentulousness.