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SUMMARYWe developed and tested a novel quantitative method for the quantification of film autoradiographs, involving a mathematical model and a dot-blot-based membrane standard scale. The exponential model introduced here, ROD ϭ p 1 (1 Ϫ exp[p 2 x]), appropriately (r 2 Ͼ0.999), describes the relation between relative optical density (ROD) and radioactivity (x) in the range between 0 and 240 gray scale values (using a 256-gray scale level digitizer). By means of this model, standard curves with distinct quenching properties can be exactly interconverted, permitting the tissue-equivalent calibration of different standard scales. The membrane standard scale employed here has several advantages, including the flexible radioactivity range, the facile and rapid preparation technique, and the compact size. The feasibility of the quantification procedure is exemplified by the comparative quantification of multiple calmodulin mRNAs in the rat brain by in situ hybridization with [ 35 S]-cRNA probes. The procedure for quantification provides a significant improvement in that the direct and exact comparison of radiolabeled species, even from different experiments, can be reliably performed. Further, the procedure can be adapted to the quantification of autoradiographs produced by other methods. (J Histochem Cytochem 46:1141-1149, 1998)

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The present paper is devoted to studying Hubbard's pendulum equation x + 10 −1ẋ + sin(x) = cos(t). By rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is lifted from the level of experimentally observed facts to the level of a theorem completely proved. A distinguished family of solutions is shown to be chaotic in the sense that on consecutive time intervals (2kπ, 2(k+1)π) (k ∈ Z) individual members of the family can freely "choose" between the following possibilities: the pendulum either crosses the bottom position exactly once clockwise or does not cross the bottom position at all or crosses the bottom position exactly once counterclockwise. The proof follows the topological index/degree approach by Mischaikow, Mrozek, and Zgliczynski. The novelty is a definition of the transition graph for which the periodic orbit lemma, the key technical result of the approach aforementioned, turns out to be a consequence of Brouwer's fixed point theorem. The role of wholly automatic versus 'trial and error with human overhead' computer procedures in detecting chaos is also discussed.