In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at various scales. Let N = 1/h, in which h is the Planck constant. First, for all integers N ∈ N, we show quantum ergodicity at logarithmical scales | log h| −α for some α > 0. Second, we show quantum ergodicity at polynomial scales h α for some α > 0, in two special cases: N ∈ S(N) of a full density subset S(N) of integers and Hecke eigenbasis for all integers.2010 Mathematics Subject Classification. 35P20, 58G25, 81Q50, 37D20, 11F25. Key words and phrases. Hyperbolic linear maps of the torus, quantum ergodicity, small scale.

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In this article, chlorosilane-modified ZSM-5 particles were incorporated into polydimethylsiloxane (PDMS) to form mixed matrix membranes (MMMs) for ethanol/ water mixture separation via pervaporation (PV). The membranes were characterized by Fourier transform infrared spectroscopy, X-ray diffraction, scanning electron microscopy, and mechanical performance testing. The maximum loading and dispersion of ZSM-5 into PDMS were improved by chlorosilane modification. To evaluate the PV performance, the MMMs were used to separate an aqueous ethanol solution. The effect of zeolite loading and operational conditions on PV performance was investigated in detail. The separation factor of the composite membranes filled with modified ZSM-5 increased considerably versus unmodified membrane, while the total flux decreased to some degree. Of all the chlorosilane-modified membranes, dodecyltrichlorosilane modified ZSM-5 filled PDMS showed the best separation factor of 15.8 for ethanol. POLYM. COM-POS., 37:1282-1291