So far in the course we have been concerned almost exclusively with functions on the set {−1, 1}n (sometimes written F2 ). What about other product spaces? As alluded to in Problem #1 on Homeworks #4 and #5, we can develop some kind of “orthogonal decomposition” for functions on any product probability space X. This will give us some subset of our Fourier analysis, and it’s useful when one does...

This result strengthens a previous theorem, again due to Furstenberg and Katznelson ([FK1]), stating that for commuting measure preserving transformations T1, . . . , Tk of a probability space (X,A, μ) and A ∈ A with μ(A) > 0 there exists n > 0 such that μ ( A∩ ⋂k i=1 T −n i A ) > 0. The case of [FK1] in which each Ti is a power of the same transformation T was proved by Furstenberg in 1977 ([F...

The essential oil of Betula pendula Roth. buds was obtained using both hydrodistillation and microdistillation techniques and their chemical compositions were analyzed using both gas chromatography (GC) and gas chromatography-mass spectrometry (GC-MS). Overall, more than 50 compounds were identified representing 80% and 92% for hydrodistillation and microdistillation, respectively. The main com...

J. Tan, A. Cuevas, D. Macdonald Department of Engineering, College of Engineering and Computer Science, Australian National University, Canberra, ACT 0200, Australia T. Trupke, R. Bardos Centre of Excellence for Advanced Silicon Photovoltaics and Photonics, University of New South Wales, Sydney 2052, Australia.. K. Roth Roth and Rau AG, Gewerbering 09337 Hohenstein-Ernstthal,OT Wuestenbrand, Ge...