In order to generate a random orthogonal matrix distributed according to Haar measure over the orthogonal group it is natural to start with a matrix of normal random variables and then factor it by the singular value decomposition. A more efficient method is obtained by using Householder transformations. We propose another alternative based on the product of n(n-1)/2 orthogonal matrices, each of which represents an angle of rotation. Some numerical comparisons of alternative methods are made.

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During the 1990s and early 2000s, populations of surface-nesting seabirds at Marion Island showed different trends, but for the majority of species numbers decreased. Reduced numbers of gentoo penguins Pygoscelis papua, eastern rockhopper penguins Eudyptes chrysocome filholi, Crozet shags Phalacrocorax [atriceps] melanogenis and probably macaroni penguins E. chrysolophus are most plausibly attributed to an altered availability of food. Decreases in numbers of dark-mantled sooty albatrosses Phoebetria fusca, light-mantled sooty albatrosses P. palpebrata, southern giant petrels Macronectes giganteus and possibly northern giant petrels M. halli may have resulted from mortality of birds in longline fisheries. However, populations of wandering Diomedea exulans and grey-headed Thalassarche chrysostoma albatrosses fluctuated around a stable level. Numbers of Subantarctic skuas Catharacta antarctica and kelp gulls Larus dominicanus breeding at Marion Island also decreased. Kerguelen Sterna virgata and Antarctic S. vittata terns remain scarce at the island. Trends for king penguins Aptenodytes patagonicus were not reliably gauged, but numbers probably remained stable or increased. There were large fluctuations in numbers of king penguin chicks surviving to the end of winter.

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