Theropod Fauna from Southern Australia Indicates High Polar Diversity and Climate-Driven Dinosaur Provinciality

The Early Cretaceous fauna of Victoria, Australia, provides unique data on the composition of high latitude southern hemisphere dinosaurs. We describe and review theropod dinosaur postcranial remains from the Aptian–Albian Otway and Strzelecki groups, based on at least 37 isolated bones, and more than 90 teeth from the Flat Rocks locality. Several specimens of medium- and large-bodied individuals (estimated up to ∼8.5 metres long) represent allosauroids. Tyrannosauroids are represented by elements indicating medium body sizes (∼3 metres long), likely including the holotype femur of Timimus hermani, and a single cervical vertebra represents a juvenile spinosaurid. Single specimens representing medium- and small-bodied theropods may be referrable to Ceratosauria, Ornithomimosauria, a basal coelurosaur, and at least three taxa within Maniraptora. Thus, nine theropod taxa may have been present. Alternatively, four distinct dorsal vertebrae indicate a minimum of four taxa. However, because most taxa are known from single bones, it is likely that small-bodied theropod diversity remains underestimated. The high abundance of allosauroids and basal coelurosaurs (including tyrannosauroids and possibly ornithomimosaurs), and the relative rarity of ceratosaurs, is strikingly dissimilar to penecontemporaneous dinosaur faunas of Africa and South America, which represent an arid, lower-latitude biome. Similarities between dinosaur faunas of Victoria and the northern continents concern the proportional representatation of higher clades, and may result from the prevailing temperate–polar climate of Australia, especially at high latitudes in Victoria, which is similar to the predominant warm–temperate climate of Laurasia, but distinct from the arid climate zone that covered extensive areas of Gondwana. Most dinosaur groups probably attained a near-cosmopolitan distribution in the Jurassic, prior to fragmentation of the Pangaean supercontinent, and some aspects of the hallmark ‘Gondwanan’ fauna of South America and Africa may therefore reflect climate-driven provinciality, not vicariant evolution driven by continental fragmentation. However, vicariance may still be detected at lower phylogenetic levels

Nearly tight bounds for testing function isomorphism

We study the problem of testing isomorphism (equivalence up to relabelling of the variables) of two Boolean functions f, g: {0, 1} n → {0, 1}. Our main focus is on the most studied case, where one of the functions is given (explicitly) and the other function may be queried. We prove that for every k ≤ n, the worst-case query complexity of testing isomorphism to a given k-junta is Ω(k) and O(k log k). Consequently, the query complexity of testing function isomorphism is e Θ(n). Prior to this work, only lower bounds of Ω(log k) queries were known, for limited ranges of k, proved by Fischer et al. (FOCS 2002), Blais and O’Donnell (CCC 2010), and recently by Alon and Blais (RANDOM 2010). The nearly tight O(k log k) upper bound improves on the e O(k 4) upper bound from Fischer et al. (FOCS 2002). Extending the lower bound proof, we also show polynomial query-complexity lower bounds for the problems of testing whether a function can be computed by a circuit of size ≤ s, and testing whether the Fourier degree of a function is ≤ d. This answers questions posed by Diakonikolas et al. (FOCS 2007). We also address two closely related problems – 1. Testing isomorphism to a k-junta with one-sided error: we prove that for any 1 < k < n − 1, the query complexity is Ω(log ` ´ n), which is almost optimal. Thi