67 research outputs found

    From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

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    The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery's work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note: text overlap with arXiv:0909.491

    New fossils of Australopithecus sediba reveal a nearly complete lower back

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    Abstract: Adaptations of the lower back to bipedalism are frequently discussed but infrequently demonstrated in early fossil hominins. Newly discovered lumbar vertebrae contribute to a near-complete lower back of Malapa Hominin 2 (MH2), offering additional insights into posture and locomotion in Australopithecus sediba. We show that MH2 demonstrates a lower back consistent with human-like lumbar lordosis and other adaptations to bipedalism, including an increase in the width of intervertebral articular facets from the upper to lower lumbar column (“pyramidal configuration”). This contrasts with recent work on lordosis in fossil hominins, where MH2 was argued to demonstrate no appreciable lordosis (“hypolordosis”) similar to Neandertals. Our three-dimensional geometric morphometric (3D GM) analyses show that MH2’s nearly complete middle lumbar vertebra is human-like in shape but bears large, cranially-directed transverse processes, implying powerful trunk musculature. We interpret this combination of features to indicate that A. sediba used its lower back in both human-like bipedalism and ape-like arboreal positional behaviors, as previously suggested based on multiple lines of evidence from other parts of the skeleton and reconstructed paleobiology of A. sediba

    Historical changes in sediments of Pyramid Lake, Nevada, USA: consequences of changes in the water balance of a terminal desert lake

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    Sediment cores from the shallow and deep basins of Pyramid Lake, Nevada, revealed variations in composition with depth reflecting changes in lake level, river inflow, and lake productivity. Recent sediments from the period of historical record indicate: (1) CaCO 3 and organic content of sediment in the shallow basin decrease at lower lake level, (2) CaCO 3 content of deep basin sediments increases when lake level decreases rapidly, and (3) the inorganic P content of sediments increases with decreasing lake volume. Variations in sediment composition also indicate several periods for which productivity in Pyramid Lake may have been elevated over the past 1000 years. Our data provide strong evidence for increased productivity during the first half of the 20th Century, although the typical pattern for cultural eutrophication was not observed. The organic content of sediments also suggests periods of increased productivity in the lake prior to the discovery and development of the region by white settlers. Indeed, a broad peak in organic fractions during the 1800's originates as an increase starting around 1600. However, periods of changing organic content of sediments also correspond to periods when inflow to the lake was probably at extremes (e.g. drought or flood) indicating that fluctuations in river inflow may be an important factor affecting sediment composition in Pyramid Lake.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43069/1/10933_2004_Article_BF00678089.pd
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