Topological entropy and secondary folding

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.Comment: 13 pages, 8 figures. pdfLaTeX with RevTeX4 macro

Past and present distribution, densities and movements of blue whales <i>Balaenoptera musculus</i> in the Southern Hemisphere and northern Indian Ocean

1Blue whale locations in the Southern Hemisphere and northern Indian Ocean were obtained from catches (303 239), sightings (4383 records of =8058 whales), strandings (103), Discovery marks (2191) and recoveries (95), and acoustic recordings.2Sighting surveys included 7 480 450 km of effort plus 14 676 days with unmeasured effort. Groups usually consisted of solitary whales (65.2%) or pairs (24.6%); larger feeding aggregations of unassociated individuals were only rarely observed. Sighting rates (groups per 1000 km from many platform types) varied by four orders of magnitude and were lowest in the waters of Brazil, South Africa, the eastern tropical Pacific, Antarctica and South Georgia; higher in the Subantarctic and Peru; and highest around Indonesia, Sri Lanka, Chile, southern Australia and south of Madagascar.3Blue whales avoid the oligotrophic central gyres of the Indian, Pacific and Atlantic Oceans, but are more common where phytoplankton densities are high, and where there are dynamic oceanographic processes like upwelling and frontal meandering.4Compared with historical catches, the Antarctic (‘true’) subspecies is exceedingly rare and usually concentrated closer to the summer pack ice. In summer they are found throughout the Antarctic; in winter they migrate to southern Africa (although recent sightings there are rare) and to other northerly locations (based on acoustics), although some overwinter in the Antarctic.5Pygmy blue whales are found around the Indian Ocean and from southern Australia to New Zealand. At least four groupings are evident: northern Indian Ocean, from Madagascar to the Subantarctic, Indonesia to western and southern Australia, and from New Zealand northwards to the equator. Sighting rates are typically much higher than for Antarctic blue whales.6South-east Pacific blue whales have a discrete distribution and high sighting rates compared with the Antarctic. Further work is needed to clarify their subspecific status given their distinctive genetics, acoustics and length frequencies.7Antarctic blue whales numbered 1700 (95% Bayesian interval 860–2900) in 1996 (less than 1% of original levels), but are increasing at 7.3% per annum (95% Bayesian interval 1.4–11.6%). The status of other populations in the Southern Hemisphere and northern Indian Ocean is unknown because few abundance estimates are available, but higher recent sighting rates suggest that they are less depleted than Antarctic blue whales.</li

Search for leptophobic Z ' bosons decaying into four-lepton final states in proton-proton collisions at root s=8 TeV

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Search for black holes and other new phenomena in high-multiplicity final states in proton-proton collisions at root s=13 TeV

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Searches for invisible decays of the Higgs boson in pp collisions at root S=7, 8, and 13 TeV

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Measurements of differential production cross sections for a Z boson in association with jets in pp collisions at root s=8 TeV

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Measurement of the mass difference between top quark and antiquark in pp collisions at root s=8 TeV

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Search for high-mass diphoton resonances in proton-proton collisions at 13 TeV and combination with 8 TeV search

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Search for heavy resonances decaying into a vector boson and a Higgs boson in final states with charged leptons, neutrinos, and b quarks

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