On Liouville decompositions in local fields

Burger, Edward B.

doi:10.1090/s0002-9939-96-03572-1

Cited by 9 publications

(10 citation statements)

References 15 publications

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“…Species of Leptostrobus (Czekanowskiaceae) also may have been a target, through which 2.5 to 5.0 mm of similar distance would have been traversed for access to micropylar fluid opposite the entry area. Modern taxa bearing tubular proboscides of this length range include the tabanid fly Mesopangonius (proboscis length ~4.5 mm) (34) and the long-tongued vespid wasp Ceramius hispanicus (~5.6 mm) (35). The third group, diminutive species of Pseudopolycentropodidae, supported proboscides ranging from 0.9 to 1.8 mm in length, with diameters sufficiently small to enable them to target plants with equally short micropylar lengths.…”

mentioning

confidence: 99%

A Probable Pollination Mode Before Angiosperms: Eurasian, Long-Proboscid Scorpionflies

Ren

Labandeira

Santiago‐Blay

et al. 2009

Science

226

177

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The head and mouthpart structures of 11 species of Eurasian scorpionflies represent three extinct and closely related families during a 62-million-year interval from the late Middle Jurassic to the late Early Cretaceous. These taxa had elongate, siphonate (tubular) proboscides and fed on ovular secretions of extinct gymnosperms. Five potential ovulate host-plant taxa co-occur with these insects: a seed fern, conifer, ginkgoopsid, pentoxylalean, and gnetalean. The presence of scorpionfly taxa suggests that siphonate proboscides fed on gymnosperm pollination drops and likely engaged in pollination mutualisms with gymnosperms during the mid-Mesozoic, long before the similar and independent coevolution of nectar-feeding flies, moths, and beetles on angiosperms. All three scorpionfly families became extinct during the later Early Cretaceous, coincident with global gymnosperm-to-angiosperm turnover.Animal pollination, most frequently accomplished by insects (1), benefits seed plants by ensuring efficient fertilization without relying on costlier, abiotic modes such as wind and water

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mentioning

confidence: 99%

A Probable Pollination Mode Before Angiosperms: Eurasian, Long-Proboscid Scorpionflies

Ren

Labandeira

Santiago‐Blay

et al. 2009

Science

226

177

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“…Now any pair (y, ξ − y) with y in the intersection consists of Liouville numbers that by construction sum up to a given ξ. The argument can be widely extended, see Rieger [39], Schwarz [43], Burger [13], [14] and Senthil Kumar, Thangadurai, Waldschmidt [42]. The second proof effectively constructs Liouville numbers x, y with the property that x + y = ξ for given ξ ∈ R. We recall this proof as well.…”

Section: Product Sets Of Liouville Numbersmentioning

confidence: 99%

“…The restriction to prime bases b is just for ease of the proof, we strongly expect the same result for arbitrary b ≥ 2. The crucial point is the lower bound n − 1 in (13). The Hausdorff dimension formula (16) dim…”

Section: Products Of Very Well Approximable Numbersmentioning

confidence: 99%

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Cartesian product sets in Diophantine approximation with large Hausdorff dimension

Schleischitz¹

2020

Preprint

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We find sets naturally occurring in Diophantine approximation whose Cartesian products exceed the expected Hausdorff dimension, that is the sum of the single dimensions. Examples include n-fold products of the set of Liouville numbers (vectors) as well as of the Diophantine numbers (vectors) with prescribed irrationality exponent, and extend to classical fractals. We also address packing dimensions of Cartesian products. Our method vastly extends ideas of Erdős.

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“…Then there exists an uncountable G δ -set E ⊆ L ∩ I such that f n (E) ⊆ L for all n ≥ 0. See also [2], [10], [25], [30] and [5] (however, as pointed out in the MathSciNet review, the proof in [5] has a small gap and it does not work in general. See Silva [31] for a recent slightly weaker result).…”

Section: 2mentioning

confidence: 99%

On a Problem Posed by Mahler

Marques¹,

Schleischitz²

2015

J. Aust. Math. Soc.

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Abstract. E. Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Kurt Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that it can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to f (z) = z q where q is a rational number.

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On Liouville decompositions in local fields

Cited by 9 publications

References 15 publications

A Probable Pollination Mode Before Angiosperms: Eurasian, Long-Proboscid Scorpionflies

A Probable Pollination Mode Before Angiosperms: Eurasian, Long-Proboscid Scorpionflies

Cartesian product sets in Diophantine approximation with large Hausdorff dimension

On a Problem Posed by Mahler

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