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A verified method for solving piecewise smooth initial value problems

Auer E., Kiel S., Rauh A.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 4

731--747

EN

In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

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Further Archaeogastropoda from the Campanian of Torallola, northern Spain

Kiel S., Bandel K.

Acta Geologica Polonica

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2002

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Vol. 52, nr 2

239-249

EN

18 archaeogastropod species are described from the Campanian of Torallola in north-eastern Spain, including 13 new species and one new genus. They are assigned to the Liotiinae, Colloniinae, Turbininae, Skeneidae and Cirridae. Our species of the Liotiinae are comparable with Recent as well as Triassic liotiids, the documented colloniids show close relations to moder species. Certain members of the Angaria-Astraea-Turbo-relation are difficult if not impossible to distinguish on shell characters alone, although these shells can be traced back into the Triassic. Three skeneiform species are assigned to Recent Skenea, but neither fossil nor recent Skoneidae are sufficiently well known, so our treatment remains preliminary. The new genus Torallochus is introduced for species having initially planispiral coiling that later changes to turriform coiling, and which are sculptured with strong ribs or spines on the periphery. The systematic position of this genus within the Trochidea remains uncertain. Among the Cirridae, the genusShikamancirrus in synonymized with the older name Sensuitrochus, and the youngest member of this genus is described. The new species are Pseudoliotina stinnesbecki, Arene mcleani, Homalopoma schroederi, Astraea iredalei,Astraea batalleria, Astraea hickmanae, Astraea? sohli, Skenea Wareni, Skenea suturata, Skenea torallensis, Torallochus rempensis, Torallochus pupiformis, and Torallochus discus.

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Trochidae (Archaeogastropoda) from the Campanian of Torallola in northern Spain

Kiel S., Bandel K.

Acta Geologica Polonica

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2001

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Vol. 51, nr 2

137-154

EN

30 species of the archaeogastropod family Trochidae are described from the Campanian of the southern Pyrenees, 15 of them are new. The new genus Amphigibbula is introduced and the genus Chilodontoidea is renamed as Hudledonta. Apparently several trochid lineages with living species can be traced back into the Late Cretaceous. This regards the Eucyclinae, Margaritinae, and Solariellinae, and the Tegulinae with some reservations. Among the groups examined here, only the members of the Trochinae appear to be of rather different character than their modern representatives. The fossil record of the Umboniinae can go far beyond that of the Late Cretaceous and connect even to Palaeozoic genera. The 15 new species are: Eucyclomphalus reminiscencius, Calliotropis torallolensis, Calliotropis seguris, Ilerdus pyrenaeus, Eucycloscala cretacea, Hudledonta nicolae, Danilia kosslerae, Margarites kasei, Margarites nielseni, Margarites kowalkei, Tectus quinteroi, Thoristella marshalli, Suavotrochus ponsi, Ethalia vinxae, and Protorotella herberti.