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  • Ultrastructure  (348)
  • CR: 5.17  (98)
  • Springer  (446)
  • American Geophysical Union (AGU)
  • Annual Reviews
  • 1980-1984  (446)
  • 1935-1939
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  • Springer  (446)
  • American Geophysical Union (AGU)
  • Annual Reviews
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  • 1
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    Development genes and evolution 192 (1983), S. 171-178 
    ISSN: 1432-041X
    Keywords: Differentiation ; Digestive tract ; Endoderm ; Organ culture ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Summary The self-differentiation potency of the endoderm of the chick embryo was investigated mainly by transmission electron microscopy. Endodermal fragments isolated from 4- to 6-day stomach or small intestine were cultured in the absence of mesenchyme and were able to differentiate in vitro into organ-specific epithelia. Endodermal fragments isolated from the stomach region differentiated into a pseudo-stratified epithelium with periodic acid Schiff-positive mucous granules in the apical cytoplasm, while those from the small intestinal region differentiated into a simple columnar epithelium with a striated border which was positive in alkaline phosphatase activity. These features are comparable with those of the mucous secretory epithelium of the normal embryonic stomach and the absorptive epithelium of normal embryonic small intestine, respectively. Next, the self-differentiation potencies were investigated of the upper and lower layers of the blastoderms, at stages 1–5 of Hamburger and Hamilton (H. and H.). Both stomach-type and small-intestine-type epithelia developed only when fragments of the lower layer isolated from the blastoderms older than stage 3 of H. and H. were cultured, suggesting that cells possessing the potency to differentiate into the stomach- and small-intestine-type epithelia exist in the definitive endoderm at the beginning of its formation.
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  • 2
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    Development genes and evolution 192 (1983), S. 42-44 
    ISSN: 1432-041X
    Keywords: Chick embryo ; Gastrulation ; Adenylate cyclase ; cAMP phosphodiesterase ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Summary The ultrastructural localization of adenylate cyclase (E.C. 4.6.1.1.) and cAMP phosphodiesterase (PDE) (E.C. 3.1.4.17.) in the ectoderm of the developmental stage 4 chick embryo was studied. Adenylate cyclase was localized in the lateral surfaces of the ectodermal cells. In the primitive streak cells the enzymatic activity was observed on all the lateral surfaces, whereas in the periphery of the blastoderm the reaction product was localized in the apical parts of the lateral plasma membranes only. cAMP PDE localized in the apical cytoplasm of the ectodermal cells, with highest activity in the globular projections.
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  • 3
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    Development genes and evolution 188 (1980), S. 65-73 
    ISSN: 1432-041X
    Keywords: Nuclear migration ; Cleavage ; Microtubules ; Ultrastructure ; Gall midge
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Summary In the eggs ofWachtliella persicariae the cleavage nuclei move relative to the surrounding ooplasm. This ‘active’ migration is caused by an organelle whose ultrastructure was studied throughout the mitotic cycle. It consists of a greatly enlarged polar cytaster derived from the mitotic apparatus, linked to the nucleus by 100 Å filaments. The microtubules of the cytaster were found only during periods of active nuclear migration, i.e., from the onset of anaphase to the early prophase of the next mitotic cycle. They are always solitary and follow the course of the astral rays, which are known to temporarily adhere to peripheral structures of the egg cell and to exert tractive forces. In contrast to the cytaster microtubules, the microtubules in the spindle are bundled and persist from early metaphase through late telophase. During ontogenesis the first migration cytaster is built up between 3 and 12 min after oviposition near the anterior egg pole, in the vicinity of the sperm nucleus. In non-inseminated eggs time lapse films show a migration cytaster to develop autonomously in a region free from nuclei, but it does not follow the normal path of the male pronucleus. In several cases the female pronucleus, which remains without a cytaster of its own, was observed to move to the cytaster generated in the absence of the male pronucleus. Whether or not it is adhering to a nucleus, the cytaster divides into two at the correct time, i.e, corresponding to the first cleavage division in fertilized eggs. In some non-inseminated eggs this type of ‘pseudocleavage’ has been observed to occur repeatedly, giving rise to an increasing number of anucleate cytasters.
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  • 4
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    Development genes and evolution 188 (1980), S. 163-177 
    ISSN: 1432-041X
    Keywords: Yolk sac ; Ultrastructure ; Embryogenesis ; Drosophila
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Summary Changes at the ultrastructural level during germ band extension in the embryo ofDrosophila melanogaster are described. Cytoplasmic connections between cells and the yolk sac are present during initial cellular movements. At this time, a continuous system of microfilaments is present adjacent to the membranes in the connections and at the periphery of the yolk sac. As germ band extension progresses, this system becomes discontinuous, and microfilaments are apparent only in the immediate vicinity of the connections. Cytoplasmic connections are disassembled at approximately the midpoint of extension; at the same time, extensive membrane associations develop between germ band cells and between these cells and adjacent yolk sac membranes. Positioning and orientation of cytoplasmic connections suggest that the yolk sac, via these connections, is actively involved in the cellular movements of early germ band extension.
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  • 5
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    Calcified tissue international 36 (1984), S. 550-555 
    ISSN: 1432-0827
    Keywords: Enamel crystals ; Length ; Shape ; Apatite ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Summary An original method for fractionating and preparing isolated crystals of homogeneous size was developed. It was demonstrated that enamel apatite crystals are at least 100 µm long. The flexibility of the very long crystallites was demonstrated. Crystal curvatures, accounting for the irregular course of the prisms through the enamel thickness, were visualized and measured. It was shown that in the deep forming enamel layer, lateral branches may grow out of the crystals and crystal fusing often occurs, inducing the crystallites to assume pyramidal shapes with their wide bases pointing toward the dentino-enamel junction and one or two tops toward Tomes' processes. During the maturation process, the two tops of the still immature crystals also fuse so that the mature crystals acquire a rodlike aspect, with parallel faces and steplike graduations along thec axis, allowing a close contact between the crystals. These results support the hypothesis that the crystallites would be continuous from the dentino-enamel junction to the surface.
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  • 6
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    Calcified tissue international 34 (1982), S. 273-279 
    ISSN: 1432-0827
    Keywords: Odontogenesis ; Ultrastructure ; Alkaline phosphatase
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Summary The ultrastructural localization and gradient of activity of alkaline phosphatase were studied with respect to cell differentiation, matrix synthesis, and matrix mineralization in the incisor and molar teeth of 4-day-old Sprague-Dawley rats. The animals were perfused intracardially at room temperature with 2.5% glutaraldehyde in 0.1M sodium cacodylate (pH 7.4) with 3–4% sucrose. The jaws were dissected, immersion-fixed for 24 h, and the incisor and molar tooth germs removed. These were demineralized in 10% EDTA in NaOH (pH 7.4) with 7% sucrose. After reactivation of the enzyme with 0.1M MgCl in Tris-maleate buffer (pH 7.4) at 4°C, the teeth were incubated for alkaline phosphatase in a medium consisting of 6 ml 3% sodiumβ-glycerophosphate, 4 ml 0.2M Tris-HCl buffer (pH 9.2), 3 ml 1.6% MgSO4, 12 ml 0.5% lead citrate (pH⋍12), and 2.1 g sucrose. The pH was adjusted to 9.2 with 0.2M HCl, the volume made up to 30 ml, and the solution centrifuged for 10 min at 5000 rpm. Control teeth were incubated in medium minus the substrate. Finally, the specimens were routinely post-fixed and embedded for sectioning and examination with a Philips 300 electron microscope. A gradient of alkaline phosphatase activity was mapped along the developing teeth in the cells of the stratum intermedium, the proximal borders of the ameloblasts, the early dentine matrix, the predentine-dentine border, matrix vesicles, and the plasma membranes of odontoblasts and subodontoblast cells. The gradient of alkaline phosphatase activity was evident in the forming tooth from the cervical loop to the crown apex and was related to the cellular events, matrix synthesis, and matrix mineralization occurring during odontogenesis.
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  • 7
    ISSN: 1432-072X
    Keywords: Arthrobacter ; Facultative methylotroph ; Amine oxidase ; Catalase ; RuMP cycle of formaldehyde fixation ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract A facultative methylotrophic bacterium was isolated from enrichment cultures containing methylamine as the sole carbon source. It was tentatively identified as an Arthrobacter species. Extracts of cells grown on methylamine or ethylamine contained high levels of amine oxidase (E.C. 1.4.3.) activity. Glucose- or choline-grown cells lacked this enzyme. Oxidation of primary amines by the enzyme resulted in the formation of H2O2; as a consequence high levels of catalase were present in methylamine-and ethylamine-grown cells. The significance of catalase in vivo was demonstrated by addition of 20 mM aminotriazole (a catalase inhibitor) to exponentially growing cells. This completely blocked growth on methylamine whereas growth on glucose was hardly affected. Cytochemical studies showed that methylamine-dependent H2O2 production mainly occurred on invaginations of the cytoplasmic membrane. Assimilation of formaldehyde which is generated during methylamine oxidation was by the FBP variant of the RuMP cycle of formaldehyde fixation. The absence of NAD-dependent formaldehyde and formate dehydrogenases indicated the operation of a non-linear oxidation sequence for formal-dehyde via hexulose phosphate synthase. Enzyme profiles of the organism grown on various substrates suggested that the synthesis of amine oxidase, catalase and the enzymes of the RuMP cycle is not under coordinate control.
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  • 8
    ISSN: 1432-072X
    Keywords: Methanogenium tatii ; Ultrastructure ; Physiology ; Glycoproteins ; DNA-DNA Homology ; Taxonomy ; Archaebacteria
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract A new coccoid methanogen, Methanogenium tatii, was isolated and characterized. The mesophilic isolate can grow on and produce methane from H2:CO2 and formate. For growth acetate is strictly required. The cell shape, the G+C content of 54 mol% and DNA-DNA homology data suggest it to be a Methanogenium species.
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  • 9
    ISSN: 1432-072X
    Keywords: Nitrobacter hamburgensis ; Nitrite oxidoreductase ; Nitrate reductase ; Molybdenum iron-sulfur protein ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract Nitrite oxidoreductase, the essential enzyme complex of nitrite oxidizing membranes, was isolated from cells of the nitrifying bacterium Nitrobacter hamburgensis. The enzyme system was solubilized and purified in the presence of 0.25% sodium deoxycholate. Nitrite oxidoreductase oxidized nitrite to nitrate in the presence of ferricyanide. The pH optimum was 8.0, and the apparent K m value for nitrite amounted to 3.6 mM. With reduced methyl-and benzylviologen nitrite oxidoreductase exhibited nitrate reductase activity with an apparent K m value of 0.9 mM for nitrate. NADH was also a suitable electron donor for nitrate reduction. The pH optimum was 7.0. Treatment with SDS resulted in the dissociation into 3 subunits of 116,000, 65,000 and 32,000. The enzyme complex contained iron, molydbenum, sulfur and copper. A c-type cytochrome was present. Isolated nitrite oxidoreductase is a particle of 95±30 Å in diameter.
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  • 10
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    Archives of microbiology 130 (1981), S. 204-212 
    ISSN: 1432-072X
    Keywords: Agmenellum quadruplicatum ; Nitrogen starvation ; Ultrastructure ; PATO poststain ; Cyanobacteria
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract The effects of nitrogen limitation on the ultrastructure of the unicellular cyanobacterium, Agmenellum quadruplicatum, were studied by thin sectioning transmission electron microscopy. Nitrogen became limiting for growth 14–15 h after transfer to nitrogen-limiting medium, but cultures retained full viability for at least 45 h. The c-phycocyanin: chlorophyll a ratio and cellular nitrogen content of the culture dropped rapidly after 14–15 h, as a progressive deterioration of major cell structures took place. Phycobilisomes were degraded first, followed by ribosomes and, then, thylakoid membranes. These structures were virtually depleted from the cells within 26 h. Intracellular polysaccharide accumulated in place of the normal cell structures throughout this period. Nitrogen limitation did not affect polyphosphate bodies, carboxysomes, lipid granules, the cell envelope, or the extra-cellular glycocalyx. All of the ultrastructural changes resulting from nitrogen limitation were reversed upon addition of nitrate to a starved culture. Most cell structures were restored within 3 h, and restoration was complete within 9 h.
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  • 11
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    Archives of microbiology 134 (1983), S. 295-298 
    ISSN: 1432-072X
    Keywords: Actinomycetes ; Streptomyces thermoviolaceus ; Sporogenesis ; Spore ornamentation ; Cupular knobs ; Sheath ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract The sporogenesis of aerial spores in Streptomyces thermoviolaceus corresponded to a common sporulation type in the genus. The sporulation septum was composed of an outer ring-shaped constriction wall and an inner interspace septum arising by the inwards growth of a double annulus. In mature spores the wall was composed of two layers, the outer one was part of the parent hyphal wall and septum material, the inner one was formed de novo. The spore chains were enclosed by the thin breakable sheath containing small rod-like elements. The ornamentation in the form of knobs, which were a characteristic feature of the species originated from the sheath. The knobs were hemispherical particles with an inner electron dense core and an outer electron transparent shell. The term “cupular knobs” was suggested for this type of tuberculate ornamentation. Frequently, the knobs became detached from the surface in which case the inner core separated easily from the shell.
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  • 12
    ISSN: 1432-072X
    Keywords: Claviceps purpurea ; Ultrastructure ; Development ; Sclerotium ; Oleosomes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract The development of sclerotia of Claviceps purpurea was investigated by light and electron microscopy. During the first days after infection sterigma and conidiospores are formed. The spores show a moderately developed vacuolar system, they are thick walled and contain about 20% lipid (related to the cell volume) embedded in glycogen. The sterigma are cylindrical unicellular hyphae with electron dense cytoplasm and isolated strongly contrasted lipid droplets. In maturing sclerotia the hyphae become septated with increasingly thick cell walls and a large lipid content. The lipid forms small droplets in young cells, while in the mature sclerotium it occurs in the form of very large drops, occupying the major part of the cell. Simultaneously the composition of the lipid is changed. The mature cells have several nuclei. They are partially connected by osmiophilic substances, forming a network of intercellular spaces.
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  • 13
    ISSN: 1432-072X
    Keywords: Mating tube ; Microtubule ; Tremella ; Ultrastructure ; Yeast
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract Ultrastructure of the mating tube formed in yeast haplont of the heterobasidiomycete Tremella mesenterica was studied by electron microscopy. Cell wall of the mating tube emerged as evagination of the inner layers, rupturing outer layers of the mother cell wall. Comparison with budding cells suggested that the tube emergence place at bud scar and the process of tube emergence was the same as that of bud emergence. Electron transparent vesicles of 0.1 μm diameter were scattered in the cytoplasm of the mating tube. Nucleus-associated organelle was located at one side of the nuclear envelope which extended towards the mating tube. A few microtubules were detected in the mating tube, but their association with a nucleus was not clear. The cytoplasmic structure of the mating tube was discussed in comparison with that of hyphae of the filamentous fungi.
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  • 14
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    Archives of microbiology 133 (1982), S. 11-19 
    ISSN: 1432-072X
    Keywords: Cyanobacteria ; Ultrastructure ; Mastigocladus laminosus ; Fischerella ; True branching
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract The morphology and ultrastructure of the thermophilic cyanobacteriumMastigocladus laminosus were examined by scanning and transmission electron microscopy. Mature cultures consisted of relatively old, wide filaments that branched frequently to form younger, thinner filaments. The cells of the younger filaments had a consistently cylindrical morphology, while those of older filaments were rounded and pleomorphic. The internal ultrastructure of the cells depended somewhat on their age. As young cells became larger and wider, their thylakoids underwent slight rearrangement and spread out toward the center of the cytoplasm. Polyphosphate bodies, carboxysomes (polyhedral bodies), and lipid-body-like structures increased in number as the cells aged, but ribosomes and cyanophycin granules were depleted. Cell division involved septum formation followed by ingrowth of the outer membrane and sheath. Cells in older filaments were separated from each other by a complete layer of sheath material. Septum formation in older cells was also seen to occur parallel to the long axis of the filament, thereby confirming that true branching took place.
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  • 15
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    Archives of microbiology 138 (1984), S. 229-232 
    ISSN: 1432-072X
    Keywords: Actinomycetes ; Streptomyces torulosus ; Morphology ; Ultrastructure ; Verrucate spores ; Knobby ornamentation ; Sheath
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract The type strain of Streptomyces torulosus Lyons and Pridham (1971) was studied by scanning- and transmission electron microscope. Spore chains were formed in spirals by aerial mycelium. The spores were connected by nozzles in which small channels could be observed. The knobby ornamentations of the spores arised on a thin fibrous sheath, enveloping the spore chains. These irregular blunt projections, called knobs, had varying diameters of 100 to 250 nm. The base of the knob, consisting of globose to flattened electron dense material, was sitting directly on the sheath. It was covered by several small vesicles of the same material. Each hollow vesicle beared a thin bowlshaped shell of electron transparent material. In general, the cupular bowls and their supporting vesicles became easily depressed on their base, but not detached from the surface of the spores. This type of knobby spore ornamentation was suggested to be designated as a verrucate spore type.
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  • 16
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    Archives of microbiology 128 (1981), S. 384-389 
    ISSN: 1432-072X
    Keywords: Didymium iridis ; Microcyst ; Excystment ; Germination ; Ultrastructure ; Mycetozoa ; Myxomycetes ; Myxamoeba
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract Microcysts of the myxomycete Didymium iridis were induced to excyst by transfer to 5mM potassium phosphate buffer. After 1 h in suspension, 90% of the microcysts had germinated into myxamoebae distinguishable by phase contrast microscopy and staining with Lugol's iodine. Both pH and osmolarity affected the kinetics of excystment. The rate and extent of excystment were decreased by cycloheximide but remained unaffected by actinomycin D, suggesting a requirement for protein synthesis but not RNA synthesis. Initially, the outer wall layers separated from the inner layer, which gradually expanded and loosened. The protoplast rehydrated and reverted to a vegetative morphology. Excysting cells were characterized by nucleolar inclusions, changes in the nuclear envelope and plasma membrane, appearance of ringed cisternal elements and microbodies in the cytoplasm, and formation of a densely fibrous zone adjacent to the site of emergence. Excysting populations have been classified into characteristic stages: mature, initiated, swollen, and pre-emergent microcysts.
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  • 17
    ISSN: 1432-072X
    Keywords: Cyanobacteria ; Ultrastructure ; Nitrogen fixation ; Water stress ; Taxonomy ; DNA ; Plasmids
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract Two strains of desiccation-tolerant coccoid cyanobacteria, Chroococcus S24, a marine form, and Chroococcus N41, a cryptoendolith isolated from a hot-desert rock, have been characterized. The mol % DNA base compositions of the strains are 47.1 and 48.9% respectively. Plasmid DNA was not detected in either strain. The pigment contents and nutritional characteristics of the strains are identical. Both lack phycoerythrinoid pigments and, in culture, behave as slow-growing halotolerant marine forms with elevated requirements for Na+, Cl−, Mg2+ and Ca2+. Sucrose was the only carbon source of those tested that supported photoheterotrophic growth. Each strain synthesizes nitrogenase under anaerobic conditions but not in air. Morphologically the two strains are indistinguishable. They are considered to be independent isolates of the same cyanobacterial species. Chroococcus N41 was studied in detail with the electron microscope. When brought to equilibrium at matric water potentials of-168 MPa and lower (to-673 MPa=c0.12a w) the protoplast shrinks, but the cells maintain the same size and diameter as those at-2,156 kPa (MN medium; control); the sheath expands and remains attached to the cell wall outer membrane by fibrils. The cell wall, cell membrane, thylakoid membranes, cyanophycin granules and carboxysomes appeared intact in desiccated cells.
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  • 18
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    Calcified tissue international 34 (1982), S. 382-390 
    ISSN: 1432-0827
    Keywords: Avian osteopetrosis ; Avian oncornavirus ; Ultrastructure ; Calcification ; Bone cells
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Summary Diaphyseal tibial bone of 12.5 – 13-day and 19-day-old embryos and 20-day-old hatched chicks infected with retrovirus MAV.2-O were examined by transmission electron microscopy. The viruses were associated with lining osteoblasts and osteocytes. Whereas the infection of the osteoblast layer seemed to be a transient stage, virus association with osteocytes was a constant and main ultrastructural feature. The viruses were found either in the osteoid or in the periosteocytic space of the bone lacunae. They arose from dense cytoplasmic areas located near the cell plasmalemma via a budding process. The newly budded virus particles often had a large tail or a fine stalk-like process lost in the extracellular space. The viruses underwent calcification by deposition of inorganic material and were incorporated in the bone trabeculae. No production of virus was observed in typical osteoclasts with well-differentiated ruffled borders. The viral-induced avian osteopetrosis seemed to result from increased bone deposition through stimulation of osteoblast and osteocyte activities, whereas osteoclastic bone resorption seemed to be undisturbed.
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  • 19
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    Calcified tissue international 33 (1981), S. 603-618 
    ISSN: 1432-0827
    Keywords: Preameloblasts ; Tooth germs ; Monkey ; Enamel ; Ultrastructure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Medicine , Physics
    Notes: Summary Cytodifferentiation of inner enamel epithelium and the adjacent connective tissue from the tip of the cervical loop to the initiation of enamel elaboration in twoMacaca species was examined. Ten- to twelve-month-old specimens were fixed by perfusion and the permanent tooth buds were prepared for transmission electron microscopy. At the cervical loop proper, inner enamel epithelium cells have lobed nuclei, a paucity of cytoplasm, and wide extracellular spaces; the basal lamina facing the dental papilla is straight. With increasing distance from the tip of the cervical loop, the following changes occur gradually: (a) preameloblasts elongate from 15 to 45 µm, and their organelles, particularly mitochondria and profiles of rough endoplasmic reticulum, become more numerous; (b) extracellular spaces decrease between preameloblasts starting at the basal (infranuclear) end; (c) the basement membrane becomes convoluted and associated with aperiodic fibers; (d) preodontoblast projections penetrate the aperiodic fibers; (e) collagen fibers subjacent to the basement membrane increase in density, with particularly thick fibers paralleling the aperiodic fibers. These modifications occur within three-fourths of the distance from the tip of the cervical loop to the mineralization front. The condensation of preodontoblasts is followed immediately by predentin synthesis. Concomitantly, the basement membrane breaks down and the aperiodic fibers are engulfed by preameloblasts. Preameloblast projections penetrate junctional predentin, contact mineralized dentin, and enamel synthesis ensues. At this stage the ameloblast is 45 µm long, the nucleus is central or basal, the Golgi apparatus has migrated apically, but the Tomes' process has not yet formed. The results indicate that odontogenesis inMacaca monkeys more closely resembles human odontogenesis than does that in the murine rodents.
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  • 20
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65M25, 35L20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The ALGOL-procedure1 char2 presented in this paper can be applied to the initial or initial-boundary value problem of a quasilinear hyperbolic differential equation of second order. A method of characteristics is combined with extrapolation to the limit. Thus, the results are very accurate. The same accuracy can also be obtained if the initial values are only piecewise smooth.
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  • 21
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    Numerische Mathematik 42 (1983), S. 173-194 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.
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  • 22
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    Numerische Mathematik 42 (1983), S. 311-322 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 N 30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents Galerkin approximations for solutions of two dimensional interface problems by solving corresponding boundary integral equations. These are obtained by simple layer potential operators only. Due to the strong ellipticity of the integral equations the Galerkin procedure converges with optimal order. Smoothness of the given data implies high convergence rates for the layers.
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  • 23
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    Numerische Mathematik 43 (1984), S. 175-198 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 L 10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders. It is shown that a solution of the scheme converges, uniformly in a perturbation parameter, to that of the continuous problem.
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  • 24
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    Numerische Mathematik 36 (1980), S. 253-266 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65LO5 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Single step exponentially fitted integration formulae of orders 4 and 6 are derived. The final approximation to the required solution is obtained via a linear combination of asymptotically less accurate solutions obtained using conventional implicit integration formulae. This linear combination is performed in such a way that the final integration formula, as well as having an increased order of accuracy, integrates a certain pre-determined ordinary differential equation exactly. The algorithms developed are illustrated by means of some numerical examples.
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  • 25
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    Numerische Mathematik 44 (1984), S. 75-102 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper examines the concepts of feedback and adaptivity for the Finite Element Method. The model problem concernsC 0 elements of arbitrary, fixed degree for a one-dimensional two-point boundary value problem. Three different feedback methods are introduced and a detailed analysis of their adaptivity is given.
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  • 26
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    Numerische Mathematik 36 (1981), S. 389-403 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N 30 ; CR: 5.17
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    Notes: Summary The paper deals with nonconforming finite element methods for the approximate solution of the interior boundary value problem for Maxwell equations in the time-harmonic case. The methods are based on penalization in the boundary conditions of total reflexion. Qualitative convergence results are obtained by a-priori estimates which are proven in the first part of this paper. The main object is to establish estimates for the global discretization error in various norms of the underlying spaces of approximating vector fields.
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    Numerische Mathematik 45 (1984), S. 105-116 
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    Notes: Summary We study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.
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    Numerische Mathematik 45 (1984), S. 283-300 
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    Keywords: AMS(MOS): 65L05 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system partitioned adaptive Runge-Kutta methods are studied. Nonstiff equations are integrated by an explicit Runge-Kutta method while an adaptive Runge-Kutta method is used for the stiff part of the system. The paper discusses numerical stability and contractivity as well as the implementation and usage of such compound methods. Test results for three partitioned stiff initial value problems for different tolerances are presented.
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    Numerische Mathematik 39 (1982), S. 449-463 
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    Keywords: AMS (MOS): 65L65 ; CR: 5.17 ; AMS (MOS): 65L65 ; CR: 5.17
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    Description / Table of Contents: Summary In [10] a general procedureV is presented to obtain spline approximations by collocation for the solutions of initial value problems for first order ordinary differential equations. In this paper the attainable order of convergence with respect to the maximum norm is characterized in dependence of the parameters involved inV; in particular the appropriate choice of the collocation points is considered.
    Notes: Zusammenfassung In [10] ist ein allgemeines VerfahrenV beschrieben, das die Lösungen von Anfangswertproblemen bei gewöhnlichen Differentialgleichungen erster Ordnung durch Splines approximiert. Die Konstruktion der Splines erfolgt hierbei mittels Kollokation. In dieser Arbeit wird die maximal erreichbare Konvergenzordnung vonV bezüglich der Maximumnorm in Abhängigkeit aller Parameter vonV charakterisiert, insbesondere wird auf die geeignete Wahl der Kollokationsknoten eingegangen.
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    Numerische Mathematik 40 (1982), S. 373-406 
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    Keywords: AMS(MOS): 65N30, 65M20 ; CR: 5.17
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    Notes: Summary We extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper [3] for time-independent space meshes. In the context of the model problem studied there, results are given relating the effectiveness of the error estimator to properties of the solution, space meshes, and manner in which the meshes change. A procedure based upon this theory is presented for the adaptive construction of time-dependent meshes. The results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems.
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    Numerische Mathematik 41 (1983), S. 255-279 
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    Keywords: AMS(MOS): 65L15 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary Some methods for evaluating the characteristic exponents in connection with Newton's iteration are applied for solving the eigenvalue problem related to the finite Hill's differential equation or, in particular, Mathieu's equation. By using these methods a high accuracy is achieved, furthermore a complete error analysis, which yields rather realistic error bounds, is possible.
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    Numerische Mathematik 41 (1983), S. 345-371 
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    Topics: Mathematics
    Notes: Summary Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
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    Numerische Mathematik 41 (1983), S. 373-398 
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    Notes: Summary The paper introduces a new semi-implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations. The existence of a quadratic asymptotic expansion in terms of the stepsize is shown. Moreover, the new discretization is analyzed in the light of well-known stability models. The efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples.
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    Notes: Summary We study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere $$\mathbb{C}$$ ). It is shown that the solutions of these difference equations are bounded forn≧0, uniformly for λh∈D.S is itself closed in $$\mathbb{C}$$ iff ∂S is free of cusps. The question is studed by means of contractivity analysis and a matrix theorem, derived from the matrix theorem of Kreiss.
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    Numerische Mathematik 42 (1983), S. 15-30 
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    Keywords: AMS: 65L10 ; 34C25 ; 34K10 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary An iterative method is presented which starting from a lower or from an upper periodic solution, provides a monotone sequence converging to a periodic solution of (1). With some restrictions on the growth off, the method extends to functional differential equations of type (1′). Two numerical examples with an “a posteriori” error analysis are given.
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    Numerische Mathematik 34 (1980), S. 41-62 
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    Topics: Mathematics
    Notes: Abstract This paper presents a new approach to the analysis of finite element methods based onC 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL 2 norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL 2 norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.
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    Numerische Mathematik 34 (1980), S. 235-246 
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    Topics: Mathematics
    Notes: Summary A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(α)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
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    Numerische Mathematik 34 (1980), S. 457-467 
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    Topics: Mathematics
    Notes: Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.
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    Numerische Mathematik 35 (1980), S. 13-20 
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    Keywords: AMS (MOS): primary 65M05 ; 65M10 ; 65M15 ; secondary: 35M05 ; CR: 5.17
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    Notes: Summary We consider the numerical solution of the Tricomi problem. Using a weak formulation based on different spaces of test and trial functions, we construct a new Galerkin procedure for the Tricomi problem. Existence, uniqueness, and uniform stability of the approximate solution is proven, and a priori error bounds are given.
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    Numerische Mathematik 35 (1980), S. 21-33 
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    Topics: Mathematics
    Notes: Summary The trapezoidal rule with deferred corrections using uncentered end formulas is shown to converge. While the proof technique is more specialized than the standard asymptotic expansion approach, it has some advantages. In addition to providing a more complete theoretical justification for current implementations of deferred corrections with the trapezoidal rule, the approach given here will hopefully apply for several other discretization methods.
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    Numerische Mathematik 35 (1980), S. 57-68 
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    Notes: Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA 0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.
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    Numerische Mathematik 35 (1980), S. 143-162 
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    Notes: Summary We study the error due to the discretization in time of a nonlinear parabolic problem by a multistep method. Error estimates are obtained if the method is of the orderp (p〉1) and stronglyA(Θ)-stable $$\left( {0〈 \Theta〈 \frac{\pi }{2}} \right)$$ . The method is also applied to the Navier-Stokes equations in two dimensions.
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    Numerische Mathematik 35 (1980), S. 127-142 
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    Keywords: AMS (MOS): 34G05, 65L05 ; CR: 5.17
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    Notes: Summary A class of approximation schemes of arbitrary accuracy, generated by a two-step recurrence relation, is devised for evolution equations of the second order. The schemes are effected via a specially constructed family of rational approximations to cos τ for τ≧0 and yield computationally efficient methods for systems of second-order ordinary differential equations and semidiscrete approximations for initial-boundary value problems for second-order hyperbolic equations.
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    Numerische Mathematik 35 (1980), S. 231-240 
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    Notes: Summary The homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.
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    Numerische Mathematik 35 (1980), S. 257-276 
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    Description / Table of Contents: Résumé Considérons une équation d'évolution parabolique linéaire associée à un opérateur linéaireA(t) dépendant du tempst. Nous développons dans cet article une méthode de discrétisation, basée sur les méthodes linéaires à pas multiples, traitant de manière implicite une partie de l'opérateurA(t) indépendante du temps, l'autre partie est traitée de manière explicite. Nous étudions la stabilité et la convergence de cette méthode.
    Notes: Summary Let us consider a linear parabolic equation which is associated with a time dependent operatorA(t). In this paper, we present a method, which is founded on linear multistep methods, which discretize a time-independent part of the operatorA(t) in an implicit way, and the other part in an explicit way. We study stability and convergence for this method.
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    Numerische Mathematik 35 (1980), S. 315-341 
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    Notes: Summary We present here some new families of non conforming finite elements in ℝ3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.
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    Numerische Mathematik 35 (1980), S. 381-404 
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    Notes: Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.
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    Numerische Mathematik 36 (1980), S. 1-25 
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    Topics: Mathematics
    Notes: Summary We begin in this paper the study of a general method of approximation of solutions of nonlinear equations in a Banach space. We prove here an abstract result concerning the approximation of branches of nonsingular solutions. The general theory is then applied to the study of the convergence of two mixed finite element methods for the Navier-Stokes and the von Kármán equations.
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    Numerische Mathematik 36 (1980), S. 33-52 
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    Notes: Summary A modified variational formulation, recently introduced by Taylor, Beresford and Wilson for solving second order problems, using the nonconforming Wilson element is here analysed. It is shown that the Patch Test is satisfied and that stresses and displacements are respectively first and second order accurate for arbitrary quadrilateral meshes.
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    Numerische Mathematik 42 (1983), S. 271-290 
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    Keywords: AMS(MOS): 65J10, 65L20, 65M10 ; CR: 5.17
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    Notes: Summary Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|≦|U(0)| (t≧0). We call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations. A givenk-step method is said to be unconditionally contractive if for each stepsizeh〉0 the numerical process is contractive. In this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity. It turns out that unconditionally contractive methods are subject to an order barrierp≦1. Further the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behaviour of methods with an orderp〉1 as well. Most theoretical results in this paper are formulated for differential equations in arbitrary Banach spaces. Applications are given to numerical methods for solving ordinary as well as partial differential equations.
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    Numerische Mathematik 43 (1984), S. 105-119 
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    Notes: Summary A new formulation of the Dirichlet problem for the biharmonic operator is presented. This gives rise to a simple numerical method to solve the above problem. Convergence is proved in the unidimensional case. Numerical results in one and two dimensional test problems are presented.
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    Description / Table of Contents: Summary This paper uses Augmented Lagrangian techniques for the numerical solution of equilibrium problems of compressible hyperelastic bodies subjected to large deformations. The resulting method is illustrated by several numerical examples.
    Notes: Résumé Cet article applique les techniques de Lagrangien Augmenté à la résolution numérique des problèmes d'équilibre de corps hyperélastiques compressibles soumis à de grandes déformations. La méthode obtenue est illustrée par plusieurs exemples numériques.
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    Numerische Mathematik 43 (1984), S. 343-360 
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    Notes: Summary Retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme. Hence one can control the global error of the modified problem but not directly the actual global error of the original problem. In this paper we give an estimate for the actual global error in terms of controllable quantities. Further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems. Along with the new definition we are led to developing a reliable basis for a step selection scheme.
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    Numerische Mathematik 43 (1984), S. 389-396 
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    Notes: Summary Stability analysis of θ-methods for functional differential equations based on the test equation $$\begin{gathered} y'(t) = ay(t - \lambda ) + by(t),t 〉 0 \hfill \\ y(t) = \psi (t),t \in \left[ { - \lambda ,0} \right] \hfill \\ \end{gathered} $$ λ〉0, is presented. It is known thaty(t)→0 ast→∞ if and only if |b|〈−a and we investigate whether this property is inherited by the numerical solution approximatingy.
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    Numerische Mathematik 43 (1984), S. 463-483 
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    Notes: Summary The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.
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    Numerische Mathematik 37 (1981), S. 157-166 
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    Topics: Mathematics
    Notes: Summary Consider the systemy′=f(x,y),y(a)=η,x∈[a,b],y∈R n wheref is continuous and Lipschitzian with respect to the second argument. Very often linear multistep variable stepsize variable formula methods (LM VSVFM's) are used to computey k≈y(xk) on the points of the grid:a=x 0〈x1〈x2〈...〈xN=b. The general LM VSVFM is based on formulae of the following type $$y_k = \sum\limits_{i = 1}^{s_k } {\alpha _i (\bar h_k ,sk)y_{k - i} } + \sum\limits_{i = 0}^{s_k } {h_{k - i} \beta _i (\bar h_k ,s_k )f(x_{k - i} ,y_{k - i} )} $$ whereh k=xk−xk−1, $$\bar h_k = (h_k ,h_{k - 1} , \ldots ,h_{k - s_k } )$$ ,s k≦k, k=1(1)N. The coefficients α i and β i depend on the lasts k+1 stepsizes and on the formula used at stepk. Only the zero-stability properties of some special classes of LM VSVFM's (as for example those based on Adams formulae) were investigated in the literature. A class of three-ordinate LM VSVFM's is defined in this paper. Some results concerning the zero-stability properties of these methods are proved. It is shown that some well-known results are simple corollaries of the results found for the three-ordinate LM VSVFM's. It is easily seen that similar results hold for the corresponding one-leg VSVFM's. Finally, the use of the theoretical results in the practical implementations is briefly discussed.
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    Numerische Mathematik 38 (1981), S. 255-261 
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    Notes: Summary This paper extends the earlier results by the author on two-dimensional free boundary problems. The main aim consists in derivation of an optimal error bound for the approximations of the free boundary.
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    Numerische Mathematik 38 (1981), S. 279-298 
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    Notes: Summary This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.
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    Numerische Mathematik 38 (1982), S. 365-382 
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    Notes: Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.
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    Numerische Mathematik 38 (1982), S. 447-453 
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    Notes: Summary As is known [4]. theC o Galerkin solution of a two-point boundary problem using piecewise polynomial functions, hasO(h 2k ) convergence at the knots, wherek is the degree of the finite element space. Also, it can be proved [5] that at specific interior points, the Gauss-Legendre points the gradient hasO(h k+1) convergence, instead ofO(h k ). In this note, it is proved that on any segment there arek−1 interior points where the Galerkin solution is ofO(h k+2), one order better than the global order of convergence. These points are the Lobatto points.
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    Numerische Mathematik 38 (1982), S. 467-471 
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    Keywords: AMS(MOS) 65L20 ; CR: 5.17
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    Notes: Summary The approximation of linear systemsy′=−A(t)y+b(t) by backward differentiation methods up to order 5 is considered. It is proved that the error does not increase if the real symmetric matrixA(t) is positive definite andA′(t) is negative semi-definite.
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    Numerische Mathematik 39 (1982), S. 15-37 
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    Keywords: AMS(MOS): 65 N 30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary We make several assumptions on a nonlinear evolution problem, ensuring the existence of a Hopf bifurcation. Under a fairly general approximation condition, we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions.
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    Numerische Mathematik 39 (1982), S. 39-50 
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    Topics: Mathematics
    Notes: Summary In this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems. Furthermore, approximations of stress intensity factors are given. The infinite element method may be considered as a certain scheme of mesh refinement, but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently, and that an approximate solution which has a singularity at the singular point can be also obtained.
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    Numerische Mathematik 39 (1982), S. 221-230 
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    Keywords: AMS(MOS): 65L05 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a “non-smooth” solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.
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    Numerische Mathematik 39 (1982), S. 309-324 
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    Keywords: AMS (MOS): 65L10 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary A method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems. Uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s. An improvement by a factor ofε n+1 can be obtained (where ɛ is the “small” parameter andn is the order of the asymptotic approximation) for a small amount of extra work. Numerical experiments are presented.
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    Numerische Mathematik 39 (1982), S. 341-350 
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    Topics: Mathematics
    Notes: Summary We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x α y′)′=f(x,y), y(0)=A, y(1)=B, 0〈α〈1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all α∈(0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM 1 based on just one evaluation off. For uniform mesh we obtain two methodsM 2 andM 3 each based on three evaluations off. For α=0,M 1 andM 2 both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h 2)-convergence established and illustrated by numerical examples.
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    Numerische Mathematik 40 (1982), S. 169-177 
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    Notes: Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.
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    Numerische Mathematik 40 (1982), S. 319-328 
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    Keywords: AMS (MOS): 65J05, 65L15 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf. [7, II, VII]) to multiparameter eigenvalue problmes, formulate in the framework of discrete approximation a convergent numerical treatment, establish algebraic bifurcation equations for the intersection points of the eigenvalue curves and illustrate this with some numerical examples.
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    Numerische Mathematik 40 (1982), S. 329-337 
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    Topics: Mathematics
    Notes: Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.
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    Numerische Mathematik 40 (1982), S. 339-371 
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    Keywords: AMS(MOS): 65N30 ; 65M20 ; CR: 5.17
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    Notes: Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.
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    Numerische Mathematik 41 (1983), S. 165-175 
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    Keywords: AMS(MOS): 65LO5 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary The extrapolated midpoint rule is a popular way to solve the initial value problem for a system of ordinary differential equations. As originally formulated by Gragg, the results are smoothed to remove the weak instability of the midpoint rule. It is shown that this smoothing is not necessary. A cheaper smoothing scheme is proposed. A way to exploit smoothing to increase the robustness of extrapolation codes is formulated.
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    Numerische Mathematik 42 (1983), S. 65-76 
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    Keywords: AMS(MOS): 65 N 20 ; CR: 5.17
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    Notes: Summary We consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces
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    Numerische Mathematik 42 (1983), S. 51-64 
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    Keywords: AMS (MOS): 65N99 ; 35L05 ; CR: 5.17
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    Notes: Summary Engquist and Majda [3] proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. (In the case of the wave equation this boundary condition is valid at all frequencies.) Here, least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. It is shown that for the wave equation this approach leads to Kreiss well-posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of Taylor- and Padé-approximations [3, 4]. Numerical examples indicate that this method works even more effectively for hyperbolic systems. The least-squares approach may be used to generate the boundary conditions automatically.
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    Numerische Mathematik 42 (1983), S. 77-95 
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    Notes: Summary Brakhage and Werner, Leis and Panich suggested to reduce the exterior Dirichlet boundary value problem for the Helmholtz equation to an integral equation of the second kind which is uniquely solvable for all frequencies by seeking the solution in the form of a combined double- and single-layer potential. We present an analysis of the appropriate choice of the parameter coupling the double- and single-layer potential in order to minimize the condition number of the integral operator.
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    Numerische Mathematik 42 (1983), S. 119-123 
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    Notes: Summary Many difference methods for the numerical solution of elliptic boundary value problems lead to systems of linear equations whose matrices areM-matrices and which therefore have nonnegative inverses. In this paper it is shown, that these difference methods are at most consistent of second order.
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    Numerische Mathematik 34 (1980), S. 171-187 
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    Notes: Summary Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called “Mehrstellenformeln’ or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.
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    Numerische Mathematik 34 (1980), S. 29-40 
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    Keywords: AMS(MOS): 65N25 ; CR: 5.17
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    Notes: Summary A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=λTu. The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.
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    Numerische Mathematik 42 (1983), S. 299-310 
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    Notes: Summary A new approach to the problem of numerically integrating stiff differential systems is described. In this approach a linear multistep method (the basic method) is split into a kind of predictor-corrector scheme, where the predictor is also implicit. If this splitting is done in an appropriate manner, the modified method has considerably better stability properties than the basic method. As a result, splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties. In particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed.
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    Numerische Mathematik 42 (1983), S. 359-377 
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    Keywords: AMS 65L05 ; 65L07 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary This paper is concerned with the stability of multistep methods for ordinary initial-value problems on grids with variable mesh-sizes. A necessary and sufficient condition for stability is given from which generalizations of recent results by Gear et al. and by Zlatev can be obtained as special cases. As an application the stability of the variable BDF-formulas is treated.
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    Numerische Mathematik 36 (1980), S. 319-331 
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    Notes: Summary Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.
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    Notes: Summary The paper deals with such estimates of the rate of convergence of difference methods, which are compatible with the smoothness of the exact solutionu ∈ W 2 m (Ω),m〉0.5, of elliptic equations with mixed derivatives: The error in the norm of the discrete Sobolev spaceW 2 s (ω), ω denoting the set of grid points, is shown to be of the orderO(|h| m−s), 0≦s〈m.
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    Numerische Mathematik 44 (1984), S. 247-259 
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    Keywords: AMS (MOS): 65L05 ; 65M05 ; 65M20 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary The aim of this paper is to study contractivity properties of two locally one-dimensional splitting methods for non-linear, multi-space dimensional parabolic partial differential equations. The term contractivity means that perturbations shall not propagate in the course of the time integration process. By relating the locally one-dimensional methods with contractive integration formulas for ordinary differential systems it can be shown that the splitting methods define contractive numerical solutions for a large class of non-linear parabolic problems without restrictions on the size of the time step.
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    Numerische Mathematik 44 (1984), S. 285-300 
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    Notes: Summary In this paper, we investigate the numerical asymptotic behavior of the finite element solutions for linear parabolic equations under some appropriate conditions. We also give some results of numerical experiments in the two dimensional problems to indicate the effectiveness of our results.
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    Numerische Mathematik 44 (1984), S. 191-200 
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    Keywords: AMS (MOS): Primary 65M05, 65M10, 65M15 ; Secondary: 35M05 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary We consider the numerical solution of the boundary value problem (1) $$(1) Lu = k(y)u_{xx} + u_{yy} - c(x,y)u = f, u|_{\Gamma _0 \cup \Gamma _1 } = 0,$$ where $$k(y) \gtreqless 0$$ for $$y \gtreqless 0$$ Г 0 andГ 1 are parts of the boundary of a bounded simply connected regionG inR 2.G is bounded fory〉0 by a piecewise smooth curveГ 0 which intersects the liney=0 at the pointsA(−1,0) andB(0, 0). Fory〈0,G is bounded by a piecewise smooth curveГ 1 throughA, which meets the characteristic of (1) issued fromB at pointC, and by the curveГ 2 which consists of the portionCB of the characteristic throughB. Using a weak formulation based on different spaces of test and trial functions, we construct a Galerkin procedure for the above boundary value problem. Existence, uniqueness and uniform stability of an approximate solution is proven and a priori error bounds are given.
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    Numerische Mathematik 45 (1984), S. 51-74 
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    Keywords: MR65 M15 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary The error of the approximate solution obtained by discretising a functional equation can be shown under certain conditions to possess an asymptotic expansion in terms of some parameter which is usually a representative step-length. We consider the case of two-parameter expansions, which is particularly relevant to parabolic equations. We derive results for the existence of the expansion and for the application of the classical difference correction and of defect correction. The theory is illustrated by the discussion of a simple parabolic problem
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    Numerische Mathematik 45 (1984), S. 201-206 
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    Topics: Mathematics
    Notes: Summary Bulirsch and Stoer have shown how to construct asymptotic upper and lower bounds on the true (global) errors resulting from the solution by extrapolation of the initial value problem for a system of ordinary differential equations. It is shown here how to do this for any one-step method endowed with an asymptotically correct local error estimator. The one-step method can be changed at every step.
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    Numerische Mathematik 37 (1981), S. 235-255 
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    Keywords: AMS(MOS): 65L05 ; 65Q05 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically. Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.
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    Numerische Mathematik 37 (1981), S. 257-277 
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    Notes: Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.
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    Numerische Mathematik 37 (1981), S. 333-337 
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    Notes: Summary A nonconforming mixed finite element method is presented for approximation of ∇w with Δw=f,w| r =0. Convergence of the order $$\left\| {\nabla w - u_h } \right\|_{0,\Omega } = \mathcal{O}(h^2 )$$ is proved, when linear finite elements are used. Only the standard regularity assumption on triangulations is needed.
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    Numerische Mathematik 37 (1981), S. 355-370 
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    Notes: Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy″=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.
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    Numerische Mathematik 37 (1981), S. 387-404 
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    Keywords: AMS(MOS): 65N20 ; CR: 5.17
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    Notes: Summary The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence. For the linear equations which arise from the discretization of the Poisson equation, a convergence factor 0,5 is established independent of the shape of the domain and of the regularity of the solution.
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    Numerische Mathematik 37 (1981), S. 405-421 
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    Notes: Abstract The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures.
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    Numerische Mathematik 38 (1982), S. 1-30 
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    Notes: Summary In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Kármán equations.
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    Numerische Mathematik 38 (1982), S. 39-52 
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    Keywords: AMS (MOS): Primary 65L10 ; Secondary 35R35 ; CR: 5.17
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    Notes: Summary An approximate method for nonlinear problems with functional constraints is considered, in which the constraint in the whole domain is replaced by the constraint on a manifold of lower dimension. The stability criterion is introduced, and convergence theorems are proved for the onedimensional problem. Numerical results for the elastic-plastic torsion problem are given.
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    Numerische Mathematik 39 (1982), S. 371-404 
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    Notes: Summary The purpose of this paper is to study the approximation of the Von Karman equations by the mixed finite element scheme of Miyoshi and to follow the solutions arcs at a neighbourhood of the first eigenvalue of the linearized problem. This last problem is solved by a continuation method.
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    Numerische Mathematik 41 (1983), S. 399-422 
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    Notes: Summary The paper presents a new theory for joint order and stepsize control in extrapolation methods. This theory defines a locally optimal order that can be determined along any trajectory to be computed. In addition, Shannon's information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems. Extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non-stiff integration by the new device. Moreover, a significant increase in reliability, robustness, and portability of the extrapolation codes is achieved.
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    Numerische Mathematik 34 (1980), S. 1-13 
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    Keywords: AMS(MOS): 34A45, 34A50, 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper Lie series are presented in Chebyshev form and applied to the iterative solution of initial value problems in differential equations. The resulting method, though algebraically complicated, is of theoretical interest as a generalisation of Taylor series methods and iterative Chebyshev methods. The theory of the method is discussed and the solutions of some simple scalar equations are analysed to illustrate the behaviour of the process.
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  • 98
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 34 (1980), S. 143-154 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L65 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the first part [1] a general procedure is presented to obtain polynomial spline approximations for the solutions of initial value problems for ordinary differential equations; furthermore a divergence theorem is proved there. Sufficient conditions for convergence of the method are given in the second part [2]. The remaining case which has not been considered in [1] and [2] is treated in the present paper. In this special case the procedure is equivalent to an unstable two-step method with special initial values; nevertheless, convergence can be proved. Finally,A 0-stability of the method as well as the influence of rounding errors are investigated.
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  • 99
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    Electronic Resource
    Springer
    Numerische Mathematik 36 (1980), S. 73-98 
    ISSN: 0945-3245
    Keywords: AMS: 35R35, 65P05, 76S05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.
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  • 100
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    Electronic Resource
    Springer
    Numerische Mathematik 42 (1983), S. 349-357 
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper theD-stability properties of some classes of perturbed collocation methods are investigated. Among others we show the implicit Runge-Kutta methods described by Ehle-Chipman to beD-stable. We also give implicit Runge-Kutta methods that are bothD-stable andA-stable but notB-stable. Finally we indicate in what senseB-stability is stronger thanA-stability together withD-stability. The results of this paper are based on recent results of Nørsett and Wanner on perturbed collocation methods.
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