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51.
Andreas E. Schroth 《Journal of Geometry》1991,42(1-2):141-147
The number of circles of a flat Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.Dedicated to Prof. J. Joussen on his 60th birthday 相似文献
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Lufthansa Technical Training GmbH (LTT) performs training courses for Lufthansa Technik AG as well as for several other international airlines. Courses of about 670 different types are offered of which several hundred take place each year. The course scheduling problem faced by LTT is to construct a yearly schedule which maximizes the profit margin incurred while meeting a variety of complex precedence, temporal, and resource-related constraints. A “good” operational schedule should also meet a number of additional subordinate objectives. We formalize the problem and develop a heuristic scheme along with several priority rules, as well as a local search algorithm to determine well-suited weights for weighted composite rules. The operational planning situation of 1996 served as our major test instance; additional test instances were constructed by modifying this data. Several computational experiments were carried out to evaluate the performance of the algorithms. It turned out that the best so-found schedule is substantially better in terms of the profit margin incurred than the solution manually constructed by LTT. 相似文献
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Andreas Vohns 《ZDM》2006,38(6):498-504
“Basic ideas” (or “fundamental ideas” etc.) have been discussed in mathematical curriculum theory for about forty years. This paper will centre on the hypothesis that this concept can only be applied successfully by using it as a category for the analysis of concrete mathematical problems. This hypothesis will be illustrated by means of a sample problem from the Austrian Standards for Mathematics Education (“Bildungsstandards”). In this example, basic ideas are used in a content matter analysis which takes students' solutions to the problem as a starting point for the creation of a potentially substantial learning environment in trigonometry. 相似文献
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Diemar A. Plattner Andreas Brunner Max Dobler Hans-Martin Müller Walter Petter Peter Zbinden Dieter Seebach 《Helvetica chimica acta》1993,76(5):2004-2033
Cyclic Oligomers of (R)-3-Hydroxybutanoic Acid: Preparation and Structural Aspects The oligolides containing three to ten (R)-3-hydroxybutanoate (3-HB) units (12-through 40-membered rings 1–8 ) are prepared from the hydroxy acid itself, its methyl ester, its lactone (‘monolide’), or its polymer (poly(3-HB), mol. wt. ca. 106 Dalton) under three sets of conditions: (i) treatment of 3-HB ( 10 ) with 2,6-dichlorobenzoyl chloride/pyridine and macrolactonization under high dilution in toluene with 4-(dimethylamino)pyridine (Fig. 3); (ii) heating a solution (benzene, xylene) of the β-lactone 12 or of the methyl ester 13 from 3-HB with the tetraoxadistanna compound 11 as trans-esterification catalyst (Fig. 4); (iii) heating a mixture of poly(3-HB) and toluene-sulfonic acid in toluene/1,2-dichloroethane for prolonged periods of time at ca. 100° (Fig. 6). In all three cases, mixtures of oligolides are formed with the triolide 1 being the prevailing component (up to 50% yield) at higher temperatures and with longer reaction times (thermodynamic control, Figs. 3–6). Starting from rac-β-lactone rac- 12 , a separable 3:1 to 3:2 mixture of the l,u- and the l,l-triolide diasteroisomers rac- 14 and rac- 1 , respectively, is obtained. An alternative method for the synthesis of the octolide 6 is also described: starting from the appropriate esters 15 and 17 and the benzyl ether 16 of 3-HB, linear dimer, tetramer, and octamer derivatives 18–23 are prepared, and the octamer 23 with free OH and CO2H group is cyclized (→ 6 ) under typical macrolactonization conditions (see Scheme). This ‘exponential fragment coupling protocol’ can be used to make higher linear oligomers as well. The oligolides 1–8 are isolated in pure form by vacuum distillation, chromatography, and crystallization, an important analytical tool for determining the composition of mixtures being 13C-NMR spectroscopy (each oligolide has a unique and characteristic chemical shift of the carbonyl C-atom, with the triolide 1 at lowest, the decolide 8 at highest field). The previously published X-ray crystal structures of triolide 1 , pentolide 3 , and hexolide 4 (two forms), as well as those of the l,u-triolide rac- 14 , of tetrolide ent- 2 , of heptolide 5 , and of two modifications of octolide 6 described herein for the first time are compared with each other (Figs. 7–10 and 12–15, Tables 2 and 5–7) and with recently modelled structures (Tables 3 and 4, Fig. 11). The preferred dihedral angles τ1 to τ4 found along the backbone of the nine oligolide structures (the hexamer and the larger ones all have folded rings!) are mapped and statistically evaluated (Fig. 16, Tables 5–7). Due to the occurrence of two conformational minima of the dihedral angle O? CO? CH2? CH (τ3 = + 151 or ?43°), it is possible to locate two types of building blocks for helices in the structures at hand: a right-handed 31 and a left-handed 21 helix; both have a ca. 6 Å pitch, but very different shapes and dispositions of the carbonyl groups (Fig. 17). The 21 helix thus constructed from the oligolide single-crystal data is essentially superimposable with the helix derived for the crystalline domains of poly(3-HB) from stretched-fiber X-ray diffraction studies. The absence of the unfavorable (E)-type arrangements around the OC? OR bond (‘cis-ester’) from all the structures of (3-HB) oligomers known so far suggests that the model proposed for a poly(3-HB)-containing ion channel (Fig. 2) must be modified. 相似文献
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