Reconstructing basic ideas in geometry—an empirical approach

“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.     

Cyclische Oligomere von (R)-3-Hydroxybuttersäure: Herstellung und strukturelle Aspekte

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.     

Phase transitions and positive definite couplings

The method of infrared bounds is extended to a large class of nearest neighbour interactions in classical spin systems. Temperature controlled bounds on fluctuations follow whenever the coupling function is a positive definite kernel. Existence of phase transitions is demonstrated for the RP ^Nmodel for d3.     

On combinatorial and projective geometry

Cross ratios constitute an important tool in classical projective geometry. Using the theory of Tutte groups as discussed in [6] it will be shown in this note that the concept of cross ratios extends naturally to combinatorial geometries or matroids. From a thorough study of these cross ratios which, among other observations, includes a new matroid theoretic version and proof of the Pappos theorem, it will be deduced that for any projective space M= _n (K) of dimension n2 of M over some skewfield K the inner Tutte group is isomorphic to the commutator factor group K ^*/[K ^*, K ^*] of K ^*K{0}. This shows not only that in case M= _n (K) our matroidal cross ratios are nothing but the classical ones. It can also be used to correlate orientations of the matroid M= _n (K) with the orderings of K. And it implies that Dieudonné's (non-commutative) determinants which, by Dieudonné's definition, take their values in K ^*/[K ^*, K ^*] as well, can be viewed as a special case of a determinant construction which works for just every combinatorial geometry.Research supported by the DFG (Deutsche Forschungsgemeinschaft).     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

An instanton-invariant for 3-manifolds

To an oriented closed 3-dimensional manifoldM withH ₁(M, )=0, we assign a ₈-graded homology groupI _*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×.