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.     

Sensitivity ofn-p scattering observables to charge-independence breaking of the3 P N-N phase shifts

The sensitivity of neutron-proton scattering observables to the recently reported charge-independence breaking of the isospinT=1³ P nucleon-nucleon interaction has been calculated. It is shown that only the analyzing powerA _y () at extreme forward angles and at =90° c.m. and the spin-correlation coefficientA _zx () at 90° c.m. exhibit a measurable sensitivity. It is concluded that the determination ofA _y at 90° c.m. is currently the most accurate approach to verify independently charge-independence breaking in the³ P nucleon-nucleon phase shifts, considering the status of present experimental techniques.     

On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces

For a separable metric space (X, d) L^p Wasserstein metrics between probability measures μ and v on X are defined by where the infimum is taken over all probability measures η on X × X with marginal distributions μ and v, respectively. After mentioning some basic properties of these metrics as well as explicit formulae for X = R a formula for the L² Wasserstein metric with X = Rⁿ will be cited from [5], [9], and [21] and proved for any two probability measures of a family of elliptically contoured distributions. Finally this result will be generalized for Gaussian measures to the case of a separable Hilbert space.     

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).     

Hyper-rook Domain Inequalities

A hyper-rook domain of an element x in the space (words of length n over alphabets with k elements) is a sphere with center x and fixed radius j in Hamming distance. The number j determines the dimension of the hyper-rook domain. The classical (and far from solved) problem of covering by rook domains (here considered as the 1-dimensional case) is the problem of finding minimal coverings of by such spheres. Very few results are known in the literature for dimensions ≥ 2. We prove in this paper certain classes of inequalities based on coverings using matrices, which give upper and lower bounds for several cases of the problem for higher dimensions.     

Replacement-Policy Development: A Case Study of Glove Replacement at a Nuclear Processing Facility

Individual/group replacement models were used to identify the minimum-cost policy for replacing protective gloves that are utilized in product fabrication at a nuclear processing facility. Since historical data on glove longevity were not available to determine the failure-rate distribution, the classical replacement model for items that fail was modified to consider glove life as a normally distributed variable. Moreover, the absence of replacement-cost required that the classical model be transformed to incorporate the ratio of single to unit-group replacement cost. Sensitivity analysis on these two input parameter values provided guidance for the development of an appropriate replacement policy.