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Nylons 8 10 and 10 12 have been synthesized and crystallized as chain-folded lamellae from 1,4-butanediol and the results compared with previous studies on Nylons 4 6 and 6 8. In 2N 2(N + 1) Nylons, the lengths of the two alkane segments are equal and two different hydrogen-bonded sheet schemes are possible: progressive or alternating shear. At room temperature, Nylons 8 10 and 10 12 adopt the progressive scheme and the adjacent re-entry folds in the crystals must be in the alkane chain segments. In contrast, Nylons 4 6 and 6 8 lamellae, crystallized from the same solvent, exhibit the alternating hydrogen bonding scheme and each adjacent re-entry fold must contain an amide group. The transition in the chemical nature of the lamellar surface, from the amide fold to the alkane fold, occurs in passing from Nylon 6 8 to 8 10. Thus, the progressive hydrogen-bonded sheet/alkane fold structure is energetically more favorable, provided the alkane-folding geometry is sufficiently relaxed; this comes with increasing alkane segment length. For each hydrogen-bonded sheet structure there are still two principal intersheet stacking modes in lamellar crystals: the progressively sheared R-phase or the alternatingly sheared β-phase, both of which have been found in the 8 10 and 10 12 Nylons. The 2N 2(N + 1) Nylons have the choice of four possible structures. The melting points of solution grown crystals of Nylons 4 6, 6 8, 8 10, and 10 12 decrease with decreasing intrachain amide density. When lamellar crystals of these Nylons are heated, the two characteristic interchain diffraction signals move together and meet at their Brill temperature; for Nylon 10 12 it appears to be close to the melting point.

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A wide-ranging and varied group of 31 even-even nylons, in the form of adjacent, re-entry chain-folded lamellar crystals, are compared and contrasted with respect to their room-temperature structures and behavior on heating. In this comparison, various relationships and trends emerge that provide a clearer understanding of the salient features that control the competitive interplay between the nylon chemistry, the crystal structure of the lamellar core, and the nature of the folds. For eveneven nylons with differing alkane segment lengths, only one type of hydrogen-bonded sheet is found; for those with equal alkane segment lengths (2N 2(N + 1) nylons), two types of hydrogen-bonded sheet are found and there is direct coupling with the fold chemistry. In some cases the fold chemistry and/or stereochemistry dictate the sheet structure (e.g., nylon 4 6), while in other cases the reverse is true (e.g., nylon 4 4). There is no chain directionality in these even-even nylons, and it transpires that the two distinct alkane segments, diamine and diacid, can independently influence the final structure and behavior on heating. Relatively high intrachain amide density molecules (e.g., nylon 2 4) need to incorporate amide units within the adjacent, re-entry folds, and the fold geometry can bear a resemblance to the folding found in apβ-sheet proteins. For molecules with short, dimethylene, diamine alkanes (2 Y nylons) the proximity of the intrachain amide units perturbs the all-trans conformation; however, for molecules with similarly short, dimethylene, diacid alkanes (X 4 nylons) the all-trans conformation occurs. Nylon isomer pairs with inverted amides (nylons X Y and Y-2 X+2) form sheets with the same hydrogen-bonded lattice parameters; however, these pairs usually exhibit different sheet stacking and behave differently on heating. Comparisons are made between the behavior on heating, including the Brill transformation, of the 31 even-even nylons.

Chain-folded single crystals of the five even−even nylons 4 4, 6 4, 8 4, 10 4, and 12 4 were grown from solution and their structures and morphologies studied using transmission electron microscopy, both imaging and diffraction. Sedimented mats were examined using X-ray diffraction. All these nylons have room temperature crystal structures that relate to that reported for nylon 6 6, yet there are differences, reflecting the differences in the amide group distribution. At room temperature, all the crystals are composed of chain-folded, hydrogen-bonded sheets; the hydrogen bonds within the sheets form a progressive shear pattern, and, in addition, the sheets themselves are sheared progressively parallel to the sheet plane so that they generate triclinic unit cells. The magnitude of this intersheet shear may differ between nylons; it is dependent on the details of the amide decoration pattern on the hydrogen-bonded sheet faces. In all five nylons studied, the two strong and characteristic diffraction signals of the room temperature triclinic structure, at spacings 0.44 nm (projected interchain/intrasheet distance) and 0.37 nm (intersheet distance), move together and merge as they do for single crystals of nylon 6 6. For each of the X 4 nylons, the Brill temperature (lowest temperature where the spacings are equal) is in the range 140−190 °C. In each case, the triclinic structure gradually transforms into a pseudohexagonal structure as the temperature rises. The melting points of solution-grown crystals of this series of even−even nylons decrease with the linear hydrogen bond density. This series of nylons is unique since in each case the chain folds must be in the diamine alkane segment.

Abstract. Using a multidimensional large sieve inequality, we obtain a bound for the mean-square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.

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Abstract. There is a modular curve X ′ (6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q that satisfy Q(E[2]) ⊆ Q (E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X ′ (6). Our motivation is two-fold: on the one hand, X ′ (6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, X ′ (6)(Q) gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q.