Chirality of toroidal molecular graphs

Symmetry properties of a class of toroidal molecular graphs, arising as covers of certain bipartite cubic Cayley graphs of dihedral groups, are studied. Although these symmetries make all vertices and all edges indistinguishable, they imply intrinsic chirality.     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

The conformation of t-butyl 2-methoxy-5, 6-dihydro-2 H-pyran-6-carboxylates and 6,6′- disubstituted 2-methoxy-5,6-dihydro-2H-pyran derivatives on the basis of 1H and 13 C NMR spectra

The ¹H and ¹³C NMR spectra of trans- and cis-tert-butyl 2-methoxy-5,6-dihydro-2H-pyran-6-carboxylates (1 and 2) and 6,6′-disubstituted 2-methoxy-5,6-dihydro-2H-pyrans (3-7) have been recorded. HH and CH coupling constants are discussed in terms of the ¹H₆?⁶H₁ conformational equilibrium. It has been found that 1 occurs exclusively in the ¹H₆ conformation, whereas its cis isomer, 2, exists in an equilibrium of both half-chair forms. 6,6′-Disubstituted 2-methoxy-5,6-dihydro-2H-pyrans 3-6 display spectral and conformational behaviour similar to that of 1, whereas 7 resembles 2 in this respect.     

Asymmetric photocycloaddition between furan and optically active ketones

The photochemical reaction between 3β-acetoxyandrost-5-en-17-one4 or (–)-methone5 and furan was preceded by isomerization of the ketones to the corresponding γδ-unsaturated aldehydes which subsequently added to furan furnishing 6-substituted 2,7-dioxabicyclo-[3.2.0]hept-3-enes,8 and9. Under the same conditions 1,2:5,6-di-O-isopropylidene α-d-ribo-hexofuranos-3-ulose12 reacted with furan without any isomerization and afforded all four stereoisomeric bicyclic photoadducts14–17. The ratio of the products resulting from the approach of the furan molecule to12 from the less hindered and from the more hindered sides was 3:1. This pointed at the decisive role of steric factors in controlling the course of photochemical cycloaddition.     

Estimation of the level and phase of the simple distortion tone the modulation domain

These experiments were designed to test the idea that nonlinearities in the auditory system can introduce a distortion component into the internal representation of the envelope of a sound, and to estimate the phase of the hypothetical distortion component. In experiment 1, a two-alternative forced-choice (2AFC) task with feedback was used to measure psychometric functions for detecting 5-Hz probe modulation of a 4-kHz sinusoidal carrier in the presence of a masker modulator with components at 50 and 55 Hz (m=0.3 for each component). Performance was measured as a function of the relative phase, delta[symbol see text], of the probe relative to the "venelope" (envelope of the envelope) of the masker. Performance was poorest for delta[symbol see text]= 135 degrees. In experiment 2, delta[symbol see text] was fixed at 135 degrees, m was set to 0.48 for each masker component, and psychometric functions for detecting probe modulation were measured using a 2AFC task without feedback. For small probe modulation depths (m approximately 0.03), the detectability index, d', was consistently negative, consistent with the existence of a weak distortion product which can "cancel" the probe modulation. The distortion component for the conditions of the experiment was estimated to have a phase of about -25 degrees relative to the venelope.     

Flow limitation in convective dominated open capillary channel flows

In this work the flow rate limitation of convective dominated flows in open capillary channels is investigated experimentally. Since irreversible flow losses are small in this type of flow, the flow variables are approximately symmetrical with respect to the middle of the channel. Therefore a very precise proof of the origin of the limitation is possible. A new technique for the determination of the key property, the speed index, is presented. The speed index confirms that also in convective dominated flows the flow limitation occurs due to a choking-effect. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)