4H-Chromenone Glycosides from Eranthis hyemalis (L.) SALISBURY

Investigation of the tubers of Eranthis hyemalis (Ranunculaceae) afforded six chromenone glycosides. Their structures have been elucidated mainly by spectroscopic (FAB-MS, 2D-NMR techniques) and chemical methods (acidic and enzymatic hydrolysis) as 9-{[(β-D -glucopyranosyl)oxy]methyl}-8,11-dihydro-5-hydroxy-2-methyl-4H-pyrano[2,3-g][1]benzoxepin-4-one ( 1 ), 9-{[(β-D -gentiobiosyl)oxy]methyl}-8,11-dihydro-5-hydroxy-2-methyl-4H-pyrano[2,3-g][1]benzoxepin-4-one( 2 ), 9-{[(β-D -glucopyranosvl)oxy]melhyl}-8,11-dihydro-5-hydroxy-2-(hydroxy-methyl)-4H-pyrano[2,3-g][1]benzoxepin-4-one( 3 ), 8-{(2E)-4-[(β-D -glucopyranosyl)oxy]-3-methylbut-2-enyl}-5,7-dihydroxy-2-methyl-4H-1-benzopyran-4-one ( 4 ), 8-{(2E)-4-[(β-D -glucopyranosyi)oxy]-3-methylbut-2-enyl}-5,7-dihydroxy-2-(hydroxymethyl)-4H-1-benzopyran-4-one ( 5 ), and 7-{[(β-D -glucopyranosy1)oxy]methyl}-2,3-dihydro-2-(l-hydroxy-1-methylethyl)-4-methoxy-5H-furo[3,2-g][1]benzopyran-5-one ( 6 ). Compound 2 exhibited negative inotropic activity.     

Reaktionen cyclischer Oxalylverbindungen, 34. Mitt.: Synthese von Dibenzoylacet-N-carboxyalkylamiden und semiempirische Rechnungen zur Keto-Enol Tautomerie

Summary 4-Benzoyl-5-phenylfuran-2,3-dione (1) and the urethanes2 combine under loss of carbon monoxide yielding the open chain dibenzoylacetic acid derivatives3 and4. 3a, b only can be cyclized to the oxazinone5. The keto-enole tautomerism3 4 is further investigated with aid of semiempirical quantum chemical calculations, based upon the molecular geometry of3a, deduced from an X-ray study.

Frau Univ. Prof. Dr. Helga Wittmann mit den besten Wünschen zum 70. Geburtstag gewidmet.     

Molecular Cage Assembly by Sn−O−Sn Bridging of Di-, Tri- and Tetranuclear Organotin Tectons: Extending the Spacing in Double Ladder Structures

Hydrolysis reactions of di- and trinuclear organotin halides yielded large novel cage compounds containing Sn−O−Sn bridges. The molecular structures of two octanuclear tetraorganodistannoxanes showing double-ladder motifs, viz., [{Me₃SiCH₂(Cl)SnCH₂YCH₂Sn(OH)CH₂SiMe₃}₂(μ-O)₂]₂ [ 1 , Y=p-(Me)₂SiC₆H₄-C₆H₄Si(Me)₂] and [{Me₃SiCH₂(I)SnCH₂YCH₂Sn(OH)CH₂SiMe₃}₂(μ-O)₂]₂ ⋅ 0.48 I₂ [ 2⋅ 0.48 I₂, Y=p-(Me)₂SiC₆H₄-C₆H₄Si(Me)₂], and the hexanuclear cage-compound 1,3,6-C₆H₃(p-C₆H₄Si(Me)₂CH₂Sn(R)₂OSn(R)₂CH₂Si(Me)₂C₆H₄-p)₃C₆H₃-1,3,6 ( 3 , R=CH₂SiMe₃) are reported. Of these, the co-crystal 2⋅ 0.48 I₂ exhibits the largest spacing of 16.7 Å reported to date for distannoxane-based double ladders. DFT calculations for the hexanuclear cage and a related octanuclear congener accompany the experimental work.     

Euclidean algorithms are Gaussian

We obtain a central limit theorem for a general class of additive parameters (costs, observables) associated to three standard Euclidean algorithms, with optimal speed of convergence. We also provide very precise asymptotic estimates and error terms for the mean and variance of such parameters. For costs that are lattice (including the number of steps), we go further and establish a local limit theorem, with optimal speed of convergence. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various other techniques: Dirichlet series, Perron's formula, quasi-powers theorems, and the saddle-point method. Such dynamical analyses had previously been used to perform the average-case analysis of algorithms. For the present (dynamical) analysis in distribution, we require estimates on transfer operators when a parameter varies along vertical lines in the complex plane. To prove them, we adapt techniques introduced recently by Dolgopyat in the context of continuous-time dynamics (Ann. Math. 147 (1998) 357).     

Micro–macro multilevel latent class models with multiple discrete individual-level variables

An existing micro–macro method for a single individual-level variable is extended to the multivariate situation by presenting two multilevel latent class models in which multiple discrete individual-level variables are used to explain a group-level outcome. As in the univariate case, the individual-level data are summarized at the group-level by constructing a discrete latent variable at the group level and this group-level latent variable is used as a predictor for the group-level outcome. In the first extension, that is referred to as the Direct model, the multiple individual-level variables are directly used as indicators for the group-level latent variable. In the second extension, referred to as the Indirect model, the multiple individual-level variables are used to construct an individual-level latent variable that is used as an indicator for the group-level latent variable. This implies that the individual-level variables are used indirectly at the group-level. The within- and between components of the (co)varn the individual-level variables are independent in the Direct model, but dependent in the Indirect model. Both models are discussed and illustrated with an empirical data example.     

A short note on harmonic functions and zero divisors on the Sierpinski fractal

We show that harmonic functions are not zero divisors in the algebra of real-valued continuous functions on the Sierpinski fractal.     

Uniquely solvable quadratic boolean equations

Quadratic boolean equations with a unique solution are characterized. A linear-time algorithm is proposed to recognize them.