A stannaethene with a planar environment of the tricoordinated tin and carbon atoms

Addition of the diarylstannylene R″₂Sn (R″ = 2−^tBu−4,5,6-Me₃C₆H) to the cryptodiborylcarbene (Me₃Si)₂C(B^tBu)₂C furnishes the stannaethene (Me₃Si)₂C(B^tBu)₂C=SnR″₂ (10). The X-ray structure analysis of 10 reveals a strictly planar environment of the tricoordinated tin and carbon atoms and a slight twisting of the Sn=C double bond.     

On the Generalized Rogers–Ramanujan Continued Fraction

On page 26 in his lost notebook, Ramanujan states an asymptotic formula for the generalized Rogers–Ramanujan continued fraction. This formula is proved and made slightly more precise. A second primary goal is to prove another continued fraction representation for the Rogers–Ramanujan continued fraction conjectured by R. Blecksmith and J. Brillhart. Two further entries in the lost notebook are examined. One of them is an identity bearing a superficial resemblance to the generating function for the generalized Rogers–Ramanujan continued fraction. Thus, our third main goal is to establish, with the help of an idea of F. Franklin, a partition bijection to prove this identity.     

Size-dependent light emission from mass-selected clusters

The emission of photons in the visible wavelength range from mass-selected Ag⁺ _n, Cu⁺ _n, Pt⁺ _n and Pd⁺ _n () clusters is observed. Photons are detected 10^-4 s after the cluster generation in a sputter source. The emission intensities display distinct variations with cluster size and material. The observations are interpreted in terms of the decay of metastable states which are excited during the high-energy sputtering process used for the generation of these clusters. Received: 28 October 1997 / Revised: 5 January 1998 / Accepted: 30 January 1998     

A new, efficient and stereoselective synthesis of tricyclic and tetracyclic compounds by samarium diiodide induced cyclisations of naphthyl-substituted arylketones--an easy access to steroid-like skeletons

In this report, we present the application of samarium diiodide induced cyclisations of naphthyl-substituted ketones towards an easy and stereoselective access to tri- and tetracyclic-functionalised compounds. Typical naphthalene derivatives were studied to investigate the scope and limitations of this novel cyclisation process. The model substrates studied demonstrate that the samarium ketyl cyclisations are essentially restricted to the formation of six-membered rings. The diastereoselectivity of these reactions is strongly influenced by the connection of the alkyl side chain to the naphthalene core. Gamma-naphth-1-yl-substituted ketones furnished cyclisation products, such as 17 or 22-26, as single diastereomers, whereas gamma-naphth-2-yl-substituted precursors gave mixtures of diastereomers--as demonstrated by the conversion of model compound 10 into tricyclic products 18 a/18 b, or that of cyclohexanone derivative 33 into tetracyclic diastereomers 34 a/34 b. Cyclic ketones as ketyl precursors furnished steroid-like tetracyclic skeletons; however, due to the cis/cis fusion of rings B/C and C/D these products have an "unnatural" bowl-like shape. Several of the cyclisation products have been identified by X-ray analyses, which not only proved the constitutions, but also the relative configurations and the preferred conformations. Steroid analogue 23 was subjected to subsequent transformations, which demonstrate that the styrene-like double bond of such compounds can be used for further structural diversification. First attempts to synthesise related azasteroids by incorporating nitrogen atoms into the ketone moiety are also reported. Thus, pyrrolidine derivatives 44 and 47 as well as piperidine derivatives 50 and 52 were subjected to samarium diiodide induced cyclisations. The expected tetracyclic products 48, 49 a/49 b, 51 and 53 a/53 b were obtained in moderate to good yields. The stereoselectivities observed follow the rules already established for the all-carbon precursors. The resulting products, bearing a nitrogen atom in ring D, are interesting azasteroid analogues with "unnatural" configuration.     

How to catch a ‘fat’ proton

We argue that high-multiplicity events in proton–proton or proton–nucleus collisions originate from large-size fluctuations of the nucleon shape. We discuss a pair of simple models of such proton shape fluctuations. A “fat” proton with a size of 3 fm occurs with observable frequency. In light of this result, collective flow behavior in the ensuing nuclear interaction seems feasible. We discuss the influence of these models on the parton structure of the proton.     

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones.

     

Circle and divisor problems,and double series of Bessel functions

In approximately 1915, Ramanujan recorded two identities involving doubly infinite series of Bessel functions. The identities were brought to the mathematical public for the first time when his lost notebook was published in 1988, and are connected with the classical, long-standing circle and divisor problems, respectively. We provide a proof of the first identity for the first time by analytically continuing a new kind of Dirichlet series. Delicate estimates of exponential sums are needed, and the new methods we introduce may be of independent interest.