Menger curvature and rectifiability in metric spaces

We show that for any metric space X the condition

     

Bremsstrahlung in alpha decay reexamined

A high-statistics measurement of bremsstrahlung emitted in the alpha decay of (210)Po has been performed, which allows us to follow the photon spectra up to energies of approximately 500 keV. The measured differential emission probability is in good agreement with our theoretical results obtained within the quasiclassical approximation as well as with the exact quantum mechanical calculation. It is shown that, due to the small effective electric dipole charge of the radiating system, a significant interference between the electric dipole and quadrupole contributions occurs, which is altering substantially the angular correlation between the alpha particle and the emitted photon.     

A Synthesis Detour to Planar‐Diastereoisomeric Ferrocene Derivatives around an Unexpected Rearrangement of ortho‐Lithiated Kagan's Template [S(S)]‐(p‐Tolylsulfinyl)ferrocene

Usually, ortho lithiation of Kagan's template 1 and quenching with electrophiles leads highly diastereoselectively to planar‐chiral 1,2‐disubstituted ferrocenes. Surprisingly, lithiation of 1 with lithium diisopropylamide (LDA) followed by addition of paraformaldehyde afforded regioisomer (+)‐{[S(S)]‐[4‐(2‐hydroxyethyl)phenyl]sulfinyl}ferrocene ( 2 ), which was converted to (+)‐{[S(S)]‐{4‐{2‐[(methylsulfonyl)oxy]ethyl}phenyl}sulfinyl}ferrocene ( 3 ) (Scheme 1). The desired diastereoisomer (l)‐1‐(hydroxymethyl)‐2‐(p‐tolylsulfinyl)ferrocene ( 5 ) in turn could also be obtained by ortho lithiation of 1 with LDA but by quenching with DMF to yield aldehyde 4 first, which then was reduced with NaBH₄ to 5 . Finally, target compound (l)‐1‐[(dimethylamino)methyl]‐2‐(p‐tolylsulfinyl)ferrocene ( 6 ) was obtained by substitution of the OH group of 5 under mild conditions or directly by ortho lithiation of 1 with lithio‐2,4,6‐triisopropylbenzene (=2,4,6‐triisopropylphenyl)lithium; LTP) followed by quenching with N,N‐dimethylmethyleneiminium chloride. At low temperatures, reaction of 1 with LDA leads, via the preferred diastereoisomeric transition state ‘exo’‐ 7 and under extrusion of a (diisopropylamine)lithium complex of type 8 , in a highly selective manner, to diastereoisomeric ortho‐lithiated chelate (l)‐ 9 (Scheme 2). The reaction of 1 to 2 is explained by a rearrangement of (l)‐ 9 to {[S(S)] [4‐(lithiomethyl)phenyl]sulfinyl}ferrocene 10 , which is acid‐catalyzed by coordinated diisopropylamine in complexes of type 8 . This rearrangement is not observed if LTP is used as base or, in case LDA is applied, if the electrophile is sufficiently reactive at low temperatures.     

On S-cyclic steiner systems

In this paper we investigate the structure of S-cyclic Steiner quadruple systems and derive a necessary condition on thier existence which might turn out to be sufficient. In the third chapter we prove a necessary condition on the existence of general S-cyclic Steiner systems. It turns out that S-cyclicityis a very restrictive property that hardly any systems with t ≠ 3 can have.     

Trajectory nets connecting all critical points of a smooth function

The disconnected components of certain trajectory nets containing all critical points of a differentiable functionf can be connected by suitably chosen contour sets of a certain associated functiong. A recursive construction yields a locally 1-dimensional connected set Ω which contains all critical points. The possible ways of tracing this set numerically are discussed. This work was supported by the Deutsche Forschungsgemeinschaft.     

On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations

In this paper we prove a general theorem stating a sufficient condition for the inverse image of a point under a continuously differentiable map from ℝ ⁿ to ℝ ^k to be connected. This result is applied to the trajectories generated by the Newton flow. Several examples demonstrate the applicability of the results to nontrivial problems. This work was supported by the Deutsche Forschungsgemeinschaft.     

Cascadic Multigrid in a Spectral-Element Context

Current research in computational fluid dynamics focuses on high-order methods, which offer a significant reduction of the computational effort for a given error bound. In low-order methods optimal complexity solvers for elliptic equations, e.g. the Helmholtz equation, are readily available, but for high-order methods these are still an area of research. This work evaluates the effectiveness of the Cascadic Multigrid Method in the context of spectral elements, comparing it to a standard Krylov subspace solver. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mitocanic Di- and Triterpenoid Rhodamine B Conjugates

The combination of the “correct” triterpenoid, the “correct” spacer and rhodamine B (RhoB) seems to be decisive for the ability of the conjugate to accumulate in mitochondria. So far, several triterpenoid rhodamine B conjugates have been prepared and screened for their cytotoxic activity. To obtain cytotoxic compounds with EC₅₀ values in a low nano-molar range combined with good tumor/non-tumor selectivity, the Rho B unit has to be attached via an amine spacer to the terpenoid skeleton. To avoid spirolactamization, secondary amines have to be used. First results indicate that a homopiperazinyl spacer is superior to a piperazinyl spacer. Hybrids derived from maslinic acid or tormentic acid are superior to those from oleanolic, ursolic, glycyrrhetinic or euscaphic acid. Thus, a tormentic acid-derived RhoB conjugate 32, holding a homopiperazinyl spacer can be regarded, at present, as the most promising candidate for further biological studies.