When maths really is rocket science

Two problems in projectiles are discussed which generate interestingloci.     

One cannot hear orbifold isotropy type

We show that the isotropy types of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. Indeed, we construct arbitrarily large families of mutually isospectral orbifolds with different isotropy types. Finally, we show that the corresponding singular strata of two isospectral orbifolds may not be homeomorphic. Received: 6 October 2005     

Generalized solutions of a class of nuclear-space-valued stochastic evolution equations

Generalized solutions are defined for stochastic evolution equations of the formdY _t =A ^* Y _t dt + dZ _t on the nuclear triplel(R ^d ) L²(R ^d ) l(R ^d ), whereA does not mapl(R ^d ) into itself. One case which is treated in detail involvesA = –(–) ^/2 ,0 < < 2. This example arises as the Langevin equation for the fluctuation limit of a system of particles migrating according to a symmetric stable process and undergoing critical branching in a random medium.The research of D. A. Dawson was supported by the Natural Sciences and Engineering Research Council of Canada. L. G. Gorostiza's research was supported in part by CONACyT Grants PCEXCNA-040319 and 140102 G203-006, Mexico.     

Crystal and molecular structure of crenatine carbonate

The crystal and molecular structure of 1-ethyl-4-methoxy-9H-pyrido[3,4b]indole (crenatine) carbonate C₁₄H₁₄N₂O·H₂CO₃, (MS, m/z 226)M _R 288.3, a-carboline alkaloid, has been determined from X-ray diffraction data. The compound crystallizes in the space group Pbca with cell parameters:a=11.616(4),b=18.450(8),c=12.992(5)Å,V=2784(2)ų,Z=8,D _calc=1.375 g cm^–3, (MoK)=0.71069Å,(Mo K)=0.94 cm^–1,F(000)=1216,R/R _w =8.2/10.3% for 1099 reflections. The ring system of the-carboline nucleus is planar. The title compound shows a two center hydrogen bond between the indole N-H group and the oxygen atom of a carbonate group. The structure does not display hydrogen bonding between-carboline groups but rather a bonding network involving the carbonate group.     

‘Sacrificial’ supramolecular assembly and pressure-induced polymerization: toward sequence-defined functionalized nanothreads

Limited supramolecular strategies have been utilized to synthesize sequence-defined polymers, despite the prominence of noncovalent interactions in materials design. Herein, we illustrate the utility of ‘sacrificial’ aryl-perfluoroaryl supramolecular synthons to synthesize sp³-hybridized nanothreads from sp²-enriched reactants. Our strategy features A–B reactant pairs in the form of a phenol:pentafluorophenol co-crystal that is preorganized for an electronically-biased and sequence-defined polymerization. The polymerization, initiated at 12 GPa, affords an alternating copolymer featuring exogenous –OH functionalities. The external substitution is confirmed through IR spectroscopy. Importantly, the inclusion of the functional unit provides the first experimental glimpse at reaction mechanism: keto–enol tautomerization that can only occur during cycloaddition is observed through IR spectroscopy. Our approach realizes the first example of a functionalized nanothread and attains sequence definition through sacrificial supramolecular preorganization and presents a further approach for de novo design of complex nanothreads.

Supramolecular synthons are exploited to synthesize –OH functionalized sp³-rich sequence-defined nanothreads using pressure-induced polymerization of a phenol:pentafluorophenol co-crystal.      

A facile approach to a d4 RuN: moiety

Replacement of chloride in (PNP)RuCl, PNP = (tBu2PCH2SiMe2)2N, by Me3SiN3 gives a pre-redox adduct that, already at -30 degrees C, releases N2 to produce the mononuclear nonplanar Ru(IV) nitride (PNP)RuN, characterized by spectroscopic and X-ray methods. DFT calculations show the planar structure to be only 1.6 kcal/mol less stable, which explains the time-averaged simplicity of the 1H NMR spectrum, as well as the large vibrational amplitude of the nitride ligand.