The chromatic number of random Borsuk graphs

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in , and a pair of vertices is connected if the Euclidean distance between them is at least 2??. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ?>0 is small and fixed, then the chromatic number is d+2 with high probability. We show that this holds even if ?→0 slowly enough. We quantify the rate at which ? can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor—if ?→0 faster than this, we show that the graph is (d+1)‐colorable with high probability.25     

Benchtop NMR analysis of piperazine-based drugs hyperpolarised by SABRE

Piperazine-based drugs, such as N-benzylpiperazine (BZP), became attractive in the 2000s due to possessing effects similar to amphetamines. Herein, BZP, in addition to its pyridyl analogues, 2-, 3-, and 4-pyridylmethylpiperidine (2-PMP, 3-PMP, and 4-PMP respectively) was subjected to the hyperpolarisation technique Signal Amplification By Reversible Exchange (SABRE) in order to demonstrate the use of this technique to detect these piperazine-based drugs. Although BZP was not hyperpolarised via SABRE, 2-PMP, 3-PMP, and 4-PMP were, with the ortho- and meta-pyridyl protons of 4-PMP showing the largest enhancement of 313-fold and 267-fold, respectively, in a 1.4-T detection field, following polarisation transfer at Earth's magnetic field. In addition to the freebase, 4-PMP.3HCl was also appraised by SABRE and was found not to polarise, however, the addition of increasing equivalents of triethylamine (TEA) produced the freebase, with a maximum enhancement observed upon the addition of 3 equivalents of TEA. Further addition of TEA led to a reduction in the observed enhancement. SABRE was also employed to polarise 4-PMP.3HCl (~20% w/w) in a simulated tablet to demonstrate the forensic application of the technique (138-fold enhancement for the ortho-pyridyl protons). The amount of 4-PMP.3HCl present in the simulated tablet was quantified via NMR using D₂O as a solvent and compared well to complimentary gas chromatography–mass spectrometry data. Exchanging D₂O for CD₃OD as the solvent utilised for analysis resulted in a significantly lower amount of 4-PMP.3HCl being determined, thus highlighting safeguarding issues linked to drug abuse in relation to determining the amount of active pharmaceutical ingredient present.     

Synthesis and physical properties of highly branched perfluorinated polymers from AB and AB2 monomers

Highly branched perfluorinated aromatic polyether copolymers were prepared from the polycondensation of the AB₂ monomer, 3,5‐bis[(pentafluorobenzyl)oxy]benzyl alcohol with a variety of fluoroaryl and alkyl bromide AB comonomers. The structures and comonomer distribution of the resulting polymers were characterized in detail. ¹H NMR data from kinetic trials illustrated that perfluoroaryl AB comonomer distribution correlated to AB comonomer sterics. ¹⁹F NMR data revealed that fluorinated AB monomers and 3‐bromo‐1‐propanol AB monomers were distributed within the AB₂ polymer backbone, while longer alkyl bromide AB monomers, 6‐bromo‐1‐hexanol, were mostly distributed along hyperbranched polymer chain ends. In general, as AB comonomer incorporation increased for nonsterically hindered copolymers, thermal decomposition onset increased and glass transition temperatures decreased. The combined data demonstrated the effect of comonomer distribution and sterics on physical properties of AB₂‐based polymer systems. The resulting materials were used to cast thin polymer films for measurement of contact angle, which were shown to be directly related to comonomer content. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 1880–1894     

Tetrazole Azasydnone (C2N7O2H) And Its Salts: High-Performing Zwitterionic Energetic Materials Containing A Unique Explosophore

This work reports the first compound containing both a tetrazole and an azasydnone ring, a unique energetic material. Several energetic salts of the tetrazole azasydnone were synthesized and characterized, leading to the creation of new secondary and primary explosives. Molecular structures are confirmed by ¹H and ¹³C NMR, IR spectroscopy, and X-ray crystallographic analysis. The high heats of formation, fast detonation velocities, and straight-forward synthesis of energetic azasydnones should capture the attention of future energetics research.     

High iso Aldehyde Selectivity in the Hydroformylation of Short‐Chain Alkenes

The hydroformylation of propene to give predominantly iso‐butanal has been achieved; class‐leading selectivity is possible even at higher temperatures that deliver fast conversion. Racemic rhodium complexes of bidentate phospholane phosphites derived from tropos‐biphenols and unusual solvent systems are the key to the selectivity observed.