Practical syntheses of [1,3,5]‐triazine dendritic molecules on solid supports

A practical and divergent synthesis of supported [1,3,5]‐triazine dendritic molecules on Wang resin, PEGA resin, SynPhase? Lanterns, and silica gel is described. The alkylamine linkers used allow derivatization with functionality for both synthetic (e.g., supported reagent and scavenger activity) and chemical biology applications. The use of supported intermediates allows differentiation of symmetric linkers without the need for protecting group chemistry. The synthetic route uses inexpensive, readily available starting materials in a straightforward and scaleable strategy. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2248–2259, 2006     

Photocrosslinked poly(vinylbenzophenone)‐core micelles via mild Friedel–Crafts benzoylation of polystyrene amphiphiles

Photocrosslinkable poly(vinylbenzophenone)‐containing polymers were synthesized via a one‐step, Friedel–Crafts benzoylation of polystyrene‐containing starting materials [including polystyrene, polystyrene‐block‐poly(tert‐butyl acrylate), polystyrene‐block‐poly(ethylene oxide), polystyrene‐block‐poly(methyl methacrylate), and polystyrene‐block‐poly(n‐butyl acrylate)] with benzoyl trifluoromethanesulfonate as a benzoylation reagent. The use of this mild reagent (which required no added Lewis acid) permitted polymers with well‐defined compositions and narrow molecular weight distributions to be synthesized. Micelles formed from one of these benzoylated polymers, [polystyrene_0.25‐co‐poly(vinylbenzophenone)_0.75]₁₁₅‐block‐poly(acrylic acid)₁₄, were then fixed by the irradiation of the micelle cores with UV light. As the irradiation time was increased, the pendent benzophenone groups crosslinked with other chains in the glassy micelle cores. Dynamic light scattering, spectrofluorimetry, and Fourier transform infrared spectroscopy were all used to verify the progress of the crosslinking reaction. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2604–2614, 2006     

SPAR: stochastic programming with adversarial recourse

We consider a general adversarial stochastic optimization model. Our model involves the design of a system that an adversary may subsequently attempt to destroy or degrade. We introduce SPAR, which utilizes mixed-integer programming for the design decision and a Markov decision process (MDP) for the modeling of our adversarial phase.     

Establishing vicinal proton-proton connectivities using two-dimensional proton zero quantum coherence NMR

Phenanthro[3,4:3′,4′]phenanthro[2,1-b]thiophene ( 1 ) served as the model system to evaluate two-dimensional proton zero quantum coherence nmr in order to establish the vicinal proton-proton connectivities. The utility of the two-dimensional proton zero quantum nmr experiment has been compared with the utility of the traditional autocorrelated proton-proton (COSY) experiment. In the case of a molecule such as 1 , where the proton chemical shifts are so highly congested, the zero quantum coherence experiment provides data not obtainable from the COSY experiment.     

Disturbing the Dyson conjecture, in a generally GOOD way

Dyson's celebrated constant term conjecture [F.J. Dyson, Statistical theory of the energy levels of complex systems I, J. Math. Phys. 3 (1962) 140-156] states that the constant term in the expansion of _∏1≦i≠j≦naj(1−xi/xj) is the multinomial coefficient (a₁+a₂+?+an)!/(a₁!a₂!?an!). The definitive proof was given by I.J. Good [I.J. Good, Short proof of a conjecture of Dyson, J. Math. Phys. 11 (1970) 1884]. Later, Andrews extended Dyson's conjecture to a q-analog [G.E. Andrews, Problems and prospects for basic hypergeometric functions, in: R. Askey (Ed.), The Theory and Application of Special Functions, Academic Press, New York, 1975, pp. 191-224]. In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding q-analogs are supplied. Finally, perturbed versions of the q-Dixon summation formula are presented.     

Samuelson maps—recent results

Samuelson maps are maps whose Jacobian matrix has nowhere vanishing leading principal minors. This paper surveys some recent results on invertibility of Samuelson maps and their decomposition into maps that fix all but one coordinate. The most noteworthy result is that real, rational, everywhere defined Samuelson maps are invertible. Proofs are sketched or omitted entirely.