A characterization of discrete unimodality with applications to variance upper bounds

Bertin and Theodorescu (1984,Statist. Probab. Lett.,2, 23–30) developed a characterization of discrete unimodality based on convexity properties of a discretization of distribution functions. We offer a new characterization of discrete unimodality based on convexity properties of a piecewise linear extension of distribution functions. This reliance on functional convexity, as in Khintchine's classic definition, leads to variance dilations and upper bounds on variance for a large class of discrete unimodal distributions. These bounds are compared to existing inequalities due to Muilwijk (1966,Sankhy, Ser. B,28, p. 183), Moors and Muilwijk (1971,Sankhy, Ser. B,33, 385–388), and Rayner (1975,Sankhy, Ser. B,37, 135–138), and are found to be generally tighter, thus illustrating the power of unimodality assumptions.     

Nitrato- and fluoroboracites M3B7O13NO3 and M3B7O13F

New boracites containing nitrato- or fluoroanions that appear to be true low-pressure phases have been synthesized at superatmospheric pressures. The M₃B₇O₁₃NO₃ compounds (M = Co, Ni, Cu, Zn, Cd) transform rapidly and reversibly in the temperature region 300–500°C between probable orthorhombic and face centered cubic symmetry, while the M₃B₇O₁₃F compounds (M = Mg, Mn, Fe, Co, Zn) appear to maintain rhombohedral symmetry up to their decomposition temperatures of 800–900°C. True high-pressure boracite-like phases containing F and Cr, Mn, Fe, or Co that decompose upon heating to M₃B₇O₁₃F have also been isolated.     

Assembly of the inorganic double helix Mg6(Et2NCO2)12 by fixation of carbon dioxide: crystal and solution structure

The homoleptic magnesium carbamato complex Mg6(Et2NCO2)12, 1 (Et2NCO2- = diethylcarbamato anion), was prepared by the reaction of dibutylmagnesium with diethylamine, followed by carboxylation using gaseous carbon dioxide. Crystallographic characterization demonstrated that 1 has the standard M6(R2NCO2)12 structure and is a double helix of MgO(x)(x = 5, 6) coordination polyhedra with Delta or Lambda stereochemistry arising from the configuration around the six-coordinate Mg2+ cations. It crystallized in the orthorhombic space group Ccca with two molecules of Delta1 and two of Lambda1 per unit cell (a = 21.548 A, b = 25.094 A, c = 15.4485(11) A, alpha = beta = gamma = 90 degrees ). Extensive solution characterization of 1 by 1-dimensional proton and 13C NMR spectroscopy and by two-dimensional 1H-[13C] NMR correlation techniques verified that the helical structure is maintained in solution. Moreover, these measurements indicated that the intramolecular dynamics of 1 relating to motions of the ethyl groups was substantially hindered in solution. Correlation of the crystallographic and NMR structural studies indicated that this arises from a combination of hindered rotation about the carbamato C-N bond and efficient packing of the ethyl groups around the Mg6O24 core. The result is an inverted-micelle-like structure for 1 in which the hydrophobic ethyl groups form a sheath largely restricting access to the hydrophilic Mg6O24 core.     

Chemical modification of a viral cage for multivalent presentation

Here we present generalized methods for chemically modifying the surface of a viral protein cage; this exploits the chemistry of native and engineered surface exposed functional groups for multivalent presentation of ligands.     

Dye-sensitized solar cells based on composite solid polymer electrolytes

The ionic conductivity of polymer electrolytes and their interfacial contact with dye-attached TiO2 particles were enhanced markedly by the addition of amorphous oligomer into polymer electrolytes, resulting in very high overall energy conversion efficiency.