Heterobimetallic lanthanide-gold coordination polymers: structure and emissive properties of isomorphous [(n)Bu4N]2[Ln(NO3)4Au(CN)2] 1-D chains

A new series of lanthanide-containing dicyanoaurate coordination polymers, [(n)Bu(4)N](2)[Ln(NO(3))(4)Au(CN)(2)] (Ln = Nd, Eu, Gd or Tb), were synthesized and structurally characterized. They form an isomorphous series, crystallizing in the space group I2(1)2(1)2(1). The structure is composed of a one dimensional zigzag of Ln-N-C-Au-C-N-Ln chains with no intra- or inter-chain aurophilic interactions. The series is related to and can be described as a reduced dimensionality analogue of the previously studied Ln[Au(CN)(2)](3)·3H(2)O. Unlike the Ln[Au(CN)(2)](3)·3H(2)O series, there is no efficient energy transfer between dicyanoaurate and the lanthanide metal centers in the complexes and they essentially act as two separate emissive chromophores.     

Synthesis and Liquid Crystal Behaviour of Further 4,4″-Disubstituted 2'-Fluoro-1,1':4',1”-Terphenyls

Further to earlier work on the liquid crystal properties of fluoro-1,1':4',1”-terphenyls¹ we have now extended this series to include terminally fluoro-and cyano-substituted 2'-fluoro-1,1':4',1″-terphenyls, chiral 2'-fluoro-1,1':4',1″-terphenyls, and esters derived from 2'-fluoro-1,1':4',1”-terphenyls and incorporating the alkylcyclobutyl group. The preparations and transition temperatures for these series of compounds are presented and their transition temperatures and mesophase types are discussed. An interesting result from this work was the appearance of an S^* _C phase for one chiral homologue and of S _c phases for the esters incorporating the alkylcyclobutyl group. The S _c formation is attributed to the presence of the fluoro-substituent.

Keywords: terphenyls, fluoroterphenyls, smectic C phases, cyclobutyl esters     

Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces

We show that for 1 < p < ??, weight w ?? A _p , and any L ²-bounded Calderón-Zygmund operator T, there is a constant C _T,p such that the weak- and strong-type inequalities $${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$ $${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$ hold, where T _? denotes the maximal truncations of T and ${\left\| w \right\|_{{A_p}}}$ denotes the Muckenhoupt A _p characteristic of w. These estimates are not improvable in the power of ${\left\| w \right\|_{{A_p}}}$ . Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hyt?nen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T _? and sufficient conditions for two-weight inequalities in L ^p for T _?. Our proof does not rely upon extrapolation.     

Partial Differential Equations That Are Hard to Classify

Semi-linear n×n systems of the form A∂u/∂x+B∂u/∂y=f can generally be solved, at least locally, provided data are imposed on non-characteristic curves. There are at most n characteristic curves and they are determined by the coefficient matrices on the left-hand sides of the equations. We consider cases where such problems become degenerate as a result of ambiguity associated with the definition of characteristic curves. In such cases, the existence of solutions requires restrictions on the data and solutions might not be unique.     

Radial oscillations of neutron stars in strong magnetic fields

The eigen frequencies of radial pulsations of neutron stars are calculated in a strong magnetic field. At low densities we use the magnetic BPS equation of state (EOS) similar to that obtained by Lai and Shapiro while at high densities the EOS obtained from the relativistic nuclear mean field theory is taken and extended to include strong magnetic field. It is found that magnetized neutron stars support higher maximum mass whereas the effect of magnetic field on radial stability for observed neutron star masses is minimal.     

Twisted smectic A phases in side chain liquid crystal polymers

The syntheses of two side chain liquid crystal polymers, a polyacrylate and a polymethacrylate, are reported. In each of the polymers the liquid-crystalline side group carries an asymmetric carbon atom, thereby making some of the liquid crystal phases formed by the polymers optically active and chiral. For the chiral polyacrylate smectic A and chiral ferroelectric smectic C phases are observed, however for the chiral polymethacrylate a cholesteric phase is detected above the smectic A phase. It is found that the smectic A to cholesteric phase transition is mediated by the formation of an intermediary twisted smectic A phase. This intermediary phase is a liquid-crystalline analogue of the Abrikosov flux phase found in Type II superconductors.     

Weighted Bounds for Variational Walsh–Fourier Series

For 1<p<?? and a weight w??A _p and a function in L ^p ([0,1],w) we show that variational sums with sufficiently large exponents of its Walsh?CFourier series are bounded in L ^p (w). This strengthens a result of Hunt?CYoung and is a weighted extension of a variation norm Carleson theorem of Oberlin?CSeeger?CTao?CThiele?CWright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lépingle.