Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures

A scale of BMO spaces appears naturally in product spaces corresponding to the different, yet equivalent, characterizations of the class of functions of bounded mean oscillation in one variable. S.-Y. Chang and R. Fefferman characterized product BMO, the dual of the (real) Hardy spaceH _Re ¹ on product domains, in terms of Carleson measures. Here we describe two other BMO spaces, one contained in and the other containing product BMO, in terms of Carleson measures and Hankel operators. Both of these spaces play a significant role in harmonic analysis of the polydisk and in multivariable operator theory. First author supported in part by a grant from the National Science Foundation. Second author supported in part by a grant from the Department of Energy.     

On the balancing of structural and acoustic performance of a sandwich panel based on topology,property, and size optimization

Balancing structural and acoustic performance of a multi-layered sandwich panel is a formidable undertaking. Frequently the gains achieved in terms of reduced weight, still meeting the structural design requirements, are lost by the changes necessary to regain acceptable acoustic performance. To alleviate this, a design method for a multifunctional load bearing vehicle body panel is proposed which attempts to achieve a balance between structural and acoustic performance. The approach is based on numerical modelling of the structural and acoustic behaviour in a combined topology, size, and property optimization in order to achieve a three dimensional optimal distribution of structural and acoustic foam materials within the bounding surfaces of a sandwich panel. In particular the effects of the coupling between one of the bounding surface face sheets and acoustic foam are examined for its impact on both the structural and acoustic overall performance of the panel. The results suggest a potential in introducing an air gap between the acoustic foam parts and one of the face sheets, provided that the structural design constraints are met without prejudicing the layout of the different foam types.     

Self-assembled palladium(II) "click" cages: synthesis, structural modification and stability

Readily synthesised and functionalised di-1,2,3-triazole "click" ligands are shown to self-assemble into coordinatively saturated, quadruply stranded helical [Pd(2)L(4)](BF(4))(4) cages with Pd(II) ions. The cages have been fully characterised by elemental analysis, HR-ESMS, IR, (1)H, (13)C and DOSY NMR, DFT calculations, and in one case by X-ray crystallography. By exploiting the CuAAC "click" reaction we were able to rapidly generate a small family of di-1,2,3-triazole ligands with different core spacer units and peripheral substituents and examine how these structural modifications affected the formation of the [Pd(2)L(4)](BF(4))(4) cages. The use of both flexible (1,3-propyl) and rigid (1,3-phenyl) core spacer units led to the formation of discrete [Pd(2)L(4)](BF(4))(4) cage complexes. However, when the spacer unit of the di-1,2,3-triazole ligand was a 1,4-substituted-phenyl group steric interactions led to the formation of an oligomeric/polymeric species. By keeping the 1,3-phenyl core spacer constant the effect of altering the "click" ligands' peripheral substituents was also examined. It was shown that ligands with alkyl, phenyl, electron-rich and electron-poor benzyl substituents all quantitatively formed [Pd(2)L(4)](BF(4))(4) cage complexes. The results suggest that a wide range of functionalised palladium(II) "click" cages could be rapidly generated. These novel molecules may potentially find uses in catalysis, molecular recognition and drug delivery.     

On the symmetry of wulfenite (Pb[MoO4]) from Mežica (Slovenia)

Wulfenite [lead(II) molybdate(VI)] is known as a scheelite structure in the I4₁/a space group. The structure of the unusual `hemimorphic' wulfenite crystals from the Mežica mine was refined in the noncentrosymmetric space group I using a Pb/Mo exchange disorder model with the approximate composition Pb_0.94Mo_0.06[MoO₄]. Pb atoms in the 2b positions are substituted by Mo at about 12%. The crystal is shown to be twinned by inversion. Hemimorphism may result from the short‐range chemical ordering of the metal atoms at the 2b positions.     

On the yrast levels in204Pb

High-spin states in²⁰⁴Pb were populated in the²⁰⁴Hg(α,4n) reaction using α-particles in the energy region 42–51 MeV. Prompt and delayedγ-rays as well as conversion electrons were studied in addition to excitation functions, angular distributions andγ-γ coincidences. In this way a stretched cascade ofγ-rays from a level at 8125.9 keV was found to feed the previously known isomeric 9^? level at 2185.7 keV. Spins and parities were established for levels up to and including a 19^? level at 6098.0 keV. The levels with c= 17^2212; and 19^? at excitation energies of 5664.3 and 6098.0 keV are likely to be due to the simplep ₁ ^2/s-1 i ₁₃ ^2/?3 andf ₅ ^2/?1 i ₁₃ ^2/?3 configurations. The agreement between calculated and experimental energies for all observed levels in the regionJ=9–19 is very good in cases where the empirical two-particle interactions used are satisfactorily well known. Above the 19^? level there are three weakly populated levels at 7402.1, 7849.2 and 8125.9 keV, which are likely to haveJ≥20. None of these energies agrees with the calculated value 7695±20 keV for the 20⁺ state of thei ₁₃ ^2/?4 configuration which has the highest angular momentum produced by the four valence neutron holes. This apparent anomaly can be understood if the yrast levels withJ≥20 have angular momentum contribution from the core. It seems likely that the states at 7402.1, 7849.2 and 8125.9 keV are due to proton core excited states of the typeπh _9/2 h ₁₁ ^2/?1 ×νp ₁ ^2/?2 i ₁₃ ^2/?2 withJ ^π=20⁺ andJ ^π=21⁺ andπh _9/2 h ₁₁ ^2/?1 ×νp ₁ ^2/?1 f ₅ ^2?1 i ₁₃ ^2/?2 withJ ^π=22⁺ or 23⁺, respectively. The state at 8126 keV has the highest energy so far directly observed in a stretched cascade ofγ-rays from the decay of a heavy nucleus produced in (α, xn) reactions.     

Enantiomeric discrimination of aromatic-containing anionic substrates using cationic cyclodextrins

Cationic α-, β-, and γ-cyclodextrins were prepared by reacting the corresponding native cyclodextrin (CD) with glycidyltrimethylammonium chloride (GTAC). The reaction conditions were varied in order to alter the degree of substitution of GTAC units on the cyclodextrin. The CD-GTAC derivatives, which retain a positive charge independent of pH, were evaluated as water-soluble chiral NMR solvating agents for anionic substrates. The CD-GTAC derivatives are considerably more effective at causing enantiomeric discrimination in the ¹H NMR spectra of aromatic-containing anionic substrates than the neutral native cyclodextrins. Derivatives with a degree of substitution of about 1.5 were more effective than those with lower or higher degree of substitution. Not one of the α-, β-, and γ-CD-GTAC derivatives was consistently the most effective at causing enantiomeric discrimination for all of the substrates examined herein.     

The role played by salts in the formation of SBA-15, an in situ small-angle X-ray scattering/diffraction study

The influence of salts (NaCl, NaBr, and NaI) on the formation of mesoporous silica SBA-15 was studied in situ by small-angle X-ray scattering and diffraction. Pluronic P104 was used as structure director. The micellar properties and the dynamics of formation were clearly dependent on the presence of salt. It was also shown that the kinetics of mesophase formation, the initial value of the cell parameters, and the extent of long-range order were all influenced by salt additions. The observations are explained to primarily originate from the influence of the anions on the ethylene oxide part of the polymer, i.e., the corona region of the Pluronic micelles. Two effects are identified: a general ion effect causing dehydration of the ethylene oxide part and consequently inducing micellar growth, and a specific ion effect that counterbalances this. The study provides the basis for understanding the means by which addition of simple Na-salts influence the formation of mesoscopically ordered silicas synthesized using nonionic surfactants as structure directors, hence advancing the knowledge base toward a more rational design of mesoporous materials.