Über eine Abschätzung der Amplituden in freien Schwingungs-problemen veränderlicher Kreisfrequenz

Summary Consider any differential equation of the form given under (1) below and, ifx(t) is any solution of (1) which is real-valued and does not vanish identically, letC be a half-waveatb of the curvex=x(t) in a (t, x)-plane [by this is meant thatx(a)=0 andx(b)=0 but |x(t)|>0 ifa<t<b]. Then ana priori upper limitation, involving the moments of the coefficient function of (1) [cf. the three integrals in formula (2) of the text, wherea=0 andb=h], is obtained for the ratiom/f. Herem denotes the maximum amplitude occurring inC, andf the area of the region surrounded by the segmentatb on thet-axis and the graph ofC in the (t, x)-plane (this region is always convex).     

Pro-porous coordination polymers of the 1,4-bis((3,5-dimethyl-1H-pyrazol-4-yl)-methyl)benzene ligand with late transition metals

Solvothermal reactions of the flexible, pyrazole-based 1,4-bis((3,5-dimethyl-1H-pyrazol-4-yl)methyl)benzene ligand (H(2)BDMPX) with late transition metal ions allowed the isolation of the four coordination compounds M(x)(BDMPX) (x = 1 for M = Zn, 1; Co, 2; Cd, 3; x = 2 for M = Cu, 4). The investigation of the thermal behavior assessed the high thermal robustness of these materials, which are stable in air at least up to 300 °C, with the Cd(II) derivative starting to decompose only around 500 °C. As retrieved by ab initio X-ray powder diffraction, the isomorphous compounds 1-3 possess a dense 3-D network featuring rhombic motifs hinged about rigid and parallel chains of tetrahedral MN(4) chromophores. As demonstrated by thermodiffractometric measurements, temperature increase triggers framework flexibility. The latter is at work also when N(2) adsorption is assayed at 77 K: 1 and 2 show permanent porosity, with BET and Langmuir specific surface areas of 515, 667 m(2)/g and 209, 384 m(2)/g, respectively. 1 and 2 thus represent an intriguing example of "porosity without pores", their pro-porous nature being explained in terms of the flexibility of the rhombic motifs, stimulated by the gas probe and facilitated by the nature of the ligand.     

Nanostructuring of CyPLOS (Cyclic Phosphate-Linked OligoSaccharides), novel saccharide-based synthetic ion transporters

Ionophores are an important class of synthetic molecules which mimic natural ion channels or carriers. Here we report the aggregation behavior in pseudo-physiological environment of three Cyclic Phosphate-Linked Oligosaccharides (CyPLOS) derivatives, synthetic ion transporters based on cyclic, phosphate-linked disaccharide skeleton differing for the nature of the tails (tetraethylene-TEG glycol and/or n-undecyl chains) attached to the C-2 and C-3 of the constitutive monosaccharides. Their aggregation behavior has been studied by a combined use of dynamic light scattering (DLS), electron paramagnetic resonance spectroscopy (EPR) and Small Angle Neutron Scattering (SANS). DLS measurements were performed to reveal the formation and size distribution of the CyPLOS aggregates. EPR measurements, by using 5-doxyl stearic acid (5-DSA) as spin-probe, showed that the aggregates are mainly due to the formation of double layers and allowed to analyze the local fluidity. Finally, SANS measurements allowed estimating the layer thickness of the double layers. Our results indicate that the three CyPLOS analogs show self-aggregation properties that depend on the different nature of the inserted tails.     

Synthesis of dihydroxy telechelic oligomers of β-butyrolactone catalyzed by titanium(IV)-alkoxides and their use as macrodiols in polyurethane chemistry

We report on a solvent-free approach for the synthesis of low molecular weight, α,ω-dihydroxy telechelic poly(β-butyrolactone). In the presence of Ti(IV) alkoxides, mixtures of β-butyrolactone and diols, like di- or triethylene glycol, were reacted in ratios between 4:1 and 10:1. The oligomerization proceeds at elevated temperatures (80–100°C). Different alkoxide substituents (R = Me, iPr, tBu) of the Ti(IV)(OR)₄ catalyst were investigated. The resulting oligomers were characterized by nuclear magnetic resonance (NMR), infra-red (IR), gel-permeation chromatography (GPC), titration, and matrix-assisted laser desorption-time-of-flight mass spectrometry (MALDI-ToF-MS) analysis. Aside from low molecular weight products, special effort was devoted to achieve high O-acyl cleavage selectivity and to circumvent the formation of unsaturated end-groups in order to form exclusively dihydroxy-telechelic oligomers. Optimized results in terms of selectivity and reaction rates were achieved at 100°C using catalyst loadings of 0.2 mol% with respect to the monomer. The molecular weights determined by GPC were in good accordance with the ratio of monomer to diol used, confirming successful oligomer formation. Polyurethanes prepared from crude macrodiols without any additional catalyst feature molecular weights up to 50,000 g/mol. The reported work serves as concept to utilize β-lactones for tailored polyol synthesis; the resulting products are suitable for polyurethane chemistry.     

Novel coordination frameworks incorporating the 4,4'-bipyrazolyl ditopic ligand

The reaction of the rigid spacer 4,4'-bipyrazole (H(2)BPZ) with late transition metals, either following conventional routes or under solvothermal conditions, afforded the coordination polymers [M(BPZ)]·Solv (M = Zn, 1; Co, 2; Cd, 3; Hg, 4; Cu, 5; Ni, 6; Pd, 7; Solv = DMF, 3; MeCN, 5 and 6; H(2)O, 7), [Cu(H(2)BPZ)(2)(NO(3))(2)] (8), and [Cd(H(2)BPZ)(CH(3)COO)(2)] (9). State-of-the-art laboratory powder diffraction methods allowed to disclose the isomorphous character of 1 and 2, as well as of 5 and 6, which feature 3D porous networks containing 1D channels of square and rhombic shape, respectively. 3, crystallizing in the relatively rare P6(1)22 space group, consists of homochiral helices of octahedral Cd(II) ions, packing in bundles mutually linked by "radial", nonplanar BPZ ligands. Finally, the dense species 8 and 9 contain parallel 2D layers of square and rectangular meshes, respectively. Thermogravimetric analyses witnessed the relevant thermal robustness of all the [M(BPZ)] materials [except the mercury(II) derivative], which are stable in air at least up to 300 °C, with the zinc(II) derivative decomposing only around 450 °C. Variable-temperature powder diffraction experiments highlighted the permanent porosity of 1-3, 5, and 6, retained along consecutive temperature cycles in all cases but 3. When probed with N(2) at 77 K, 1-3 and 5-7 showed Brunauer-Emmett-Teller and Langmuir specific surface areas in the ranges 314(2)-993(11) and 509(16)-1105(1) m(2)/g, respectively.