Inequalities for Volumes of Simplices in Terms of Their Faces

By generalizing some well-known results, we first obtain an inequality involving the volume and product of s-contents of s-faces of an n-simplex. Using this we generalize two inequalities maximizing the volume of one or two simplices in terms of their edge lengths.     

Probing alkali coordination environments in alkali borate glasses by multinuclear magnetic resonance

Five series of binary alkali borate glasses were prepared to compare the alkali dependence of network and modifier short-range order. ¹¹B magic-angle spinning (MAS) NMR reveals that the fraction of four-coordinate boron depends strongly upon alkali type at high-alkali concentrations: heavier alkalis favour the formation of non-bridging oxygens, whereas lithium borates contain a much higher concentration of tetrahedral boron units. The alkali modifiers were observed directly by MAS NMR to measure the change in chemical shift with composition. All alkali peaks shift to higher frequency with increasing loading, indicative of decreasing average coordination numbers. Relative to their known chemical shift ranges, the heavier alkalis exhibit the greatest shifts, whereas the lithium shifts are subtle. This is interpreted in terms of the availability of charged and partially charged coordinating oxygens in the network. Moreover, the ¹³³Cs chemical shifts plateau at 40 mol%, implying that the Cs⁺ coordination number reaches a lower limit at this composition. This work demonstrates that NMR instrumentation and methodology have reached a level where even challenging nuclei like ³⁹K and ⁸⁷Rb can be probed to yield structural information in glasses.     

Synthesis and X-ray structural analysis of (2<Emphasis Type="Italic">Z</Emphasis>)-3(4-hydroxyphenyl)-2-pyridin-4-ylacrylonitrile and (2<Emphasis Type="Italic">Z</Emphasis>)-3-(3,5-di-<Emphasis Type="Italic">tert</Emphasis>-butyl-4-hydroxyphenyl)2-pyridin-4-ylacrylonitrile

Synthesis and X-ray structural investigations have been carried out for the two title compounds C₁₄H₁₀N₂O (3a) and C₂₂H₂₆N₂O (3b). Compound 3a crystallizes in the monoclinic space group P2₁/c, with a = 3.843(1) Å, b = 24.618(5) Å, c = 11.318(2) Å, = 92.61(3), V = 1069.7(4) Å,³ and Z = 4. Compound 3b crystallizes in the triclinic space group P-1, with a = 9.004(2) Å, b = 9.447(2) Å, c = 11.713(2) Å, = 76.70(3), = 83.12(3), = 82.16(3), V = 956.5(3) Å,³ and Z = 2. Both stilbazole derivatives have Z-geometry about the ethylene bridge which links the heterocyclic and aromatic rings. The molecular skeleton of 3a is slightly non-planar: the dihedral angles between the acrylonitrile linkage and the pyridine ring, and between this linkage and the p-hydroxyphenyl ring are 7.2(2) and 4.1(2), respectively. The molecular skeleton of 3b is less planar: the values of similar dihedral angles are 17.0(2) and 20.8(2), respectively. In the crystal of 3a, the molecules are packed in stacks along the a axis with head-to-head orientation. Intermolecular hydrogen bonds O=H s N and C=H s N link molecules into sheets parallel to (100) plane. In the crystals of 3b, the molecules have a head-to-tail orientation and intermolecular hydrogen bonds O=H s N link the molecules into infinite chains along [01-1] direction.     

Antimony(III) fluoride: inclusion complexes with crown ethers

Four crystalline molecular complexes between antimony(III) fluoride and 18-membered crown ethers have been obtained and their structures investigated by single crystal X-ray diffraction techniques: [18-crown-6·SbF₃], C₁₂H₂₄F₃O₆Sb,P2₁2₁2₁,a=8.328(4),b=11.573(4),c=18.094(4),V=1744(1)³,Z=4; [benzo-18-crown-6·SbF₃], C₁₆H₂₄F₃O₆Sb,P2₁/n,a=10.490(2),b=13.714(1),c=13.442(2), =101.94(1)°,V=1892(1)³,Z=4; [cis-syn-cis-dicyclohexano-18-crown-6·SbF₃·CH₃OH], C₂₁H₄₀F₃O₇Sb,P2₁/n,a=8.270(4),b=23.386(3),c=12.772(1), =96.31(2)°,V=2455(1)³,Z=4; [cis-anti-cis-dicyclohexano-18-crown-6·SbF₃], C₂₀H₃₆F₃O₆Sb,Pna2₁,a=21.091(8),b=12.829(5),c=8.437(3),V=2283(2)³,Z=4. All species are the perching-type complexes with the antimony fluoride above the cavity and the metal lone pair pointed toward the center of the crown ring. The antimony atom interacts with all six crown oxygen atoms with Sb–O distances of 2.837(2)–3.344(2) . The antimony atom is displaced from the least square plane of the crown oxygen atoms at the distances of 1.288–1.383 .     

Low temperature organocopper-mediated two-component cross coupling/cycloisomerization approach toward N-fused heterocycles

Organocopper reagents smoothly react with heterocyclic propargyl mesylates at low temperature to produce N-fused heterocycles. The copper reagent plays a "double duty" in this novel cascade transformation, which proceeds via an SN2' substitution followed by a subsequent cycloisomerization step.     

Surfactant-induced amorphous aggregation of tobacco mosaic virus coat protein: a physical methods approach

The interactions of non-ionic surfactant Triton X-100 and the coat protein of tobacco mosaic virus, which is an established model for both ordered and non-ordered protein aggregation, were studied using turbidimetry, differential scanning calorimetry, isothermal titration calorimetry, and dynamic light scattering. It was found that at the critical aggregation concentration (equal to critical micelle concentration) of 138 x 10(-6) M, Triton X-100 induces partial denaturation of tobacco mosaic virus coat protein molecules followed by protein amorphous aggregation. Protein aggregation has profound ionic strength dependence and proceeds due to hydrophobic sticking of surfactant-protein complexes (start aggregates) with initial radii of 46 nm. It has been suggested that the anionic surfactant sodium dodecyl sulfate forms mixed micelles with Triton X-100 and therefore reverses protein amorphous aggregation with release of protein molecules from the amorphous aggregates. A stoichiometric ratio of 5 was found for Triton X-100-sodium dodecyl sulfate interactions.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Graphs with many r-cliques have large complete r-partite subgraphs

Let r2 and c>0. Every graph on n vertices with at least cn^rcliques on r vertices contains a complete r-partite subgraphwith r–1 parts of size c^rlog n and one part of size greaterthan n^1–cr–1. This result implies a quantitativeform of the Erdös–Stone theorem.