Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

Fluorenyl-containing sulfonated poly(aryl ether ether ketone ketone)s (SPFEEKK) for fuel cell applications

A series of fluorenyl-containing sulfonated poly(aryl ether ether ketone ketone)s (SPFEEKK) were synthesized via aromatic nucleophilic substitution polymerization. The sulfonation content (SC) was controlled by the feed ratios of sulfonated and nonsulfonated monomers. Flexible and strong membranes in the sulfonic acid form were obtained from cast membranes in the sodium salt forms by treatment with acid. The thermal properties, water uptake, swelling ratio, water state, oxidative stability, proton conductivity and methanol permeability were investigated. All the polymers had proton conductivities greater than 1 × 10⁻² S/cm at room temperature, and the conductivity values of m-SPFEEKK-80 and p-SPFEEKK-80 were up to 1.86 × 10⁻¹ and 1.78 × 10⁻¹ S/cm at 100 °C. This series of polymers also possessed good dimensional stability in water and low methanol crossover.     

Do halogen and methyl substituents have electronic effects on the structures of simple disilanes? An experimental and theoretical study of the molecular structures of the series X3SiSiMe3 (X = H, F, Cl and Br)

The gas-phase molecular structures of a series of halogen-substituted disilanes [X₃SiSiMe₃ (X = H, F, Cl and Br)], 1,1,1-trimethyldisilane (H₃SiSiMe₃), 1,1,1-trifluoro-2,2,2-trimethyldisilane (F₃SiSiMe₃), 1,1,1-trichloro-2,2,2-trimethyldisilane (Cl₃SiSiMe₃) and 1,1,1-tribromo-2,2,2-trimethyldisilane (Br₃SiSiMe₃), have been determined in the gas phase by electron diffraction. Ab initio calculations at the HF and MP2 level were used to support the experimental investigation using the SARACEN method. All of the investigated structures were determined to adopt a staggered structure with C _3v symmetry. The effect of substitution on the Si–Si bond and the Si–Si–X bond angle was investigated and these results were compared to results obtained from a recent study of halogen-substituted disilanes [X₃SiSiXMe₂ (X = F, Cl, Br and I)] to consider the effect of the methyl groups on the substituted disilanes.     

Vacuum Ultraviolet Spectrum of IBr

Rydberg transitions for the interhalogen IBr, are reported in the wavelength region 160-130 nm.     

Nitrogen Doping and Carbon Coating Affects Substrate Selectivity of TiO2 Photocatalytic Organic Pollutant Degradation

A series of carbon-coated, nitrogen-doped titanium dioxide photocatalysts was produced and characterized. N-doped TiO₂ powder samples were prepared using a sol-gel method and subsequently used for making doped-TiO₂ thin films on glass substrates. Carbon layers were coated on the films by a thermal decomposition method using catechol. Diffuse reflectance spectra and Mott-Schottky analyses of the samples proved that nitrogen doping and carbon coating can slightly lower the band gap of TiO₂, broaden its absorption to visible light and enhance its n-type character. According to photocatalytic tests against model contaminants, carbon-coated nitrogen-doped TiO₂ films have better performance than simple TiO₂ on the degradation of Rhodamine B dye molecules, but are poorly effective for degrading 4-chlorophenol molecules. Several possible explanations are proposed for this result, supported by scavenging experiments. This reveals the importance of a broad substrate scope when assessing new photocatalytic materials for water treatment, something which is often overlooked in many literature studies.     

Synthesis,Structure, and DFT Analysis of the THF Solvate of 2-Picolyllithium: A 2-Picolyllithium Solvate with Significant Carbanionic Character

Previous studies of different solvates of 2-methylpyridyllithium (2-picolyllithium) have uncovered electronic structures corresponding to aza-allyl and enamido resonance forms of the metallated pyridine-based compounds. Here, we report the synthesis and characterization of [2-CH₂Li(THF)₂C₅H₄N], a new THF solvate. X-ray crystallographic studies reveal a dimeric arrangement featuring a non-planar eight-membered [NCCLi]₂ ring, in which the primary cation-anion interaction is between the central Li atom and the C atom of the deprotonated methyl group [length, 2.285(2) Å], suggesting a new carbanionic resonance structure for this 2-picolyllithium series. The significant carbanionic character of [2-CH₂Li(THF)₂C₅H₄N] was confirmed by gas-phase DFT calculations [B3LYP/6-311+G(d)] with the calculated electron density interrogated by means of quantum theory of atoms in molecules (QTAIM) and natural bond orbital (NBO) analyses. For comparison these computational analyses were also performed on the literature structures of [2-CH₂Li(2-Picoline)C₅H₄N] and [2-CH₂Li(PMDETA)C₅H₄N]. In a reactivity study, [2-CH₂Li(THF)₂C₅H₄N] was found to undergo nucleophilic addition to pyridine to generate dipyridylmethane in a good yield.