In industrial processes, formaldehyde is mainly handled in aqueous solutions, which often contain methanol. In these solutions, formaldehyde forms predominantly adducts with the solvents.In aqueous solutions, methylene glycol and poly(oxymethylene) glycols are formed, in methanolic solutions hemiformal and poly(oxymethylene) hemiformals. As both the formation of poly-(oxymethylene) glycol and of poly(oxymethylene) hemiformal are slow compared to typical residence times in separation equipment, reliable information on kinetics of these reactions is essential for process design. Two independent methods were applied to obtain this information: NMR spectroscopy and high-resolution densimetry. The experiments were carried out at temperatures between 273 and 334 K and pH between 2 and 9. Both for poly(oxymethylene) glycol formation and poly(oxymethylene) hemiformal formation, the minimal reaction rate occurs between pH 3 and 5. At 293 K, the inverse rate constant 1/k at this minimum is about 6 min for poly(oxymethylene) glycol formation and about 110 h for poly(oxymethylene) hemiformal formation. The rate constants determined with NMR spectroscopy and densimetry generally agree well. Previously reported discrepancies between results from both methods are explained by the fact that rate constants of poly(oxymethylene) glycol formation depend strongly on the solvent water or deuterium oxide. Reaction kinetics of poly(oxymethylene) glycol and poly-(oxymethylene) hemiformal formation in the mixed-solvent system with water and methanol predicted from results obtained in the single-solvent systems are in good agreement with experimental data. is large Ocmg¿cfa % 650 at 293 K), the inverse reaction, the degradation of methylene glycol, is slow (at room temperature typically 1/&*mg « 1 min).
The edges of the complete graph $K_n$ are coloured so that no colour appears more than $\lceil cn\rceil$ times, where $c < 1/32$ is a constant. We show that if $n$ is sufficiently large then there is a Hamiltonian cycle in which each edge is a different colour, thereby proving a 1986 conjecture of Hahn and Thomassen. We prove a similar result for the complete digraph with $c < 1/64$. We also show, by essentially the same technique, that if $t\geq 3$, $c < (2t^2(1+t))^{-1}$, no colour appears more than $\lceil cn\rceil$ times and $t|n$ then the vertices can be partitioned into $n/t$ $t-$sets $K_1,K_2,\ldots,K_{n/t}$ such that the colours of the $n(t-1)/2$ edges contained in the $K_i$'s are distinct. The proof technique follows the lines of Erdős and Spencer's modification of the Local Lemma.
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.
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