A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in ?², there is a line ? such that in both line sets, for both halfplanes delimited by ?, there are $\sqrt{n}$ lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are $\sqrt{n/3}$ of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erd?s–Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to graph drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labeled lines that are universal for all n-vertex labeled planar graphs. In contrast, the main result by Pach and Toth (J. Graph Theory 46(1):39–47, 2004), has, as an easy consequence, that every set of n (unlabeled) lines is universal for all n-vertex (unlabeled) planar graphs.     

Improving the Accuracy of Small-Molecule Crystal Structures Solved from Powder X-Ray Diffraction Data by Using External Sources

Structural analysis using powder X-ray diffraction data has overcome many obstacles and nowadays is readily applicable for structural analysis of all types of compounds and materials. Being less straightforward than single crystal diffraction, it requires significant users’ input and consequently, implementation of standardized tools to assess the accuracy of crystal structures. This article discusses potential errors in crystal structure solution and refinement of small-molecule structures obtained from PXRD data. Moreover, it proposes how accuracy of these structures can be improved by using high-quality PXRD data, complementary external analytical techniques, knowledge stored in crystal structure databases, as well as an approach to search the parameter space to avoid local minima in testing different sets of geometry restraints.     

Nonionic surfactant-selective electrode and its application for determination in real solutions

A liquid membrane nonionic surfactant sensitive electrode has been prepared, based on a new barium pseudocationic complex of a highly ethoxylated fatty alcohol polyglycol ether and tetraphenylborate as sensing material. The complex has been incorporated into the plasticized PVC-membrane and used as sensing material.The electrode exhibited positive linear non-Nernstian response toward different nonionic surfactants and sub-Nernstian response toward tetraphenylborate with the lower detection limit of 3.3 × 10⁻⁷ mol dm⁻³ in barium chloride solution.The interfering effect of some alkaline, alkaline earth, and heavy metal cations, has been demonstrated by displaying their calibration curves compared with that of Triton X-100.The electrode has been used as an end-point indicator for potentiometric titration of analytical and technical grade polyethoxylated nonionic surfactants, modelled detergent products, and commercial detergents.     

Chemistry of Spontaneous Alkylation of Methimazole with 1,2-Dichloroethane

Spontaneous S-alkylation of methimazole (1) with 1,2-dichloroethane (DCE) into 1,2-bis[(1-methyl-1H-imidazole-2-yl)thio]ethane (2), that we have described recently, opened the question about its formation pathway(s). Results of the synthetic, NMR spectroscopic, crystallographic and computational studies suggest that, under given conditions, 2 is obtained by direct attack of 1 on the chloroethyl derivative 2-[(chloroethyl)thio]-1-methyl-1H-imidazole (3), rather than through the isolated stable thiiranium ion isomer, i.e., 7-methyl-2H, 3H, 7H-imidazo[2,1-b]thiazol-4-ium chloride (4a, orthorhombic, space group Pnma), or in analogy with similar reactions, through postulated, but unproven intermediate thiiranium ion 5. Furthermore, in the reaction with 1, 4a prefers isomerization to the N-chloroethyl derivative, 1-chloroethyl-2,3-dihydro-3-methyl-1H-imidazole-2-thione (7), rather than alkylation to 2, while 7 further reacts with 1 to form 3-methyl-1-[(1-methyl-imidazole-2-yl)thioethyl]-1H-imidazole-2-thione (8, monoclinic, space group P 2₁/c). Additionally, during the isomerization of 3, the postulated intermediate thiiranium ion 5 was not detected by chromatographic and spectroscopic methods, nor by trapping with AgBF₄. However, trapping resulted in the formation of the silver complex of compound 3, i.e., bis-{2-[(chloroethyl)thio]-1-methyl-1H-imidazole}-silver(I)tetrafluoroborate (6_, monoclinic, space group P 2₁/c), which cyclized upon heating at 80 °C to 7-methyl-2H, 3H, 7H-imidazo[2,1-b]thiazol-4-ium tetrafluoroborate (4b, monoclinic, space group P 2₁/c). Finally, we observed thermal isomerization of both 2 and 2,3-dihydro-3-methyl-1-[(1-methyl-1H-imidazole-2-yl)thioethyl]-1H-imidazole-2-thione (8), into 1,2-bis(2,3-dihydro-3-methyl-1H-imidazole-2-thione-1-yl)ethane (9), which confirmed their structures.     

Synthesis,structure and properties of eight novel molybdenum(VI) complexes of the types: [MoO2LD] and [{MoO2L}2D] (L = thiosemicarbazonato ligand,D = N-donor molecule)

Eight new molybdenum(VI) complexes with 4-(diethylamino)salicylaldehyde and 2-hydroxy-3-methoxybenzaldehyde thiosemicarbazones have been prepared. They were characterized as mononuclear [MoO₂LD] or dinuclear [{MoO₂L}₂D] complexes. In all the compounds the MoO₂²⁺ core is coordinated by a tridentate ONS thiosemicarbazonato ligand and by the N-donor molecule (imidazole, pyridine or γ-picoline). All the complexes were characterized by chemical analysis, IR spectroscopy and thermogravimetry. Three of the mononuclear complexes, dioxo[(2-hydroxy-3-methoxybenzaldehyde thiosemicarbazonato)(pyridine)]molybdenum(VI), dioxo[(2-hydroxy-3-methoxybenzaldehyde thiosemicarbazonato)(γ-picoline)]molybdenum(VI) and dioxo[(2-hydroxy-3-methoxybenzaldehyde thiosemicarbazonato)(imidazole)]molybdenum(VI) were also characterized by single-crystal X-ray structural analysis. A spectrophotometric method for the determination of molybdenum based on extraction of ion-pairs formed by the cationic surfactant and the [MoO(SCN)₄]⁻ anion is described.     

A Polynomial Bound for Untangling Geometric Planar Graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002) asked if every n-vertex geometric planar graph can be untangled while keeping at least n ^ε vertices fixed. We answer this question in the affirmative with ε=1/4. The previous best known bound was \(\Omega(\sqrt{\log n/\log\log n})\). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least \(\Omega(\sqrt{n})\) vertices fixed, while the best upper bound was \(\mathcal{O}((n\log n)^{2/3})\). We answer a question of Spillner and Wolff (http://arxiv.org/abs/0709.0170) by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than \(3(\sqrt{n}-1)\) vertices fixed.     

Synthesis, NMR and X-ray structure analysis of macrolide aglycons

Macrolide aglycons (E)-9-hydroxyimino-6-O-methylerythronolide A (4), 9a-aza-9-deoxo-9,9-dihydro-9a,11-O-dimethyl-9a-homoerythronolide A (5) and 9a-aza-9-deoxo-9,9-dihydro-9a-homoerythronolide A (6) were prepared by multistep syntheses. A conformational study of these new macrolide aglycons was performed using single crystal X-ray crystallography to gain information about the solid state, while a combination of NMR spectroscopy and molecular modelling was employed to study the solution structures. The crystal structures were found to be stabilised by a complex network of hydrogen bonds and van der Waals interactions. To some extent, the same building motif of infinite molecular chains held together by O?CH···O hydrogen bonds was present in the crystal structure of all three compounds. Thorough analysis and comparison of the obtained solid state structures with their solution counterparts showed no significant differences between them, confirming the constrained flexibility of the macrocyclic ring. Moreover, in all three compounds, in both solution and solid state, the macrolactone ring adopts energetically more favoured folded-out conformations.     

Di­iodo­bis(1‐methyl‐1,3‐imidazolium‐2‐thiol­ato‐S)­mercury(II)

The reaction of 1‐methyl‐1,3‐imidazole‐2‐thione (meimtH) with mercury(II) iodide in methanol in a 2:1 molar ratio resulted in the formation of single crystals of the title compound, [HgI₂(C₄H₆N₂S)₂]. The Hg atom is coordinated by two I [2.7809 (9) and 2.7999 (8) Å] and two thione S atoms [2.520 (3) and 2.576 (3) Å] with irregular tetrahedral coordination geometry. The NH groups of the imidazole ring take part in intra‐ and intermolecular hydrogen bonds with I atoms [N?I 3.596 (8) and 3.611 (9) Å, respectively] joining mol­ecules into infinite chains parallel to the z axis.