Multiple Coverings of the Plane with Triangles

We prove that any 43-fold covering of the plane with translates of a triangle can be decomposed into two coverings.     

The remainder in Weyl's law for random two-dimensional

We prove that the average order of the remainder in counting the number of points of a random lattice inside a disc of radius . Our proof is spectral in nature. Submitted: July 2001, Revised: August 2001, Revised: March 2002.     

A Law of Large Numbers for the Zeroes of Heine–Stieltjes Polynomials

We determine the limiting density of the zeroes of Heine–Stieltjes polynomials (or of any set of points satisfying the conclusion of Heine–Stieltjes Theorem) in the thermodynamic limit and use this to prove a strong law of large numbers for the zeroes.     

Synthesis and Odor of Chiral Partial Structures of Khusimone. Part 1

Khusimone (1), one of the main odor-donating compounds of vetiver oil is subject of the following study on structure/odor relationship. The omittance of the ethano bridge of the tricyclic khusimone leads to a bicyclic system. The stereoselective approach to this degraded structure is described, and the olfactory properties are studied. The key step of the synthesis of the hydrindane nucleus is based on a highly diastereoselective conjugate addition to a chiral oxo-cyclopentene-2-carboxylate.     

Infrared spectra of M+NO 3 − contact ion pairs in aprotic solvents and matrix isolated in the corresponding glasses

Studies of the vibrational spectra of matrix-isolated M⁺NO ₃ ^– ion pairs have been extended to glassy aprotic solvents. The deuterated form of the solvents DMSO, THF, and ACN have windows through the 7- nitrate ionv ₃(e) mode infrared region, so it was possible to clearly observe the splitting of the degeneracy of this mode,v ₃, produced by the contacting, but solvated, alkali metal cation. Primary attention has been directed to the extent to which this splitting is reduced relative to the argon matrix values. This reduction, which reflects electron-density transfer from the solvating molecules to the ion pairs, is comparable to that observed for H₂O and NH₃ matrices as the splitting is reduced to 20–35% of the argon-matrix values. The extent of reduction ofv ₃ for the different solvents has been related to Gutmann's donicity number scale with the correlation holding well for solvent molecules of comparable size, DMSO, THF and DMF, but breaking down for the smaller linear ACN, apparently because of more molecules in the cation solvation sphere. The matrix data have also been used, through comparison with spectra for saturated liquid solutions of Li⁺NO ₃ ^– , to show that the contact ion pair is the dominant species in liquid THF and ACN, whereas the ions are largely solvent separated in DMSO.     

Hydrolysis of the Peptide Bond in N-Acetylated L-Methionylglycine Catalyzed by Various Palladium(II) Complexes: Dependence of the Hydrolytic Reactions on the Nature of the Chelate Ligand in cis-[Pd( L)(H 2O) 2] 21 Complexes

Summary. Hydrolytic reactions between various palladium(II) complexes of the type cis-[Pd(L)(H₂O)₂]²⁺ in which L is ethylenediamine (en), 1,2-propylenediamine (1,2-pn), isobutylenediamine (ibn), 1,2-diaminocyclohexane (1,2-dach), N-methylethylenediamine (Meen), N,N,N,N-tetramethylethylenediamine (Me₄en), S-methyl L-cysteine (MeS-L-HCys), L-methionine (L-HMet), and 2,5-dithiahexane (dth) and dipeptide N-acetylated L-methionylglycine (MeCOMet-Gly) were studied by ¹H NMR spectroscopy. The reactions were carried out in the pH range 2.0–2.5 and at 50°C. In all these reactions, palladium(II) complex bound to a methionine residue effects the regioselective cleavage of the amide bond involving the carboxylic group of methionine. We found that the rate of hydrolysis and mechanism of this reaction are strongly dependent from the nature of the chelate ligand L in palladium(II) complexes of the type cis-[Pd(L)(H₂O)₂]²⁺.     

Formation and stability of N-heterocyclic carbenes in water: the carbon acid pKa of imidazolium cations in aqueous solution

We report second-order rate constants kDO (M-1 s-1) for exchange for deuterium of the C(2)-proton of a series of simple imidazolium cations to give the corresponding singlet imidazol-2-yl carbenes in D2O at 25 degrees C and I = 1.0 (KCl). Evidence is presented that the reverse protonation of imidazol-2-yl carbenes by solvent water is limited by solvent reorganization and occurs with a rate constant of kHOH = kreorg = 10(11) s-1. The data were used to calculate reliable carbon acid pK(a)s for ionization of imidazolium cations at C(2) to give the corresponding singlet imidazol-2-yl carbenes in water: pKa = 23.8 for the imidazolium cation, pKa = 23.0 for the 1,3-dimethylimidazolium cation, pKa = 21.6 for the 1,3-dimethylbenzimidazolium cation, and pKa = 21.2 for the 1,3-bis-((S)-1-phenylethyl)benzimidazolium cation. The data also provide the thermodynamic driving force for a 1,2-hydrogen shift at a singlet carbene: K12 = 5 x 10(16) for rearrangement of the parent imidazol-2-yl carbene to give neutral imidazole in water at 298 K, which corresponds to a favorable Gibbs free energy change of 23 kcal/mol. We present a simple rationale for the observed substituent effects on the thermodynamic stability of N-heterocyclic carbenes relative to a variety of neutral and cationic derivatives that emphasizes the importance of the choice of reference reaction when assessing the stability of N-heterocyclic carbenes.     

Asymmetric synthesis of [7]helicene-like molecules

[reaction: see text] A new approach to nonracemic [7]helicene-like molecules has been developed. Stereoselective Co(I)-mediated [2 + 2 + 2] cycloisomerization of aromatic triynes containing an asymmetric carbon atom produces [7]helicene-like scaffolds in diastereomeric ratios up to 100:0. This central-to-helical chirality transfer can be controlled by the absolute configuration at the asymmetric center and by the presence of carbon substituents.     

An exact algorithm for the capacitated shortest spanning arborescence

We are given a complete and loop-free digraphG=(V, A), whereV={1,...,n} is the vertex set,A={(i, j) :i, j V} the arc set, andr V is a distinguishedroot vertex. For each arc (i, j) A, letc _ij be the associatedcost, and for each vertexi, letq _i 0 be the associateddemand (withq _r =0). Moreover, a nonnegativebranch capacity, Q, is defined.A Capacitated Shortest Spanning Arborescence rooted at r (CSSA _r ) is a minimum cost partial digraph such that: (i) each vertexj r has exactly one entering arc; (ii) for each vertexj r, a path fromr toj exists; (iii) for each branch leaving vertexr, the total demand of the vertices does not exceed the branch capacity,Q. A variant of theCSSA _r problem (calledD-CSSA _r ) arises when the out-degree of the root vertex is constrained to be equal to a given valueD. These problems are strongly NP-hard, and find practical applications in routing and network design. We describe a new Lagrangian lower bound forCSSA _r andD-CSSA _r problems, strengthened in a cutting plane fashion by iteratively adding violated constraints to the Lagrangian problem. We also present a new lower bound based on projection leading to the solution of min-cost flow problems. The two lower bounds are then combined so as to obtain an overall additive lower bounding procedure. The additive procedure is then imbedded in a branch-and-bound algorithm whose performance is enhanced by means of reduction procedures, dominance criteria, feasibility checks and upper bounding. Computational tests on asymmetric and symmetric instances from the literature, involving up to 200 vertices, are given, showing the effectiveness of the proposed approach.     

Organopolysiloxanes as Chemically Sensitive Coatings for Optical Fibers

Various types of UV-curable organically modified siloxanes have been synthesized by the sol-gel method with the aim of fabricating chemically sensitive coatings for silica optical fibers. The refractive index of the coating material can be tailored in the range from 1.46 to 1.56 and sensitivity towards CO2 is achieved by incorporation of amino groups. The interaction of the cured layers with CO2 or with hydrocarbons has been studied in immersion experiments. Both the reaction of CO2 with incorporated amino groups and the penetration of hydrocarbons into the layer induce changes of the light absorption coefficient and the refractive index of the coating which are detected by measuring the output light intensity from the fiber.