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.     

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.     

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.     

Statistical properties of chaos demonstrated in a class of one-dimensional maps

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.     

Ab initio prediction of vibrational spectra: A database approach

Efforts to develop a database of quadratic force fields for organic molecules are described. The database is based on systematic ab initio calculations, scaled to reproduce the experimentally observed frequencies. The choice of the theoretical method, the basis sets, geometries, internal coordinates and the scaling procedure are discussed. A key point in the procedure is the automatic generation of the internal valence coordinates. This is also very advantageous for geometry optimization. The database should permit the prediction of vibrational frequencies for most organic molecules to 10–20 cm⁻¹, together with semiquantitative intensities. The accuracy is sufficient to identify unknown compounds from a list of reasonable candidates.     

Optimal Joint Distributions of Several Random Variables with Given Marginals

Solutions of a singularly perturbed vector boundary-value problem are studied under the principle assumption that the trivial solution of the unperturbed equation is stable in certain senses. This is accomplished by constructing special invariant regions in which solutions display the kind of nonuniformity known as boundary-layer behavior.