An efficient spectral method for ordinary differential equations with rational function coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple 3-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e., matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation , is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

     

Synthesis, structural characterization, and computational studies of novel diiodine adducts with the heterocyclic thioamides N-methylbenzothiazole-2-thione and benzimidazole-2-thione: implications with the mechanism of action of antithyroid drugs

Reaction of N-methylbenzothiazole-2-thione (C8H7NS2 or NMBZT) with diiodine produced the charge-transfer (ct) complex [(NMBZT).I2] (1). NMBZT reacts with diiodine in the presence of FeCl3 in a molar ratio of 3:6:1 and forms the ionic complex [[(NMBZT)2I+].[FeCl4]-] (2) together with [[(NMBZT)2I+].[I7]-] (2a) iodonium salt. The reaction of benzimidazole-2-thione (C7H6N2S or MBZIM) with diiodine on the other hand results in the formation of the ct [[(MBZIM)2I]+[I3]-].[(MBZIM).I2] (3) compound. The compounds have been characterized by elemental analyses, DTA-TG, FT-Raman, FT-IR, UV-vis, and 1H NMR spectroscopies, and X-ray crystal structure determinations. Compound 1, C8H7I2NS2, is orthorhombic with a space group Pna2(1) and a = 12.5147(13) angstroms, b = 22.536(3) angstroms, c = 4.2994(5) angstroms, and Z = 4. Compound 2, C16H14Cl4FeIN2S4, is monoclinic, space group C2/c, a = 35.781(2) angstroms, b = 7.4761(5) angstroms, c = 18.4677(12) angstroms, beta = 107.219(1) degrees, and Z = 8. Compound 3, C21H18I6N6S3, monoclinic, space group P2(1)/n, a = 14.0652(11) angstroms, b = 22.536(3) angstroms, c = 4.2994(5) angstroms, beta = 99.635(7) degrees, and Z = 4, consists of two component moieties cocrystallized, one neutral which contains the benzimidazole-2-thione (MBIZM) ligand bonded with an iodine atom through sulfur, forming a compound with a "spoke" structure [(MBZIM)I2] 3a, while the other is the ionic complex [[(MBZII)2I+].[I3]-] (3b). The X-ray crystal structure of 1 shows a bond between the thione-sulfur atom and one of the iodine atoms in an essentially planar arrangement. In the cation of 2, an iodine is coordinated by two thione-sulfur atoms in a linear arrangement but the molecule is not planar. For the first time in the solid state a spoke-ionic mixed complex has been characterized in 3. One component of the structure is a molecular diiodine adduct, i.e., [(MBZIM)I2] (3a), with a linear coordination geometry in a decidedly planar arrangement, and the other component is an ionic adduct [[(MBZIM)2I]+.[I3]-] (3b) with the cation having an arrangement similar to that found for 1. Theoretical calculations using density functional (DFT) and ab initio Hartree-Fock theory have been carried out for 1 and 3a,b. The results are consistent with the experimental data. Conclusions on the behavior of a thioamide, when used as an antithyroid drug, have also been made.     

Lower bounds for the zeros of analytic functions

Summary In the present work the problem of finding lower bounds for the zeros of an analytic function is reduced by a Hilbert space technique to the well-known problem of finding upper bounds for the zeros of a polynomial. Several lower bounds for all the zeros of analytic functions are thus found, which are always better than the well-known Carmichael-Mason inequality. Several numerical examples are also given and a comparison of our bounds with well-known bounds in literature and/or the exact solution is made.     

A volume‐consistent discrete formulation of aggregation population balance equations

In this paper, a new finite volume scheme for the numerical solution of the pure aggregation population balance equation, or Smoluchowski equation, on non‐uniform meshes is derived. The main feature of the new method is its simple mathematical structure and high accuracy with respect to the number density distribution as well as its moments. The new method is compared with the existing schemes given by Filbet and Laurençot (SIAM J. Sci. Comput., 25 (2004), pp. 2004–2028) and Forestier and Mancini (SIAM J. Sci. Comput., 34 (2012), pp. B840–B860) for selected benchmark problems. It is shown that the new scheme preserves all the advantages of a conventional finite volume scheme and predicts higher‐order moments as well as number density distribution with high accuracy. Copyright © 2015 John Wiley & Sons, Ltd.     

Algorithmic dimensionality reduction for molecular structure analysis

Dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion. This has been used to help visualize complex energy landscapes, to extend the time scales of simulation, and to improve the efficiency of optimization. Until recently, linear approaches for dimensionality reduction have been employed. Here, we investigate the efficacy of several automated algorithms for nonlinear dimensionality reduction for representation of trans, trans-1,2,4-trifluorocyclo-octane conformation--a molecule whose structure can be described on a 2-manifold in a Cartesian coordinate phase space. We describe an efficient approach for a deterministic enumeration of ring conformations. We demonstrate a drastic improvement in dimensionality reduction with the use of nonlinear methods. We discuss the use of dimensionality reduction algorithms for estimating intrinsic dimensionality and the relationship to the Whitney embedding theorem. Additionally, we investigate the influence of the choice of high-dimensional encoding on the reduction. We show for the case studied that, in terms of reconstruction error root mean square deviation, Cartesian coordinate representations and encodings based on interatom distances provide better performance than encodings based on a dihedral angle representation.     

9-Substituted acridines. Demethoxylation of 9-methoxyacridine and dechlorination of 9-chloroacridine in hydroxylic solvents

In methanol-water mixtures containing sodium or potassium hydroxide (up to 2.00M) the demethoxylation of 9-methoxyacridine to 9-acridone is of first order in both the free form of 9-methoxyacridine and the hydroxyl ion. The rate of the reaction is increased with an increase in the concentration of water. Sodium perchlorate has a small retarding effect on the reaction. In methanol-water mixtures containing perchloric or hydrochloric acid (up to 3.45M) the demethoxylation is of first order with respect to the protonated form of 9-methoxyacridine. The rate of the reaction decreases with an increase in the concentration of the acid or of sodium perchlorate, but when the concentration of water is increased (≈?1.7 to 50M) and that of the acid is not changed, it reaches a maximum value in mixtures containing 8 to 10M-water. The dechlorination of 9-chloroacridine to 9-methoxyacridine in methanol containing sodium hydroxide or methoxide (up to 0.31 M) is of first order in both the 9-chloroacridine and the hydroxyl or methoxyl ions.     

Speciation of Fe(II) and Fe(III) by the modified ferrozine method,FIA-spectrophotometry,and flame AAS after cloud-point extraction

A method has been developed for the simultaneous determination of traces of Fe(III) and Fe(II) in water by on-line coupling of spectrophotometry with flame atomic absorption spectrometry (FAAS). The method involves cloud-point extraction (CPE) of both species with ammonium pyrrolidinecarbodithioate (APDC) under standard conditions, which facilitates the in situ complexation and extraction of both species. Differentiation of the oxidation states of iron is achieved by using mathematical equations to overcome the interference of Fe(III) in the spectrophotometric determination of Fe(II) when they are both present in the same solution. In this manner the time-consuming and labor-intensive steps of preoxidation of Fe(II) or reduction of Fe(III) are eliminated. By preconcentrating a 10-mL sample solution detection limits as low as 7 microg L(-1), were obtained after a single-step extraction procedure. The relative standard deviation (n=4, 30 microg L(-1)) was 2.6 % and 1.8 % for spectrophotometry and FAAS, respectively. Recoveries in the range of 96-105 % were obtained by analysis of spiked real samples. The method was further verified by analyzing a certified reference material (IMEP-9); for this the recovery was 98.5 %.