Kinetics and mechanism of the formation of poly[(1,1,3,3-tetramethyldisiloxanyl)ethylene] and poly(methyldecylsiloxane) by hydrosilylation

The kinetics of the formation of poly(carbosiloxane), as well as of alkyl-substituted poly(siloxane), by Karstedt's catalyst catalyzed hydrosilylation were investigated. Linear poly(carbosiloxane), poly[(1,1,3,3-tetramethyldisiloxanyl)ethylene], (PTMDSE), was obtained by hydrosilylation of 1,3-divinyltetramethyldisiloxane (DVTMDS) and 1,1,3,3-tetramethyldisiloxane (TMDS), while alkyl-substituted poly(siloxane), poly(methyldecylsiloxane), (PMDS), was synthesized by hydrosilylation of poly(methylhydrosiloxane) (PMHS) and 1-decene. To investigate the kinetics of PTMDSE formation, two series of experiments were performed at reaction temperatures ranging from 25 to 56 °C and with catalyst concentrations ranging from 7.0 × 10⁻⁶ to 3.1 × 10⁻⁵ mol Pt/mol CHCH₂. A series of experiments was performed at reaction temperatures ranging from 28 to 48 °C, with catalyst concentrations of 7.0 ×10⁻⁶ mol of Pt per mol of CHCH₂, when kinetics of PMDS formation was investigated. All reactions were carried out in bulk, with equimolar amounts of the reacting Si H and CHCH₂ groups. The course of the reactions was monitored by following the disappearance of the Si H bands using quantitative infrared spectroscopy. The results obtained showed typical first order kinetics for the PTMDSE formation, consistent with the proposed reaction mechanism. In the case of PMDS an induction period occurred at lower reaction temperatures, but disappeared at 44 °C and the rate of Si H conversion also started to follow the first-order kinetics. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 2246–2258, 2007     

Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

Bipyridinophane‐containing conjugated polymers modulated with an intramolecular aromatic C?H/π interaction

Bipyridinophane–fluorene conjugated copolymers have been synthesized via Suzuki and Heck coupling reactions from 5,8‐dibromo‐2,11‐dithia[3]paracyclo[3](4,4′)‐2,2′‐bipyridinophane and suitable fluorene precursors. Poly[2,7‐(9,9‐dihexylfluorene)‐co‐alt‐5,8‐(2,11‐dithia[3]paracyclo[3](4,4′)‐2,2′‐bipyridinophane)] ( P7 ) exhibits large absorption and emission redshifts of 20 and 34 nm, respectively, with respect to its planar reference polymer Poly[2,7‐(9,9‐dihexylfluorene)‐co‐alt‐1,4‐(2,5‐dimethylbenzene)] ( P11 ), which bears the same polymer backbone as P7 . These spectral shifts originate from intramolecular aromatic C? H/π interactions, which are evidenced by ultraviolet–visible and ¹H NMR spectra as well as X‐ray single‐crystal structural analysis. However, the effect of the intramolecular aromatic C? H/π interactions on the spectral shift in poly[9,9‐dihexylfluorene‐2,7‐yleneethynylene‐co‐alt‐5,8‐(2,11‐dithia[3]paracyclo[3](4,4′)‐2,2′‐bipyridinophane)] ( P10 ) is much weaker. Most interestingly, the quenching behaviors of these two conjugated polymers are largely dependent on the polymer backbone. For example, the fluorescence of P7 is efficiently quenched by Cu²⁺, Co²⁺, Ni²⁺, Zn²⁺, Mn²⁺, and Ag⁺ ions. In contrast, only Cu²⁺, Co²⁺, and Ni²⁺ ions can partially quench the fluorescence of P10 , but much less efficiently than the fluorescence of P7 . The static Stern–Volmer quenching constants of Cu²⁺, Co²⁺, and Ni²⁺ ions toward P7 are of the order of 10⁶ M^?1, being 1300, 2500, and 37,300 times larger than those of P10 , respectively. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 4154–4164, 2006     

A marker method for the solution of the damped Burgers' equation

A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On disjoint configurations in infinite graphs

For a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of ?₀ disjoint copies of A is referred to as ?₀A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all n??, but ?₀A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G₁, G₂,… such that G_i is not a minor of G_j for all i ≠ j. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all n ? ?, then ?₀A is a minor of G, too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016     

Linearized and rational approximation method for solving non‐linear Burgers' equation

A new numerical method called linearized and rational approximation method is presented to solve non‐linear evolution equations. The utility of the method is demonstrated for the case of differentiation of functions involving steep gradients. The solution of Burgers' equation is presented to illustrate the effectiveness of the technique for the solution of non‐linear evolution equations exhibiting nearly discontinuous solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

A Laplace-like method for solving linear difference equations

This paper is concerned with the problem of solving linear difference equations of ordern with constant coefficients and with given initial conditions in which the variable runs not only through the integers but over . The main idea is the introduction of a suitable commutative ring of functions with discrete convolution as multiplication rule which works, although it is not a field. The existence of inverses is studied and, after the introduction of suitable functions, the problem is reduced by means of a Laplace-like relation to an algebraic equation. Examples of application are given. Finally some remarks make the connection with the Operational Calculus of Mikusinski and with the Operational Calculus of Fenyö. The advantages of this method lie in the fact that it is applicable to functions others than the step functions, in its simplicity from the theoretical point of view, in its usefulness even when computation is required and in its formal similarity to the classical treatment of linear differential equations with constant coefficients.     

On a pair of functional equations of combinatorial interest

The equations of the title appear in the author's paper Chromatic Sums for Rooted Planar Triangulations, V: Special Equations. (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy ₂ of two variablesy ₁ andz. A parameter is involved. The main result is that for = 2 cos (2/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.The present work is basically an expository paper on the theory given in Chromatic Sums, V, but it includes some new results and many simplifications.