Binary and Ternary Copper(II) Complexes with NIPAAm/IA Copolymers and Aminoacids: Interpretation of UV‐Visible Spectra and Cyclic Voltammograms

The complex formations of homopolymers and copolymers of poly(N‐isopropyl acrylamide) (PNIPAAm), polyacrylamide (PAAm) and poly(itaconic acid) (PIA) with Cu(II) ions in aqueous solutions were studied by using UV‐visible spectra in the region of 200–1100 nm and cyclic voltammograms between ?0.800 V and 0.600 V. According to the optical and electrochemical spectra of the polymer‐ and copolymer‐Cu(II) complexes and their ternary complexes with alanine, i.e., absorptions and the shifts in the wavelength of the maximum absorptions, currents and potentials of the peaks in the pH range of 3–12, the intensities of anodic and cathodic peak currents of polymers containing IA groups decrease with increasing pH and they show maximum absorptions at lower wavelengths than do the homopolymers used in this work. The peak point of the visible band shifts from 800 to 650 nm, with increasing pH, while the intensity of the third anodic peak observed after pH=4 increases in the case of both Ala‐Cu(II) and its ternary solution with P(NIPAAm‐co‐IA, 9.8 mol%). Both the pH‐dependent shifts of maximum absorptions and the appearence of the third anodic peaks as the pH raised were interpreted as a presence of tetracoordinated Cu(II) complexes in the solution and on the electrode surface, involving combined carboxyl, amide and amine interaction.     

Effect of local variations of the laminar flame speed on the global finger-flame acceleration scenario

Numerous formulations describing the dynamics and morphology of corrugated flames, including the scenarios of flame acceleration, are based on a “geometrical consideration”, where the wrinkled-to-planar flame velocities ratio, S_w /S_L , is evaluated as the scaled flame surface area, while the entire combustion chemistry is immersed into the planar flame speed S_L , which is assumed to be constant. However, S_L may experience noticeable spatial/temporal variations in practice, in particular, due to pressure/temperature variations as well as non-uniform distribution of the equivalence ratio and/or that of combustible or inert dust impurities. The present work initiates the systematic study of the impact of the local S_L -variations on the global flame evolution scenario. The variations are assumed to be imposed externally, in a manner being a free functional of the formulation. Specifically, the linear, parabolic and hyperbolic spatial S_L -distributions are incorporated into the formulations of finger flame acceleration in pipes, and they are compared to the case of constant S_L . Both two-dimensional channels and cylindrical tubes are considered. The conditions promoting or moderating flame acceleration are identified, and the revisited equations for the flame shape, velocity and acceleration rate are obtained for various S_L -distributions. The theoretical findings are validated by the computational simulations of the reacting flow equations, with agreement between the theory and modelling demonstrated.     

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

This article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

A rational approximation based on Bernstein polynomials for high order initial and boundary values problems

We introduce a new method to solve high order linear differential equations with initial and boundary conditions numerically. In this method, the approximate solution is based on rational interpolation and collocation method. Since controlling the occurrence of poles in rational interpolation is difficult, a construction which is found by Floater and Hormann [1] is used with no poles in real numbers. We use the Bernstein series solution instead of the interpolation polynomials in their construction. We find that our approximate solution has better convergence rate than the one found by using collocation method. The error of the approximate solution is given in the case of the exact solution f ∈ C^d+2[a, b].     

A taylor collocation method for solving high‐order linear pantograph equations with linear functional argument

A numerical method based on the Taylor polynomials is introduced in this article for the approximate solution of the pantograph equations with constant and variable coefficients. Some numerical examples, which consist of the initial conditions, are given to show the properties of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:1628–1638, 2011     

Stereoselective synthesis of optically active cyclopenta[c]pyrans and cyclopenta[c]furans by the intramolecular Pauson–Khand reaction

An intramolecular Pauson–Khand reaction of enynes derived from homoallyl, homopropargyl, and allyl alcohols is described. 2-Heteroaryl-substituted homoallyl, homopropargyl, and allyl alcohols are easily and efficiently resolved through enzymatic resolution in high ee (91–99%) and with a known stereochemistry. Each enantiomerically enriched enyne derived from homoallyl and homopropargyl alcohols affords the conformationally most stable diastereomeric cyclopenta[c]pyran ring system as the sole product, whereas enantiomerically enriched enynes derived from allyl alcohols give a diastereomeric cis:trans mixture of the cyclopenta[c]furan ring system.     

Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization

A new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can be applied to find descent directions of nonsmooth functions. The preliminary results of numerical experiments with unconstrained nonsmooth optimization problems as well as the comparison of the proposed method with the nonsmooth optimization solver DNLP from CONOPT-GAMS and the derivative-free optimization solver CONDOR are presented.     

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.