SYMMETRICAL ORTHOGONAL POLYNOMIALS AND THE ASSOCIATED ORTHOGONAL L-POLYNOMIALS

We show how symmetric orthogonal polynomials can be linked to polynomials associated with certain orthogonal L-polynomials. We provide some examples to illustrate the results obtained Finally as an application, we derive information regarding the orthogonal polynomials associated with the weight function (1 + kx(2))(1 - x(2))(-1/2), k > 0.

Companion orthogonal polynomials: some applications

In this paper the recurrence relations of symmetric orthogonal polynomials whose measures are related to each other in a certain way are considered. Many of the relations satisfied by the coefficients of the recurrence relations are exposed. The results are applied to obtain, for example, information regarding certain Sobolev orthogonal polynomials and regarding the measures of certain orthogonal polynomial sequences with twin periodic recurrence coefficients. (C) 2001 IMACS. Published by Elsevier B.V. B.V. All rights reserved.

Monotonicity of the zeros of orthogonal polynomials through related measures

Relation between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), is well known. We use this relation to study the monotonicity properties of the zeros of generalized orthogonal polynomials. As examples, the Jacobi, Laguerre and Charlier polynomials are considered. (c) 2005 Elsevier B.V. All rights reserved.

On linear combinations of L-orthogonal polynomials associated with distributions belonging to symmetric classes

This paper deals with the classes S-3(omega, beta, b) of strong distribution functions defined on the interval [beta(2)/b, b], 0 < beta < b <= infinity, where 2 omega epsilon Z. The classification is such that the distribution function psi epsilon S-3(omega, beta, b) has a (reciprocal) symmetry, depending on omega, about the point beta. We consider properties of the L-orthogonal polynomials associated with psi epsilon S-3(omega, beta, b). Through linear combination of these polynomials we relate them to the L-orthogonal polynomials associated with some omega epsilon S-3(1/2, beta, b). (c) 2004 Elsevier B.V. All rights reserved.

Another connection between orthogonal polynomials and L-orthogonal polynomials

We consider a connection that exists between orthogonal polynomials associated with positive measures on the real line and orthogonal Laurent polynomials associated with strong measures of the class S-3 [0, beta, b]. Examples are given to illustrate the main contribution in this paper. (c) 2006 Elsevier B.V. All rights reserved.

Polar Multiplicities and Euler obstruction of the stable types in weighted homogeneous map germs from C-n to C-3 n >= 3

In this article we show that for corank 1, quasi-homogeneous and finitely determined map germs f : (C-n, 0)-> (C-3, 0), n >= 3 one can obtain formulae for the polar multiplicities defined on the following stable types of f, f(Delta(f) and f(Sigma(n-2,1)(f), in terms of the weights and degrees of f. As a consequence we show how to compute the Euler obstruction of such stable types, also in terms of the weights and degrees of f.

Manganese(III) porphyrins: Catalytic activity and intermediate studies in homogeneous systems

We studied how solvent, stirring method, PhIO/MnP molar ratio, presence of water and axial ligand affect the catalytic activities of Mn(TPP)Cl, Mn(MNPP)Cl, Mn(TDCPP)Cl and Mn(TFPP)Cl in the oxidation of cyclohexane by PhIO. A study of the catalytic intermediates in the reaction between Mn(TPP)Cl or Mn(TDCPP)Cl and PhIO was also carried out by UV-Vis and EPR spectroscopies. The reaction of Mn(TPP)Cl with PhIO showed the formation of a mixture of species Mn-IV(OP+ and Mn-V(O)P as intermediates, which were confirmed by the deconvolution of the UV-Vis spectra. Addition of imidazole as cocatalyst favoured the formation of the intermediate species Mn-V(O)P, evidenced by the UV-Vis band at 408 nm. The corresponding EPR spectra gave evidence that in the presence of imidazole, Mn-IV(OP+ species are formed only in very low amounts. For Mn(TDCPP)Cl the dominating intermediate species is Mn-IV(OP+. Addition of imidazole to halogen-substituted MnP systems does not result in increase of the C-ol yields because very stable bis-imidazole-MnP complexes are formed. Anchoring of such MnP on imidazole propyl gel (IPG) results in better catalytic activity because in this case, the catalyst is mono-coordinated to the support and imidazole favours the formation of the intermediate species Mn-V(O)P.

Homogeneous photodegradation of CI Reactive Blue 4 using a photo-Fenton process under artificial and solar irradiation

The oxidation of C.I. Reactive Blue 4 (RB4) by photo-Fenton process mediated by lerrioxalate was investigated under artificial and solar irradiation. The RB4 degradation in acidic medium (pH 2.5) was evaluated by the decrease in Total Organic Carbon (TOC) content and color, measured by the decrease in chromophore absorption band (600 nm). The influence of ferrioxalate and H2O2 concentrations on the dye degradation was studied and best results were obtained using 1.0 mM ferrioxalate and 10 nM of hydrogen peroxide. Under these experimental conditions, 80% of TOC and 100% of color removal were obtained for a 0.1 mM RB4 dye in 35 min of solar irradiation. (c) 2006 Elsevier Ltd. All rights reserved.

Blumenthal's theorem for Laurent orthogonal polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B-n(x) = (x - beta(n))beta(n-1)(x) - alpha(n)xB(n-2)(x), with positive recurrence coefficients alpha(n+1),beta(n) (n = 1, 2,...). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where alpha(n) --> alpha and beta(n) --> beta and show that the zeros of beta(n) are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials. (C) 2002 Elsevier B.V. (USA).

Szego polynomials and the truncated trigonometric moment problem

We show how Szego polynomials can be used in the theory of truncated trigonometric moment problem.

Companion orthogonal polynomials

We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions d phi(1)(x) and d phi(2)(x) such that d phi(2)(x) = (I + kx(2))d phi(1)(x). AS applications of these properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained.

Chain sequences and symmetric generalized orthogonal polynomials

in this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K-n((lambda.,M,k)) associated with the probability measure dphi(lambda,M,k;x), which is the Gegenbauer measure of parameter lambda + 1 with two additional mass points at +/-k. When k = 1 we obtain information on the polynomials K-n((lambda.,M)) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K-n((lambda,M,k)) in relation to M and k are also given. (C) 2002 Elsevier B.V. B.V. All rights reserved.

Chebyshev-Laurent polynomials and weighted approximation

Let (a, b) subset of (0, infinity) and for any positive integer n, let S-n be the Chebyshev space in [a, b] defined by S-n:= span{x(-n/2+k),k= 0,...,n}. The unique (up to a constant factor) function tau(n) is an element of S-n, which satisfies the orthogonality relation S(a)(b)tau(n)(x)q(x) (x(b - x)(x - a))(-1/2) dx = 0 for any q is an element of Sn-1, is said to be the orthogonal Chebyshev S-n-polynomials. This paper is an attempt to exibit some interesting properties of the orthogonal Chebyshev S-n-polynomials and to demonstrate their importance to the problem of approximation by S-n-polynomials. A simple proof of a Jackson-type theorem is given and the Lagrange interpolation problem by functions from S-n is discussed. It is shown also that tau(n) obeys an extremal property in L-q, 1 less than or equal to q less than or equal to infinity. Natural analogues of some inequalities for algebraic polynomials, which we expect to hold for the S-n-pelynomials, are conjectured.

Synthesis and properties of branched polyethylene/high-density polyethylene blends using a homogeneous binary catalyst system composed of early and late transition metal complexes

Branched polyethylene/high-density polyethylene blends (BPE/HDPE) with a wide range of molecular weights, melt flow indexes (MFI), and intrinsic viscosity were prepared using the homogeneous binary catalyst system composed by Ni(alpha-diimine)Cl-2 (1) (alpha-diimine = 1,4-bis(2,6-diisopropylphenyl)-acenaphthenediimine) and {Tp(Ms*)} TiCl3 (2) (Tp(Ms*)=hydridobis(3-mesitylpyrazol-1-yl)(5-mesityl-pyrazol-1-yl)) activated with MAO and/or TIBA in hexane at two different polymerization temperatures (30 and 55 degreesC) and by varying the nickel loading molar fraction (x(Ni)). At all Temperatures, a non-linear correlation between the x(Ni) and the productivity was observed, suggesting the occurrence of a synergistic effect between the nickel and the titanium catalyst precursors, which is more pronounced at 55 degreesC. The molecular weight of the BPE/HDPE blends considerably decreases with increasing Al/M molar ratio. The melt flow indexes (MFI) and intrinsic viscosities (eta) are strongly affected by x(Ni), but the melting temperatures are nearly constant, 132 +/- 3 degreesC. Dynamic mechanical thermal analysis (DMTA) shows the formation of different polymeric materials where the stiffness vanes according, to the x(Ni) and temperature used in the polymerization reaction. The surface morphology of the BPE/HDPE blends studied by scanning electron microscopy (SEM) revealed a low miscibility between the PE phases resulting in the formation of a sandwich structure after etching with o-xylene.

Factors which affect the catalytic activity of iron(III) meso tetrakis(2,6-dichlorophenyl)porphyrin chloride in homogeneous system

An optimization study of the reaction conditions of Fe(TDCPP)Cl when it is used as catalyst in the hydroxylation of cyclohexane by iodosylbenzene (PhIO) has been carried out, It was found that Fe(TDCPP)Cl follows the classical PhIO mechanism described for Fe(TPP)Cl, which involves the monomeric active species Fe-IV(O)P-+. (I). In the optimized condition ([Fe(TDCPP) = 3.0 X 10(-4) mol l(-1) in 1,2-dichloroethane (DCE); ultrasound stirring at 0 degrees C; PhIO/FeP molar ratio = 100), this FeP led to a yield of cyclohexanol (C-ol) of 96% and a turnover number of 96, Therefore, Fe(TDCPP)Cl may be considered a good biomimetic model and a very stable, resistant and selective catalyst, which yields C-ol as the sole product. DCE showed to be a better solvent than dichloromethane (DCM), 1 DCE:1 MeOH mixture or acetonitrile (ACN). Since the Fe-IV(O)P-+. is capable of abstracting hydrogen atom from DCM, MeOH or ACN, the solvent competes with the substrate. Presence of O-2 lowers the yield of C-ol, as it can further oxidize this alcohol to carboxylic acid in the presence of radicals, Presence of H2O also causes a decrease in the yield, since it converts the active species I into Fe-IV(OH)P, which cannot oxidize cyclohexane. Addition of excess imidazole or OH- to the system results in a decrease in the yield of C-ol, due to the formation of the hexacoordinated complexes Fe(TDCPP)Im(2)(+) (low-spin, beta(2) = 2.5 X 10(8) mol(-2) l(2)) and Fe(TDCPP)(OH)(2)(-) (high-spin, beta(2) = 6.3 X 10(7) mol(-2) l(2)), the formation of both Fe(TDCPP)Im(2)(+) and Fe(TDCPP)(OH)(2)(-) complexes were confirmed by EPR studies. The catalytic activities of Fe(TDCPP)C and Fe(TFPP)Cl were compared, the unusually high yields of C-ol with Fe(TFPP)Cl obtained when ultrasound, DCM and O-2 atmosphere were used, suggest that a parallel mechanism involving the mu-oxo dimer form, O-2 and radicals may also be occurring with this FeP, besides the PhIO mechanism.