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Abstract  

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l 2-norm for some general Mercer kernel matrices are provided. As an example, we give the l 2-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l 2-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.

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Abstract  

The reaction between [M(DMBG)2nH2O ((1) M:Ni, n = 0; (4) M:Cu, n = 1), ammonia/hydrazine and formaldehyde in methanol resulted in new complexes of type [ML]·nH2O ((2) M:Ni, L:L1, n = 0; (3) M:Ni, L:L2, n = 0, (5) M:Cu, L:L1, n = 0 and (6) M:Cu, L:L2, n = 3; HDMBG: N,N-dimethylbiguanide, L1 = ligand resulted from ammonia system and L2 = ligand resulted from hydrazine system). The features of complexes have been assigned from microanalytical, IR and UV–Vis data. The thermal transformations of compounds are complex processes according to TG and DTG curves including melting, phase transition, dehydration, oxidative condensation of –C=N– units as well as thermolysis processes. The final products of decomposition are the most stable metal oxides.

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Abstract  

The thermodynamic and thermal properties of [Cu(L)2·Cl2], [Ni(L)2]·Cl2, [Co(L)2·Cl2]; L=1,2-bis(o-aminophenoxy)ethane (BAFE), complexes have been investigated. The thermal decomposition of the complexes took place in two distinct steps in endothermic reaction up to 700°C. The activation energy E, the entropy change S #, enthalpy H change and Gibbs free energy change G # were calculated from the results of thermogravimetry analysis (TG) and heat capacity from the results of differential scanning calorimetry (DSC). It was found that the thermal stabilities and activation energies of the complexes follow the order Ni(II)>Cu(II)>Co(II) and E Co<E Ni<E Cu, respectively.

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Abstract  

The complexes of piperidine dithiocarbamate, 2-aminopyridine dithiocarbamate and organotin(IV) of the type R3Sn(L1), R2Sn(L1)2, R3Sn(L2), R2Sn(L2)2, [R=C6H5CH2 (benzyl), p-ClC6H4CH2 (p-chlorobenzyl), L 1=sodium piperidine dithiocarbamate and L 2=sodium 2-aminopyridine dithiocarbamate] have been synthesised and characterised by spectral studies (IR, UV, 1H NMR). Thermogravimetric (TG) and differential thermal analytical (DTA) studies have beeen carried out for these complexes and from the TG curves, the order and apparent activation energy for the thermal decomposition reactions have been elucidated. The various thermal studies have been correlated with some structural aspects of the complexes concerned. From DTA curves, the heat of reaction has been calculated.

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Abstract  

The new mixed ligand complexes with formulae Co(4-bpy)2L2⋅2H2O (I), Cu(4-bpy)2L2⋅H2O (II) and Cd(4-bpy)L2⋅H2O (III) (4-bpy=4,4'-bipyridine, L=CCl3COO) were prepared. Analysis of the IR spectra indicate that 4-bpy is coordinated with metal ions and carboxylates groups bond as bidentate chelating ligands. The electronic spectra are in accordance with pseudo-octahedral environment around the central metal ion in the Co(II) and Cu(II) complexes. The thermal decomposition of the synthesized complexes was studied in air. A coupled TG-MS system was used to analyse the principal volatile thermal decomposition products of Co(II) and Cu(II) complexes. Corresponding metal oxides were identified as a final product of pyrolysis with intermediate formation of metal chlorides.

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Abstract  

The main aim of this paper is to prove that the maximal operator σ * α of the (C, α) means of the cubical partial sums of the two-dimensional Walsh-Fourier series is bounded from the Hardy space H 2/(2+α) to the space weak-L 2/(2+α).

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In this paper we prove that the maximal operator of the Marcinkiewicz-Fejér means of the 2-dimensional Vilenkin-Fourier series is not bounded from the Hardy space H 2/3 ( G 2 ) to the space L 2/3 ( G 2 ).

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Abstract

Müntz–Legendre polynomials L n(Λ;x) associated with a sequence Λ={λ k} are obtained by orthogonalizing the system in L 2[0,1] with respect to the Legendre weight. Under very mild conditions on Λ, we establish the endpoint asymptotics close to x=1. The main result is

ea
where and J 0 is the Bessel function of order 0.

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Abstract  

We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L 2 norm.

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Abstract  

A series of ternary complexes of the types M2L′2L″2;ML′2L″2 (M=Fe, Cu, Zn; L′=2-oxocyclopentane dithiocarboxylate; L″=pyridine, morpholine) and CuL′2H2O was prepared afresh. Except the iron complex, all are dimer and complexation is through the dithio moiety of the ligand L′. Their thermal decomposition was carried out in air at heating rate 10°C min−1 and it revealed that the dehydration of the aqua complex follows the same path as the carboxylates and the pyridine complexes have the tendency to follow one-step decomposition. The copper complexes are less thermally stable. The overall thermal stability of the 2-oxocyclopentanedithiocarboxylato complexes of the three metals with the volatile ligands was found to be in the order: (CuLmorph)2< CuL2H2O<(CuLpy)2<(ZnLmorph)2<(ZnLpy)2<FeL2py2. The thermogravimetric properties of the complexes have been studied and the data were subjected to kinetic analysis. The values of n, E, A and ΔS# have been approximated and compared. Any formation of bridged structure is not indicated in the first step case.

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