SUMMARYA general numerical procedure is presented for the e cient computation of corner singularities, which appear in the case of non-smooth domains in three-dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin-Petrov ÿnite element method. A quadratic eigenvalue problem (P + Q + 2 R) u = 0 is obtained, with explicitly analytically deÿned matrices P; Q; R. Moreover, the three matrices are found to have optimal structure, so that P; R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for ÿnding all eigenvalues in the interval Re( ) ∈ (−0:5; 1:0) (no eigenpairs can be 'lost') as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much ÿll-in. To illustrate the e ectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on di erent geometrical parameters and the material properties.

Abstract:The formation and extraction of ion-associate complexes between the vanadium(V) -4-(2-thiazolylazo)resorcinol (TAR) anionic chelate and the cations of some mono-and ditetrazolium salts {3- (4,5-dimethyl-2-thiazol-tetrazolium] chloride (Tetrazolium blue chloride) and 3,3'-(3,3'-dimetoxy-4,4'-biphenylene)bis[2-(4-nitrophenyl)-5-phenyl-2H-tetrazolium chloride] (Nitro blue tetrazolium chloride)} have been studied. The optimum extraction conditions have been found. The composition of the V-TARmonotetrazolium and V-TAR-ditetrazolium complexes extracted into chloroform has been determined to be 1:2:3 and 2:4:3 respectively. The extraction, distribution and association constants, and the recovery factors have been calculated. The relationship between the molecular weight of tetrazolium cations, and the association constants of their complexes has been discussed. The special behavior of the tetrazolium cations, containing -NO 2 groups has been noticed. The effects of foreign ions and reagents on the extraction of vanadium with TAR and the best tetrazolium salt -MTT have been studied. A sensitive, selective, simple and fast method for the determination of vanadium has been developed.

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SUMMARYA general numerical procedure is presented for constructing the terms in asymptotical expansions near 3D corners in linear elasticity, without geometrical restrictions to Lipschitzian domains. The method is based on a weak formulation of the problem and a higher order Galerkin-Petrov approximation technique on the unit sphere. It results in a quadratic eigenvalue problem, which is solved iteratively by the Arnoldi method. An a posteriori error estimator as well as a suitable technique for adaptive mesh reÿnement is proposed for controlling the discretization error. Some benchmark tests show that this method is robust and very accurate. For illustrating the application to non-Lipschitzian domains new results for the Neumann problem on a non-Lipschitzian polyhedron are obtained for di erent geometrical parameters and materials.

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