Existing methods for solving Poisson's equation using basis functions can be made considerably more exible, yet simpler, by using non-orthogonal basis functions. The rst part of this paper describes how such basis sets can be constructed. The second part gives explicit formulas (up to quadratures and matrix inversions) for basis functions being spherical harmonics times radial functions; two ba...

The interaction of Magnesium hydrate at the phosphate oxygen atom of the pyrimidine nucleotides (CMP,UMP,dTMP) were studied at the Hartree-Fock level Theory. We used LANL2DZ basis set for Mg and 6-31g* basis set for atoms.The basis set superposition error (BSSE) begins to converge for used Method/basis set. The gauge-invariant atomic orbital (GIAO) method and the continuous-set-of-gauge-transfo...

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a “good” additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are consid...

We present analytical calculation of the first seven moments of the heavy quark vacuum polarization function in the three-loop order. The obtained results are compared against the asymptotic formulas following from the threshold singulari-ties. We also discuss the µ dependence of the moments within the BLM procedure.

Central subsets of a discrete semigroup S have very strong combinatorial properties which are a consequence of the Central Sets Theorem. Call a set a C-set if it satisfies the conclusion of the Central Sets Theorem. We obtain here a reasonably simple characterization of C-sets in an arbitrary discrete semigroup.

We present analytical calculation of the rst seven moments of the heavy quark vacuum polarization function in the three-loop order. The obtained results are compared against the asymptotic formulas following from the threshold singulari-ties. We also discuss the dependence of the moments within the BLM procedure.

In the paper we introduce the notion of Bi Unique Range Sets for derivatives of meromorphic functions and with the aid of the same we improve all previous results regarding derivatives of set sharing.

Introduction. The two sections of this note are independent, but they are related by the fact that both use the results of [5 ] to obtain information on the properties of weakly compact sets in Banach spaces. In the first section we prove some results on a class of compact sets which is believed to include all weakly compact subsets of Banach spaces. We are interested in the properties of the n...