Decomposing complete multipartite graphs into closed trails of arbitrary even lengths

We prove that any complete multipartite graph with parts of even size can be decomposed into closed trails with prescribed even lengths. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:455‐462, 2011     

Syntheses,spectral, thermal and structural characterization of dinuclear and polynuclear copper(II) orotate complexes, [Cu2(HOr)2(H2O)4] and [Cu(μ-HOr)(ba)2]n

The novel catena-poly-μ-orotatobis(butylamine)copper(II), [Cu(μ-HOr)(ba)₂]_n (1), and diaqua(orotato)copper(II), [Cu₂(HOr)₂(H₂O)₄] (2), complexes have been prepared and characterized by elemental analysis, magnetic measurements, FT-IR spectroscopy, EPR spectroscopy, thermal analysis and X-ray diffraction. Both complexes crystallize in the monoclinic space group, C2/c in 1 and P2₁/n in 2. In the complexes, the copper(II) ion is chelated by a deprotonated pyrimidine nitrogen atom and a carboxylate oxygen atom of the orotate. While the coordination sphere around Cu(II) is completed by two N atoms from butylamine groups and a carboxylic O atom in the axial position from a neighboring molecule in 1, the square-planar environment of Cu(II) is completed by two water atoms and one axial position is occupied by the carbonyl oxygen atom from the symmetry related molecule in 2. The coordination sphere should be described as a square pyramid and (4+1)-geometry in 1 and 2, respectively. While complex 1 shows a polymeric arrangement, compound 2 has a dimeric arrangement. The non-covalent Cu(II)-π binding force is very important for stabilizing the crystal structure of 2. The thermal decomposition of the complexes has been predicted by the help of thermal analysis (TG, DTG and DTA).     

On a family of nonlocal difference schemes

We consider a weighted difference scheme approximating the heat equation with nonlocal boundary conditions. We analyze the behavior of the spectrum of the main finite-difference operator depending on the parameters occurring in the boundary conditions. We state inequalities whose validity is necessary and sufficient for the stability of the difference scheme with respect to the initial data.     

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.     

Experimental studies of low-frequency bistatic reverberation in a shallow sea

Experimental data are presented on time and frequency dependences of the reverberation level for bistatic transmission and reception at low acoustic frequencies. The data are obtained from the studies carried out in a coastal shallow-water region and in the central region of the Barents Sea with explosion-generated and pulsed cw signals. By using the simplest computational model, approximate estimates are obtained for the effective coefficient of spatial attenuation and the effective scattering coefficient in the frequency band 40–400 Hz.