Weight choosability of graphs

Suppose the edges of a graph G are assigned 3‐element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different? We prove that the answer is positive for several classes of graphs, including complete graphs, complete bipartite graphs, and trees (except K₂). The argument is algebraic and uses permanents of matrices and Combinatorial Nullstellensatz. We also consider a directed version of the problem. We prove by an elementary argument that for digraphs the answer to the above question is positive even with lists of size two. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 242–256, 2009     

(Cyclopentadienyl){(N,N-dimethylaminoethyl)cyclopentadienyl} complexes of zirconium: Crystal structure of [(η-C5H5)(η-C5H4CH2CH2NHMe2)ZrCl2]2[ZrCl6]

[(η⁵-C₅H₅)ZrCl₃] reacts with [C₅H₄CH₂CH₂NMe₂]Li yielding the coordination polymer [(C₅H₅)(C₅H₄CH₂CH₂NMe₂)ZrCl₂]_n (1) as a brown solid which is very sensitive to moisture. The reaction of 1 with 1.35 equivalent of HCl (methanolic solution) yields pale yellow green crystals of [(η⁵-C₅H₅)(η⁵-C₅H₄CH₂CH₂NHMe₂)ZrCl₂]₂[ZrCl₆] (2). Compound 2 was fully characterized on the basis of NMR data and X-ray crystal structure analysis. The formation of this product indicates the elimination of C₅H₄CH₂CH₂NMe₂ as well as C₅H₅ ligands from the Zr(IV) metal centre.     

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The semiempirical study on the addition of the chiral ammonium hypophosphite to an aldehyde

The PM3 and AM1 semiempirical computations were performed in order to explain the stereochemistry of the addition of the chiral α-methylbenzylammonium hypophosphite to an aldehyde, which is stereoselective to 100%. Both mechanisms: one considering the intermediate formation of α-hydroxy phosphonous acids followed by the nucleophilic substitution with a chiral amine and the second considering the formation of a Schiff base followed by the addition of hypophosphorous acid to an azomethine bond were taken. © 2004 Wiley Periodicals, Inc. Heteroatom Chem 15:162–168, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/hc.10230