From Isomorphic Rooted Trees to Isometric Ultrametric Spaces

For every finite ultrametric space X we can put in correspondence its representing tree T_X. We found conditions under which the isomorphism of representing trees T_X and T_Y implies the isometricity of ultrametric spaces X and Y having the same range of distances.     

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C³ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C¹ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.     

Dynamics and pattern formation in invasive tumor growth

We study the in vitro dynamics of the malignant brain tumor glioblastoma multiforme. The growing tumor consists of a dense proliferating zone and an outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior: one cell line invades in a spherically symmetric manner, but another gives rise to branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space are determined. The results are in a qualitative agreement with experimental findings for the two types of cells.     

Synthesis,characterization and epoxidation of hexakis-4-(2-(4-((β-methallyl)oxy)phenyl)propan-2-yl)phenoxycyclotriphosphazene

4-(2-(4-((β-Methallyl)oxy)phenyl)propan-2-yl)phenol was prepared via the reaction of methallyl chloride with bisphenol A and used for the synthesis of hexakis-4-(2-(4-((β-methallyl)oxy)phenyl)propan-2-yl)phenoxycyclotriphosphazene. It was revealed that the latter undergoes the Claisen rearrangement and can be also epoxidized by 3-chloroperbenzoic acid. The obtained epoxide was cured by a treatment with isophorone diamine. The decomposition and glass transition temperatures of cured resin were also estimated (275 and 130?°C, respectively) by DSC and TGA methods.     

Decoration of 1,3-oxazole by alkyl substituents via CH activation (microreview)

Chemistry of Heterocyclic Compounds - Microreview describes recent advances in the synthesis of alkyl-substituted 1,3-oxazoles via CH activation reaction. The latest selected examples...     

Synthesis and structure of complexes of phosphorus pentachloride with 4‐dimethylaminopyridine and n‐methylimidazole

Complexes of phosphorus pentachloride with 4‐dimethylaminopyridine and N‐methylimidazole were synthesized. The molecular structure of the phosphorus pentachloride complex with N‐methylimidazole was determined by single‐crystal X‐ray diffraction. In the cationic part of the complex, the phosphorus atom possesses four P Cl bonds within the range 2.109–2.148 Å and two cis‐P–N bonds (1.811 and 1.832 Å) with N‐methylimidazole and exhibits slightly distorted octahedral coordination with angles at phosphorus atom in the range 87.57°–91.50°. The relative stability of the cis and trans conformations of the complex was studied by DFT calculations. The chemical properties and reactivity of the compounds obtained are determined; their utility as condensing agents in the synthesis of amides from acids and amines was shown. © 2008 Wiley Periodicals, Inc. Heteroatom Chem 19:171–177, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20392