On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences

We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the k-level has subquadratic (O(n^2-1/2s) complexity. This answers one of the main open problems from the author’s previous paper [DCG 29, 375-393 (2003)], which provided a weaker upper bound for a restricted class of curves only (graphs of degree-s polynomials). When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved k-level results for most of the curve families studied earlier, including a near-O(n^3/2 bound for parabolas.     

A variant of the pure quaternionic representation of the Navier-Stokes equation

It is shown that the three-dimensional Navier-Stokes equation for incompressible fluid can be represented as an equation including only quaternions as co-ordinate operators and variables. Only the quaternion operations such as a quaternion product and a quaternion conjugation over these variables are used.     

Synthesis of alkylindoles (review)

Methods for obtaining alkylindoles, excluding the Fischer synthesis and direct alkylation, developed during the last 15 years are correlated.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 9, pp. 1155–1172, September, 1987.     

A trust region method for solving the decentralized static output feedback design problem

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n ²) time sequential algorithm and anO(n ²/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.     

Characterization of nanocrystals in laser condensates deposited through vanadium evaporation in an oxygen atmosphere

The influence of the substrate temperature T _sub (20–360°C) and the oxygen pressure P(O₂) (5 × 10⁻³−0.13 Pa) in an evaporation chamber on the structure and phase composition of films prepared through laser sputtering of a vanadium target is investigated by electron diffraction and in situ transmission electron microscopy (with the use of the bend extinction contour technique for determining the bending of the crystal lattice). It is demonstrated that the oxygen content in the films increases with an increase in the oxygen pressure P(O₂) at a fixed substrate temperature T _sub and decreases with an increase in the substrate temperature T _sub at a fixed oxygen pressure P(O₂). The conditions responsible for the formation and composition of the crystalline (VO_0.9) and amorphous (V₂O₃) phases in the films are determined. It is established that the phase composition of the film depends on the angle of condensation of the vapor-plasma flow. The crystallization of the V₂O₃ amorphous phase is accompanied by an increase in the density by 9.2%. It is revealed that the V₂O₃ spherulites growing in the amorphous film have a bent crystal lattice. The bending of the crystal lattice can be as large as ∼42 deg/μm.

     

Modified Coumarins. 9. Synthesis of Amino-Acid Derivatives of 3-(2,3,5-Trimethyl-7-oxofuro[3,2-g]chromen-6-yl)propanoic Acid

Furocoumarins modified by amino acids were prepared by condensation of the N-hydroxysuccinimide ester of 3-(2,3,5-trimethyl-7-oxofuro[3,2-g]chromen-6-yl)propanoic acid with amino acids.     

Radial and angular correlation in heliumlike ions

In the framework of nonrelativistic variational formalism a new type of basis set is proposed, to estimate separately the effect of radial and angular correlations on the ground‐state energy for helium isoelectronic sequence H^? to Ar¹⁶⁺. Effect of radial correlation is incorporated by using multiexponential functions arising from product basis sets suitably formed out of Slater‐type one‐particle orbitals. The angular correlation can be switched on by incorporating an expansion in terms of basis involving interparticle coordinates. With a set of six‐term Slater‐type one‐particle basis and five‐term interparticle expansion, the ground‐state energy of helium is estimated as ?2.9037236 (a.u.) compared with the multiterm variational estimates ?2.9037244 (a.u.) due to Pekeris and Thakkar and Smith and Drake. Matrix elements of different operators in the ground state have been calculated and found to be in good agreement with available accurate results. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

The synthesis of 6-trifluoroethyl-l-lysine: a method to introduce functionality at C-6 of l-lysine

Described is a method of introducing trifluoroalkyl groups at C-6 of lysine. This chemistry has the potential to introduce a variety of functionality at C-6 of lysine.