20-relative neighborhood graphs are hamiltonian

For any two points p and q in the Euclidean plane, define LUN_pq = { v | v ∈ R², d_pv < d_pq and d_qv < d_pq}, where d_uv is the Euclidean distance between two points u and v . Given a set of points V in the plane, let LUN_pq(V) = V ∩ LUN_pq. Toussaint defined the relative neighborhood graph of V, denoted by RNG(V) or simply RNG, to be the undirected graph with vertices V such that for each pair p,q ∈ V, (p,q) is an edge of RNG(V) if and only if LUN_pq (V) = ?. The relative neighborhood graph has several applications in pattern recognition that have been studied by Toussaint. We shall generalize the idea of RNG to define the k-relative neighborhood graph of V, denoted by kRNG(V) or simply kRNG, to be the undirected graph with vertices V such that for each pair p,q ∈ V, (p,q) is an edge of kRNG(V) if and only if | LUN_pq(V) | < k, for some fixed positive number k. It can be shown that the number of edges of a kRNG is less than O(kn). Also, a kRNG can be constructed in O(kn₂) time. Let E_c = {e_pq| p ∈ V and q ∈ V}. Then G_c = (V,E_c) is a complete graph. For any subset F of E_c, define the maximum distance of F as max_epq∈Fd_pq. A Euclidean bottleneck Hamiltonian cycle is a Hamiltonian cycle in graph G_c whose maximum distance is the minimum among all Hamiltonian cycles in graph G_c. We shall prove that there exists a Euclidean bottleneck Hamiltonian cycle which is a subgraph of 20RNG(V). Hence, 20RNGs are Hamiltonian.     

Asymptotics of Dirichlet Spectrum on Some Class of Noncompact Domains

We investigate the asymptotic behaviour of the counting function of Dirichlet eigenvalues on some class of noncompact manifolds. We prove that in cases when the volume or the perimeter (the volume of the boundary) of the manifold is infinite, some additional (nonclassical) terms appear in the precise asymptotics. The coefficients at the classical terms in those are regularized in some special way volume (resp. perimeter) of the manifold.     

Nitroxide‐mediated radical polymerization in miniemulsion: Bimolecular termination in monomer‐free model systems

Bimolecular termination in nitroxide‐mediated radical polymerization in miniemulsion has been investigated through the heating of a polystyrene–2,2,6,6‐tetramethylpiperidinyl‐1‐oxy macroinitiator and its 4‐hydroxy‐2,2,6,6‐tetramethylpiperidinyl‐1‐oxy analogue in an aqueous toluene dispersion with sodium dodecyl benzenesulfonate as a surfactant at 125 °C. The level of bimolecular termination by combination, evaluated from the high‐molecular‐weight shoulder, was higher in miniemulsion than in solution and increased with decreasing particle size. Quantitative analysis revealed that these results cannot be rationalized solely by nitroxide partitioning to the aqueous phase. The results are explained by an interface effect, by which nitroxide is adsorbed or located at the aqueous–organic interface. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 4995–5004, 2007     

An implicit characteristic-flux-averaging method for the euler equations for real gases

A formulation of an implicit characteristic-flux-averaging method for the unsteady Euler equations with real gas effects is presented. Incorporation of a real gas into a general equation of state is achieved by considering the pressure as a function of density and specific internal energy. The Ricmann solver as well as the flux-split algorithm are modified by introducing the pressure derivatives with respect to density and internal energy. Expressions for calculating the values of the flow variables for a real gas at the cell faces are derived. The Jacobian matrices and the eigenvectors are defined for a general equation of state. The solution of the system of equations is obtained by using a mesh-sequencing method for acceleration of the convergence. Finally, a test case for a simple form of equation of state displays the differences from the corresponding solution for an ideal gas.     

Interference inspection of the aspherical components of an objective for nanolithography

Methods and schemes to inspect the EUV mirror shape are developed on the basis of a point diffraction interferometer with computer processing of interferograms. A measurement accuracy to within 0.001 of a wavelength in the visual range is achieved.     

The effect of olefinic terminal chains on the mesomorphic properties of 4,4'-disubstituted diphenyldiacetylenes

New diphenyldiacetylenes of the type with A, B = H and/or F; m = 0, 1; n = 1-4; and X = C n H 2n + 1 , F, CF 3 or CN were synthesized and their mesomorphic properties determined by hot stage polarizing microscopy and DSC. When m = 0, all of these compounds showed only a nematic phase except when X = CF 3 when both nematic and smectic A phases were seen. Both clearing and melting temperatures were higher than those reported for substitution with the corresponding alkyl chains but the much larger increase in clearing temperatures produced considerably wider nematic phases. Eutectic mixtures of a few of these olefins yielded nematic materials also having much wider temperature ranges and higher clearing temperatures than the eutectic mixtures of the alkyl compounds, while retaining their high birefringence and low viscosities. Such materials are of interest for beam-steering devices.

Four of the diacetylenes with m = 1 ( A, B = H) were also prepared ( X = C 6 H 13 , F, n = 2, 3). When X was C 6 H 13 ( n = 2), the nematic range was smaller in the 2- than in the 1-olefin but wider than in the alkyl series. When X = F, either no nematic phase or a monotropic one was observed, whereas the 1-olefins gave a much wider nematic phase. Both transition temperatures were lower than those for the corresponding 1-olefin and alkyl analogues. The compound with X = C 6 H 13 and n = 2 had a melting temperature below room temperature.     

New Heat-Resistant Polyimidates Derived from Bisphenols and Mono- and Dicarboxylic Acid Imidoyl Chlorides

New polyimidates were prepared from bisphenols and mono- and dicarboxylic acid imidoyl chlorides. The structures of the polymers were determined, and their physicochemical, chemical, and thermal properties were studied. The possibility of preparing film and compression materials with good mechanical characteristics from these polymers was examined.     

Numerical study of simultaneous natural convection heat transfer from both surfaces of a uniformly heated thin plate with arbitrary inclination

 Numerical studies were conducted to investigate the natural convection heat transfer around a uniformly heated thin plate with arbitrary inclination in an infinite space. The numerical approach was based on the finite volume technique with a nonstaggered grid arrangement. For handling the pressure–velocity coupling the SIMPLE-algorithm was used. QUICK scheme and first order upwind scheme were employed for discretization of the momentum and energy convective terms respectively. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 4.8×10⁶ to 1.87×10⁸. Local and average heat transfer characteristics were compared with regarding to the inclination angle. The empirical expressions for local and average Nusselt number were correlated. It has been found that for inclination angle less than 10^∘, the flow and heat transfer characteristics are complicated and the average Nusselt number can not be correlated by one equation while for inclination angle larger than 10^∘, the average Nusselt number can be correlated into an elegant correlation. Received on 18 April 2001 / Published online: 29 November 2001