The minimum span of L(2,1)-labelings of certain generalized Petersen graphs

In the classical channel assignment problem, transmitters that are sufficiently close together are assigned transmission frequencies that differ by prescribed amounts, with the goal of minimizing the span of frequencies required. This problem can be modeled through the use of an L(2,1)-labeling, which is a function f from the vertex set of a graph G to the non-negative integers such that |f(x)-f(y)|? 2 if xand y are adjacent vertices and |f(x)-f(y)|?1 if xand y are at distance two. The goal is to determine the λ-number of G, which is defined as the minimum span over all L(2,1)-labelings of G, or equivalently, the smallest number k such that G has an L(2,1)-labeling using integers from {0,1,…,k}. Recent work has focused on determining the λ-number of generalized Petersen graphs (GPGs) of order n. This paper provides exact values for the λ-numbers of GPGs of orders 5, 7, and 8, closing all remaining open cases for orders at most 8. It is also shown that there are no GPGs of order 4, 5, 8, or 11 with λ-number exactly equal to the known lower bound of 5, however, a construction is provided to obtain examples of GPGs with λ-number 5 for all other orders. This paper also provides an upper bound for the number of distinct isomorphism classes for GPGs of any given order. Finally, the exact values for the λ-number of n-stars, a subclass of the GPGs inspired by the classical Petersen graph, are also determined. These generalized stars have a useful representation on Möebius strips, which is fundamental in verifying our results.     

Selective chemical vapor deposition synthesis of double-wall carbon nanotubes on mesoporous silica

Double-wall carbon nanotubes (DWNTs) have been selectively synthesized over Fe/Co loaded mesoporous silica by catalytic chemical vapor deposition of alcohol. Several silica materials with desired pore diameter and morphology have been investigated for the DWNT growth. The diameter distribution and selectivity of the DWNT are found to depend on the reaction temperature, pore size, and thermal stability of the support material. A high-yield synthesis of DWNTs has been achieved at 900 degrees C over high-temperature stable mesoporous silica. The outer diameter of DWNTs is found to be in the range of 1.5-5.4 nm with a "d" spacing of 0.38 +/- 0.02 nm between inner and outer layers, which is much larger than those of multiwall carbon nanotubes.     

Infrared and Raman spectral interpretations of the ferroelectric phases of NaH3(SeO3)2 and NaD3(SeO3)2

The infrared and Raman spectra of NaH₃(SeO₃)₂ and NaD₃(SeO₃)₂ have been recorded from 24 to 300°K. The interpretation, assignments, and analysis of the spectral studies are presented on the paraelectric α phase (proton disordered), ferroelectric β phase (proton ordered) and ferroelectric γ phase (proton ordered). A discussion of a newly proposed proton-triggered phase transition mechanism and a possible origin of the hydrogen-bonded OH stretching region of KH₂PO₄-type ferroelectrics is given.     

A facile method for the stereoselective preparation of (1E,3E)-4-substituted-1-amino-1,3-dienes via 1,4-elimination

The 1,4-elimination reaction of 1-amino-4-methoxy-(2Z)-alkenes is shown to proceed with high (1E,3E)-stereoselectivities to afford the corresponding 4-substituted-1-amino-1,3-dienes in good yield. The scope and stereochemical features of the synthetic method are described.     

Determination of phenylalanine isotope ratio enrichment by liquid chromatography/time- of-flight mass spectrometry

The application of time-of-flight mass spectrometry to isotope ratio measurements has been limited by the relatively low dynamic range of the time-to-digital converter detectors available on commercial LC/ToF-MS systems. Here we report the measurement of phenylalanine isotope ratio enrichment by using a new LC/ToF-MS system with wide dynamic range. Underivatized phenylalanine was injected onto a C18 column directly with 0.1% formic acid/acetonitrile as the mobile phase. The optimal instrument parameters for the time-of-flight mass spectrometer were determined by tuning the instrument with a phenylalanine standard. The accuracy of the isotope enrichment measurement was determined by the injection of standard solutions with known isotope ratios ranging from 0.02% to 9.2%. A plot of the results against the theoretical values gave a linear curve with R2 of 0.9999. The coefficient of variation for the isotope ratio measurement was below 2%. The method is simple, rapid, and accurate and presents an attractive alternative to traditional GC/MS applications.     

Crystal Structure and Property of Perovskite-Type Oxides Containing Ion Vacancy

Studies on the role of oxygen vacancy in structural change of nonstoichiometric perovskites and a property of oxygen-deficient perovskite-related K₂NiF₄ compounds are reviewed.The structural changes on which the authors focused are cation ordering and lattice distortion. The relationship between the distortion and oxygen vacancy was investigated by comparing the structures of Sr₂(Sr_1-xM_x)TaO_6-d (M = Ca²⁺ and Nd³⁺) solid solutions. It was found that distortion of a perovskite-type lattice decreased with an increasing amount of oxygen vacancies. In order to investigate the relationship between the cation ordering on octahedral sites and oxygen vacancy, structures of stoichiometric Sr_2-xLa_xCo_1-yTa_1+yO₆ and oxygen-deficient Sr_2-xLa_xMg_1-yTa_1+yO_6-d solid solutions were compared. The authors' work reveals that the cation ordering affects the amount of oxygen vacancies in addition to cation charge and size.