On the pagenumber of trivalent Cayley graphs

Book embedding of graphs is one of the graph layout problem. It is useful for the multiprocessor network layout or the fault-tolerant processor arrays. We show that the trivalent Cayley graphs proposed by Vadapalli and Srimani can be embedded in five pages, and show some additional results on cube-connected cycles.     

具有唯一4度最大度点的图的全色数

一个图Ｇ的全色数χ_T（Ｇ）是使得Ｖ（Ｇ）∪Ｅ（Ｇ）中相邻或相关联元素均染不同颜色的最少颜色数．文中证明了，若图Ｇ只有唯一的一个４度最大度点，则χ_T（Ｇ）＝Δ（Ｇ）＋１．     

Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes.     

Diagrams,orientations, and varieties

One of the central problems in the theory of ordered sets is to describe the orientations of the covering graph of an ordered set. We show that the particular operation called inversion, together with the classical constructions of retraction and product, provide a context for the classification of all such orientations. AMS subject classifications (1980). 06A10, 05C75, 05C20.     

Structural features of zinc hydroxyfluoride

Zinc hydroxyfluoride (ZnOHF), obtained by coprecipitation of a zinc salt in aqueous HF, exhibits a variable stoichiometry Zn(OH)_2−xF_x, where x is tuneable from 0.63 to 0.87 by controlling the pH of the solution. The structure was determined from Rietveld refinements using X-ray powder diffraction data. Crystallizing with the orthorhombic symmetry (SG : Pna2₁), the ZnOHF-type structure exhibits two different anionic sites. A bond valence analysis shows that fluorine exclusively occupies the anionic site that has shorter contacts to zinc. This site is split into two partially occupied sites, one corresponding to the position of a fluoride ion and the other to the position of a hydroxide ion. Bond valence calculations show that the split site model gives a more accurate picture of the local coordination of the anions on this site.     

Structures and synthesis of framework Rb and Cs uranyl arsenates and their relationships with their phosphate analogues

Two hydrated uranyl arsenates, Cs₂(UO₂)[(UO₂)(AsO₄)]₄(H₂O)₂ (CsUAs) and Rb₂(UO₂)[(UO₂)(AsO₄)]₄(H₂O)_4.5 (RbUAs), were synthesized by hydrothermal methods. Intensity data were collected at room temperature using MoKα radiation and a CCD-based area detector. The crystal structure of RbUAs was solved by direct methods, whereas the structure model of the phosphate Cs₂(UO₂)[(UO₂)(PO₄)]₄(H₂O)₂ was used for CsUAs; both were refined by full-matrix least-squares techniques on the basis of F² to agreement indices (CsUAs, RbUAs) wR₂=0.061,0.041, for all data, and R₁=0.032,0.021, calculated for 5098, 4991 unique observed reflections (|F_o|>4σ_F), respectively. The compound CsUAs is orthorhombic, space group Cmc2₁, Z=4, a=15.157(2), b=14.079(2), c=13.439(2) Å, V=2867.9(1) Å³. RbUAs is monoclinic, space group C2/m, Z=4, a=13.4619(4), b=15.8463(5), c=14.0068(4) Å, β=92.311(1)°, V=2985.52(2) Å³. The structures consist of sheets of arsenate tetrahedra and uranyl pentagonal bipyramids, with composition [(UO₂)(AsO₄)]⁻, that are topologically identical to the uranyl silicate sheets in uranophane-beta. These sheets are connected by a uranyl pentagonal bipyramid in the interlayer that shares corners with two arsenate tetrahedra on each of two adjacent sheets and whose fifth equatorial vertex is an H₂O group, resulting in an open framework with alkali metal cations in the larger cavities of the structures. CsUAs is isostructural with its phosphate analogue, and has two Cs atoms and a H₂O group in its structural cavities. RbUAs is not isostructural with its phosphate analogue, although it has a homeotypic framework. Its structural cavities are occupied by three Rb atoms and four H₂O groups; one Rb position and three of the interstitial H₂O groups are half-occupied. The partial occupancies of these positions probably result from the accommodation of the larger As atoms (relative to P) in the framework and resultant larger cavities.     

Anion substitution effects on structure and magnetism in the chromium chalcogenide Cr5Te8—Part I: Cluster glass behavior in trigonal Cr(1+x)Q2 with basic cell (Q=Te, Se; Te:Se=7:1)

The effect of substitution of the anion Te by Se in non-stoichiometric Cr₅Te₈ has been investigated with respect to its crystal structure, magnetic properties, and electronic structure. The compounds Cr_(1+x)Q₂ (Q=Te, Se; Te:Se=7:1; (1+x)=1.234(6), 1.264(6), 1.300(7)) were synthesized at elevated temperatures followed by quenching the samples to room temperature. The crystal structures have been refined with X-ray powder diffraction data with the Rietveld method in the trigonal space group with lattice parameters a=3.8651(1)-3.8831(1) Å and c=5.9917(2)-6.0528(2) Å. The structure is related to the NiAs structure with full and deficient metal layers stacking alternatively along the c-axis. The irreversibility in the field-cooled/zero-field-cooled magnetization suggests that the substitution effects of one Te by one Se is strong enough to cause cluster-glass behavior, from ferromagnetic Cr₅Te₈ to cluster-glass Cr_(1+x)Q₂. Non-saturation magnetizations at 5.5 T and the magnetic relaxation results further support the existence of cluster-glass behavior. Accompanying SPR-KKR (spin-polarized relativistic Korringa-Kohn-Rostoker) band structure calculations strongly support the observation that the Cr(1) sites are preferentially occupied by Cr atoms and predict that these compounds are metallic. Results for the spin-resolved DOS and magnetic moments on each crystallographic sites are presented.     

A combined diffraction and dielectric properties investigation of Ba3MnNb2O9 complex perovskites

A combined synthesis, diffraction and dielectric properties investigation of the dependence (and effect) of Mn²⁺/Nb⁵⁺ ordering in Ba₃MnNb₂O₉ (BMN) upon annealing atmosphere and processing conditions has been carried out. Annealing in different atmospheres was not found to significantly alter either nominal stoichiometry or structure type. The obtained structure type (disordered metrically cubic or ordered trigonal) as well as the measured electrical properties (in particular, the dielectric loss) were, however, found to be sensitive to the synthesis route. Samples obtained via solid-state reaction were found to be predominantly of 1:2 Mn²⁺/Nb⁵⁺ ordered, trigonal structure type whereas samples obtained via an aqueous solution route were found to be of a Mn²⁺/Nb⁵⁺ ‘disordered’, metrically cubic structure type. All solid-state synthesized samples showed reasonable dielectric properties. The microwave dielectric constant and dielectric quality factor, Q, at 8 GHz of an as-synthesized BMN sample were 38 and 100, respectively. By contrast, the dielectric loss of the metrically cubic, Mn²⁺/Nb⁵⁺ ‘disordered’ samples obtained via an aqueous solution synthesis process were significantly worse.     

Global Forcing Number of Benzenoid Graphs

A global forcing set in a simple connected graph G with a perfect matching is any subset S of E(G) such that the restriction of the characteristic function of perfect matchings of G on S is an injection. The number of edges in a global forcing set of the smallest cardinality is called the global forcing number of G. In this paper we prove several results concerning global forcing sets and numbers of benzenoid graphs. In particular, we prove that all catacondensed benzenoids and catafused coronoids with n hexagons have the global forcing number equal to n, and that for pericondensed benzenoids the global forcing number is always strictly smaller than the number of hexagons.