Copper redox behavior, structure and properties of copper lead borate glasses

Copper oxidation states, structure and properties of xCuO · (50-x)PbO · 50B₂O₃ glasses were investigated. Both infrared (IR) and ¹¹B magic angle spinning nuclear magnetic resonance (MAS NMR) spectroscopies were employed to determine the tetrahedral BO₄ fraction in the glasses as a function of CuO content. IR study indicates that the replacement of Pb²⁺ by Cu²⁺ ions increases the BO₃ units by converting BO₄- containing groups into ring type metaborate groups. The oxidation states of copper ions in the glasses have been studied using both X-ray photoelectron spectroscopy (XPS) and the wet chemical method. For high CuO containing (?30 mol%) glasses, high Cu⁺ ion concentrations (Cu⁺/Cu_tot.>0.3) result in a relatively slow disproportionation of B⁴-containing groups because of the small coordination number of Cu⁺ compared to Cu²⁺ ions. Effects of both glass structure and redox states of copper ions on glass properties including density, Vickers’ hardness, coefficient of thermal expansion, and chemical durability have been discussed.     

Structure and crystallization of potassium titanium phosphate glasses containing B2O3 and SiO2

Transparent glasses composition of which can be expressed by the formula: (100−x) · (K₂O · 2TiO₂ · P₂O₅) · x(K₂O · 2B₂O₃ · 7SiO₂), where x=5, 10, 15 and 20 mol% (KTP-xKBS), were obtained by melt quenching technique. The structure and crystallization behavior of these glasses have been examined by Fourier transform infrared spectroscopy, differential thermal analysis and X-ray diffraction. In spite of their nominal composition, the studied glasses exhibit a similar oxygen polyhedra distribution. However, significant differences were found in the trigonal BO₃ units amount. During DTA runs all the examined glasses devitrify in two steps. In the former, very small crystals of an unknown crystalline phase are produced. In KTP-5KBS and KTP-10KBS glasses anatase phase was also detected. Attempts were made in order to identify the unknown phase (UTP) for which a AB₃(XO₄)₂(OH)₆ Crandallite-type structure was proposed where the A, B and X sites were occupied by K, Ti and/or Al, and P, respectively. In the second devitrification step the crystallization of the KTiOPO₄ phase occurs while the UTP phase previously formed disappears. Isothermal heat treatments performed at temperature just above T_g have allowed one to obtain transparent crystal-glass nanocomposites, formed by crystalline nanostructure of the UTP phase uniformly dispersed in the amorphous matrix.     

Microstructure and optical properties of (Pb,Ca)TiO3 films grown on Si(1 0 0) substrates by a simple sol-gel process

Amorphous and polycrystalline (Pb_0.76Ca_0.24)TiO₃ (PCT) thin films deposited on an Si(1 0 0) substrate have been prepared by a simple sol-gel process. The microstructure and surface morphologies of the thin films have been studied by X-ray diffraction (XRD) and atomic force microscopy (AFM). The polycrystalline PCT film on the Si(1 0 0) substrate has a tetragonal perovskite structure with grain size from 60 to 110 nm. AFM reveals smooth surfaces and root mean square (rms) roughness of 0.17 and 4.4 nm for amorphous and polycrystalline films, respectively. The refractive index n and extinction coefficient k of the amorphous and polycrystalline thin films was obtained by spectroscopic ellipsometry as a function of the photon energy in the range from 2.0 to 5.4 eV. The maximum n and direct bandgap energies of amorphous and polycrystalline thin films were 2.66 and 4.11 eV, 2.64 and 3.84 eV, respectively.     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

The poset structures admitting the extended binary Golay code to be a perfect code

Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.     

Characterizations of postman sets

Using results by McKee and Woodall on binary matroids, we show that the set of postman sets has odd cardinality, generalizing a result by Toida on the cardinality of cycles in Eulerian graphs. We study the relationship between T-joins and blocks of the underlying graph, obtaining a decomposition of postman sets in terms of blocks. We conclude by giving several characterizations of T-joins which are postman sets.     

Critically prime interval orders

We prove that a prime interval order, not necessarily finite, has no prime upper cover if and only if it has width two. On the other hand, we prove that every finite prime N-free order has at least one prime upper cover.     

Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces Lp(Rn;X) of X -valued functions on Rn. We characterize Kato's square root estimates and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative Lp space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the Lp theory of square roots of elliptic operators, as well as an Lp version of Carleson's inequality.     

Noncommutative maximal ergodic theorems for positive contractions

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ○T?τ and such that the numerical range of T as an operator on L₂(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu's noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1<p?∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations.     

On the Cauchy-Rassias stability of a generalized additive functional equation

Let X and Y be Banach spaces and f:X→Y an odd mapping. We solve the following generalized additive functional equation