Crystal Structure of BaNb6.3(1)Ti3.6(1)O16 containing Nb6O12, fused Ti3O13 Clusters and Ti3O10 Units

A new phase, BaNb_6.3(1)Ti_3.6(1)O₁₆, has been synthesised. Electron diffraction studies indicate an hexagonal substructure with unit cell parameters a ≈ 8.9 Å and c ≈ 9.5 Å. In some of the ED patterns superstructure reflections are present, indicating a supercell with a = √3 · a_sub and c = c_sub. However, X‐ray single‐crystal diffraction studies of a crystallite yielding reflections corresponding to the supercell revealed it to be monoclinic, with the unit cell parameters a = 26.811(2) Å, b = 15.4798(2) Å, c = 9.414(2) Å, β = γ = 90° and α = 90.0(3)°. The average crystal structure was refined, using the subcell with a = 8.937(2) Å, b = 15.479(2) Å, c = 9.414(2) Å, β = γ = 90° and α = 90.0(3)°, space group Cm11, and Z = 4, to R_I = 3.24% and R_wI = 3.44%. The structure can be described as an hexagonal close packing layers of Nb₆ octahedra, Ba, and O atoms (A1, A2) and layers of O atoms (B1, B2), appearing in the packing sequence: A1B1A2B2. The Nb₆ octahedra are found in isolated Nb₆O₁₂O₆ clusters, and the Ti atoms in Ti₃O₁₃ and Ti₃O₁₀ units in octahedral and tetrahedral voids formed by O atoms, respectively. The Ti positions were found to be only partly occupied. Microanalysis indicates that some Nb atoms are located in the Ti₃ triangles. A model is presented that interprets these not fully occupied Ti₃ triangles as a result of a superimposing of three different structures. Two of these consist of two fused Ti₃O₁₃ units, forming an Ti₆O₁₉ unit, and a Ti₃O₁₀ unit, while the third consists of alternating Ti₃O₁₃ units.     

Two new phosphate langbeinites,Rb2YbTi(PO4)3 and Rb2Yb0.32Ti1.68(PO4)3, investigated at 293 and 150 K

The rubidium ytterbium titanium phosphates Rb₂YbTi(PO₄)₃, (I), and Rb₂Yb_0.32Ti_1.68(PO₄)₃, (II), have been structurally characterized from X‐ray data collected at both 293 and 150 K. Compound (II) is blue owing to the presence of mixed‐valence titanium (41% Ti³⁺ + 59% Ti⁴⁺). Both (I) and (II) belong to the langbeinite structure type, with mixed Yb/Ti populations in the two crystallographically independent octahedral sites (of symmetry 3). Ytterbium favours one of these sites, where about two‐thirds of the Yb atoms are found. The O‐atom displacement parameters are large in both compounds at both temperatures.     

Azin­yl sulfides. LXXXI. 14‐Phen­yl‐7‐thia­pyrano[2,3‐c;6,5‐c′]diquinoline‐14‐carbonitrile

The mol­ecules of the title compound, C₂₆H₁₅N₃S, have a penta­cyclic ring system which is almost planar, with the central ring in a flattened boat conformation. The folding angle between the two quinoline rings is 6.75 (7)°. The 14‐phen­yl substituent is in a quasi‐axial conformation, while the 14‐cyano substituent is in a quasi‐equatorial conformation with respect to the thio­pyran ring. The S⋯C—C_phen­yl and S⋯C—C_CN angles are 116.8 (2) and 129.3 (2)°, respectively. The plane of the phen­yl group is nearly coplanar with the plane bis­ecting the dihedral angle of the penta­cyclic ring system.     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

Sorting Using Complete Subintervals and the Maximum Number of Runs in a Randomly Evolving Sequence

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0’s, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1’s? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n ^1/2, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n ^1/3. We also treat some variations, including priority queues and sock-sorting. The proofs use methods originally developed for random graphs.     

Dynamical Systems Associated with Crossed Products

In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C ^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A′. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.     

A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n^?1/3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Two size regimes of methanol clusters produced by adiabatic expansion

Free neutral methanol clusters produced by adiabatic expansion have been studied by photoelectron spectroscopy and line shape modeling. The results show that clusters belonging to two distinct size regimes can be produced by changing the expansion conditions. While the larger size regime can be well described by line shapes calculated for clusters consisting of hundreds of molecules, the smaller size regime corresponds to methanol oligomers, predominantly of cyclic structure. There is little contribution from dimers to the spectra.     

Layer-by-layer assembly of mucin and chitosan--Influence of surface properties, concentration and type of mucin

Bovine submaxillary mucin (BSM) and chitosan were used to build layer-by-layer structures on solid substrates. The build-up was monitored using in situ ellipsometry to obtain time resolved values of the thickness and adsorbed amount. Additionally surface morphology during build-up was studied by atomic force microscopy (AFM). It was found that the adsorbed amount of the film increases approximately linearly with each deposition cycle on hydrophobized silica whereas construction on silica was found not to be possible at the experimental conditions used. We conclude that sufficient amount of the first mucin layer is crucial for the subsequent multilayer formation. The complex build-up kinetics on hydrophobized silica is characterized by adsorption and redissolution processes and the overall growth is the sum of both processes. AFM imaging on hydrophobized silica also confirmed the presence of redissolution processes and chitosan addition led to a reduction both in the number of surface aggregates and in the roughness of the surface. The present work also shows that by adjusting the relative concentrations of the polyelectrolytes it is possible to change the growth rate considerably. The final structures after deposition of 8 bilayers were found to have a high content of water and film stability test revealed that a substantial amount dissolves when increasing electrolyte concentration or pH of the ambient solution. Human mucin from saliva (MUC5B) was also used to create multilayers with chitosan on hydrophobized silica and it was revealed that no redissolution appears to be present in this system.     

The Sorption Behavior of Wood Studied by Means of an Improved Cup Method

In transport models for wood, sorption is an essential parameter. Sorption is the balancing process between the two phases of water present in wood below the fiber saturation point, namely water vapor in the lumens and bound water in the cell walls. To gain better insight into the physical background of transport processes, a special experimental test setup—the improved cup method—is presented. It allows for separation of sorption from other processes. In this test, a diffusion cup contains a thin specimen of wood, with one side facing outwards to a climate chamber and the opposite side facing inwards the cup. In contrast to the common cup method, the herein presented method uses a data logger for relative humidity and temperature placed inside the cup. The use of thin cross-cut specimens allows for explicit separation of the different processes occurring during transient moisture transport. Mass changes were determined and relative humidity inside the cups was measured for eight specimens of Norway spruce with different specimen thicknesses. Relative humidity was increased in three uniform steps in the test chamber from 4.0 up to 76.5%. The results obtained with this special test setup indicate that the sorption process is different than assumed in previous publications. This emphasizes the need of improved modeling approaches.