Baxter-Guttmann-Jensen conjecture for power series in directed percolation problem

A probabilistic model of a flow of fluid through a random medium,percolation model, provides a typical example of statistical mechanical problems which are easy to describe but difficult to solve. While the percolation problem on undirected planar lattices is exactly solved as a limit of the Potts models, there still has been no exact solution for the directed lattices. The most reliable method to provide good approximations is a numerical estimation using finite power-series expansion data of the infinite formal power series for percolation probability. In order to calculate higher-order terms in power series, Baxter and Guttmann [6] and Jensen and Guttmann [33] proposed an extrapolation procedure based on an assumption that thecorrection terms, which show the difference between the exact infinite power series and approximate finite series, are expressed as linear combinations of the Catalan numbers.In this paper, starting from a brief review on the directed percolation problem and the observation by Baxter, Guttmann, and Jensen, we state some theorems in which we explain the reason why the combinatorial numbers appear in the correction terms of power series. In the proof of our theorems, we show several useful combinatorial identities for the ballot numbers, which become the Catalan numbers in a special case. These identities ensure that a summation of products of the ballot numbers with polynomial coefficients can be expanded using the ballot numbers. There is still a gap between our theorems and the Baxter-Guttmann-Jensen observation, and we also give some conjectures.As a generalization of the percolation problem on a directed planar lattice, we present two topics at the end of this paper: The friendly walker problem and the stochastic cellular automata in higher dimensions. We hope that these two topics as well as the directed percolation problem will be of much interest to researchers of combinatorics.     

Graph presentations for moments of noncentral Wishart distributions and their applications

We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and 2 × 2 Wishart distributions by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials.     

Stability problems for Cantor stochastic differential equations

We consider driftless stochastic differential equations and the diffusions starting from the positive half line. It is shown that the Feller test for explosions gives a necessary and sufficient condition to hold pathwise uniqueness for diffusion coefficients that are positive and monotonically increasing or decreasing on the positive half line and the value at the origin is zero. Then, stability problems are studied from the aspect of Hölder-continuity and a generalized Nakao–Le Gall condition. Comparing the convergence rate of Hölder-continuous case, the sharpness and stability of the Nakao–Le Gall condition on Cantor stochastic differential equations are confirmed. Furthermore, using the Malliavin calculus, we construct a smooth solution to degenerate second order Fokker–Planck equations under weak conditions on the coefficients.     

Transparent and flexible field emission display device based on single‐walled carbon nanotubes

A fully transparent and flexible field emission device (FED) has been demonstrated. Single‐walled carbon nanotubes (SWCNTs) coated on arylite substrate were used as electron emitters for the FED and a novel metavanadate phosphor coated on the SWCNTs/arylite film was used as transparent and flexible screen. The SWCNTs/arylite based emitters and the SWCNTs/arylite/metal‐vanadate‐based phosphor showed a transmittance value of 92.6% and 54%, respectively. The assembled device also showed satisfactory transparency and flexibility as well as producing significant current. Metavanadate phosphor is considered to be an excellent candidate due to its superior luminescence properties and easy fabrication onto transparent and flexible conductive substrate at room temperature while retaining reasonable transparency of the substrate. Thus, its transparency and flexibility will open the door to next‐generation FEDs. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reduced crystallization time of YBCO in a fluorine-free MOD process using uv-lamp irradiation

We have studied the crystallization time dependence of the epitaxial YBCO films (t = 0.8 μm) grown on CeO₂-buffered SrTiO₃ substrates by fluorine-free metal–organic deposition using uv-lamp irradiation (uv-MOD). As increasing the time (T₀) for heat treatment at the reaction temperature (760 °C) from 0 to 90 min, J_c and the YBCO 0 0 l XRD intensity are steeply increased and reach their maximum values at T₀ = 10 min. This suggests that the heat treatment required for YBCO crystallization is significantly shortened in uv-MOD compared to conventional all-pyrolytic F-free MOD processes, which consume T₀ = 90–150 min for crystallizing 0.4–0.5-μm-thick films. Scanning electron microscope measurement revealed a drastic change in surface morphology between T₀ = 8 and 10 min, showing a good correspondence to the J_c and XRD data which suggest that the epitaxial growth reaches the film surface at the very early stage in the heat treatment.     

Liquid ionization mass spectrometry of some triorganotin carboxylates.

and ESI, in which [M + H]+ were not observed or the spectra were complicated. The liquid ionization mass spectra of triorganotin carboxylates varied with solvents and sample concentrations. For instance, the fragment ions [M + (C4H9)3Sn]+ of dimeric ions were observed with chloroform used as a solvent, while the [M + H]+ were observed as the base peak using ethylene dichloride. Spectra useful for the differentiation of isomers [CgH7O3Sn(C4Hg)3] were obtained by the formation of characteristic adduct ions, such as [M + EA + H]+ and [M + 2EA + H]+, with a reagent like 2-aminoethanol. Collision-induced dissociation (CID) spectra observed by ESI and LPI mass spectrometry were similar and provided less information than adduct ions did.