This paper is the companion article to [Ann. Probab. 39 (2011) 779–856]. We consider a discrete elliptic equation on the d-dimensional lattice Zd with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (co...

We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed us also to extract the limiting curves when n → ∞. In this limit we have also obtained the isolated limiting points for these square-lattice strips and checked some...

Cellulose sulfuric acid is an efficient metal-free catalyst for the synthesis of 1,3,5-trisubstituted pyrazoles via the condensation of 1,3-diketones and hydrazines. The reaction was carried out in Solvent-free condition at room temperature and the products were isolated in good to excellent yields. Mild reaction conditions, as well as ease of operation and workup are some advantages of the pro...

A mild and efficient method has been developed for the preparation of substituted coumarins from reaction of various phenols with different β-ketoesters via Pechmann condensation in the presence of Brønsted acidic ionic liquid (N-(4-sulfonic acid) butyl triethylammonium hydrogen sulfate) as an effective catalyst under solvent-free conditions. Different phenols reacted with ethyl acetoacetate to...

A green and efficient one pot, three component protocol for synthesis of naphthopyranopyrimidines by cyclocondensation of β-naphthol, aldehyde, and 6-amino-1,3-dimethyluracil using L-proline as a beneficial catalyst with high catalytic activity under solvent-free conditions at 100 °C is described. In this study, several types of aromatic aldehyde, containing electron-withdrawing groups as well ...

In this paper, static and free vibrations behaviors of the orthotropic functionally graded material (FGM) plates resting on the two-parameter elastic foundation are analyzed by the a mesh-free method based on the first order shear deformation plate theory (FSDT). The mesh-free method is based on moving least squares (MLS) shape functions and essential boundary conditions are imposed by transfer...

We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. The equation happens to possess a Lyapunov or “free energy” functional. This functional consists of the term |∆h|2 that represents the surfac...

We analyse boundary conformal field theories on random surfaces using the conformal gauge approach of David, Distler and Kawai. The crucial point is the choice of boundary conditions on the Liouville field. We discuss the Weyl anomaly cancellation for Polyakov’s non-critical open bosonic string with Neumann, Dirichlet and free boundary conditions. Dirichlet boundary conditions on the Liouville ...

A green and efficient one pot, three component protocol for synthesis of naphthopyranopyrimidines by cyclocondensation of β-naphthol, aldehyde, and 6-amino-1,3-dimethyluracil using L-proline as a beneficial catalyst with high catalytic activity under solvent-free conditions at 100 °C is described. In this study, several types of aromatic aldehyde, containing electron-withdrawing groups as well ...

In this paper, the vibrations of thin plate in modified couple stress thermoelastic medium by using Kirchhoff- Love plate theory has been investigated. The governing equations of motion and heat conduction equation for Lord Shulman (L-S) [1] theory are written with the help of Kirchhoff- Love plate theory. The thermoelastic damping of micro-beam resonators is analyzed by using the normal mode a...