In this article, small scale effects on free vibration analysis of non-uniform nanobeams is discussed. Small scale effects are modelled after Eringen’s nonlocal elasticity theory while the non-uniformity is presented by exponentially varying width among the beams length with constant thickness. Analytical solution is achieved for free vibration with different boundary conditions. It is shown th...

Oxalic acid dihydrate as a green, mild and efficient catalyst for the one-pot three-component Biginelli synthesis of 3,4-dihydropyrimidin-2-(1H)-one derivatives from the reaction between β-keto esters (methyl or ethyl acetoacetate), aromatic aldehyde (benzaldehye derivatives) and urea or thiourea under thermal and solvent-free conditions with excellent yields and short reaction time is st...

This article describe the applicability of 1,4-disulfopiperazine-1,4-diium chloride ([Piper-(SO3H)2].2Cl) as a green, versatile and Brönsted acidic ionic catalyst in the promotion of the synthesis of phthalazine derivatives via one-pot three component reaction between aromatic aldehydes, 1,3-diketone derivatives and phthalhydrazide under solvent-free reaction cond...

A practical and green method for the synthesis of 5H-dibenzo[b,i]xanthene-tetraones and spiro[dibenzo[b,i]xanthene-13,3'-indoline]-pentaones by the condensation of 2-hydroxynaphthalene-1,4-dione with aldehydes or isatins, respectively, in the presence of a catalytic amount of cellulose sulfuric acid (CSA) under solvent-free conditions at 100 °C is described. The significant features of this pro...

An efficient and simple one-pot approach for the synthesis of 3,4-dihydropyrimidin-2-(1H)-one derivatives using antimony trichloride (SbCl3) as a mild catalyst by means of three-component Biginelli reaction between β-keto esters, aldehyde derivatives and urea/thiourea under thermal and solvent-free conditions with excellent yields and short reaction times is reported. This m...

CdO nanoparticles efficiently catalyzes the condensation of aromatic aldehydes with 2-aminophenol at room temperature to afford 2-aryl benzoxazole derivatives by grinding method. The reactions proceed under heterogeneous and mild conditions to provide 2-aryl benzoxazoles in excellent yields (87-97 %) with high purity under solvent free condition. The reaction requires short time (5-23 minutes) ...

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...

Euclidean n-component φ theories whose Hamiltonians areO(n) symmetric except for quadratic symmetry breaking boundary terms are studied in the film geometry R×[0, L]. The boundary terms imply the Robin boundary conditions ∂nφα = c̊ (j) α φα at the boundary planes Bj=1 at z = 0 and Bj=2 at z = L. Particular attention is paid to the cases in which mj of the n variables c̊ (j) α associated with plan...

We study the Pöschl-Teller equation in complex domain and deduce infinite families of TQ and Bethe ansatz equations, classified by four integers. In all these models the form of T is very simple, while Q can be explicitly written in terms of the Heun function. At particular values there is a interesting interpretation in terms of finite lattice spin2 XXZ quantum chain with ∆ = cos π L (for free...

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 ...