A tiling of a surface is a decomposition of the surface into pieces, i.e. tiles, which cover it without gaps or overlaps. In this paper some special polygonal tiling of sphere, nellipsoid, cylinder, and torus as the most abundant shapes of fullerenes are investigated.

Purpose: The purpose of this paper is to develop a set of identities for EULER type sums of products of harmonic numbers and reciprocal binomial coefficients.Method: We use analytical methods to obtain our results.Results: We obtain identities for variant EULER sums of the type ¥Sn=1H2n/n (n+k)k, and its finite counterpart, which generalize some results obtained by other authors.Conclusions: Identities are successfully achieved for the sums under investigation. Some published results have been successfully generalized.

This paper presents the results of a two phase EULER-EULER simulation of seepage flow through granular media for upward flow conditions in soils. The simulation first deals with the fluidization process in a stable granular soil. The first fluidization in the sand column is studied and the stable fluidization condition is reached by the development of particle repulsion and the effect of inertia on the resistance of the granular bed. Considering Richardson's expansion law, porosity changes with the flow rate variation are described based on the rarefaction waves. In this study the variation of water and soil phases before and after the critical hydraulic gradient are evaluated. The results of the simulation reveal that the variation of soil pressure before and after the critical hydraulic gradient are inverse and the Terzaghi's effective stress theory is valid only to the onset of the first fluidization. In the second part of the paper, the fluidization and internal erosion processes in the artesian conditions in internally unstable soils are discussed. In internally unstable soils the finer soil particles are able to move through the constrictions of the coarse particles. This may lead to internal erosion and as a result the critical hydraulic gradient drastically increases. The erosion phenomenon is described using the theory of continuum mechanics for the soil consisting of stationary grains, movable grains and the fluid. The continuity and momentum equations are derived for the soil and based on this theory a numerical two phase model for the erosion and fluidization of particles are presented. The results reveal that by the commencement of erosion, the porosity of the soil decreases and the flux of eroded particles increases with time.

The issue of the new elastic terms discovered in the nonlinear dynamic model of an enhanced nonlinear 3D EULER-Bernoulli beam is discussed. While the elastic orientation is negligible, the nonlinear dynamic model governing tension-compression, torsion and two spatial bendings is presented. Considering this model, some new elastic terms can be identified in the variation of elastic potential energy in each bending motion equation, and in each transverse shear force. Due to the new terms, each term of a bending equation and a transverse shear force, finds a counterpart in the other bending equation and transverse shear force, but the equations remain asymmetric. The new terms have arisen, since variation of strains and variation of elastic potential energy are derived from exact strains and exact deformations regarding considerable elastic orientation, then the elastic orientation is neglected. The new terms perish in the nonlinear 3D EULER-Bernoulli beam theory, since elastic orientation is neglected first, then variation of strains and variation of elastic potential energy are derived from the approximated strains.

The accuracy and efficiency of the elements proposed by finite element method (FEM) considerably depend on theinterpolating functions namely shape functions used to formulate the displacement field within the element. In thepresent study, novel functions, namely basic displacements functions (BDFs), are introduced and exploited forstructural analysis of nanobeams using finite element method based on Eringen’ s nonlocal elasticity and EULER-Bernoulli beam theory. BDFs are obtained through solving the governing differential equation of motion of nanobeamsusing the power series method. Unlike the conventional methods which are almost categorized as displacement-basedmethods, the flexibility basis of the method ensures true satisfaction of equilibrium equations at any interior point ofthe element. Accordingly, shape functions and structural matrices are achieved in terms of BDFs by application ofmerely mechanical principles. In order to evaluate the competency and accuracy of the proposed method with differentboundary conditions, several numerical examples with various boundary conditions are scrutinized. Carrying outseveral numerical examples, the results in stability analysis, free longitudinal vibration and free transverse vibrationshow a complete accordance with those in literature.

In this paper we introduce EULER method for solving one dimensional fuzzy differential inclusion. Fuzzy reachable set can be approximated by EULER method with complete analysis. The method is illustrated by numerical example.