An arithmetical property of powers of Salem numbers

Let ζ be a nonzero real number and let α be a Salem number. We show that the difference between the largest and smallest limit points of the fractional parts of the numbers ζαn, when n runs through the set of positive rational integers, can be bounded below by a positive constant depending only on α if and only if the algebraic integer α−1 is a unit.     

Counting occurrences of 231 in an involution

We study the generating function for the number of involutions on n letters containing exactly r?0 occurrences of 231. It is shown that finding this function for a given r amounts to a routine check of all involutions of length at most 2r+2.     

Dyck paths and restricted permutations

This paper is devoted to characterize permutations with forbidden patterns by using canonical reduced decompositions, which leads to bijections between Dyck paths and S_n(321) and S_n(231), respectively. We also discuss permutations in S_n avoiding two patterns, one of length 3 and the other of length k. These permutations produce a kind of discrete continuity between the Motzkin and the Catalan numbers.     

Numerical investigation of hydrodynamics behaviour of melt layer during laser cutting of steel

Understanding of the melt layer hydrodynamic behaviour during laser-cutting process under gas jet assistance is of high importance for cut quality control. In the present work, a numerical model is developed to calculate the three-dimensional behaviour of the melt flow on the kerf front, while an inert gas jet interacts with the melt film. Fluent CFD code is used to solve the governing hydrodynamic equations by finite volume method. The results show that the melt flow on the kerf front reveals a strong instability, which depends on the cutting speed and on the gas jet velocity. Global flow behaviour (gas and molten metal flows) computed using a laminar model, reveals oscillations of the gas–metal liquid interface, which is assimilated to Kelvin–Helmholtz instability. The origin of this instability is discussed in terms of instabilities in thermal dynamics and hydrodynamics. Instability in thermal dynamics is related to the localized melting, while the instability in hydrodynamics is governed by forces balance between gas and resistant surface tension.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

Identities involving Narayana polynomials and Catalan numbers

We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.     

The PI Index of Polyomino Chains of 4<Emphasis Type="Italic">k</Emphasis>-Cycles

The Padmakar-Ivan (PI) index of a graph G is the sum over all edges uv of G of the number of edges which are not equidistant from the vertices u and v. In this paper we compute the PI index of polyomino chains of 4k-cycles and establish bounds for it.