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We prove that the equation x2 − kxy + y2 + x = 0 with k ∊ N+ has an infinite number of positive integer solutions x and y if and only if k = 3. For k = 3 the quotient SHAPE="SOL" ALIGN="C" STYLE="S">xy is asymptotically equal to (3 + √5)/2 or (3 − √5)/2. Results of the paper are based on data obtained via Computer Algebra System (derive 5). Some derive procedures presented in the paper made it possible to discover interesting regularities concerning simple continued fractions of certain numbers. [Copyright &y& Elsevier]

Abstract: In this paper we give a method for peturbation of solutions of linear homogeneous differential equation of the second order. [Copyright &y& Elsevier]

Abstract: In this paper, we give necessary and sufficient conditions for generalized solution and periodicity of the difference equation with -periodic coefficients, where , . Also, we obtain that the generalized solution is periodic with -period. [Copyright &y& Elsevier]

Abstract: In this paper we give some sufficient conditions in terms of regularly generated sequences for a sequence to be subsequentially convergent. [ABSTRACT FROM AUTHOR]

Abstract: Molodtsov [D. Molodtsov, Soft set theory–First results, Comput. Math. Appl. 37 (1999) 19–31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft -ideals and -idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) -ideals in BCI-algebras. We provide relations between fuzzy -ideals and -idealistic soft BCI-algebras. [Copyright &y& Elsevier]

Abstract: In this paper, we obtained rich solutions for the discrete complex cubic Ginzburg–Landau equation by means of the extended tanh-function approach. These solutions include chirpless bright soliton, chirpless dark soliton, triangular function solutions and some solutions with alternating phases, and so on. Meanwhile, the range of parameters where some exact solution exists is given. [Copyright &y& Elsevier]

Abstract: In this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra . Necessary and sufficient conditions are obtained for the matrix equation AX = C and the following systems to have bisymmetric solutions, and the system to have centrosymmetric solutions. The expressions of such solutions of the matrix and the systems mentioned above are also given. Moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the partial difference equation , where k and l are two positive integers, {pm,n} is a real double sequence. Some new oscillation criteria for this equation are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, some new oscillation criteria are obtained for the first-order delay difference equation , where k,l1,l2,, ls are positive integer numbers, f is a function. Our results improve the known results in the literature. And some examples are given to demonstrate the advantage of our results. [Copyright &y& Elsevier]

Abstract: The scheduling problems studied in this paper concern the two-stage no-wait flowshops with parallel machines under the objective function of the minimization of the maximum completion time. A new heuristic algorithm, i.e., the minimum deviation algorithm, is developed to solve the problems. In order to evaluate the average case performance of the algorithm, we design numerical experiments to compare the effectiveness of the algorithm with that of the other approximation algorithms. Extensive simulations are conducted under different shop conditions, and the results statistically show that the minimum deviation algorithm performs well under most of the situations. [Copyright &y& Elsevier]

In this paper, nonlinear dispersive variants of the generalized KdV equation are investigated. The compact and the noncompact physical structures are studied. Two general formulas for the focusing branch and the defocusing branch, that are of substantial interest, are formally derived. [Copyright &y& Elsevier]

Some new oscillation criteria are established in this paper for the second-order nonlinear neutral delay differential equation [r(t)(x(t) + p(t)x(t−τ))′]′ + q(t)f(x(t−)δ)) = 0, where τ and δ are nonnegative constants, r, p, q ∊ C([t0, ∞), R), and f ∊ C(R, R). These results are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. Our results are sharper than some previous results about neutral and delay differential equations. In particular, several examples that dwell upon the sharp conditions of our results are also included. [Copyright &y& Elsevier]

In this paper, the analysis of the dynamic system for iterative improvement of a solution is discussed, and the new iterative improvement of a solution is proposed based on the dynamic system. We have proved that the new iterative improvement of the solution is convergent unconditionally. The numerical experiments illustrate that the new iterative improvement of the solution is more effective for an ill-conditional system of linear equations. [Copyright &y& Elsevier]