RATIONAL APPROXIMATIONS OF (1 + x)a AND APPLICATIONS TO THUE INEQUALITIES
We study some Pade approximations to (1 + x)a and we give their lower bound using Rickert's theorem, where a and x are rational numbers in the open interval (0, 1) with (denominator of x) > (numerator of x)2×(denominator of a)3/2. Thenwe give an effective bound for the solutions of the Thue inequalities |Xn-(1+x)αY n| ≤ k for a positive integer n and rational numbers α, x with n ≥ 2 and 0 n, x k. As an application we solve these inequalities for some special cases. Our result depends on the methods of Chudnovsky [C], Rickert [R] and Wakabayashi [W1, W2].
Software engineers often need to identify and correct design defects, i.e., recurring design problems that hinder development and maintenance by making programs harder to comprehend and/or evolve. While detection of design defects is an actively researched area, their correction- mainly a manual and time-consuming activity- is yet to be extensively investigated for automation. In this paper, we propose an automated approach for suggesting defect-correcting refactorings using relational concept analysis (RCA). The added value of rca consists in exploiting the links between formal objects which abound in a software re-engineering context. We validated our approach on instances of the Blob design defect taken from four different open-source programs.