The complexity of normal form rewrite sequences for Associativity

The complexity of a particular term-rewrite system is considered: the rule of associativity (x*y)*z --> x*(y*z). Algorithms and exact calculations are given for the longest and shortest sequences of applications of --> that result in normal form (NF). The shortest NF sequence for a term x is always n-drm(x), where n is the number of occurrences of * in x and drm(x) is the depth of the rightmost leaf of x. The longest NF sequence for any term is of length n(n-1)/2.Comment: 5 page

Tropical totally positive matrices

We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowshi

Extreme close approaches in hierarchical triple systems with comparable masses

We study close approaches in hierarchical triple systems with comparable masses using full N-body simulations, motivated by a recent model for type Ia supernovae involving direct collisions of white dwarfs (WDs). For stable hierarchical systems where the inner binary components have equal masses, we show that the ability of the inner binary to achieve very close approaches, where the separation between the components of the inner binary reaches values which are orders of magnitude smaller than the semi-major axis, can be analytically predicted from initial conditions. The rate of close approaches is found to be roughly linear with the mass of the tertiary. The rate increases in systems with unequal inner binaries by a marginal factor of $\lesssim 2$ for mass ratios ${0.5<m_1/m_2<1}$ relevant for the inner white-dwarf binaries. For an average tertiary mass of $\sim 0.3 M_{\odot}$ which is representative of typical M-dwarfs, the chance for clean collisions is $\sim 1$% setting challenging constraints on the collisional model for type Ia's.Comment: Python code included as ancillary fil

Evaluating cross-organizational ERP requirements engineering practices: a focus group study

This focus group study presents our first validation of practices for engineering the coordination requirements in cross-organizational Enterprise Resource Planning (ERP) projects. The study evaluates 13 practices addressing a variety of coordination aspects crucial to ERP projects. These practices are results in previously published research publications by the first author. The practices are formulated in response to practitioners' needs at ERP adopting organizations. The proposed practices have now reached the stage where we need some independent feedback as to the extent to which they fit the realities of practitioners. We perform this validation by means of a qualitative research approach, namely the focus group method. Current software engineering literature provides few examples of using focus groups in the evaluation of good software development practices. Because of this, providing reflections on our focus-group-based validation experiences will be of value to both the research community and practitioners