Derivative based global sensitivity measures

The method of derivative based global sensitivity measures (DGSM) has recently become popular among practitioners. It has a strong link with the Morris screening method and Sobol' sensitivity indices and has several advantages over them. DGSM are very easy to implement and evaluate numerically. The computational time required for numerical evaluation of DGSM is generally much lower than that for estimation of Sobol' sensitivity indices. This paper presents a survey of recent advances in DGSM concerning lower and upper bounds on the values of Sobol' total sensitivity indices $S\_{i}^{tot}$. Using these bounds it is possible in most cases to get a good practical estimation of the values of $S\_{i}^{tot}$. Several examples are used to illustrate an application of DGSM

Object grammars and bijections

AbstractA new systematic approach for the specification of bijections between sets of combinatorial objects is presented. It is based on the notion of object grammars. Object grammars give recursive descriptions of objects and generalize context-free grammars. The study of a particular substitution in these object grammars confirms once more the key role of Dyck words in the domain of enumerative and bijective combinatorics

Steep polyominoes, q-Motzkin numbers and q-Bessel functions

AbstractWe introduce three definitions of q-analogs of Motzkin numbers and illustrate some combinatorial interpretations of these q-numbers. We relate the first class of q-numbers to the generating function for steep parallelogram polyominoes according to their width, perimeter and area. We show that this generating function is the quotient of two q-Bessel functions. The second class of q-Motzkin numbers counts the steep staircase polyominoes according to their area, while the third one enumerates the inversions of steep Dyck words. These enumerations allow us to illustrate various techniques of counting and q-counting