The generating polynomial of all $n$--permutations  with respect to the number of alternating runs possesses a root at $-1$ of  multiplicity $\lfloor (n-2)/2\rfloor$ for$n\ge2$. This fact can be deduced by combining the  David--Barton formula for Eulerian polynomials with the Foata--Schützenberger $\gamma$--decomposition of these  polynomials. Recently, Bóna provided a group--action proof of this  result. In this talk, I propose an alternative approach based on the  Hetyei--Reiner action on binary trees, which yields a new combinatorial  interpretation of Bóna’s quotient polynomial. Furthermore, we  extend our study to analogous results for permutations of types~B and~D. As a  consequence of our bijective framework, we also obtain combinatorial proofs  of David--Barton type identities for permutations of types~A and~B.This talk is based on a joint work with Yunze Wang  and Jiang Zeng.

Testingand formal methods are two essential approaches to ensuring softwarereliability. Software testing can only reveal defects, it cannot prove theirabsence. Formal methods, grounded in mathematical logic, can rigorously verifythe correctness of software, but their strictness often limits the degree ofautomation. How to combine the strengths of both approaches while mitigatingtheir limitations, in order to achieve automated software verification andvalidation, is an attractive research problem. In this talk, I will introducean advanced software quality assurance technique called Testing-Based FormalVerification (TBFV), which effectively integrates specification-based testingwith formal verification of program correctness. I will outline the fundamentalprinciples, specific techniques, and challenges of TBFV. A key feature of TBFVis its ability to automatically verify the correctness of all program pathsexplored during testing, while also potentially discovering new paths. Thisdual capability supports both software verification and validation. Once widelyadopted, TBFV is expected to significantly reduce testing time and cost,alleviate developers’ workload, and greatly enhance software reliability andquality.

We present some special properties of classicalHamiltonian systems and iterated Hamiltonian systems in finite dimension,recently discussed in some works by the author and his collaborator, CornelMurea (Univ. Mulhouse, France).

One application is a new local representation ofmanifolds, using iterated Hamiltonian systems, in arbitrary dimension and co-dimension.It provides a constructive extension of the implicit functions theorem. Indimension two, under certain assumptions, the representation is global. Thisallows a complete treatment of the considered questions, in dimension two.