Dirac--Lie systems and Schwarzian equations

A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the Dirac's description of constrained systems, we introduce and analyse a particular class of Lie systems on Dirac manifolds, called Dirac--Lie systems, which are associated with `Dirac--Lie Hamiltonians'. Our results enable us to investigate constants of the motion, superposition rules, and other general properties of such systems in a more effective way. Several concepts of the theory of Lie systems are adapted to this `Dirac setting' and new applications of Dirac geometry in differential equations are presented. As an application, we analyze traveling wave solutions of Schwarzian equations, but our methods can be applied also to other classes of differential equations important for Physics.Comment: 41 page

Fault Localization in Multi-Threaded C Programs using Bounded Model Checking (extended version)

Software debugging is a very time-consuming process, which is even worse for multi-threaded programs, due to the non-deterministic behavior of thread-scheduling algorithms. However, the debugging time may be greatly reduced, if automatic methods are used for localizing faults. In this study, a new method for fault localization, in multi-threaded C programs, is proposed. It transforms a multi-threaded program into a corresponding sequential one and then uses a fault-diagnosis method suitable for this type of program, in order to localize faults. The code transformation is implemented with rules and context switch information from counterexamples, which are typically generated by bounded model checkers. Experimental results show that the proposed method is effective, in such a way that sequential fault-localization methods can be extended to multi-threaded programs.Comment: extended version of paper published at SBESC'1