Lie Group Analysis of the Time-delayed Inviscid Burgers' Equation

In this paper, we discuss group analysis of rst-order delay partial di erential equations and use it to obtain symmetries of the Invis- cid Burgers' equation with delay, its kernel and extensions of the kernel. We obtain a Lie type invariance condition for rst-order delay partial di erential equations by using Taylor's theorem for a function of several variables. We obtain the symmetries admitted by this delay partial di er- ential equation. Further, we obtain representations of analytic solutions and the reduced equations from the symmetries.

Non-ambiguous trees: new results and generalization

International audience We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differ- ential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain q-versions of our formula. And we generalize NATs to higher dimension.

Analysis of a vibrating heat engine using PACTOLUS

The purpose of this paper is to investigate the theoretical feasibility of a proposed vibrating heat engine. This engine, intended for small cryogenic refrigerators, is hoped to be more reliable than existing ones. The thermodynamics of the process is described and analyzed, and a final differ ential equation relating the pressure and volume of the gas in the engine is obtained. This equation is solved and its parameters are investigated using the PACTOLUS analog simulator on an IBM 1710 computer. The results of these calculations are shown, and from them have been calcu lated the heat engine efficiency as a function of its pa rameters.