This article is to discuss the automatic continuity properties and the representation of disjointness preserving linear mappings on certain normal Fre´chet algebras of complex-valued functions. This class of operators is defined by the condition that any pair of functions with disjoint cozero sets is mapped to functions with disjoint cozero sets, and subsumes the class of local operators. It turns out that such operators are always continuous outside some finite singularity set of the underlying topological space. Our main emphasis is on disjointness preserving operators from Fre´chet algebras of differentiable functions. Such operators are shown to admit a canonical representation that involves weighted composition for the derivatives. This result extends the classical characterization of local operators as linear partial differential operators.1991 Mathematics Subject Classification. Primary 47B38. Secondary 47B48, 46H40, 46J15.

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About a century ago, the French artillery commandant Charbonnier envisioned an intriguing result on the trajectory of a projectile that is moving under the forces of gravity and air resistance. In 2000, Groetsch discovered a significant gap in Charbonnier’s work and provided a valid argument for a certain special case. The goal of the present article is to establish a rigorous new approach to the full result. For this, we develop a theory of those functions which can be sandwiched, in a natural way, by a pair of quadratic polynomials. It turns out that the convexity or concavity of the derivative plays a decisive role in this context.

This article concerns the problem of maximizing the horizontal range of a projectile that is launched from atop a tower and is subject only to gravity and linear air resistance. Here the surface to which the projectile is launched is represented by a continuous function whose intersection with every flight path is a single point. In this general setting, tools from real analysis need to be employed to gain detailed information about the shape of the graph and the maximum of the distance function. In particular, the article provides geometric insight into the maximal lateral displacement of the projectile and the corresponding optimal launch angle.

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