David J Gaebler

A discrete brachistochrone is the fastest piecewise linear ramp between fixed endpoints with a given number of segments. This article introduces a new conceptual framework for discrete brachistochrones, proves their two fundamental symmetry properties, and examines the manner in which they converge to the cycloid (the well-known continuous brachistochrone) as the number of sides tends to infinity.

Since it began in the 1970s, the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup. Several authors have proved the existence of dilations, but have usually only achieved unital dilations of specific examples, while general theorems have tended to produce non-unital dilations. A unique approach due to Jean-Luc Sauvageot leads to a unital dilation in a general setting, but leaves unclear the continuity of the dilation semigroup. The major purpose of this paper, therefore, is to further develop Sauvageot's theory in order to prove the existence of continuous unital dilations. The dilation is characterized by a combinatorial property akin to free independence. This property is implicit in some of Sauvageot's original calculations, but a secondary goal of this paper is to present it as its own object of study, which is done in Section 4. This work is based on the author's Ph.D. thesis.