Set Theory or Higher Order Logic to Represent Auction Concepts in Isabelle?

When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL system, and we have produced verified code for combinatorial Vickrey auctions. A fundamental question in this was how to represent some basic concepts: since set theory is available inside Isabelle/HOL, when introducing new definitions there is often the issue of balancing the amount of set-theoretical objects and of objects expressed using entities which are more typical of higher order logic such as functions or lists. Likewise, a user has often to answer the question whether to use a constructive or a non-constructive definition. Such decisions have consequences for the proof development and the usability of the formalization. For instance, sets are usually closer to the representation that economists would use and recognize, while the other objects are closer to the extraction of computational content. In this paper we give examples of the advantages and disadvantages for these approaches and their relationships. In addition, we present the corresponding Isabelle library of definitions and theorems, most prominently those dealing with relations and quotients.Comment: Preprint of a paper accepted for the forthcoming CICM 2014 conference (cicm-conference.org/2014): S.M. Watt et al. (Eds.): CICM 2014, LNAI 8543, Springer International Publishing Switzerland 2014. 16 pages, 1 figur

The development of a treadle pump: Lessons from the South African experience

Manual pumps / Design / South Africa

The conduction pathway of potassium channels is water free under physiological conditions.

Ion conduction through potassium channels is a fundamental process of life. On the basis of crystallographic data, it was originally proposed that potassium ions and water molecules are transported through the selectivity filter in an alternating arrangement, suggesting a "water-mediated" knock-on mechanism. Later on, this view was challenged by results from molecular dynamics simulations that revealed a "direct" knock-on mechanism where ions are in direct contact. Using solid-state nuclear magnetic resonance techniques tailored to characterize the interaction between water molecules and the ion channel, we show here that the selectivity filter of a potassium channel is free of water under physiological conditions. Our results are fully consistent with the direct knock-on mechanism of ion conduction but contradict the previously proposed water-mediated knock-on mechanism

Generalised Umbral Moonshine

Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon which connects finite groups and distinguished modular objects. In this paper we introduce the notion of generalised umbral moonshine, which includes the generalised Mathieu moonshine [Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data for it. A central role is played by the deformed Drinfel'd (or quantum) double of each umbral finite group $G$, specified by a cohomology class in $H^3(G,U(1))$. We conjecture that in each of the 23 cases there exists a rule to assign an infinite-dimensional module for the deformed Drinfel'd double of the umbral finite group underlying the mock modular forms of umbral moonshine and generalised umbral moonshine. We also discuss the possible origin of the generalised umbral moonshine

Split radius-form blocks for tube benders

Two-piece, radius-form block permits accurate forming and removing of parts with more than a 180 degree bend. Tube bender can shape flexible metal tubing in applications dealing with plumbing, heating, and pressure transmission lines