Euler Characteristic in Odd Dimensions

It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes

Clarifying spatial distance measurement

We examine length measurement in curved spacetime, based on the 1+3-splitting of a local observer frame. This situates extended objects within spacetime, in terms of a given coordinate which serves as an external reference. The radar metric is shown to coincide with the spatial projector, but these only give meaningful results on the observer's 3-space, where they reduce to the metric. Examples from Schwarzschild spacetime are given.Comment: 6 pages, 0 figures, submitted to the proceedings of the 2018 Marcel Grossmann conference, Rome. v2 has minor rewording and typo correction

Cosmic cable

I investigate the relativistic mechanics of an extended "cable" in an arbitrary static, spherically symmetric spacetime. Such hypothetical bodies have been proposed as tests of energy and thermodynamics: by lowering objects toward a black hole, scooping up Hawking radiation, or mining energy from the expansion of the universe. I review existing work on stationary cables, which demonstrates an interesting "redshift" of tension, and extend to a case of rigid motion. By using a partly restrained cable to turn a turbine, the energy harvested is up to the equivalent of the cable's rest mass, concurring with the quasistatic case. Still, the total Killing energy of the system is conserved.Comment: 6 pages, 0 figures, submitted to the proceedings of the 2018 Marcel Grossmann conference in Rom

The impact of artificial intelligence on jobs and work in New Zealand

Artificial Intelligence (AI) is a diverse technology. It is already having significant effects on many jobs and sectors of the economy and over the next ten to twenty years it will drive profound changes in the way New Zealanders live and work. Within the workplace AI will have three dominant effects. This report (funded by the New Zealand Law Foundation) addresses: Chapter 1 Defining the Technology of Interest; Chapter 2 The changing nature and value of work; Chapter 3 AI and the employment relationship; Chapter 4 Consumers, professions and society. The report includes recommendations to the New Zealand Government

On the alleged simplicity of impure proof

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim