Generalised Mertens and Brauer-Siegel Theorems

In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to make it tend to infinity and thus to point out the link between it and the famous Brauer-Siegel theorem. Using this we deduce an explicit version of the generalised Brauer-Siegel theorem under GRH, and a unified proof of this theorem for asymptotically exact families of almost normal number fields

LL(1) Parsing with Derivatives and Zippers

In this paper, we present an efficient, functional, and formally verified parsing algorithm for LL(1) context-free expressions based on the concept of derivatives of formal languages. Parsing with derivatives is an elegant parsing technique, which, in the general case, suffers from cubic worst-case time complexity and slow performance in practice. We specialise the parsing with derivatives algorithm to LL(1) context-free expressions, where alternatives can be chosen given a single token of lookahead. We formalise the notion of LL(1) expressions and show how to efficiently check the LL(1) property. Next, we present a novel linear-time parsing with derivatives algorithm for LL(1) expressions operating on a zipper-inspired data structure. We prove the algorithm correct in Coq and present an implementation as a parser combinators framework in Scala, with enumeration and pretty printing capabilities.Comment: Appeared at PLDI'20 under the title "Zippy LL(1) Parsing with Derivatives

From a property tax to a land tax – who wins, who loses?

Whilst the theoretical case in favour of a tax on the value of land (a land tax) is well established, examples of its implementation in practice are relatively few in number. Where a land tax is levied, it is often part of a suite of land and property taxes that includes transfer taxes, wealth taxes betterment and recurrent taxes on land and property. Rarely is a land tax the sole mechanism for taxing real estate. Yet there is no shortage of land tax supporters, even in countries where other forms of real estate tax have a long history. England is one such country, where real estate taxes have existed since the 17th century in one form or another. Despite coming close at the beginning of the 19th century, governments on the left, right and in the centre ground of political discourse have not implemented a land tax. In the land tax debate throughout this period, there was an absence of empirical research to underpin the positions adopted by either proponents of a land tax or defenders of the status quo. It was not until 1964 that a small pilot exercise was undertaken to investigate the implications of introducing a land tax in England. This seems odd given that frequently cited criticisms of a land tax centre on its practical difficulties. This paper, therefore, looks at some of the consequences of switching from recurrent real estate taxes that are based on the value of land and improvements to one that is based on the value of land only. Focusing on one local authority area in the south east of England, the paper answers the following questions: how might the valuation of land be undertaken in a developed economy where most transactional evidence relates to land together with improvements, and what are the revenue implications of switching from a tax where the liability falls on the land owner rather than the property occupier in first instance. In particular, who are the winners and losers and does expansion of the tax base to include agricultural land uses significantly increase tax revenue