Reflected Brownian motion in generic triangles and wedges

Consider a generic triangle in the upper half of the complex plane with one side on the real line. This paper presents a tailored construction of a discrete random walk whose continuum limit is a Brownian motion in the triangle, reflected instantaneously on the left and right sides with constant reflection angles. Starting from the top of the triangle, it is evident from the construction that the reflected Brownian motion lands with the uniform distribution on the base. Combined with conformal invariance and the locality property, this uniform exit distribution allows us to compute distribution functions characterizing the hull generated by the reflected Brownian motion.Comment: LaTeX, 38 pages, 14 figures. This is the outcome of a complete rewrite of the original paper. Results have been stated more clearly and the proofs have been elucidate

Godot was Always There: Repetition and the Formation of Customary International Law

Rules of customary law figure prominently in today’s law and policy. Across policy fields, courts and policy-makers are called to interpret and apply customary law. However, it is still a bit of a mystery how rules of customary law emerge and how they can be identified in the first place. In this paper, I set out why the mystery of customary law is bound to remain unresolved. Customary law cannot be treated as a body of rules ‘out there’, ready for application by domestic, regional or global authorities. Instead, it is part of a process of global cooperation where rules of customary law emerge and grow because they are restated. Rules of customary law only exist if they are successfully presented as already there

The Phenomenon of Yearbooks in International Law:An Introduction

In 1970 the first Netherlands Yearbook of International Law (NYIL) was published. The current Volume is No. 50, which means that the Yearbook has now been with us for half a century. The current General Editors decided not to let this moment pass unnoticed, and have devoted this entire Volume to an analysis of the phenomenon of Yearbooks in international law as such. Indeed, not many academic disciplines have Yearbooks, so why do we? What is the added value of having a Yearbook alongside the abundance of international law journals, regular monographs and edited volumes that are produced each year? Does the existence of Yearbooks tell us something about who we are, or who we think we are, or what we have to contribute to the world

Continuity and Change in Legal Positivism

Institutional theory of law (ITL) reflects both continuity and change of Kelsen's legal positivism. The main alteration results from the way ITL extends Hart's linguistic turn towards ordinary language philosophy (OLP). Hart holds – like Kelsen – that law cannot be reduced to brute fact nor morality, but because of its attempt to reconstruct social practices his theory is more inclusive. By introducing the notion of law as an extra-linguistic institution ITL takes a next step in legal positivism and accounts for the relationship between action and validity within the legal system. There are, however, some problems yet unresolved by ITL. One of them is its theory of meaning. An other is the way it accounts for change and development. Answers may be based on the pragmatic philosophy of Charles Sanders Peirce, who emphasises the intrinsic relation between the meaning of speech acts and the process of habit formation

Constraining modelled global vegetation dynamics and carbon turnover using multiple satellite observations

The response of land ecosystems to future climate change is among the largest unknowns in the global climate-carbon cycle feedback. This uncertainty originates from how dynamic global vegetation models (DGVMs) simulate climate impacts on changes in vegetation distribution, productivity, biomass allocation, and carbon turnover. The present-day availability of a multitude of satellite observations can potentially help to constrain DGVM simulations within model-data integration frameworks. Here, we use satellite-derived datasets of the fraction of absorbed photosynthetic active radiation (FAPAR), sun-induced fluorescence (SIF), above-ground biomass of trees (AGB), land cover, and burned area to constrain parameters for phenology, productivity, and vegetation dynamics in the LPJmL4 DGVM. Both the prior and the optimized model accurately reproduce present-day estimates of the land carbon cycle and of temporal dynamics in FAPAR, SIF and gross primary production. However, the optimized model reproduces better the observed spatial patterns of biomass, tree cover, and regional forest carbon turnover. Using a machine learning approach, we found that remaining errors in simulated forest carbon turnover can be explained with bioclimatic variables. This demonstrates the need to improve model formulations for climate effects on vegetation turnover and mortality despite the apparent successful constraint of simulated vegetation dynamics with multiple satellite observations

Monte Carlo study of the hull distribution for the q=1 Brauer model

We study a special case of the Brauer model in which every path of the model has weight q=1. The model has been studied before as a solvable lattice model and can be viewed as a Lorentz lattice gas. The paths of the model are also called self-avoiding trails. We consider the model in a triangle with boundary conditions such that one of the trails must cross the triangle from a corner to the opposite side. Motivated by similarities between this model, SLE(6) and critical percolation, we investigate the distribution of the hull generated by this trail (the set of points on or surrounded by the trail) up to the hitting time of the side of the triangle opposite the starting point. Our Monte Carlo results are consistent with the hypothesis that for system size tending to infinity, the hull distribution is the same as that of a Brownian motion with perpendicular reflection on the boundary.Comment: 21 pages, 9 figure