Diffusion constants and martingales for senile random walks

We derive diffusion constants and martingales for senile random walks with the help of a time-change. We provide direct computations of the diffusion constants for the time-changed walks. Alternatively, the values of these constants can be derived from martingales associated with the time-changed walks. Using an inverse time-change, the diffusion constants for senile random walks are then obtained via these martingales. When the walks are diffusive, weak convergence to Brownian motion can be shown using a martingale functional limit theorem.Comment: 17 pages, LaTeX; the proof of Proposition 2.3 has been simplified, and an error in the proof of Theorem 2.4 has been correcte

Design and implementation of a method forthe synthesis of travel diary data

Transport has been called the maker and breaker of cities. A maker of cities because of its vital role in the bringing together of people, goods and services. A breaker of cities because of its e®ects on the quality of the living environment. In balancing both aspects, a detailed cost-bene¯t analysis of transport policy is imperative. However, compared to the analysis of (external) costs, the analysis of (external) bene¯ts is hampered by the low availability of travel diary data

Rotor-router aggregation on the layered square lattice

In rotor-router aggregation on the square lattice Z^2, particles starting at the origin perform deterministic analogues of random walks until reaching an unoccupied site. The limiting shape of the cluster of occupied sites is a disk. We consider a small change to the routing mechanism for sites on the x- and y-axes, resulting in a limiting shape which is a diamond instead of a disk. We show that for a certain choice of initial rotors, the occupied cluster grows as a perfect diamond.Comment: 11 pages, 3 figures

A Guide to Stochastic Loewner Evolution and its Applications

This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Loewner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Loewner's method, explaining how this method can be used to describe families of random curves. Then we define SLE and discuss some of its properties. We also explain how the connection can be made between SLE and the discrete models whose scaling limits it describes, or is believed to describe. Finally, we have included a discussion of results that were obtained from SLE computations. Some explicit proofs are presented as typical examples of such computations. To understand SLE sufficient knowledge of conformal mapping theory and stochastic calculus is required. This material is covered in the appendices.Comment: 80 pages, 22 figures, LaTeX; this version has 5 minor corrections to the text and improved hyperref suppor

Existence and uniqueness of the stationary measure in the continuous Abelian sandpile

Let \Lambda be a finite subset of Z^d. We study the following sandpile model on \Lambda. The height at any given vertex x of \Lambda is a positive real number, and additions are uniformly distributed on some interval [a,b], which is a subset of [0,1]. The threshold value is 1; when the height at a given vertex exceeds 1, it topples, that is, its height is reduced by 1, and the heights of all its neighbours in \Lambda increase by 1/2d. We first establish that the uniform measure \mu on the so called "allowed configurations" is invariant under the dynamics. When a < b, we show with coupling ideas that starting from any initial configuration of heights, the process converges in distribution to \mu, which therefore is the unique invariant measure for the process. When a = b, that is, when the addition amount is non-random, and a is rational, it is still the case that \mu is the unique invariant probability measure, but in this case we use random ergodic theory to prove this; this proof proceeds in a very different way. Indeed, the coupling approach cannot work in this case since we also show the somewhat surprising fact that when a = b is rational, the process does not converge in distribution at all starting from any initial configuration.Comment: 22 page

Reconstruction of tax balance sheets based on IFRS information: A case study of listed companies within Austria, Germany, and the Netherlands

The internationalisation of financial accounting and the European Commission's ambition to harmonise corporate taxation have raised the question whether IFRS accounts could be used for tax purposes. In order to quantify the effect of an IFRS-based taxation on corporate tax burdens in different EU member states, we estimate firms' tax equity using notes on income taxes in IFRS financial statements of companies listed in Austria, Germany, and the Netherlands. The difference between estimated tax equity and IFRS-equity, adjusted for the effect resulting from the recognition of deferred taxes, indicates the effect of using IFRS as a tax base on corporate tax burden. We find that estimated tax equity is mostly lower than IFRSequity, indicating that an IFRS-based taxation would often increase the corporate tax burden. The median of estimated tax equity is 5.6% (Austria), 6.4% (Germany) and 9.0% (the Netherlands) below IFRS-equity. Our results suggest that using IFRS for the determination of taxable income would often increase corporate tax burden. However, an IFRS-based taxation does not always induce higher equity as often argued in the literature. In 307 of 1.113 totally analysed firm-years, estimated tax equity exceeds IFRS-equity. Analysing IFRS-tax differences on a balance sheet caption level, we find that the most important differences can be observed for intangibles and provisions. We find for all three analysed countries that IFRS-tax differences relating to inventories, receivables, and liabilities are typically small. We also approximate the total stock of unused tax losses and the amount of useable tax losses which can provide additional information about the management's estimates of future earnings. We find that deferred tax assets for unused tax losses are depreciated to a substantial extent, indicating that companies often assume insufficient future taxable income to utilise the total stock of tax loss carry-forwards. --

Statistical Modelling of Pre-Impact Velocities in Car Crashes

The law wants to determine if any party involved in a car crash is guilty. The Dutch court invokes the expertise of the Netherlands Forensic Institute (NFI) to answer this question. We discuss the present method of the NFI to deter- mine probabilities on pre-impact car velocities, given the evidence from the crash scene. A disadvantage of this method is that it requires a prior distribution on the velocities of the cars involved in the crash. We suggest a different approach, that of statistical significance testing, which can be carried out without a prior. We explain this method, and apply it to a toy model. Finally, a sensitivity analysis is performed on a simple two-car collision model

Exact Solutions for Loewner Evolutions

In this note, we solve the Loewner equation in the upper half-plane with forcing function xi(t), for the cases in which xi(t) has a power-law dependence on time with powers 0, 1/2 and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If xi(t)=2*(kappa*t)^1/2, the trace is a straight line set at an angle to the real axis. If xi(t)=2*(kappa*(1-t))^1/2, the behavior of the trace as t approaches 1 depends on the coefficient kappa. Our calculations give an explicit solution in which for kappa<4 the trace spirals into a point in the upper half-plane, while for kappa>4 it intersects the real axis. We also show that for kappa=9/2 the trace becomes a half-circle. The third case with forcing xi(t)=t gives a trace that moves outward to infinity, but stays within fixed distance from the real axis. We also solve explicitly a more general version of the evolution equation, in which xi(t) is a superposition of the values +1 and -1.Comment: 20 pages, 7 figures, LaTeX, one minor correction, and improved hyperref

The asymptotics of group Russian roulette

We study the group Russian roulette problem, also known as the shooting problem, defined as follows. We have $n$ armed people in a room. At each chime of a clock, everyone shoots a random other person. The persons shot fall dead and the survivors shoot again at the next chime. Eventually, either everyone is dead or there is a single survivor. We prove that the probability $p_n$ of having no survivors does not converge as $n\to\infty$, and becomes asymptotically periodic and continuous on the $\log n$ scale, with period 1.Comment: 26 pages, 1 figure; Mathematica notebook and output file (calculated exact bounds) are included with the source file