Random unfriendly seating arrangement in a dining table

A detailed study is made of the number of occupied seats in an unfriendly seating scheme with two rows of seats. An unusual identity is derived for the probability generating function, which is itself an asymptotic expansion. The identity implies particularly a local limit theorem with optimal convergence rate. Our approach relies on the resolution of Riccati equations.Comment: 23 pages with 12 figures in Advances in Applied Mathematics, 201

Processor Allocation for Optimistic Parallelization of Irregular Programs

Optimistic parallelization is a promising approach for the parallelization of irregular algorithms: potentially interfering tasks are launched dynamically, and the runtime system detects conflicts between concurrent activities, aborting and rolling back conflicting tasks. However, parallelism in irregular algorithms is very complex. In a regular algorithm like dense matrix multiplication, the amount of parallelism can usually be expressed as a function of the problem size, so it is reasonably straightforward to determine how many processors should be allocated to execute a regular algorithm of a certain size (this is called the processor allocation problem). In contrast, parallelism in irregular algorithms can be a function of input parameters, and the amount of parallelism can vary dramatically during the execution of the irregular algorithm. Therefore, the processor allocation problem for irregular algorithms is very difficult. In this paper, we describe the first systematic strategy for addressing this problem. Our approach is based on a construct called the conflict graph, which (i) provides insight into the amount of parallelism that can be extracted from an irregular algorithm, and (ii) can be used to address the processor allocation problem for irregular algorithms. We show that this problem is related to a generalization of the unfriendly seating problem and, by extending Tur\'an's theorem, we obtain a worst-case class of problems for optimistic parallelization, which we use to derive a lower bound on the exploitable parallelism. Finally, using some theoretically derived properties and some experimental facts, we design a quick and stable control strategy for solving the processor allocation problem heuristically.Comment: 12 pages, 3 figures, extended version of SPAA 2011 brief announcemen

The Washroom Game

This article analyses a game where players sequentially choose either to become insiders and pick one of finitely many locations or to remain outsiders. They will only become insiders if a minimum distance to the next player can be assured; their secondary objective is to maximise the minimal distance to other players. This is illustrated by considering the strategic behaviour of men choosing from a set of urinals in a public lavatory. However, besides very similar situations (e.g. settling of residents in a newly developed area, the selection of food patches by foraging animals, choosing seats in waiting rooms or lines in a swimming pool), the game might also relevant to the problem of placing billboards attempting to catch the attention of passers-by or similar economic situations. In the non-cooperative equilibrium, all insiders behave as if they cooperated with each other and minimised the total number of insiders. It is shown that strategic behaviour leads to an equilibrium with substantial underutilization of available locations. Increasing the number of locations tends to decrease utilization. The removal of some locations which leads to gaps can not only increase relative utilization but even absolute maximum capacity.Efficient design of facilities; location games; privacy concerns; strategic entry prevention; unfriendly seating arrangement; urinal problem

Per-site occupancy in the discrete parking problem

We consider the classical discrete parking problem, in which cars arrive uniformly at random on any two adjacent sites out of n sites on a line. An arriving car parks if there is no overlap with previously parked cars, and leaves otherwise. This process continues until there is no more space available for cars to park, at which point we may compute the jamming density En/n, which represents the expected fraction of occupied sites. We extend the classical results by not just considering the total expected number of cars parked, but also the probability of each site being occupied by a car. This we then use to provide an alternative derivation of the jamming density

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

Random maximal independent sets and the unfriendly theater seating arrangement problem

AbstractPeople arrive one at a time to a theater consisting of m rows of length n. Being unfriendly they choose seats at random so that no one is in front of them, behind them or to either side. What is the expected number of people in the theater when it becomes full, i.e., it cannot accommodate any more unfriendly people? This is equivalent to the random process of generating a maximal independent set of an m×n grid by randomly choosing a node, removing it and its neighbors, and repeating until there are no nodes remaining. The case of m=1 was posed by Freedman and Shepp [D. Freedman, L. Shepp, An unfriendly seating arrangement (problem 62-3), SIAM Rev. 4 (2) (1962) 150] and solved independently by Friedman, Rothman and MacKenzie [H.D. Friedman, D. Rothman, Solution to: An unfriendly seating arrangement (problem 62-3), SIAM Rev. 6 (2) (1964) 180–182; J.K. MacKenzie, Sequential filling of a line by intervals placed at random and its application to linear adsorption, J. Chem. Phys. 37 (4) (1962) 723–728] by proving the asymptotic limit 12−12e2. In this paper we solve the case m=2 and prove the asymptotic limit 12−14e. In addition, we consider the more general case of m×n grids, m≥1, and prove the existence of asymptotic limits in this general setting. We also make several conjectures based upon Monte Carlo simulations

The Aesthetic Philosophy of John Cage and the Visual Arts of the Twentieth Century

This thesis presents a biographical analysis of the role of American composer, writer, and artist John Cage (1912-1992) in the evolution of visual arts in the twentieth century. The origins of Cage\u27s aesthetic stance are discussed, particularly his melding of Marcel Duchamp\u27s Dada orientation with philosophical positions derived from the study of Zen Buddhism. The influence of His views on painters, sculptors, and performance artists of the postwar period is documented, along with the aesthetic foundations of his own work in the visual arts

Failures to Self-Locate: Counterfactual Ontologies in Contemporary Theatre and Physics

Failures to Self-Locate examines the overlooked influence of quantum mechanics on the development of contemporary theatre aesthetics. Physicists began openly grappling with the ramifications of quantum theory in 1926. The same year, Bertolt Brecht announced his theatre for a scientific age as an arena for atomic man. Unsatisfied with the metaphysical implications of the first formulation of quantum mechanics, known now as the Copenhagen interpretation, physicists and philosophers of science spent the twentieth century advocating, developing, and testing alternative interpretations of the atomic realm. Throughout that same period, the Western stage witnessed a resonant series of developments on Brechts aesthetic project. Placing the interpretations of quantum mechanics in dialogue with contemporary theatre from North America and Europe, this dissertation uncovers how, after an initial point of direct contact between Brecht and physicists, physics and theatre have developed similar ontological paradigms to interpret experiments and performances respectively. In physics, these paradigms fall into two distinct camps: those that salvage strict determinism at the expense of a singular world (collapse-free interpretations of quantum mechanics) and those that safeguard our worlds uniqueness by accepting fundamental stochasticity in reality (collapse interpretations of quantum mechanics). Experimental evidence supports both options, and so these groups must also explain the apparent validity of the other. Theatremakers actively investigated a similar ontological issue, exacerbated by Brechtian stage techniques and centred on the storied divide between reality and representation. Where the physicists navigated between determinism and locality, playwrights return to the ancient tension between fate and free will. Those crosscurrents may bring ruin to the classical protagonist, but the quantum protagonist experiences one framework (e.g., free will) while secretly being ruled by the other (e.g., determinism). So positioned, these protagonists fail to self-locate among their myriad possibilities. This dissertation maps the resonances between the scientific quest to reconcile determinism and stochasticity and the theatrical quest to reconcile free will and fate within the quantum theoretical paradigm, by analyzing the scientific and theatrical output through the lens of counterfactual analysis

Repurposed Spaces in Berlin and Johannesburg

Berlin and Johannesburg are repurposed cities, which were spatially designed to enforce a defunct social order, and which now must be used in new ways by new residents. Through a reading of several sites in each city, I examine how repurposed urban spaces and their inhabitants shape one another