Let T be a fixed triangle and consider an odd number of translated copies of T in the plane. We show that the set of points in the plane that belong to an odd number of triangles has an area of at least half of the area of T . This result is best possible. We resolve also the more general case of a trapezoid and discuss related problems.

We investigate the intrinsic motivation of individuals to report, and thereby sanction, fellow group members who lie for personal gain. We further explore the changes in lying and reporting behavior that result from giving individuals a say in who joins their group. We find that enough individuals are willing to report lies such that in fixed groups lying is unprofitable. However, we also find ...

Supersymmetric vacua ('universes') of string/M theory may be identified with certain critical points of a holomorphic section (the 'superpotential') of a Hermitian holomor-phic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points ∇s = 0 ...

This paper studies the impact of ad-avoidance behavior in media markets. We consider a situation where viewers can avoid advertisement messages. As the media market is a two-sided market, increased ad-avoidance reduces advertisers’ value of placing an ad. We contrast two financing regimes, free-to-air and pay-TV. We find that a higher viewer responsiveness to advertising decreases revenues and ...

Supersymmetric vacua (‘universes’) of string/M theory may be identified with certain critical points of a holomorphic section (the ‘superpotential’) of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points ∇s = 0 o...

In this paper, we investigate the statistical properties of the waiting times between two successive price changes above a fixed threshold (or a fixed point) for the indices of Shanghai Stock Exchange and Shenzhen Stock Exchange. The database is from the indices of Shanghai and Shenzhen in the 10-year period from January 1997 to December 2006, and the empirical research shows that the distribut...

This paper presents a general model to study the medium access control (MAC) layer performance and equilibrium of WLANs consisting of nodes with different MAC parameters (e.g., backoff window size). Our model can be used in general 802.11-based WLANs since it captures the important factors such as non-uniform backoff behavior, channel errors and unsaturated traffic. We first formalize the perfo...

We consider K-semialgebras for a commutative semiring K that are at the same time Σ-algebras and satisfy certain linearity conditions. When each finite system of guarded polynomial fixed point equations has a unique solution over such an algebra, then we call it an iterative multi-linear K-Σ-semialgebra. Examples of such algebras include the algebras of Σ-tree series over an alphabet A with coe...