Lower Bounds for L1L_1 Discrepancy

We find the best asymptotic lower bounds for the coefficient of the leading term of the $L_1$ norm of the two-dimensional (axis-parallel) discrepancy that can be obtained by K.Roth's orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, complex and harmonic analysis.Comment: a slightly different version of the article is accepted to "Mathematika

Predicting Collaboration: Risk, Power, and Dependence in the Gulf of Maine

Collaboration among natural resource organizations and users is touted by researchers as an effective approach to managing common pool resources. To understand how collaboration works, previous studies in organizational theory have identified three variables: power, dependence, and risk. Relationships between actors can be represented by these qualifications of resources or threats and may predict if those relationships are in conflict or asymmetric in power. In this study, the Gulf of Maine transboundary fishery management network relied upon a dyadic influenced survey to quantitatively capture the perception of communication ties between organizations. Four kinds of dependence and three types of risk were captured by respondent responses to be used in predictive and descriptive analysis. The patterns presented a network with low risk and high levels of dependence. Dependence and risk were able to significantly predict whether a relationship was in conflict or whether a relationship had feelings of power, with legitimacy and performance as higher rated indicators. The results suggest that policy makers and network designers should foster legitimacy and shun performance failures when evaluating the relationships among management networks

On the Validity of the 0-1 Test for Chaos

In this paper, we present a theoretical justification of the 0-1 test for chaos. In particular, we show that with probability one, the test yields 0 for periodic and quasiperiodic dynamics, and 1 for sufficiently chaotic dynamics

Herman's Theory Revisited

We prove that a $C^{2+\alpha}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\delta<\alpha\le1$, is $C^{1+\alpha-\delta}$-smoothly conjugate to a rigid rotation. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.Comment: 10 page