Realizations of the Associahedron and Cyclohedron

We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott-Taubes polytope) and compare them with the permutahedron of type A and B, respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type A_n or B_n as the only input data and which specializes to a procedure presented by J.-L. Loday for a certain orientation of A_n. The described realizations have cambrian fans of type A and B as normal fans. This settles a conjecture of N. Reading for cambrian lattices of these types.     

Dynamics of a single peak of the Rosensweig instability in a magnetic fluid

To describe the dynamics of a single peak of the Rosensweig instability a model is proposed which approximates the peak by a half-ellipsoid atop a layer of magnetic fluid. The resulting nonlinear equation for the height of the peak leads to the correct subcritical character of the bifurcation for static induction. For a time-dependent induction the effects of inertia and damping are incorporated. The results of the model show qualitative agreement with the experimental findings, as in the appearance of period doubling, trebling, and higher multiples of the driving period. Furthermore, a quantitative agreement is also found for the parameter ranges of frequency and induction in which these phenomena occur.     

Modeling Maher's Attribution Theory of Delusions as a Cusp Catastrophe

Maher's (1974, 1988) attribution theory of delusions was reformulated as a cusp catastrophe model and tested using 193 percipients' paranormal beliefs, paranormal experiences, and fear of the paranormal as measured by Kumar et al.'s (1994) Anomalous Experiences Inventory, and their tolerance of ambiguity as measured by McDonald's (1970) AT-20. The magnitude of the difference between percipients' fear and belief was used as the dependent variables, whereas experience and tolerance of ambiguity were used as asymmetry and bifurcation variables, respectively. Using the GEMCAT II software it was shown that this cusp provided a superior fit to the data than competing linear, fold, and cusp models. Consistent with Maher's formulation, two basic states were found to exist: one in which fear dominates belief, and a second in which belief dominates fear. Further, fear of the paranormal increased with greater intolerance of ambiguity while belief decreased with increasing tolerance. The model explains why paranormal beliefs are resistant to change in the face of counterargument or contradictory evidence.     

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