Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C¹⁺ conjugacy classes of C^1+H Cantor exchange systems that are C^1+H fixed points of renormalization and (ii) C¹⁺ conjugacy classes of C^1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C^1+α Cantor exchange system, with bounded geometry, that is a C^1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set.     

CONFLICT AND COOPERATION IN UTILIZING A COMMON PROPERTY RESOURCE

Noncooperative games are used to demonstrate that, while free riding is always an option, the “tragedy of the commons” is not inevitable. When the decision to cooperate or free ride is considered in a dynamic setting, there is no intermediate case where some cooperate and others free ride. The game is only stabilized in either full cooperation or full defection. The important factor in obtaining a cooperative outcome is the critical number of players that decide to cooperate. The concept of commitment is used to demonstrate the necessary and sufficient conditions for full cooperation. Although the analysis is based on a shared water resource, it can be extended to other natural resources with common costs and private benefits, especially in the cases where there are no international authorities or treaties that internalize the externalities involved in privatizing the resource.     

Congruences,equational theories and lattice representations

We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.To László Fuchs on the occasion of his 70th birthday     

Differential graded Lie algebras and singularities of level sets of momentum mappings

The germ of an analytic varietyX at a pointxX is said to bequadratic if it is bi-analytically isomorphic to the germ of a cone defined by a system of homogeneous quadratic equations at the origin. Arms, Marsden and Moncrief show in [2] that under certain conditions the analytic germ of a level set of a momentum mapping is quadratic. We discuss related ideas in a more algebraic context by associating to an affine Hamiltonian action a differential graded Lie algebra, which in the presence of an invariant positive complex structure, is formal in the sence of [5].Dedicated to the memory of Bruce Reinhart     

Velocity inversion: A case study in infinite-dimensional optimization

The goal of seismic velocity inversion is the estimation of seismic wave velocities inside the earth by attempting to predict, in a least-error sense, seismic waveforms measured at its surface. We present velocity inversion as a case study in the various infinite-dimensional pathologies which may afflict practically important problems of distributed parameter identification, treated as optimization problems in function spaces. These features differentiate various problem formulations far beyond the degree one would expect for finite- (small-) dimensional problems. We illustrate this differentiation by comparing the characteristics of three different least-squares formulations of velocity inversion.