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26. CJM 2001 (vol 53 pp. 715)

Cushman, Richard; Śniatycki, Jędrzej
 Differential Structure of Orbit Spaces We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds. Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed. Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifoldsCategories:37J15, 58A40, 58D19, 70H33

27. CJM 2001 (vol 53 pp. 382)

Pivato, Marcus
 Building a Stationary Stochastic Process From a Finite-Dimensional Marginal If $\mathfrak{A}$ is a finite alphabet, $\sU \subset\mathbb{Z}^D$, and $\mu_\sU$ is a probability measure on $\mathfrak{A}^\sU$ that looks like'' the marginal projection of a stationary stochastic process on $\mathfrak{A}^{\mathbb{Z}^D}$, then can we extend'' $\mu_\sU$ to such a process? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when $D=1$, we provide some sufficient conditions and some necessary conditions for $\mu_\sU$ to be extendible for $D>1$, and show that, in general, the problem is not formally decidable. Categories:37A50, 60G10, 37B10
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