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26. CJM 2004 (vol 56 pp. 553)

 Cohomology Ring of Symplectic Quotients by Circle Actions In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be formally Hamiltonian''. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result. Categories:53D20, 53D30, 37J10, 37J15, 53D05

27. CJM 2004 (vol 56 pp. 449)

Demeter, Ciprian
 The Best Constants Associated with Some Weak Maximal Inequalities in Ergodic Theory We introduce a new device of measuring the degree of the failure of convergence in the ergodic theorem along subsequences of integers. Relations with other types of bad behavior in ergodic theory and applications to weighted averages are also discussed. Category:37A05

28. CJM 2004 (vol 56 pp. 115)

Kenny, Robert
 Estimates of Hausdorff Dimension for the Non-Wandering Set of an Open Planar Billiard The billiard flow in the plane has a simple geometric definition; the movement along straight lines of points except where elastic reflections are made with the boundary of the billiard domain. We consider a class of open billiards, where the billiard domain is unbounded, and the boundary is that of a finite number of strictly convex obstacles. We estimate the Hausdorff dimension of the nonwandering set $M_0$ of the discrete time billiard ball map, which is known to be a Cantor set and the largest invariant set. Under certain conditions on the obstacles, we use a well-known coding of $M_0$ \cite{Morita} and estimates using convex fronts related to the derivative of the billiard ball map \cite{StAsy} to estimate the Hausdorff dimension of local unstable sets. Consideration of the local product structure then yields the desired estimates, which provide asymptotic bounds on the Hausdorff dimension's convergence to zero as the obstacles are separated. Categories:37D50, 37C45;, 28A78

29. CJM 2003 (vol 55 pp. 247)

Cushman, Richard; Śniatycki, Jędrzej
 Differential Structure of Orbit Spaces: Erratum This note signals an error in the above paper by giving a counter-example. Categories:37J15, 58A40, 58D19, 70H33

30. CJM 2003 (vol 55 pp. 3)

Baake, Michael; Baake, Ellen
 An Exactly Solved Model for Mutation, Recombination and Selection It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from. Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of M\"obius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions. Keywords:population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, MÃ¶bius inversionCategories:92D10, 34L30, 37N30, 06A07, 60J25

31. CJM 2002 (vol 54 pp. 897)

Fortuny Ayuso, Pedro
 The Valuative Theory of Foliations This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'H\^opital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results. Categories:12J20, 13F30, 16W60, 37F75, 34M25

32. 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

33. 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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