Expand all Collapse all | Results 26 - 33 of 33 |
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)
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)
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)
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)
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 inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 |
31. CJM 2002 (vol 54 pp. 897)
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)
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 manifolds Categories:37J15, 58A40, 58D19, 70H33 |
33. CJM 2001 (vol 53 pp. 382)
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 |