14 research outputs found
On Hausdorff dimension of the set of closed orbits for a cylindrical transformation
We deal with Besicovitch's problem of existence of discrete orbits for
transitive cylindrical transformations
where is an
irrational rotation on the circle \T and \varphi:\T\to\R is continuous,
i.e.\ we try to estimate how big can be the set
D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}.
We show that for almost every there exists such that the
Hausdorff dimension of is at least . We also provide a
Diophantine condition on that guarantees the existence of
such that the dimension of is positive. Finally, for some
multidimensional rotations on \T^d, , we construct smooth
so that the Hausdorff dimension of is positive.Comment: 32 pages, 1 figur
Which States Matter? An Application of an Intelligent Discretization Method to Solve a Continuous POMDP in Conservation Biology
When managing populations of threatened species, conservation managers seek to make the best conservation decisions to avoid extinction. Making the best decision is difficult because the true population size and the effects of management are uncertain. Managers must allocate limited resources between actively protecting the species and monitoring. Resources spent on monitoring reduce expenditure on management that could be used to directly improve species persistence. However monitoring may prevent sub-optimal management actions being taken as a result of observation error. Partially observable Markov decision processes (POMDPs) can optimize management for populations with partial detectability, but the solution methods can only be applied when there are few discrete states. We use the Continuous U-Tree (CU-Tree) algorithm to discretely represent a continuous state space by using only the states that are necessary to maintain an optimal management policy. We exploit the compact discretization created by CU-Tree to solve a POMDP on the original continuous state space. We apply our method to a population of sea otters and explore the trade-off between allocating resources to management and monitoring. We show that accurately discovering the population size is less important than management for the long term survival of our otter population