863 research outputs found
Continuous images of Cantor's ternary set
The Hausdorff-Alexandroff Theorem states that any compact metric space is the
continuous image of Cantor's ternary set . It is well known that there are
compact Hausdorff spaces of cardinality equal to that of that are not
continuous images of Cantor's ternary set. On the other hand, every compact
countably infinite Hausdorff space is a continuous image of . Here we
present a compact countably infinite non-Hausdorff space which is not the
continuous image of Cantor's ternary set
Diffraction of return time measures
Letting denote an ergodic transformation of the unit interval and letting
denote an observable, we construct the
-weighted return time measure for a reference point as
the weighted Dirac comb with support in and weights at , and if is non-invertible, then we set the
weights equal to zero for all . Given such a Dirac comb, we are
interested in its diffraction spectrum which emerges from the Fourier transform
of its autocorrelation and analyse it for the dependence on the underlying
transformation. For certain rapidly mixing transformations and observables of
bounded variation, we show that the diffraction of consists of a
trivial atom and an absolutely continuous part, almost surely with respect to
. This contrasts what occurs in the setting of regular model sets arising
from cut and project schemes and deterministic incommensurate structures. As a
prominent example of non-mixing transformations, we consider the family of
rigid rotations with rotation
number . In contrast to when is mixing, we observe
that the diffraction of is pure point, almost surely with respect to
. Moreover, if is irrational and the observable is Riemann
integrable, then the diffraction of is independent of . Finally,
for a converging sequence of rotation
numbers, we provide new results concerning the limiting behaviour of the
associated diffractions.Comment: 11 pages, 2 figure
Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-genus Graphs
We give an -size extended formulation
for the spanning tree polytope of an -vertex graph embedded on a surface of
genus , improving on the known -size extended formulations
following from Wong and Martin.Comment: v3: fixed some typo
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Lyceum: internet voice groupware for distance learning
This paper describes the design, implementation and deployment of Lyceum, a groupware system providing students and tutors with real time voice conferencing and visual workspace tools, over the standard internet. Lyceum uses a Java client/server architecture to tackle a formidable set of networking requirements: multi-way voice communication with synchronous shared displays, scalable to hundreds of simultaneous users, running over normal modem connections via unknown internet service providers, on home PCs. Additionally, the design had to support multiple courses with different requirements. We describe the interdisciplinary requirements analysis, and iterative design process, by which an academic course team was able to specify and evaluate prototypes. We present the systemís architecture, describe the technical successes and failures from Lyceumís first large scale deployment, and summarise its affordances for interaction and learning
On the asymptotics of the -Farey transfer operator
We study the asymptotics of iterates of the transfer operator for
non-uniformly hyperbolic -Farey maps. We provide a family of
observables which are Riemann integrable, locally constant and of bounded
variation, and for which the iterates of the transfer operator, when applied to
one of these observables, is not asymptotic to a constant times the wandering
rate on the first element of the partition . Subsequently, sufficient
conditions on observables are given under which this expected asymptotic holds.
In particular, we obtain an extension theorem which establishes that, if the
asymptotic behaviour of iterates of the transfer operator is known on the first
element of the partition , then the same asymptotic holds on any
compact set bounded away from the indifferent fixed point
Using the online cross-entropy method to learn relational policies for playing different games
By defining a video-game environment as a collection of objects, relations, actions and rewards, the relational reinforcement learning algorithm presented in this paper generates and optimises a set of concise, human-readable relational rules for achieving maximal reward. Rule learning is achieved using a combination of incremental specialisation of rules and a modified online cross-entropy method, which dynamically adjusts the rate of learning as the agent progresses. The algorithm is tested on the Ms. Pac-Man and Mario environments, with results indicating the agent learns an effective policy for acting within each environment
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