721 research outputs found
Topological entropy and blocking cost for geodesics in riemannian manifolds
For a pair of points in a compact, riemannian manifold let
(resp. ) be the number of geodesic segments with length
joining these points (resp. the minimal number of point obstacles
needed to block them). We study relationships between the growth rates of
and as . We derive lower bounds on
in terms of the topological entropy and its fundamental group. This
strengthens the results of Burns-Gutkin \cite{BG06} and Lafont-Schmidt
\cite{LS}. For instance, by \cite{BG06,LS}, implies that is
unbounded; we show that grows exponentially, with the rate at least
.Comment: 13 page
Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces
We establish the background for the study of geodesics on noncompact
polygonal surfaces. For illustration, we study the recurrence of geodesics on
-periodic polygonal surfaces. We prove, in particular, that almost all
geodesics on a topologically typical -periodic surface with boundary are
recurrent.Comment: 34 pages, 13 figures. To be published in V. V. Kozlov's Festschrif
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Cocaine Addiction as a Homeostatic Reinforcement Learning Disorder
Drug addiction implicates both reward learning and homeostatic regulation mechanisms of the brain. This has stimulated 2 partially successful theoretical perspectives on addiction. Many important aspects of addiction, however, remain to be explained within a single, unified framework that integrates the 2 mechanisms. Building upon a recently developed homeostatic reinforcement learning theory, the authors focus on a key transition stage of addiction that is well modeled in animals, escalation of drug use, and propose a computational theory of cocaine addiction where cocaine reinforces behavior due to its rapid homeostatic corrective effect, whereas its chronic use induces slow and long-lasting changes in homeostatic setpoint. Simulations show that our new theory accounts for key behavioral and neurobiological features of addiction, most notably, escalation of cocaine use, drug-primed craving and relapse, individual differences underlying dose-response curves, and dopamine D2-receptor downregulation in addicts. The theory also generates unique predictions about cocaine self-administration behavior in rats that are confirmed by new experimental results. Viewing addiction as a homeostatic reinforcement learning disorder coherently explains many behavioral and neurobiological aspects of the transition to cocaine addiction, and suggests a new perspective toward understanding addiction
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
We study the recurrence and ergodicity for the billiard on noncompact
polygonal surfaces with a free, cocompact action of or . In the
-periodic case, we establish criteria for recurrence. In the more difficult
-periodic case, we establish some general results. For a particular
family of -periodic polygonal surfaces, known in the physics literature
as the wind-tree model, assuming certain restrictions of geometric nature, we
obtain the ergodic decomposition of directional billiard dynamics for a dense,
countable set of directions. This is a consequence of our results on the
ergodicity of \ZZ-valued cocycles over irrational rotations.Comment: 48 pages, 12 figure
Insecurity for compact surfaces of positive genus
A pair of points in a riemannian manifold is secure if the geodesics
between the points can be blocked by a finite number of point obstacles;
otherwise the pair of points is insecure. A manifold is secure if all pairs of
points in are secure. A manifold is insecure if there exists an insecure
point pair, and totally insecure if all point pairs are insecure.
Compact, flat manifolds are secure. A standing conjecture says that these are
the only secure, compact riemannian manifolds. We prove this for surfaces of
genus greater than zero. We also prove that a closed surface of genus greater
than one with any riemannian metric and a closed surface of genus one with
generic metric are totally insecure.Comment: 37 pages, 11 figure
Ergodic directions for billiards in a strip with periodically located obstacles
We study the size of the set of ergodic directions for the directional
billiard flows on the infinite band with periodically placed
linear barriers of length . We prove that the set of ergodic
directions is always uncountable. Moreover, if is rational
the Hausdorff dimension of the set of ergodic directions is greater than 1/2.
In both cases (rational and irrational) we construct explicitly some sets of
ergodic directions.Comment: The article is complementary to arXiv:1109.458
Adsorption Way of the Loss of Moon's Atmosphere
Theory on gas adsorption by lunar surface to explain loss of lunar atmospher
Affine diffeomorphisms of translation surfaces : periodic points, fuchsian groups, and arithmeticity
Addendum to: Capillary floating and the billiard ball problem
We compare the results of our earlier paper on the floating in neutral
equilibrium at arbitrary orientation in the sense of Finn-Young with the
literature on its counterpart in the sense of Archimedes. We add a few remarks
of personal and social-historical character.Comment: This is an addendum to my article Capillary floating and the billiard
ball problem, Journal of Mathematical Fluid Mechanics 14 (2012), 363 -- 38
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