538 research outputs found
Hausdorff dimension of three-period orbits in Birkhoff billiards
We prove that the Hausdorff dimension of the set of three-period orbits in
classical billiards is at most one. Moreover, if the set of three-period orbits
has Hausdorff dimension one, then it has a tangent line at almost every point.Comment: 10 pages, 1 figur
Where Fail-Safe Default Logics Fail
Reiter's original definition of default logic allows for the application of a
default that contradicts a previously applied one. We call failure this
condition. The possibility of generating failures has been in the past
considered as a semantical problem, and variants have been proposed to solve
it. We show that it is instead a computational feature that is needed to encode
some domains into default logic
Eigenfunctions for smooth expanding circle maps
We construct a real-analytic circle map for which the corresponding
Perron-Frobenius operator has a real-analytic eigenfunction with an eigenvalue
outside the essential spectral radius when acting upon -functions.Comment: 10 pages, 2 figure
Algebraic entropy for algebraic maps
We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations
Conjugating Biotin to Ruthenium(II) Arene Units via Phosphine Ligand Functionalization
Two-step functionalization of 4-diphenylphosphino benzoic acid with biotin afforded 2-(biotinyloxy)ethyl 4-(diphenylphosphanyl)benzoate (LP), that was subsequently used to synthesize the Ru(II) arene complexes [RuCl2(η6-p-cymene)(LP)] (1), [Ru(C2O4)(η6-p-cymene)(LP)] (2) and [Ru(curc)(η6-p-cymene)(LP)]NO3 ([3]NO3), the latter incorporating curcumin (curcH) as an additional bioactive fragment. [Ru(curc)(η6-p-cymene)(PPh3)]NO3 ([4]NO3) was also prepared as a reference compound. Compounds 2 and [3]NO3 exhibited excellent stability in water/DMSO solution while being slowly activated in the cell culture medium over 72 hours. Together with LP, they were therefore assessed for their antiproliferative activity towards a panel of cancer cell lines, with different levels of biotin transporter expression. The apparent affinity of the compounds towards avidin varies, and their antiproliferative activity does not correlate with biotin transporter expression, although it is systematically enhanced when biotin-free cell culture medium is used
The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics
We prove that the distributional limit of the normalised number of returns to
small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical
systems is compound Poisson. The returns to small balls around a fixed point in
the phase space correspond to the occurrence of rare events, or exceedances of
high thresholds, so that there is a connection between the laws of Return Times
Statistics and Extreme Value Laws. The fact that the fixed point in the phase
space is a repelling periodic point implies that there is a tendency for the
exceedances to appear in clusters whose average sizes is given by the Extremal
Index, which depends on the expansion of the system at the periodic point.
We recall that for generic points, the exceedances, in the limit, are
singular and occur at Poisson times. However, around periodic points, the
picture is different: the respective point processes of exceedances converge to
a compound Poisson process, so instead of single exceedances, we have entire
clusters of exceedances occurring at Poisson times with a geometric
distribution ruling its multiplicity.
The systems to which our results apply include: general piecewise expanding
maps of the interval (Rychlik maps), maps with indifferent fixed points
(Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic
Large deviations for non-uniformly expanding maps
We obtain large deviation results for non-uniformly expanding maps with
non-flat singularities or criticalities and for partially hyperbolic
non-uniformly expanding attracting sets. That is, given a continuous function
we consider its space average with respect to a physical measure and compare
this with the time averages along orbits of the map, showing that the Lebesgue
measure of the set of points whose time averages stay away from the space
average decays to zero exponentially fast with the number of iterates involved.
As easy by-products we deduce escape rates from subsets of the basins of
physical measures for these types of maps. The rates of decay are naturally
related to the metric entropy and pressure function of the system with respect
to a family of equilibrium states. The corrections added to the published
version of this text appear in bold; see last section for a list of changesComment: 36 pages, 1 figure. After many PhD students and colleagues having
pointed several errors in the statements and proofs, this is a correction to
published article answering those comments. List of main changes in a new
last sectio
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