336 research outputs found
Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
This paper gives -dimensional analogues of the Apollonian circle packings
in parts I and II. We work in the space \sM_{\dd}^n of all -dimensional
oriented Descartes configurations parametrized in a coordinate system,
ACC-coordinates, as those real matrices \bW with \bW^T
\bQ_{D,n} \bW = \bQ_{W,n} where is the -dimensional Descartes quadratic
form, , and \bQ_{D,n} and
\bQ_{W,n} are their corresponding symmetric matrices. There are natural
actions on the parameter space \sM_{\dd}^n. We introduce -dimensional
analogues of the Apollonian group, the dual Apollonian group and the
super-Apollonian group. These are finitely generated groups with the following
integrality properties: the dual Apollonian group consists of integral matrices
in all dimensions, while the other two consist of rational matrices, with
denominators having prime divisors drawn from a finite set depending on the
dimension. We show that the the Apollonian group and the dual Apollonian group
are finitely presented, and are Coxeter groups. We define an Apollonian cluster
ensemble to be any orbit under the Apollonian group, with similar notions for
the other two groups. We determine in which dimensions one can find rational
Apollonian cluster ensembles (all curvatures rational) and strongly rational
Apollonian sphere ensembles (all ACC-coordinates rational).Comment: 37 pages. The third in a series on Apollonian circle packings
beginning with math.MG/0010298. Revised and extended. Added: Apollonian
groups and Apollonian Cluster Ensembles (Section 4),and Presentation for
n-dimensional Apollonian Group (Section 5). Slight revision on March 10, 200
Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group
Apollonian circle packings arise by repeatedly filling the interstices
between four mutually tangent circles with further tangent circles. We observe
that there exist Apollonian packings which have strong integrality properties,
in which all circles in the packing have integer curvatures and rational
centers such that (curvature)(center) is an integer vector. This series
of papers explain such properties. A {\em Descartes configuration} is a set of
four mutually tangent circles with disjoint interiors. We describe the space of
all Descartes configurations using a coordinate system \sM_\DD consisting of
those real matrices \bW with \bW^T \bQ_{D} \bW = \bQ_{W} where
\bQ_D is the matrix of the Descartes quadratic form and \bQ_W of the quadratic form
. There are natural group actions on the
parameter space \sM_\DD. We observe that the Descartes configurations in each
Apollonian packing form an orbit under a certain finitely generated discrete
group, the {\em Apollonian group}. This group consists of integer
matrices, and its integrality properties lead to the integrality properties
observed in some Apollonian circle packings. We introduce two more related
finitely generated groups, the dual Apollonian group and the super-Apollonian
group, which have nice geometrically interpretations. We show these groups are
hyperbolic Coxeter groups.Comment: 42 pages, 11 figures. Extensively revised version on June 14, 2004.
Revised Appendix B and a few changes on July, 2004. Slight revision on March
10, 200
Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings
Apollonian circle packings arise by repeatedly filling the interstices
between four mutually tangent circles with further tangent circles. Such
packings can be described in terms of the Descartes configurations they
contain. It observed there exist infinitely many types of integral Apollonian
packings in which all circles had integer curvatures, with the integral
structure being related to the integral nature of the Apollonian group. Here we
consider the action of a larger discrete group, the super-Apollonian group,
also having an integral structure, whose orbits describe the Descartes
quadruples of a geometric object we call a super-packing. The circles in a
super-packing never cross each other but are nested to an arbitrary depth.
Certain Apollonian packings and super-packings are strongly integral in the
sense that the curvatures of all circles are integral and the
curvaturecenters of all circles are integral. We show that (up to
scale) there are exactly 8 different (geometric) strongly integral
super-packings, and that each contains a copy of every integral Apollonian
circle packing (also up to scale). We show that the super-Apollonian group has
finite volume in the group of all automorphisms of the parameter space of
Descartes configurations, which is isomorphic to the Lorentz group .Comment: 37 Pages, 11 figures. The second in a series on Apollonian circle
packings beginning with math.MG/0010298. Extensively revised in June, 2004.
More integral properties are discussed. More revision in July, 2004:
interchange sections 7 and 8, revised sections 1 and 2 to match, and added
matrix formulations for super-Apollonian group and its Lorentz version.
Slight revision in March 10, 200
Analysis of airplane boarding via space-time geometry and random matrix theory
We show that airplane boarding can be asymptotically modeled by 2-dimensional
Lorentzian geometry. Boarding time is given by the maximal proper time among
curves in the model. Discrepancies between the model and simulation results are
closely related to random matrix theory. We then show how such models can be
used to explain why some commonly practiced airline boarding policies are
ineffective and even detrimental.Comment: 4 page
Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation
Let T be the Pascal-adic transformation. For any measurable function g, we
consider the corrections to the ergodic theorem sum_{k=0}^{j-1} g(T^k x) - j/l
sum_{k=0}^{l-1} g(T^k x). When seen as graphs of functions defined on
{0,...,l-1}, we show for a suitable class of functions g that these quantities,
once properly renormalized, converge to (part of) the graph of a self-affine
function. The latter only depends on the ergodic component of x, and is a
deformation of the so-called Blancmange function. We also briefly describe the
links with a series of works on Conway recursive recursive sequenc
Limit Distributions of Self-Normalized Sums
If Xi are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of Sn(2) =(∑ni=1 Xi)/(∑nj=1Xj2)1/2 as n→∞ has density f(t) = (2π)−1/2 exp(−t2/2) by the central limit theorem and the law of large numbers. If the tails of Xi are sufficiently smooth and satisfy P(Xi \u3e t) ∼ rt−α and P(Xi \u3c −t) ∼ lt−α as t→∞, where 0 \u3c α \u3c 2, r \u3e 0, l \u3e 0, Sn(2) still has a limiting distribution F even though Xi has infinite variance. The density f of F depends on α as well as on r/l. We also study the limiting distribution of the more general Sn(p) = (∑ni=1Xi)/(∑nj=1 |Xj|p)1/p where Xi are i.i.d. and in the domain of a stable law G with tails as above. In the cases p = 2 (see (4.21)) and p = 1 (see (3.7)) we obtain exact, computable formulas for f(t) = f(t,α,r/l), and give graphs of f for a number of values of α and r/l. For p = 2, we find that f is always symmetric about zero on (−1,1), even though f is symmetric on (−∞,∞) only when r = l
Adaptive density estimation for stationary processes
We propose an algorithm to estimate the common density of a stationary
process . We suppose that the process is either or
-mixing. We provide a model selection procedure based on a generalization
of Mallows' and we prove oracle inequalities for the selected estimator
under a few prior assumptions on the collection of models and on the mixing
coefficients. We prove that our estimator is adaptive over a class of Besov
spaces, namely, we prove that it achieves the same rates of convergence as in
the i.i.d framework
Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group
Apollonian circle packings arise by repeatedly filling the intersticesbetween four mutually tangent circles with further tangent circles.We observe that there exist Apollonian packings which have strong integralityproperties, in which all circles in the packing have integer curvatures andrational centers such that (curvature) (center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system consisting of those real matrices with where is the matrix of the Descartes quadratic form and of the quadratic form . On the parameter space the group acts on the left, and acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group . The right action of (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space while the left action of is defined only on the parameter space. We observe thatthe Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of , which we call the Apollonian group. This group consists of integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups in , the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the super-Apollonian group which is the group generated by the Apollonian anddual Apollonian groups together. These groups also consist of integer matrices. We show these groups are hyperbolic Coxeter groups.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41356/1/454_2005_Article_1196.pd
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