45,303 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
Fitness-based network growth with dynamic feedback
This article is a preprint of a paper that is currently under review with Physical Review E.We study a class of network growth models in which the choice of attachment by new nodes is governed by intrinsic attractiveness, or tness, of the existing nodes. The key feature of the models is a feedback mechanism whereby the distribution from which fitnesses of new nodes are drawn is dynamically updated to account for the evolving degree distribution. It is shown that in the case of linear mapping between fitnesses and degrees, the models lead to tunable stationary powerlaw degree distribution, while in the non-linear case the distributions converge to the stretched exponential form
Random harmonic analysis program, L221 (TEV156). Volume 1: Engineering and usage
A digital computer program capable of calculating steady state solutions for linear second order differential equations due to sinusoidal forcing functions is described. The field of application of the program, the analysis of airplane response and loads due to continuous random air turbulence, is discussed. Optional capabilities including frequency dependent input matrices, feedback damping, gradual gust penetration, multiple excitation forcing functions, and a static elastic solution are described. Program usage and a description of the analysis used are presented
Experimental validation of clock synchronization algorithms
The objective of this work is to validate mathematically derived clock synchronization theories and their associated algorithms through experiment. Two theories are considered, the Interactive Convergence Clock Synchronization Algorithm and the Midpoint Algorithm. Special clock circuitry was designed and built so that several operating conditions and failure modes (including malicious failures) could be tested. Both theories are shown to predict conservative upper bounds (i.e., measured values of clock skew were always less than the theory prediction). Insight gained during experimentation led to alternative derivations of the theories. These new theories accurately predict the behavior of the clock system. It is found that a 100 percent penalty is paid to tolerate worst-case failures. It is also shown that under optimal conditions (with minimum error and no failures) the clock skew can be as much as three clock ticks. Clock skew grows to six clock ticks when failures are present. Finally, it is concluded that one cannot rely solely on test procedures or theoretical analysis to predict worst-case conditions
Dynamics of a two-level system strongly coupled to a high-frequency quantum oscillator
Recent experiments on quantum behavior in microfabricated solid-state systems
suggest tantalizing connections to quantum optics. Several of these experiments
address the prototypical problem of cavity quantum electrodynamics: a two-level
system coupled to a quantum harmonic oscillator. Such devices may allow the
exploration of parameter regimes outside the near-resonance and weak-coupling
assumptions of the ubiquitous rotating-wave approximation (RWA), necessitating
other theoretical approaches. One such approach is an adiabatic approximation
in the limit that the oscillator frequency is much larger than the
characteristic frequency of the two-level system. A derivation of the
approximation is presented and the time evolution of the two-level-system
occupation probability is calculated using both thermal- and coherent-state
initial conditions for the oscillator. Closed-form evaluation of the time
evolution in the weak-coupling limit provides insight into the differences
between the thermal- and coherent-state models. Finally, potential experimental
observations in solid-state systems, particularly the Cooper-pair
box--nanomechanical resonator system, are discussed and found to be promising.Comment: 16 pages, 11 figures; revised abstract; some text revisions; added
two figures and combined others; added references. Submitted to Phys. Rev.
Managing the Regulatory State: The Experience of the Bush Administration
This Article traces the history of Presidential management of the regulatory state up to the administration of President George W. Bush. It focuses on the latter\u27s implementation of smarter regulation, an approach to regulation based on unfunded mandates on the private sector implemented through the Office of Management and Budget, an organization within the Executive Office of the President. It finds cost-benefit analysis an essential, yet often neglected, tool for implementing efficient and effective regulations. It concludes the policies promoted under President Bush\u27s OMB have effectively cut costs by streamlining the rule-making process and discouraging adopting new federal rules, but cautions there is still a sea of overlapping regulations and conflict over turf among agencies causing the administrative state to steadily rise in cost
Environmental Impact on the Southeast Limb of the Cygnus Loop
We analyze observations from the Chandra X-ray Observatory of the southeast
knot of the Cygnus Loop supernova remnant. In this region, the blast wave
propagates through an inhomogeneous environment. Extrinsic differences and
subsequent multiple projections along the line of sight rather than intrinsic
shock variations, such as fluid instabilities, account for the apparent
complexity of the images. Interactions between the supernova blast wave and
density enhancements of a large interstellar cloud can produce the
morphological and spectral characteristics. Most of the X-ray flux arises in
such interactions, not in the diffuse interior of the supernova remnant.
Additional observations at optical and radio wavelengths support this account
of the existing interstellar medium and its role in shaping the Cygnus Loop,
and they demonstrate that the southeast knot is not a small cloud that the
blast wave has engulfed. These data are consistent with rapid equilibration of
electron and ion temperatures behind the shock front, and the current blast
wave velocity v_{bw} approx 330 km/s. Most of this area does not show strong
evidence for non-equilibrium ionization conditions, which may be a consequence
of the high densities of the bright emission regions.Comment: To appear in ApJ, April 1, 200
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