1,253 research outputs found
A Greedy Partition Lemma for Directed Domination
A directed dominating set in a directed graph is a set of vertices of
such that every vertex has an adjacent vertex
in with directed to . The directed domination number of , denoted
by , is the minimum cardinality of a directed dominating set in .
The directed domination number of a graph , denoted , which is
the maximum directed domination number over all orientations of
. The directed domination number of a complete graph was first studied by
Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this
paper we prove a Greedy Partition Lemma for directed domination in oriented
graphs. Applying this lemma, we obtain bounds on the directed domination
number. In particular, if denotes the independence number of a graph
, we show that .Comment: 12 page
Dimension and rank for mapping class groups
We study the large scale geometry of the mapping class group, MCG. Our main
result is that for any asymptotic cone of MCG, the maximal dimension of locally
compact subsets coincides with the maximal rank of free abelian subgroups of
MCG. An application is an affirmative solution to Brock-Farb's Rank Conjecture
which asserts that MCG has quasi-flats of dimension N if and only if it has a
rank N free abelian subgroup. We also compute the maximum dimension of
quasi-flats in Teichmuller space with the Weil-Petersson metric.Comment: Incorporates referee's suggestions. To appear in Annals of
Mathematic
Geometry of the Complex of Curves I: Hyperbolicity
The Complex of Curves on a Surface is a simplicial complex whose vertices are
homotopy classes of simple closed curves, and whose simplices are sets of
homotopy classes which can be realized disjointly. It is not hard to see that
the complex is finite-dimensional, but locally infinite. It was introduced by
Harvey as an analogy, in the context of Teichmuller space, for Tits buildings
for symmetric spaces, and has been studied by Harer and Ivanov as a tool for
understanding mapping class groups of surfaces.
In this paper we prove that, endowed with a natural metric, the complex is
hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an
explanation of why the Teichmuller space has some negative-curvature properties
in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space
fails most obviously in the regions corresponding to surfaces where some curve
is extremely short. The complex of curves exactly encodes the intersection
patterns of this family of regions (it is the "nerve" of the family), and we
show that its hyperbolicity means that the Teichmuller space is "relatively
hyperbolic" with respect to this family. A similar relative hyperbolicity
result is proved for the mapping class group of a surface.
We also show that the action of pseudo-Anosov mapping classes on the complex
is hyperbolic, with a uniform bound on translation distance.Comment: Revised version of IMS preprint. 36 pages, 6 Figure
Implications of Corporate Capital Structure Theory for Banking Institutions
This paper applies some recent advances in corporate capital structure theory to the determination of optimal capital in banking. The effects of corporate and personal taxes, government regulation, the technology of producing deposit services and the costs of bankruptcy and agency problems are all discussed in the context of the U.S. commercial banking system. The analysis suggests explanations for why commercial banks tend to have relatively less capital than nonfinancial firms, why commercial bank leverage has tended to increase over time and why large banks tend to have relatively less capital than small banks.
Time-varying Huygens' meta-devices for parametric waves
Huygens' metasurfaces have demonstrated almost arbitrary control over the
shape of a scattered beam, however, its spatial profile is typically fixed at
fabrication time. Dynamic reconfiguration of this beam profile with tunable
elements remains challenging, due to the need to maintain the Huygens'
condition across the tuning range. In this work, we experimentally demonstrate
that a time-varying metadevice which performs frequency conversion can steer
transmitted or reflected beams in an almost arbitrary manner, with fully
dynamic control. Our time-varying Huygens' metadevice is made of both electric
and magnetic meta-atoms with independently controlled modulation, and the phase
of this modulation is imprinted on the scattered parametric waves, controlling
their shapes and directions. We develop a theory which shows how the scattering
directionality, phase and conversion efficiency of sidebands can be manipulated
almost arbitrarily. We demonstrate novel effects including all-angle beam
steering and frequency-multiplexed functionalities at microwave frequencies
around 4 GHz, using varactor diodes as tunable elements. We believe that the
concept can be extended to other frequency bands, enabling metasurfaces with
arbitrary phase pattern that can be dynamically tuned over the complete 2\pi
range
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