284 research outputs found
A transfer principle and applications to eigenvalue estimates for graphs
In this paper, we prove a variant of the Burger-Brooks transfer principle
which, combined with recent eigenvalue bounds for surfaces, allows to obtain
upper bounds on the eigenvalues of graphs as a function of their genus. More
precisely, we show the existence of a universal constants such that the
-th eigenvalue of the normalized Laplacian of a graph
of (geometric) genus on vertices satisfies where denotes the maximum valence of
vertices of the graph. This result is tight up to a change in the value of the
constant , and improves recent results of Kelner, Lee, Price and Teng on
bounded genus graphs.
To show that the transfer theorem might be of independent interest, we relate
eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple
graph models, and discuss an application to the mesh partitioning problem,
extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang
to arbitrary meshes.Comment: Major revision, 16 page
Optimal Kullback-Leibler Aggregation via Information Bottleneck
In this paper, we present a method for reducing a regular, discrete-time
Markov chain (DTMC) to another DTMC with a given, typically much smaller number
of states. The cost of reduction is defined as the Kullback-Leibler divergence
rate between a projection of the original process through a partition function
and a DTMC on the correspondingly partitioned state space. Finding the reduced
model with minimal cost is computationally expensive, as it requires an
exhaustive search among all state space partitions, and an exact evaluation of
the reduction cost for each candidate partition. Our approach deals with the
latter problem by minimizing an upper bound on the reduction cost instead of
minimizing the exact cost; The proposed upper bound is easy to compute and it
is tight if the original chain is lumpable with respect to the partition. Then,
we express the problem in the form of information bottleneck optimization, and
propose using the agglomerative information bottleneck algorithm for searching
a sub-optimal partition greedily, rather than exhaustively. The theory is
illustrated with examples and one application scenario in the context of
modeling bio-molecular interactions.Comment: 13 pages, 4 figure
Riemann Surfaces and 3-Regular Graphs
In this thesis we consider a way to construct a rich family of compact
Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with
orientation, we construct a finite-area hyperbolic Riemann surface by gluing
triangles according to the combinatorics of the graph. We then compactify this
surface by adding finitely many points.
We discuss this construction by considering a number of examples. In
particular, we see that the surface depends in a strong way on the orientation.
We then consider the effect the process of compactification has on the
hyperbolic metric of the surface. To that end, we ask when we can change the
metric in the horocycle neighbourhoods of the cusps to get a hyperbolic metric
on the compactification. In general, the process of compactification can have
drastic effects on the hyperbolic structure. For instance, if we compactify the
3-punctured sphere we lose its hyperbolic structure.
We show that when the cusps have lengths > 2\pi, we can fill in the horocycle
neighbourhoods and retain negative curvature. Furthermore, the last condition
is sharp. We show by examples that there exist curves arbitrarily close to
horocycles of length 2\pi, which cannot be so filled in. Such curves can even
be taken to be convex.Comment: M.Sc. Thesis (Technion, Israel Institute of Technology), 55 Pages,
Under the Direction of Robert Brook
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