33,988 research outputs found
Spin Network States in Gauge Theory
Given a real-analytic manifold M, a compact connected Lie group G and a
principal G-bundle P -> M, there is a canonical `generalized measure' on the
space A/G of smooth connections on P modulo gauge transformations. This allows
one to define a Hilbert space L^2(A/G). Here we construct a set of vectors
spanning L^2(A/G). These vectors are described in terms of `spin networks':
graphs phi embedded in M, with oriented edges labelled by irreducible unitary
representations of G, and with vertices labelled by intertwining operators from
the tensor product of representations labelling the incoming edges to the
tensor product of representations labelling the outgoing edges. We also
describe an orthonormal basis of spin networks associated to any fixed graph
phi. We conclude with a discussion of spin networks in the loop representation
of quantum gravity, and give a category-theoretic interpretation of the spin
network states.Comment: 19 pages, LaTe
Principal manifolds and graphs in practice: from molecular biology to dynamical systems
We present several applications of non-linear data modeling, using principal
manifolds and principal graphs constructed using the metaphor of elasticity
(elastic principal graph approach). These approaches are generalizations of the
Kohonen's self-organizing maps, a class of artificial neural networks. On
several examples we show advantages of using non-linear objects for data
approximation in comparison to the linear ones. We propose four numerical
criteria for comparing linear and non-linear mappings of datasets into the
spaces of lower dimension. The examples are taken from comparative political
science, from analysis of high-throughput data in molecular biology, from
analysis of dynamical systems.Comment: 12 pages, 9 figure
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
EN
Data complexity measured by principal graphs
How to measure the complexity of a finite set of vectors embedded in a
multidimensional space? This is a non-trivial question which can be approached
in many different ways. Here we suggest a set of data complexity measures using
universal approximators, principal cubic complexes. Principal cubic complexes
generalise the notion of principal manifolds for datasets with non-trivial
topologies. The type of the principal cubic complex is determined by its
dimension and a grammar of elementary graph transformations. The simplest
grammar produces principal trees.
We introduce three natural types of data complexity: 1) geometric (deviation
of the data's approximator from some "idealized" configuration, such as
deviation from harmonicity); 2) structural (how many elements of a principal
graph are needed to approximate the data), and 3) construction complexity (how
many applications of elementary graph transformations are needed to construct
the principal object starting from the simplest one).
We compute these measures for several simulated and real-life data
distributions and show them in the "accuracy-complexity" plots, helping to
optimize the accuracy/complexity ratio. We discuss various issues connected
with measuring data complexity. Software for computing data complexity measures
from principal cubic complexes is provided as well.Comment: Computers and Mathematics with Applications, in pres
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