28,239 research outputs found
Chern classes of reductive groups and an adjunction formula
In this paper, I construct noncompact analogs of the Chern classes of
equivariant vector bundles over complex reductive groups. For the tangent
bundle, these Chern classes yield an adjunction formula for the Euler
characteristic of complete intersections in reductive groups. In the case where
the complete intersection is a curve, this formula gives an explicit answer for
the Euler characteristic and the genus of the curve.Comment: LATeX, 26 pages; added references, corrected typo
Combinatorial models of expanding dynamical systems
We define iterated monodromy groups of more general structures than partial
self-covering. This generalization makes it possible to define a natural notion
of a combinatorial model of an expanding dynamical system. We prove that a
naturally defined "Julia set" of the generalized dynamical systems depends only
on the associated iterated monodromy group. We show then that the Julia set of
every expanding dynamical system is an inverse limit of simplicial complexes
constructed by inductive cut-and-paste rules.Comment: The new version differs substantially from the first one. Many parts
are moved to other (mostly future) papers, the main open question of the
first version is solve
Conjugate Projective Limits
We characterize conjugate nonparametric Bayesian models as projective limits
of conjugate, finite-dimensional Bayesian models. In particular, we identify a
large class of nonparametric models representable as infinite-dimensional
analogues of exponential family distributions and their canonical conjugate
priors. This class contains most models studied in the literature, including
Dirichlet processes and Gaussian process regression models. To derive these
results, we introduce a representation of infinite-dimensional Bayesian models
by projective limits of regular conditional probabilities. We show under which
conditions the nonparametric model itself, its sufficient statistics, and -- if
they exist -- conjugate updates of the posterior are projective limits of their
respective finite-dimensional counterparts. We illustrate our results both by
application to existing nonparametric models and by construction of a model on
infinite permutations.Comment: 49 pages; improved version: revised proof of theorem 3 (results
unchanged), discussion added, exposition revise
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
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