59 research outputs found
Thurston's pullback map on the augmented Teichm\"uller space and applications
Let be a postcritically finite branched self-cover of a 2-dimensional
topological sphere. Such a map induces an analytic self-map of a
finite-dimensional Teichm\"uller space. We prove that this map extends
continuously to the augmented Teichm\"uller space and give an explicit
construction for this extension. This allows us to characterize the dynamics of
Thurston's pullback map near invariant strata of the boundary of the augmented
Teichm\"uller space. The resulting classification of invariant boundary strata
is used to prove a conjecture by Pilgrim and to infer further properties of
Thurston's pullback map. Our approach also yields new proofs of Thurston's
theorem and Pilgrim's Canonical Obstruction theorem.Comment: revised version, 28 page
Minimal immersions of closed surfaces in hyperbolic three-manifolds
We study minimal immersions of closed surfaces (of genus ) in
hyperbolic 3-manifolds, with prescribed data , where
is a conformal structure on a topological surface , and is a holomorphic quadratic differential on the surface . We
show that, for each for some , depending only on
, there are at least two minimal immersions of closed surface
of prescribed second fundamental form in the conformal structure
. Moreover, for sufficiently large, there exists no such minimal
immersion. Asymptotically, as , the principal curvatures of one
minimal immersion tend to zero, while the intrinsic curvatures of the other
blow up in magnitude.Comment: 16 page
Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces
Using Polyakov's functional integral approach with the Liouville action
functional defined in \cite{ZT2} and \cite{LTT}, we formulate quantum Liouville
theory on a compact Riemann surface X of genus g > 1. For the partition
function and for the correlation functions with the stress-energy tensor
components , we
describe Feynman rules in the background field formalism by expanding
corresponding functional integrals around a classical solution - the hyperbolic
metric on X. Extending analysis in \cite{LT1,LT2,LT-Varenna,LT3}, we define the
regularization scheme for any choice of global coordinate on X, and for
Schottky and quasi-Fuchsian global coordinates we rigorously prove that one-
and two-point correlation functions satisfy conformal Ward identities in all
orders of the perturbation theory. Obtained results are interpreted in terms of
complex geometry of the projective line bundle \cE_{c}=\lambda_{H}^{c/2} over
the moduli space , where c is the central charge and
is the Hodge line bundle, and provide Friedan-Shenker \cite{FS}
complex geometry approach to CFT with the first non-trivial example besides
rational models.Comment: 67 pages, 4 figures (Typos corrected as in the published version
On the appearance of Eisenstein series through degeneration
Let be a Fuchsian group of the first kind acting on the hyperbolic
upper half plane , and let be the
associated finite volume hyperbolic Riemann surface. If is parabolic,
there is an associated (parabolic) Eisenstein series, which, by now, is a
classical part of mathematical literature. If is hyperbolic, then,
following ideas due to Kudla-Millson, there is a corresponding hyperbolic
Eisenstein series. In this article, we study the limiting behavior of parabolic
and hyperbolic Eisenstein series on a degenerating family of finite volume
hyperbolic Riemann surfaces. In particular, we prove the following result. If
corresponds to a degenerating hyperbolic element, then a
multiple of the associated hyperbolic Eisenstein series converges to parabolic
Eisenstein series on the limit surface.Comment: 15 pages, 2 figures. This paper has been accepted for publication in
Commentarii Mathematici Helvetic
Growth Based Morphogenesis of Vertebrate Limb Bud
Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea “growth-based morphogenesis.” We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs
Signal transduction, cell division, differentiation and development: towards unifying mechanisms for pattern formation in plants
Fitness distributions and GA hardness
Considerable research effort has been spent in trying to formulate a good definition of GA-Hardness. Given an instance of a problem, the objective is to estimate the performance of a GA. Despite partial successes current definitions are still unsatisfactory. In this paper we make some steps towards a new, more powerful way of assessing problem difficulty based on the properties of a problem's fitness distribution. We present experimental results that strongly support this idea
Arbiter Meta-Learning with Dynamic Selection of Classifiers and its Experimental Investigation
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