166,222 research outputs found
Hilbert spaces built on a similarity and on dynamical renormalization
We develop a Hilbert space framework for a number of general multi-scale
problems from dynamics. The aim is to identify a spectral theory for a class of
systems based on iterations of a non-invertible endomorphism.
We are motivated by the more familiar approach to wavelet theory which starts
with the two-to-one endomorphism in the one-torus \bt, a
wavelet filter, and an associated transfer operator. This leads to a scaling
function and a corresponding closed subspace in the Hilbert space
L^2(\br). Using the dyadic scaling on the line \br, one has a nested family
of closed subspaces , n \in \bz, with trivial intersection, and with
dense union in L^2(\br). More generally, we achieve the same outcome, but in
different Hilbert spaces, for a class of non-linear problems. In fact, we see
that the geometry of scales of subspaces in Hilbert space is ubiquitous in the
analysis of multiscale problems, e.g., martingales, complex iteration dynamical
systems, graph-iterated function systems of affine type, and subshifts in
symbolic dynamics. We develop a general framework for these examples which
starts with a fixed endomorphism (i.e., generalizing ) in a
compact metric space . It is assumed that is onto, and
finite-to-one.Comment: v3, minor addition
Wasserstein Regression
The analysis of samples of random objects that do not lie in a vector space
is gaining increasing attention in statistics. An important class of such
object data is univariate probability measures defined on the real line.
Adopting the Wasserstein metric, we develop a class of regression models for
such data, where random distributions serve as predictors and the responses are
either also distributions or scalars. To define this regression model, we
utilize the geometry of tangent bundles of the space of random measures endowed
with the Wasserstein metric for mapping distributions to tangent spaces. The
proposed distribution-to-distribution regression model provides an extension of
multivariate linear regression for Euclidean data and function-to-function
regression for Hilbert space valued data in functional data analysis. In
simulations, it performs better than an alternative transformation approach
where one maps distributions to a Hilbert space through the log quantile
density transformation and then applies traditional functional regression. We
derive asymptotic rates of convergence for the estimator of the regression
operator and for predicted distributions and also study an extension to
autoregressive models for distribution-valued time series. The proposed methods
are illustrated with data on human mortality and distributional time series of
house prices
The geometry of hyperbolic lines in polar spaces
In this paper we consider partial linear spaces induced on the point set of a
polar space, but with as lines the hyperbolic lines of this polar space. We
give some geometric characterizations of these and related spaces. The results
have applications in group theory, in the theory of Lie algebras and in graph
theory
Two constructions with parabolic geometries
This is an expanded version of a series of lectures delivered at the 25th
Winter School ``Geometry and Physics'' in Srni.
After a short introduction to Cartan geometries and parabolic geometries, we
give a detailed description of the equivalence between parabolic geometries and
underlying geometric structures.
The second part of the paper is devoted to constructions which relate
parabolic geometries of different type. First we discuss the construction of
correspondence spaces and twistor spaces, which is related to nested parabolic
subgroups in the same semisimple Lie group. An example related to twistor
theory for Grassmannian structures and the geometry of second order ODE's is
discussed in detail.
In the last part, we discuss analogs of the Fefferman construction, which
relate geometries corresponding different semisimple Lie groups
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