2,907 research outputs found
The Theory of Quasiconformal Mappings in Higher Dimensions, I
We present a survey of the many and various elements of the modern
higher-dimensional theory of quasiconformal mappings and their wide and varied
application. It is unified (and limited) by the theme of the author's
interests. Thus we will discuss the basic theory as it developed in the 1960s
in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore
the connections with geometric function theory, nonlinear partial differential
equations, differential and geometric topology and dynamics as they ensued over
the following decades. We give few proofs as we try to outline the major
results of the area and current research themes. We do not strive to present
these results in maximal generality, as to achieve this considerable technical
knowledge would be necessary of the reader. We have tried to give a feel of
where the area is, what are the central ideas and problems and where are the
major current interactions with researchers in other areas. We have also added
a bit of history here and there. We have not been able to cover the many recent
advances generalising the theory to mappings of finite distortion and to
degenerate elliptic Beltrami systems which connects the theory closely with the
calculus of variations and nonlinear elasticity, nonlinear Hodge theory and
related areas, although the reader may see shadows of this aspect in parts
A theoretical framework for supervised learning from regions
Supervised learning is investigated, when the data are represented not only by labeled points but also labeled regions of the input space. In the limit case, such
regions degenerate to single points and the proposed approach changes back to the classical learning context. The adopted framework entails the minimization
of a functional obtained by introducing a loss function that involves such regions. An additive regularization term is expressed via differential operators that model
the smoothness properties of the desired input/output relationship. Representer
theorems are given, proving that the optimization problem associated to learning
from labeled regions has a unique solution, which takes on the form of a linear
combination of kernel functions determined by the differential operators together
with the regions themselves. As a relevant situation, the case of regions given
by multi-dimensional intervals (i.e., “boxes”) is investigated, which models prior
knowledge expressed by logical propositions
Pointwise differentiability of higher order for sets
The present paper develops two concepts of pointwise differentiability of
higher order for arbitrary subsets of Euclidean space defined by comparing
their distance functions to those of smooth submanifolds. Results include that
differentials are Borel functions, higher order rectifiability of the set of
differentiability points, and a Rademacher result. One concept is characterised
by a limit procedure involving inhomogeneously dilated sets.
The original motivation to formulate the concepts stems from studying the
support of stationary integral varifolds. In particular, strong pointwise
differentiability of every positive integer order is shown at almost all points
of the intersection of the support with a given plane.Comment: Description of subsequent work added to the introduction, references
and affiliations updated, typographical corrections made; 34 page
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