5,725 research outputs found
On the p-Connectedness of Graphs – a Survey
A graph is said to be p-connected if for every partition of its vertices into two non-empty, disjoint, sets some chordless path with three edges contains vertices from both sets in the partition. As it turns out, p-connectedness generalizes the usual connectedness of graphs and leads, in a natural way, to a unique tree representation for arbitrary graphs.
This paper reviews old and new results, both structural and algorithmic, about p-connectedness along with applications to various graph decompositions
Representation of non-semibounded quadratic forms and orthogonal additivity
In this article we give a representation theorem for non-semibounded
Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint
operator. The main assumptions are closability of the Hermitean quadratic form,
the direct integral structure of the underlying Hilbert space and orthogonal
additivity. We apply this result to several examples, including the position
operator in quantum mechanics and quadratic forms invariant under a unitary
representation of a separable locally compact group. The case of invariance
under a compact group is also discussed in detail
A New Method of Extension of Local Maps of Banach Spaces. Applications and Examples
A known classical method of extension of smooth local maps of Banach spaces
uses smooth bump functions. However, such functions are absent in the majority
of infinite-dimensional Banach spaces. This is an obstacle in the development
of local analysis, in particular in the questions of extending local maps onto
the whole space. We suggest an approach that substitutes bump functions with
special maps, which we call blid maps. It allows us to extend smooth local maps
from non-smooth spaces, such as . As an example of
applications, we show how to reconstruct a map from its derivatives at a point,
for spaces possessing blid maps. We also show how blid maps can assist in
finding global solutions to cohomological equations having linear
transformation of argument.Comment: arXiv admin note: substantial text overlap with arXiv:1703.0629
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