8,585 research outputs found
Aspects of connectedness in metric frames.
Doctor of Philosophy in Mathematics. University of KwaZulu-Natal, Durban, 2019.Abstract available in PDF file
The connected Vietoris powerlocale
The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set with respect to a Vietoris metric. \ud
Application to the point-free real line gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem. \ud
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The work is topos-valid (assuming natural numbers object). Vc is a geometric constructio
Succinctness in subsystems of the spatial mu-calculus
In this paper we systematically explore questions of succinctness in modal
logics employed in spatial reasoning. We show that the closure operator,
despite being less expressive, is exponentially more succinct than the
limit-point operator, and that the -calculus is exponentially more
succinct than the equally-expressive tangled limit operator. These results hold
for any class of spaces containing at least one crowded metric space or
containing all spaces based on ordinals below , with the usual
limit operator. We also show that these results continue to hold even if we
enrich the less succinct language with the universal modality
Infinite networks and variation of conductance functions in discrete Laplacians
For a given infinite connected graph and an arbitrary but fixed
conductance function , we study an associated graph Laplacian ;
it is a generalized difference operator where the differences are measured
across the edges in ; and the conductance function represents the
corresponding coefficients. The graph Laplacian (a key tool in the study of
infinite networks) acts in an energy Hilbert space computed
from . Using a certain Parseval frame, we study the spectral theoretic
properties of graph Laplacians. In fact, for fixed , there are two versions
of the graph Laplacian, one defined naturally in the space of , and
the other in . The first is automatically selfadjoint, but the
second involves a Krein extension. We prove that, as sets, the two spectra are
the same, aside from the point 0. The point zero may be in the spectrum of the
second, but not the first.
We further study the fine structure of the respective spectra as the
conductance function varies; showing now how the spectrum changes subject to
variations in the function .Comment: 32 pages, 3 figure
Projections and idempotents with fixed diagonal and the homotopy problem for unit tight frames
We investigate the topological and metric structure of the set of idempotent
operators and projections which have prescribed diagonal entries with respect
to a fixed orthonormal basis of a Hilbert space. As an application, we settle
some cases of conjectures of Larson, Dykema, and Strawn on the connectedness of
the set of unit-norm tight frames.Comment: New title and introductio
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