707 research outputs found
Characterizing Block Graphs in Terms of their Vertex-Induced Partitions
Given a finite connected simple graph with vertex set and edge
set , we will show that
the (necessarily unique) smallest block graph with vertex set whose
edge set contains is uniquely determined by the -indexed family of the various partitions
of the set into the set of connected components of the
graph ,
the edge set of this block graph coincides with set of all -subsets
of for which and are, for all , contained
in the same connected component of ,
and an arbitrary -indexed family of
partitions of the set is of the form for some
connected simple graph with vertex set as above if and only if,
for any two distinct elements , the union of the set in
that contains and the set in that contains coincides with
the set , and holds for all .
As well as being of inherent interest to the theory of block graphs, these
facts are also useful in the analysis of compatible decompositions and block
realizations of finite metric spaces
Characterizing block graphs in terms of their vertex-induced partitions
Block graphs are a generalization of trees that arise in areas such as metric graph theory, molecular graphs, and phylogenetics. Given a finite connected simple graph with vertex set and edge set , we will show that the (necessarily unique) smallest block graph with vertex set whose edge set contains is uniquely determined by the -indexed family \Pp_G =\big(\pi_v)_{v \in V} of the partitions of the set into the set of connected components of the graph . Moreover, we show that an arbitrary -indexed family \Pp=(\p_v)_{v \in V} of partitions \p_v of the set is of the form \Pp=\Pp_G for some connected simple graph with vertex set as above if and only if, for any two distinct elements , the union of the set in \p_v that contains and the set in \p_u that contains coincides with the set , and \{v\}\in \p_v holds for all . As well as being of inherent interest to the theory of block graphs,these facts are also useful in the analysis of compatible decompositions of finite metric spaces
Landscapes of data sets and functoriality of persistent homology
The aim of this article is to describe a new perspective on functoriality of
persistent homology and explain its intrinsic symmetry that is often
overlooked. A data set for us is a finite collection of functions, called
measurements, with a finite domain. Such a data set might contain internal
symmetries which are effectively captured by the action of a set of the domain
endomorphisms. Different choices of the set of endomorphisms encode different
symmetries of the data set. We describe various category structures on such
enriched data sets and prove some of their properties such as decompositions
and morphism formations. We also describe a data structure, based on coloured
directed graphs, which is convenient to encode the mentioned enrichment. We
show that persistent homology preserves only some aspects of these landscapes
of enriched data sets however not all. In other words persistent homology is
not a functor on the entire category of enriched data sets. Nevertheless we
show that persistent homology is functorial locally. We use the concept of
equivariant operators to capture some of the information missed by persistent
homology
Fault tolerance in space-based digital signal processing and switching systems: Protecting up-link processing resources, demultiplexer, demodulator, and decoder
Fault tolerance features in the first three major subsystems appearing in the next generation of communications satellites are described. These satellites will contain extensive but efficient high-speed processing and switching capabilities to support the low signal strengths associated with very small aperture terminals. The terminals' numerous data channels are combined through frequency division multiplexing (FDM) on the up-links and are protected individually by forward error-correcting (FEC) binary convolutional codes. The front-end processing resources, demultiplexer, demodulators, and FEC decoders extract all data channels which are then switched individually, multiplexed, and remodulated before retransmission to earth terminals through narrow beam spot antennas. Algorithm based fault tolerance (ABFT) techniques, which relate real number parity values with data flows and operations, are used to protect the data processing operations. The additional checking features utilize resources that can be substituted for normal processing elements when resource reconfiguration is required to replace a failed unit
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