418 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
Operating envelope charts for the Langley 0.3-meter transonic cryogenic wind tunnel
To take full advantage of the unique Reynolds number capabilities of the 0.3-meter Transonic Cryogenic Tunnel (0.3-m TCT) at the NASA Langley Research Center, it was designed to accommodate test sections other than the original, octagonal, three-dimensional test section. A 20- by 60-cm two-dimensional test section was installed in 1976 and was extensively used, primarily for airfoil testing, through the fall of 1984. The tunnel was inactive during 1985 so that a new test section and improved high speed diffuser could be installed in the tunnel circuit. The new test section has solid adaptive top and bottom walls to reduce or eliminate wall interference for two-dimensional testing. The test section is 33- by 33-cm in cross section at the entrance and is 142 cm long. In the planning and running of past airfoil tests in the 0.3-m TCT, the use of operating envelope charts have proven very useful. These charts give the variation of total temperature and pressure with Mach number and Reynolds number. The operating total temperature range of the 0.3-m TCT is from about 78 K to 327 K with total pressures ranging from about 17.5 psia to 88 psia. This report presents the operating envelope charts for the 0.3-m TCT with the adaptive wall tes t section installed. They were all generated based on a 1-foot chord model. The Mach numbers vary from 0.1 to 0.95
High Reynolds number tests of the CAST 10-2/DOA 2 airfoil in the Langley 0.3-meter transonic cryogenic tunnel, phase 1
A wind tunnel investigation of an advanced technology airfoil, the CAST 10-2/DOA 2, was conducted in the Langley 0.3 meter Transonic Cryogenic Tunnel (0.3 m TCT). This was the first of a series of tests conducted in a cooperative National Aeronautics and Space Administration (NASA) and the Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt e. V. (DFVLR) airfoil research program. Test temperature was varied from 280 K to 100 K to pressures from slightly above 1 to 5.8 atmospheres. Mach number was varied from 0.60 to 0.80, and the Reynolds number (based on airfoil chord) was varied from 4 x 10 to the 8th power to 45 x 10 to the 6th power. This report presents the experimental aerodynamic data obtained for the airfoil and includes descriptions of the airfoil model, the 0.3 m TCT, the test instrumentation, and the testing procedures
Lassoing and corraling rooted phylogenetic trees
The construction of a dendogram on a set of individuals is a key component of
a genomewide association study. However even with modern sequencing
technologies the distances on the individuals required for the construction of
such a structure may not always be reliable making it tempting to exclude them
from an analysis. This, in turn, results in an input set for dendogram
construction that consists of only partial distance information which raises
the following fundamental question. For what subset of its leaf set can we
reconstruct uniquely the dendogram from the distances that it induces on that
subset. By formalizing a dendogram in terms of an edge-weighted, rooted
phylogenetic tree on a pre-given finite set X with |X|>2 whose edge-weighting
is equidistant and a set of partial distances on X in terms of a set L of
2-subsets of X, we investigate this problem in terms of when such a tree is
lassoed, that is, uniquely determined by the elements in L. For this we
consider four different formalizations of the idea of "uniquely determining"
giving rise to four distinct types of lassos. We present characterizations for
all of them in terms of the child-edge graphs of the interior vertices of such
a tree. Our characterizations imply in particular that in case the tree in
question is binary then all four types of lasso must coincide
spectra in elementary cellular automata and fractal signals
We systematically compute the power spectra of the one-dimensional elementary
cellular automata introduced by Wolfram. On the one hand our analysis reveals
that one automaton displays spectra though considered as trivial, and on
the other hand that various automata classified as chaotic/complex display no
spectra. We model the results generalizing the recently investigated
Sierpinski signal to a class of fractal signals that are tailored to produce
spectra. From the widespread occurrence of (elementary) cellular
automata patterns in chemistry, physics and computer sciences, there are
various candidates to show spectra similar to our results.Comment: 4 pages (3 figs included
Minimum triplet covers of binary phylogenetic X-trees
Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any `triplet cover' of the tree (based on the fact that each non-leaf vertex is the median vertex of three leaves). In this paper we describe a number of new results concerning triplet covers of minimum size. In particular, we characterize such covers in terms of an associated graph being a 2-tree. Also, we show that minimum triplet covers are `shellable' and thereby provide a set of pairs for which the inter-leaf distance values will uniquely determine the underlying tree and its associated branch lengths
Recognizing Treelike k-Dissimilarities
A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of
size k subsets of X to the real numbers. Such maps naturally arise from
edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is
defined to be the total length of the smallest subtree of T with leaf-set Y .
In case k = 2, it is well-known that 2-dissimilarities arising in this way can
be characterized by the so-called "4-point condition". However, in case k > 2
Pachter and Speyer recently posed the following question: Given an arbitrary
k-dissimilarity, how do we test whether this map comes from a tree? In this
paper, we provide an answer to this question, showing that for k >= 3 a
k-dissimilarity on a set X arises from a tree if and only if its restriction to
every 2k-element subset of X arises from some tree, and that 2k is the least
possible subset size to ensure that this is the case. As a corollary, we show
that there exists a polynomial-time algorithm to determine when a
k-dissimilarity arises from a tree. We also give a 6-point condition for
determining when a 3-dissimilarity arises from a tree, that is similar to the
aforementioned 4-point condition.Comment: 18 pages, 4 figure
Blocks and Cut Vertices of the Buneman Graph
Given a set \Sg of bipartitions of some finite set of cardinality at
least 2, one can associate to \Sg a canonical -labeled graph \B(\Sg),
called the Buneman graph. This graph has several interesting mathematical
properties - for example, it is a median network and therefore an isometric
subgraph of a hypercube. It is commonly used as a tool in studies of DNA
sequences gathered from populations. In this paper, we present some results
concerning the {\em cut vertices} of \B(\Sg), i.e., vertices whose removal
disconnect the graph, as well as its {\em blocks} or 2-{\em connected
components} - results that yield, in particular, an intriguing generalization
of the well-known fact that \B(\Sg) is a tree if and only if any two splits
in \Sg are compatible
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