The metric matrix theory is an important research object of metric measure geometry and it can be used to characterize the geometric structure of a set. For intuitionistic fuzzy sets (IFS), we ...defined metric information matrices (MIM) of IFS by using the metric matrix theory. We introduced the Gromov–Hausdorff metric to measure the distance between any two MIMs. We then constructed a kind of metric information matrix distance knowledge measure for IFS. The proposed distance measures have the ability to measure the distance between two incomplete intuitionistic fuzzy sets. In order to reduce the information confusion caused by the disorder of MIM, we defined a homogenous metric information matrix distance by rearranging MIM. Some theorems are given to show the properties of the constructed distance measures. At the end of the paper, some numerical experiments are given to show that the proposed distances can recognize different patterns represented by IFS.
On the Gomory–Hu Inequality Petrov, Evgenii A.; Dovgoshey, Aleksei A.
Journal of mathematical sciences (New York, N.Y.),
04/2014, Volume:
198, Issue:
4
Journal Article
Peer reviewed
It was proved by R. Gomory and T. Hu in 1961 that, for every finite nonempty ultrametric space (
X, d
), the inequaliy
, where Sp(
X
) = {
d
(
x, y
) :
x, y
∈
X, x ≠ y
}
,
holds. We characterize the ...spaces
X
for which the equality is attained by the structural properties of some graphs and show that the set of isometric types of such
X
is dense in the Gromov–Hausdorff space of the isometric types of compact ultrametric spaces.
We present a new class of metrics for unrooted phylogenetic
X
-trees inspired by the Gromov–Hausdorff distance for (compact) metric spaces. These metrics can be efficiently computed by linear or ...quadratic programming. They are robust under NNI operations, too. The local behaviour of the metrics shows that they are different from any previously introduced metrics. The performance of the metrics is briefly analysed on random weighted and unweighted trees as well as random caterpillars.
We study a class of graph foliated spaces, or graph matchbox manifolds, initially constructed by Kenyon and Ghys. For graph foliated spaces we introduce a quantifier of dynamical complexity which we ...call its level. We develop the fusion construction, which allows us to associate to every two graph foliated spaces a third one which contains the former two in its closure. Although the underlying idea of the fusion is simple, it gives us a powerful tool to study graph foliated spaces. Using fusion, we prove that there is a hierarchy of graph foliated spaces at infinite levels. We also construct examples of graph foliated spaces with various dynamical and geometric properties.
The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré duality complexes (PD complexes). The problem is that an arbitrary generalized manifold X ...is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincaré duality with coefficients in the group ring Λ (Λ-complexes). Standard homology theory implies that X is a Z-PD complex. Therefore by Browder's theorem, X has a Spivak normal fibration which in turn, determines a Thom class of the pair (N,∂N) of a mapping cylinder neighborhood of X in some Euclidean space. Then X satisfies the Λ-Poincaré duality if this class induces an isomorphism with Λ-coefficients. Unfortunately, the proof of Browder's theorem gives only isomorphisms with Z-coefficients. It is also not very helpful that X is homotopy equivalent to a finite complex K, because K is not automatically a Λ-PD complex. Therefore it is convenient to introduce Λ-PD structures. To prove their existence on X, we use the construction of 2-patch spaces and some fundamental results of Bryant, Ferry, Mio, and Weinberger. Since the class of all Λ-PD complexes does not contain all generalized manifolds, we appropriately enlarge this class and then describe (i.e. recognize) generalized manifolds within this enlarged class in terms of the Gromov–Hausdorff metric.
We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree ...prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov-Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.
We investigate the local structure of the space
M
consisting of isometry classes of compact metric spaces, endowed with the Gromov–Hausdorff metric. We consider finite metric spaces of the same ...cardinality and suppose that these spaces are in general position, i.e., all nonzero distances in each of the spaces are distinct, and all triangle inequalities are strict. We show that sufficiently small balls in
M
centered at these spaces and having the same radii are isometric. As consequences, we prove that the cones over such spaces (with the vertices at the single-point space) are isometric; the isometry group of each sufficiently small ball centered at a general position
n
points space,
n
≥ 3, contains a subgroup isomorphic to the symmetric group
S
n
.
In the article the matching algorithms of region images are analyzed. The work also presents their advantages and disadvantages. The comparing images algorithm of regions is developed on the basis of ...the measured chords. The comparison regions algorithms are used to evaluate segmentation algorithms in the Gromov-Hausdorff metric. The algorithm of metric evaluation is developed as an example of biomedical images segmentation.
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