homology group

homology group

[hə′mäl·ə·jē ‚grüp]
(mathematics)
Associated to a topological space X, one of a sequence of Abelian groups Hn (X) that reflect how n-dimensional simplicial complexes can be used to fill up X and also help determine the presence of n-dimensional holes appearing in X. Also known as Betti group.
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(iv) As a result of [partial derivative] * [partial derivative] = 0, one can define the pth homology group [H.sub.p](K) as the quotient of the p-cycles [Z.sub.p], elements of [C.sub.p] which are mapped to 0 by [partial derivative], and p-boundaries [B.sub.p], which is the image of [C.sub.p+1] under [partial derivative].
The homology group [H.sub.*]([M.sub.K]; Z) = [H.sup.0]([M.sub.K]; Z) [[direct sum] [H.sub.1]([M.sub.K]; Z) has the basis {[p], [[mu]]}, where [p] is the homology class of a point and [[mu]] is that of the meridian of K.
The 13 selected peer-reviewed papers explore such aspects of logic as an analogy between cardinal characteristics and highness properties of oracles, a non-uniformly C-productive sequence and non-constructive disjunctions, the characterization of the second homology group of stationary type in a stable theory, some questions concerning ab initio generic structures, realizability and existence property of a constructive set theory with types, a goal-directed unbounded coalitional game and its complexity, large cardinals and higher degree theory, and degree spectra of equivalence relations.
It is known from [4] and [7] that in this case the homology group [H.sub.r](X, Y, Z) = 0 if r [greater than or equal to] n + q and, is torsion free if r = n + q - 1.
In this setting, the concept of size function coincides with the dimension of the 0-th multidimensional persistent homology group, i.e., the 0-th rank invariant (Carlsson and Zomorodian, 2007).
and transfer of seven species of the genus Pseudomonas homology group II to the new genus with type species Burkholderia cepacia (Palleroni & Holmes 1981) comb.
(2) Compute geometric realizations of a set of generators that form a basis of the first and second homology groups of K (for the zeroth homology group, this is trivial) in O([n.sup.2][bar]g) and O(n) time (and space), respectively, where [bar]g is an invariant of K such that [bar]g [is less than] n always.
* [H.sup.[kappa].sub.q] (X, A) = [Z.sup.[kappa].sub.q](X, A)/[B.sup.[kappa].sub.q] (X, A) is called the qth digital relative simplicial homology group.
Changing a basis of the first homology group [H.sub.1]([h.sub.1]) (resp.
Calderbank, Hanlon and Robinson [3] extended these results by considering the action of the symmetric group [G.sub.n-1] on the top homology group of the order complex of [[PI].sup.d.sub.n] - {[??]}.
heilmannii" type 2 were highly related and formed a distinct cluster within the rRNA homology group III (i.e., the Helicobacter phylogenetic branch) of rRNA super-family VI (data not shown).
In fact, it computes this homotopy group as a homology group of another space (simplicial set): [K.sub.4].