(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].