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Betti number Cohen-Macaulay.

In fact some authors [8, 10, 15, 16] call dim [H.sup.i](L, [C.sup.x]) the i-th

Betti number of L, and, by means of (1.2), if we have bounds on the second

Betti number of a nilpotent Lie algebra, then we have bounds on its Schur multiplier, and viceversa.

Adriano Marzullo, assistant professor of mathematics, published his paper, "On the Periodicity of the First

Betti Number of the Semigroup Rings under Translations," in the Journal of the Ramanujam Mathematical Society.

Figure 5 shows the 0-dimensional and 1-dimensional

Betti number curves as a function of time, T, for this experiment.

The topics include the Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one, a plane sextic with finite fundamental group, the topology of abelian pencils on curves, the middle

Betti number of certain singularities with critical locus a hyperplane, standard bases and algebraic local cohomology for zero dimensional ideals, and a universal bivariant theory and cobordism groups.

Similarly, the jth

Betti number, [[Beta].sub.j], and the Euler characteristic,

Equivalently, the vectors in Cut([SIGMA]) support sets of facets whose deletion increases the codimension-1

Betti number, and the vectors in Flow([SIGMA]) support nontrivial rational homology classes.

A simplicial complex [DELTA] of dimension d is said to be a (k-)homology d- sphere if, for all faces F [member of] [DELTA] (including the empty face 0), one has [[beta].sub.i] ([lk.sub.[DELTA]] (F)) = 0 for i [not equal to] d - #F and [[beta].sub.d-#F] ([lk.sub.[DELTA]] (F)) = 1, where [[beta].sub.i] (A) = [dim.sub.k] [[??].sub.i] ([DELTA]; k) is the ith

Betti number of [DELTA] over k.

with Br[(X).sub.2]/J[(C).sub.2] [congruent to] [(Z/2Z).sup.n] and n = 2(1 + [b.sub.2](Y) - [b.sub.0](C)) - [rho] where [rho] = rank [NS(X)] and [b.sub.i] is the ith

Betti number.

Hochster's formula for computing the

Betti numbers topologically (see, e.g., [MS05, Theorem 9.2]), when applied to [I.sub.G] and the "nice" grading by Pic(G), says that for each j [member of] Pic(G) the graded

Betti number [[beta].sub.i,j](R/[I.sub.G]) is the dimension of the ith reduced homology of the simplicial complex [[DELTA].sub.j] = {supp(E): 0 [less than or equal to] E [less than or equal to] D' e [member of] [absolute value of j]} where [absolute value of j] denotes the linear system of j [member of] Pic(G).

Since the Euler characteristic of P is inherently related to both the combinatorial and topological structure of P, we will also be interested in studying the (reduced)

Betti numbers of P (over a field k), which are defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].