Betti number

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Betti number

[′bāt·tē ‚nəm·bər]

(mathematics)

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Key Words: Glued graph Height Big height Krull dimension Projective dimension Linear resolution Betti number Cohen-Macaulay.

SOME PROPERTIES OF GLUED GRAPHS AT COMPLETE CLONE IN THE VIEW OF ALGEBRAIC COMBINATORICS

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.

Some restrictions on the Betti numbers of a nilpotent lie algebra

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.

College Town

Figure 5 shows the 0-dimensional and 1-dimensional Betti number curves as a function of time, T, for this experiment.

An assessment of the effects of micron-particle aggregation on the performance of zinc-silica composite coatings using Betti numbers

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.

Singularities--Niigata--Toyama; 2007; proceedings

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

Computing Homology Groups of Simplicial Complexes in [R.sup.3]

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.

Cuts and flows of cell complexes

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.

On r-stacked triangulated manifolds

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.

Real elliptic surfaces with double section and real tetragonal curves

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).

Divisors on graphs, connected flags, and syzygies

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].