Betti number

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Betti number

[′bāt·tē ‚nəm·bər]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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We investigate how to change some properties such as height big height Krull dimension Betti numbers by gluing of two graphs at complete clone.
Jessup, New bounds on Betti numbers of nilpotent Lie algebras, Comm.
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].
By adopting the evaluation method of Betti numbers; the mixing effect can be effectively quantified under different stirring conditions.
The Euler Number [sub.[chi]](X)] and Betti numbers [[beta].sub.i](X) for i=0,1,2 are conventionally used as topological characteristics of X.
He begins with manifolds, tensors and exterior forms and progresses to such topics as the integration of differential forms and the Lie derivative, the Poincare Lemma and potentials, Monkowski space, covariant differentiation and curvature, relativity, Betti numbers and De Rham's theorem, harmonic forms, the Aharonov-Bohm effect, and Yang- Mills fields.
Such subspaces are of course realizable in the real world and, further, the zeroth, first, and second Betti numbers (i.e., ranks of the corresponding homology groups) have intuitive geometric interpretations as the number of components, tunnels, and voids, respectively.
This will imply the exponential growth of the Betti numbers of the space of long links, and even of long links modulo l-fold product of the space of long knots (which is a retract of the space of long links, as we will see in this paper.)
(a) Enumerative criterion: We give a simple criterion for the r-stackedness in terms of h-vectors and Betti numbers for homology manifolds with boundary (Theorem 3.1).