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#( and If
#( a property of Delaunay triangulations ): If there is a circle with two of the input points on its boundary which contains no other input points, the line between those two points is an edge of every Delaunay triangulation.
#( Going up ) If B is an integral extension of A, then the extension satisfies the going-up property ( and hence the lying over property ), and the incomparability property.
#( Going down ) If B is an integral extension of A, and B is a domain, and A is integrally closed in its field of fractions, then the extension ( in addition to going-up, lying-over and incomparability ) satisfies the going-down property.

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#( Rationality ) ζ ( X, s ) is a rational function of T = q < sup >− s </ sup >.
#( Functional equation and Poincaré duality ) The zeta function satisfies < dl >< dd ></ dl > or equivalently < dl >< dd ></ dl > where E is the Euler characteristic of X.

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#( the cycle property of minimum spanning trees ): For any cycle C in the graph, if the weight of an edge e of C is larger than the weights of other edges of C, then this edge cannot belong to a MST.
#( TR 3 ) Given a map between two morphisms, there is a morphism between their mapping cones ( which exist by axiom ( TR 1 )), that makes everything commute.
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The diagonal lemma also requires that there be a systematic way of assigning to every formula θ a natural number #( θ ) called its Gödel number.
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#( Riemann hypothesis ) | α < sub > i, j </ sub >| = q < sup > i / 2 </ sup > for all and all j.

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Betti and numbers
De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology.
These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians.
There is a constant C = C ( n ) such that if M is a compact connected n-dimensional Riemannian manifold with positive sectional curvature then the sum of its Betti numbers is at most C.
It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on.
and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Artin for attacking the Weil conjectures, as outlined in.
For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1.
( Complex projective space gives the relevant Betti numbers, which nearly determine the answer.
The functional equation for the zeta function follows from Poincaré duality for l-adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between l-adic and ordinary cohomology for complex varieties.
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces.
For the most reasonable finite-dimensional spaces ( such as compact manifolds, finite simplicial complexes or CW complexes ), the sequence of Betti numbers is 0 from some points onwards ( Betti numbers vanish above the dimension of a space ), and they are all finite.
The term " Betti numbers " was coined by Henri Poincaré after Enrico Betti.
A torus has one connected component, two circular holes ( the one in the center and the one in the middle of the " tube "), and one three-dimensional void ( the inside of the " tube ") yielding Betti numbers of 1, 2, 1.
The first few Betti numbers have the following intuitive definitions:
The ( rational ) Betti numbers b < sub > k </ sub >( X ) do not take into account any torsion in the homology groups, but they are very useful basic topological invariants.

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