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Topic: Betti number

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	chromatic number
		In graph theory, the minimum number of colors needed to color (the vertices of) a connected graph so that no two adjacent vertices are colored the same.
		In topology, the maximum number of regions that can be drawn on a surface in such a way that each region has a border in common with every other region.
		The chromatic number of a square, tube, or sphere, for example, is 4; in other words, it is impossible to place more than four differently-colored regions on one of these figures so that any pair has a common boundary.
	www.daviddarling.info /encyclopedia/C/chromatic_number.html (285 words)

	Betti number - Wikipedia, the free encyclopedia
		The Betti number sequence for a circle is 1, 1, 0, 0, 0,...;
		The Betti number sequence for a two-torus is 1, 2, 1, 0, 0, 0,...;
		The Betti number sequence for a three-torus is 1, 3, 3, 1, 0, 0, 0,...
	en.wikipedia.org /wiki/Betti_number (688 words)

	Enrico Betti Summary
		Betti succeeded in proving that the one-dimensional connectivity number is the same as the three-dimensional connectivity number.
		Betti was also the first to show that the quintic function—a function in which a variable is raised to the fifth power—can be solved using integrals of elliptic functions.
		Enrico Betti (21 October 1823 - 11 August 1892) was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers.
	www.bookrags.com /Enrico_Betti (785 words)

	Letter B
		A consequence is that a complex number also provides the simplest form of a spinor (PL) and the basis of a recursive generation of "the arithmetic of Clifford numbers" or multivectors (PL).
		A surd is a root of a number.
		Mixed-based notation of numbers as represented on a biquinary abacus (PL).
	members.fortunecity.com /jonhays/letterB.htm (1785 words)

	Betti biography
		Betti studied mathematics and physics at the University of Pisa and while there he was taught by Mossotti.
		Betti graduated with a degree in mathematics in 1846 and, following this, he was appointed as an assistant at the university.
		Betti published a memoir on topology in 1871 which contained what we now call the "Betti numbers".
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Betti.html (1189 words)

	Maple Code
		For the (3,0) Hodge number of the quintic threefold use h(3,0,5).
		For the (3,0) Hodge number of the z-squared eigenspace of a quintic threefold, use h(3,0,5,2).
		The dimension of the middle primitive cohomology vector space is given by b0(n,d) where n is the dimension and d is the degree.
	www.math.utah.edu /~carlson/research/maple/hodgepack.shtml (293 words)

	Betti number
		The Betti number is the maximum number of cuts that can be made without dividing the surface into two separate pieces.
		The Betti number of a square is 0 because it is impossible to crosscut without leaving two pieces.
		A torus, or donut shape, has a Betti number of 2.
	www.daviddarling.info /encyclopedia/B/Betti_number.html (218 words)

	[No title]
		A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space.
		There is a result due to Milnor[1] that gives an upper bound, k(2k-1)^{n-1}, for the sum of the (mod 2) Betti numbers of V, in particular this gives a bound for the zeroth Betti number which equals the number of connected components.
		There is a famous old paper of Milnor, part of which was done independently by Thom, in which he establishes that the total of the Betti numbers of V is at most k(2k-1)^(n-1), independent of m.
	www.math.niu.edu /~rusin/known-math/01_incoming/hilb16 (878 words)

	[No title]
		C(epsilon), for example, is one more than the number of MST edges that are longer than epsilon, and I(epsilon) is the the number of NNG edges that are longer than epsilon.
		The k-th Betti number, b_k, is defined to be the rank of H_k, so it gives us the number of non-equivalent non-bounding k-dimensional cycles.
		For subsets of 3-dimensional space, b_1 is (roughly speaking) the number of open-ended tunnels, and b_2 is the number of enclosed voids.
	www.cs.colorado.edu /~lizb/topology.html (2571 words)

	Saugata Basu (Site not responding. Last check: 2007-11-02)
		On the number of homotopy types of fibres of a definable map, (with N. Vorobjov), to appear in the Journal of the London Mathematical Society.
		Computing the first Betti number and the connected components of semi-algebraic sets (with R. Pollack and M-F. Roy), to appear in Foundations of Computational Mathematics, (preliminary version appeared in Proceedings of STOC, 2005).
		On the number of cells defined by a family of polynomials on a variety (with R. Pollack and M.-F. Roy), Mathematika, 43 (1996) 120-126.
	www.math.gatech.edu /~saugata (1279 words)

	Betti number (Site not responding. Last check: 2007-11-02)
		For the most reasonable spaces(such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some points onwards, and consists of natural numbers.
		The Betti numbers do not take into account any torsion in thehomology groups, but they are very useful basic topological invariants.
		The Betti number sequence for a circle is 1, 1, 0, 0, 0,...; for a two- torus is 1, 2,1, 0, 0, 0,..., and for a three- torus is 1, 3, 3, 1, 0, 0, 0,...
	www.therfcc.org /betti-number-35909.html (343 words)

	PlanetMath: genus of topological surface (Site not responding. Last check: 2007-11-02)
		Remark 1 The previous theorem is the reason why genus is sometimes referred to as “the number of handles”.
		Cross-references: homeomorphic, connected sum, property, simple closed curves, cardinality, Betti number, integer, boundary, manifold, connected, compact, equivalent, topologically equivalent, closed, invariant, complete, topology, surfaces, topological invariant
		This is version 26 of genus of topological surface, born on 2002-08-15, modified 2006-09-11.
	planetmath.org /encyclopedia/Genus2.html (242 words)

	[No title]
		The Betti number gives the number of p-dimensional holes of the complex K, and the number p gives the number of p-dimensional turns of K. The physical interpretation which we shall give to the holes inside of the state space of a system will be that of instability fields of the system under consideration.
		In the above discussed example a number was assigned to a quality, that is, a Betti number was assigned to a topological space.
		A stage of development of an organism cannot be considered as a state of the dynamical system, because the number of components of the system and its structure vary from one stage to the next.
	www.panmere.com /rosen/mhout/doc00003.doc (3550 words)

	Euler characteristic - Wikipedia, the free encyclopedia
		In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological space's shape or structure.
		For closed smooth manifolds, the Euler characteristic coincides with the Euler number, i.e., the Euler class of its tangent bundle evaluated on the fundamental class of a manifold.
		Any contractible space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0.
	en.wikipedia.org /wiki/Euler_characteristic (1168 words)

	Reidemeister torsion in circle-valued Morse theory (Site not responding. Last check: 2007-11-02)
		This project started when Cliff Taubes pointed out that his "Seiberg-Witten=Gromov" correspondence suggests that the Seiberg-Witten invariant of a three-manifold with positive first Betti number should be computable by counting gradient flow lines and closed orbits of a harmonic circle-valued function.
		We show that by suitably counting closed orbits and flow lines between critical points of the gradient of a circle-valued Morse function on a manifold, one recovers a form of topological Reidemeister torsion.
		On a three-manifold with positive first Betti number, we conjecture that a finer version of the Morse theory invariant is equal to the Seiberg-Witten invariant, by analogy with Taubes' ``Seiberg-Witten = Gromov'' theorem in four dimensions.
	math.berkeley.edu /~hutching/pub/rt.html (529 words)

	[No title]
		Jordan proved that the number of circuits in a complete independent set is a topological invariant of the surface.
		In 1871, Enrico Betti (1823-92) published a memoir on topology which included what are now labeled "Betti numbers", so named by Henri Poincaré; (x-y), who was inspired to work in topology by Betti's work.
		For each dimension of possible simplexes in a complex, P. defined the number of independent cycles of that dimension as the Betti number of that dimension.
	members.fortunecity.com /jonhays/tophistory.htm (1270 words)

	Computational Homology Project (Site not responding. Last check: 2007-11-02)
		Betti numbers are part of the information contained in the homology groups of a topological space which intuitively measure the number of connected components, the number of holes, and the number of enclosed cavities in low dimensions.
		To illustrate what these two Betti numbers measure, we have colored the cubes in shades of red according to which connected component of the space the cube belongs.
		=1059 is that this set has a large number of tunnels, like the interior of a cylinder, that wind through the space.
	www.math.gatech.edu /~chomp/software (1604 words)

	Springer Online Reference Works
		Cohomology of algebraic varieties with coefficients in a field of characteristic zero, with formal properties required to obtain the Lefschetz formula for the number of fixed points.
		is a prime number distinct from the characteristic of the field
		, which may also be interpreted as the number of fixed points of the endomorphism
	eom.springer.de /w/w097600.htm (292 words)

	silvello betti - ResearchIndex document query
		It measures the intersection homology Betti numbers of the toric variety associated with a
		Betti numbers Let k be a eld, and consider the
		polynomial A(t) det(tl[h,namely the Betti number b(L) bn-1 (Lj,equals the number of
	citeseer.ist.psu.edu /cis?q=Silvello+Betti (739 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		Let A be the free abelian group of rank 2 with generators a,b.
		Form the split extension G of A by C with this action, so A is a normal subgroup of G with G/A isomorphic C. This is a finitely presented group with first Betti number 0 and second Betti number 1.
		An example of a finitely generated torsion-free virtually abelian group (all finitely generated virtually abelian groups are finitely presented) with first Betti number 0 and second Betti number 1 is given in Example 4.7 on p.
	www.lehigh.edu /~dmd1/pl213 (198 words)

	Bryan Clair
		We prove upper bounds on the Betti numbers for regular coverings of X, sublinear in the order of covering.
		The bounds are sensitive to the Novikov-Shubin invariants of X, and are improved in the presence of a spectral gap.
		-Betti numbers of a residually amenable covering space are the limit of the L
	euler.slu.edu /~clair/math.html (487 words)

	Klein 2-Geometry V \| The n-Category Café
		My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser.
		Several indications suggested the same holds for the zeroth Betti number.
		First, if we require sub 2-vector spaces to have complements, and we knew what happens to Betti numbers when 2-vector spaces are added, this would set us the right way concerning the issue in the previous paragraph.
	golem.ph.utexas.edu /category/2006/09/klein_2geometry_v.html (4229 words)

	Seaman: The third Betti number of a positively pinched riemannian six manifold
		Seaman: The third Betti number of a positively pinched riemannian six manifold
		The third Betti number of a positively pinched riemannian six manifold.
		then its third (real) Betti number is zero.
	www.numdam.org /numdam-bin/item?id=AIF_1986__36_2_83_0 (131 words)

	betti - OneLook Dictionary Search
		We found 2 dictionaries with English definitions that include the word betti:
		Tip: Click on the first link on a line below to go directly to a page where "betti" is defined.
		Phrases that include betti: betti number, ugo betti, betti cohomology, betti reaction, betti ugo, more...
	www.onelook.com /?w=betti (86 words)

	HSX Prediction Market: MovieStocks® : The Notorious Bettie Page
		Gretchen Mol will play pin-up queen Bettie Page in a biopic directed by Mary Harron.
		The Ballad of Bettie Page centers on the fetish model who took the '50s by storm with her brazen style and uncompromising poses.
		Protected by U.S. Patent Number 5,950,176 U.S. Patent Number 6,505,174 and other pending patents.
	movies.hsx.com /servlet/SecurityDetail?symbol=BETTI (137 words)

	Richard Pollack at MSRI - Betti number bounds and their applications (Site not responding. Last check: 2007-11-02)
		Richard Pollack at MSRI - Betti number bounds and their applications
		Richard Pollack - Betti number bounds and their applications
		A PDF version of the lecture notes is available here.
	www.msri.org /publications/ln/msri/2003/canddgeom/pollack/1/index.html (30 words)

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