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Topic: Euler characteristic

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Euler numbers

Eulers formula in complex analysis

Betti number

Homology (mathematics)

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Leonhard Euler

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Singular homology

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	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.
		The Euler characteristic was originally formulated for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids.
		The concept of the Euler characteristic of a bounded finite poset is another generalization, important in combinatorics.
	en.wikipedia.org /wiki/Euler_characteristic (1138 words)

	ipedia.com: Euler characteristic Article (Site not responding. Last check: 2007-10-20)
		The Euler characteristic of a simply-connected manifold such as a sphere or a plane is 2.
		With this definition, circles and squaress have Euler characteristic 0 and solid balls have Euler characteristic 1.
		For Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature, see the Gauss-Bonnet theorem for two-dimensional case and generalized Gauss-Bonnet theorem for general case.
	www.ipedia.com /euler_characteristic.html (676 words)

	Euler's Formula
		Levitt, "The Euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes", Disc.
		Rocek and Williams, "On the Euler characteristic of piecewise linear manifolds", Phys.
		Abstract: The Dehn-Sommerville relations and the corresponding equations for the angle sums are used to derive two expressions for the Euler characteristic of a simplicial manifold, firstly in terms of the numbers of even dimensional subsimplices, and secondly in terms of even-dimensional deficit angles.
	www.ics.uci.edu /~eppstein/junkyard/euler/refs.html (1096 words)

	Euler characteristic (Site not responding. Last check: 2007-10-20)
		DC MetaData for: A notion of Euler characteristic for fractals...
		On the equivalence of the tube and Euler characteristic methods for the distribu...
		The K-homology class of the Euler characteristic operator is trivial, by Jonatha...
	www.scienceoxygen.com /math/673.html (262 words)

	Glossary: Euler Characteristic (Site not responding. Last check: 2007-10-20)
		The Euler characteristic of a closed surface is a topological invariant that can be computed in several ways.
		Two important ones are by counting critical points (the Euler characteristic is the number of maxima and minima minus the number of saddles) and by counting vertices, edges and faces of a polyhedral surface (the Euler characteristic is the number of vertices and faces minus the number of edges).
		That is, given the Euler characteristic and orientability of a surface, the topological type of the surface is determined.
	www.geom.uiuc.edu /docs/dpvc/Glossary/EulerChar.html (111 words)

	Descartes 4 (Site not responding. Last check: 2007-10-20)
		In honor of Euler's discovery that this number is always 2 for a convex polyhedron, it is called the Euler characteristic of the space.
		The alternating sum is 0, which is the Euler characteristic of this surface.
		The point of the topological invariance of the Euler characteristic is that any dissection of that surface will give the same answer.
	math.sunysb.edu /~tony/whatsnew/column/descartes-0899/descartes4.html (327 words)

	Euler characteristic
		In algebraic topology we learn that the Euler characteristic of a space is the alternating sum of the "ranks of the its rational homology groups", and we learn how to compute these using any way of chopping up the space into convex polytopes (or "cells").
		The Euler characteristic is really a generalization of cardinality that allows negative numbers as values, so this other stuff is a further generalization that allows non-integral cardinalities.
		Here's the abstract of my talk: Euler Characteristic versus Homotopy Cardinality Just as the Euler characteristic of a space is the alternating sum of the dimensions of its rational cohomology groups, the homotopy cardinality of a space is the alternating product of the cardinalities of its homotopy groups.
	www.lns.cornell.edu /spr/2003-07/msg0052809.html (623 words)

	The Euler Characteristic of an Annulus is Zero (Site not responding. Last check: 2007-10-20)
		The Euler characteristic of an annulus is zero: X = V - E + F = 8 - 12 + 4 = 0
		Edge A-E is used twice but it is counted only one time for the purpose of calculating the Euler characteristic -- as each inclined edge in Figure 3 was counted once, though each was shared by two polygons.
		The Euler characteristic of an annulus is zero: X = V - E + F = 8 - 9 + 1 = 0
	www.joegeluso.com /math/euler-characteristic-of-an-annulus.html (453 words)

	Boundary corrections for the expected Euler characteristic (Site not responding. Last check: 2007-10-20)
		This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of `holes'.
		For high thresholds the Euler characteristic is a measure of the number of peaks.
		The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region.
	www.math.mcgill.ca /keith/morse/morse.abstract.html (255 words)

	Euler characteristic (Site not responding. Last check: 2007-10-20)
		The Euler characteristic of a polyhedron is V - E + F where V, E, and F are respectively the numbers of vertices, edges, and faces.
		(The first word is pronounced "oiler"; see Leonhard Euler).
		All is still licensed under the GNU FDL.
	www.termsdefined.net /eu/euler-characteristic.html (467 words)

	The Euler Characteristic of a Torus is Zero (Site not responding. Last check: 2007-10-20)
		It may be helpful to refer to the web page that discusses the Euler characteristic of an annulus before proceeding here.
		The quantities needed to calculate the Euler characteristic of the object in Figure 2 are listed in the table below.
		The Euler characteristic of a torus is zero: X = V - E + F = 16 - 26 + 10 = 0
	www.joegeluso.com /math/euler-characteristic-of-a-torus.html (340 words)

	Characteristic - Wikipedia, the free encyclopedia
		Characteristic is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.
		An I-V or current-voltage characteristic is the current in a circuit as a function of the applied voltage
		This is a disambiguation page: a list of articles associated with the same title.
	en.wikipedia.org /wiki/Characteristic (113 words)

	"The euler characteristic of a category" (Site not responding. Last check: 2007-10-20)
		For example, the Julia set of any rational function f seems to have an Euler characteristic, a number giving basic information about the dynamical behaviour of f.
		But to define the Euler characteristic of such spaces, we first need to define the Euler characteristic of a category.
		We'll see, for instance, that the Euler characteristic of the category of finite sets and bijections is e = 2.718...
	www.maths.gla.ac.uk /events/seminars/mkabs.php?id=649 (149 words)

	[No title]
		Generalized orbifold Euler characteristics A generalization of physicists' orbifold Euler characteristic (1-2) was give* *n in the introduction in (1-3).
		We recall that for a finite dimensional manifold admi* tting a torus action, it is well known that Euler* characteristic of the fixed point s* ub- manifold is the same as the Euler* characteristic of the original manifold.
		The Euler characteristic of 12 HIROTAKA TAMANOI the fixed point subset under this torus action is precisely given by O(d)p(M; G* *), as in (2-20).
	hopf.math.purdue.edu /Tamanoi/orbifold.txt (7112 words)

	Positive and negative Euler characteristic (Site not responding. Last check: 2007-10-20)
		, since the Euler characteristic of the sphere is 2.
		If they form an orbifold with positive orbifold Euler characteristic, they come from a pattern of symmetry on the sphere.
		Every orbifold with negative orbifold Euler characteristic comes from a pattern of symmetry in the hyperbolic plane with bounded fundamental domain.
	www.geom.uiuc.edu /docs/doyle/mpls/handouts/node36.html (264 words)

	Elementary Amenable Groups and 4-Manifolds with Euler Characteristic 0 (Site not responding. Last check: 2007-10-20)
		We extend earlier work relating asphericity and Euler characteristic for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups.
		In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical.
		Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends ot is virtually an extension of Z by a subgroup of Q, or the manifold is aspherical and the group is virtually poly-Z of Hirsch length 4.
	anziamj.austms.org.au /JAMSA/V50/Part1/Hillman.html (160 words)

	The Euler characteristic of an orbifold
		It is important to keep in mind the distinction between the topological Euler characteristic and the orbifold Euler characteristic.
		The effect on the orbifold Euler characteristic is to subtract
		To complete the connection between orbifold Euler characteristic and symmetry patterns, we would have to verify that each of the possible configurations of parts with orbifold Euler characteristic 0 actually does come from a symmetry pattern in the plane.
	comp.uark.edu /~strauss/sym.2/sym.4.5.html (802 words)

	Validity of the expected Euler characteristic heuristic, Jonathan Taylor, Akimichi Takemura, Robert J. Adler
		We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process f.
		Takemura, A. and Kuriki, S. On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains.
		Worsley, K. Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics.
	projecteuclid.org /getRecord?id=euclid.aop/1120224584 (560 words)

	R: Expected Euler Characteristic for a 3D Random Field (Site not responding. Last check: 2007-10-20)
		Calculates the Expected Euler Characteristic for a 3D Random Field thesholded a level u.
		The Euler Characteristic chi_u (Adler, 1981) is a topological measure that essentially counts the number of isolated regions of the random field above the threshold u minus the number of 'holes'.
		Thus the Type I error of the test can be controlled through knowledge of the Expected Euler characteristic.
	pbil.univ-lyon1.fr /library/AnalyzeFMRI/html/EC.3D.html (246 words)

	Citebase - The universal Euler characteristic for varieties of characteristic zero
		The universal Euler characteristic for varieties of characteristic zero
		Using the weak factorization theorem we give a simple presentation for the value group of the universal Euler characteristic with compact support for varieties of characteristic zero and describe the value group of the universal Euler characteristic of pairs.
		This gives a new proof for the existence of natural Euler characteristics with values in the Grothendieck group of Chow motives.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0111062 (835 words)

	AProPo -- 32 Euler Characteristic (Site not responding. Last check: 2007-10-20)
		The problem is easy if Δ is given by a list of all of its simplices.
		For fixed dimension, one can enumerate all simplices of Δ and compute the Euler characteristic in polynomial time.
		Currently the fastest way to compute the Euler characteristic is to first generate (the Hasse diagram of) V = {S : S is an intersection of facets of Δ} in time O(min{n,m} · α·
	www.zib.de /pfetsch/apropo/euler_characteristic.html (172 words)

	Euler Characteristic versus Homotopy Cardinality (Site not responding. Last check: 2007-10-20)
		Just as the Euler characteristic of a space is the alternating sum of the dimensions of its rational cohomology groups, the homotopy cardinality of a space is the alternating product of the cardinalities of its homotopy groups.
		The Euler characteristic is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive fractional values.
		The two quantities have many of the same properties, but it's hard to tell if they're the same, since like Jekyll and Hyde, they're almost never seen together: there are very few spaces for which the Euler characteristic and homotopy cardinality are both well-defined.
	math.ucr.edu /home/baez/cardinality (218 words)

	Approximate Euler characteristic, dimension, (ResearchIndex)
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		Abstract: We de ne the notion of approximate Euler characteristic of de nable sets of a rst order structure.
		Also, a structure admitting a non-trivial approximate Euler...
	citeseer.ist.psu.edu /703269.html (155 words)

	About "Positive and negative Euler characteristic (Geometry and the Imagination)" (Site not responding. Last check: 2007-10-20)
		Positive and negative Euler characteristic (Geometry and the Imagination)
		In fact, if the order of symmetry is k, then the Euler characteristic of the quotient orbifold is 2/k, since the Euler characteristic of the sphere is 2.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/view/2739.html (79 words)

	About "The Euler Characteristic of an Orbifold (Geometry and the Imagination)" (Site not responding. Last check: 2007-10-20)
		About "The Euler Characteristic of an Orbifold (Geometry and the Imagination)"
		The Euler Characteristic of an Orbifold (Geometry and the Imagination)
		Suppose we have a symmetric pattern in the plane.
	mathforum.org /library/view/2738.html (72 words)

	AProPo -- 32 Euler Characteristic (Site not responding. Last check: 2007-10-20)
		For fixed dimension, one can enumerate all simplices of
		and compute the Euler characteristic in polynomial time.
		Currently the fastest way to compute the Euler characteristic is to first generate (the Hasse diagram of) V = {S : S is an intersection of facets of
	www.zib.de /pfetsch/apropo/HTML3/euler_characteristic.html (130 words)

	Find in a Library: Euler characteristic structure and weight homology
		Find in a Library: Euler characteristic structure and weight homology
		To find this item in a library, enter a postal code, state, province, or country in the field above.
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/d08b481632feca0ea19afeb4da09e526.html (45 words)

	Citebase - Unusual formulae for the Euler characteristic
		Citebase - Unusual formulae for the Euler characteristic
		Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices.
		It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristic.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0201178 (252 words)

	Publikationen (Site not responding. Last check: 2007-10-20)
		Furthermore, we present two non-isometric but isospectral solids that cannot be distinguished by the spectra of their bodies and present evidence that the spectra of their boundary shells can tell them apart.
		Moreover, we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated eigenvalues.
		This fact not only confirms the accuracy of our computed eigenvalues, but also underlines the geometrical importance of the spectrum.
	www.gdv.uni-hannover.de /research/publications.php (5310 words)

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