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	Torus - Wikipedia, the free encyclopedia
		In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle.
		Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole.
		The first homology group of the torus is isomorphic to the fundamental group (since the fundamental group is abelian).
	en.wikipedia.org /wiki/Torus (908 words)

	Torus - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-01)
		In geometry, a torus (pl. tori) is a doughnut shaped solid of revolution generated by revolving a circle about an axis coplanar with the circle.
		The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the circle.
		This is due in part to the fact that in any compact Lie group one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension.
	www.encyclopedia-online.info /Torus (347 words)

	Maximal torus - Wikipedia, the free encyclopedia
		In a compact Lie group G there is to be found a maximal torus T; that is, a closed subgroup that is a torus, and of the largest possible dimension.
		For example, the Lie group SO(3) of rotations in three dimensions has as maximal torus T a circle group (a 1-torus, that is).
		According to general theory, all the maximal tori form a single conjugacy class of subgroups.
	en.wikipedia.org /wiki/Maximal_torus (296 words)

	Torus (Site not responding. Last check: 2007-11-01)
		In geometry, a torus (pl. tori) is a solid of revolution generated by revolving a circle about an axis coplanar with the circle.
		The sphere is aspecial case of the torus obtained when the axis of rotation is a diameter of thecircle.
		This is due in part to the fact that in any compact Lie group one canalways find a maximal torus ; that is, a closed subgroup which is a torus of the largest possible dimension.
	www.therfcc.org /torus-106452.html (327 words)

	Current Projects: Torus Hitting Times
		A consequence of a theorem on Q and L in [3] is that for the torus (Z
		Also, in the torus, a vertex on the boundary is connected to the vertex on the opposite boundary.
		Maximal mean commute time is the maximum over all mean commute times of all pairs of vertices.
	math.iit.edu /~rellis/comb/torus/torus.html (782 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		By a maximal 2-discrete torus T~ in a 2-compact group X, we mean the discrete approximation [9, 6.4] to a maximal 2-compact torus ^Tin X [9, 8.9] [10, 2.15].
		Any torus T gives a lattice ß1T ; conversely, a lattice L gives a torus T (1) L. These two constructions are inverse to one another up to natural isomorphism, and induce an equivalence between the category of tori (with continuous homomorphisms) and the category of lattices.
		A reflection torus is a torus T together with a finite subgroup W of Aut (T) which is generated by the reflections it contains.
	hopf.math.purdue.edu /Dwyer-Wilkerson/normaltorus/NT.txt (14335 words)

	Torus
		In geometry, a torus (pl. tori) is a doughnut -shaped surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does...
		An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left...
		An (ordinary) torus is a surface having genus one, and therefore possessing a single " hole...
	www.logicjungle.com /wiki/Torus (286 words)

	Definition of maximal
		to the maximally even diatonic scale but are not maximally even with regards to the chromatic scale.
		Then the whole set has ''a'' as a single maximal element that is not the greatest element.
		3:...le [[Frattini subgroup]], the intersection of the maximal subgroups.
	www.wordiq.com /search/maximal.html (739 words)

	Maximal torus (Site not responding. Last check: 2007-11-01)
		In the theory of Lie groups in mathematics especially those that are compact a role is played by the torus groups.
		In a compact Lie group G there is to be found a maximal torus T ; that is a closed subgroup that is a torus and of largest possible dimension.
		In those one can easily find explicit parameter angles the maximal torus: that is commuting one-parameter of rotations exhibiting the torus as a of circle groups.
	www.freeglossary.com /Maximal_torus (632 words)

	[No title]
		For SL(n,C) the obvious choice of maximal torus consists of diagonal matrices * 0 0 0 0 * 0 0 0 0 * 0 0 0 0 * where the diagonal entries are unit complex numbers that multiply to one.
		Now, a maximal flag is a rather fancy type of figure, built from a bunch of simpler ones satisfying a bunch of incidence relations.
		The smallest parabolic subgroup is B itself, and G/B is the space of "maximal flags".
	math.ucr.edu /home/baez/twf_ascii/week178 (3254 words)

	Characteristic class - Wikipedia, the free encyclopedia
		The characteristic class approach to curvature invariants was a particular reason to make a theory, to prove a general Gauss-Bonnet theorem.
		When the theory was put on an organised basis around 1950 (with the definitions reduced to homotopy theory) it became clear that the most fundamental characteristic classes known at that time (the Stiefel-Whitney class, the Chern class, and the Pontryagin classes) were reflections of the classical linear groups and their maximal torus structure.
		What is more, the Chern class itself was not so new, having been reflected in the Schubert calculus on Grassmannians, and the work of the Italian school of algebraic geometry.
	en.wikipedia.org /wiki/Characteristic_class (755 words)

	Torus (Site not responding. Last check: 2007-11-01)
		In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle.
		The surface area and interior volume of this torus are given by :
		The torus can also be described as a quotient of the Euclidean plane under the identifications :(''x'',''y'') ~ (''x''+1,''y'') ~ (''x'',''y''+1) Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon
	torus.area51.ipupdater.com (567 words)

	Abstracts
		An amoeba is the image of a curve under this torus action.
		A basic problem in algebraic geometry is the construction of moduli spaces -- spaces parametrizing a fixed type of variety (e.g., curves of genus g) or some type of object on a fixed variety (e.g., vector bundles on a fixed surface).
		When the variety in question is affine space, and the group is a torus, the GIT quotient is called a toric variety.
	math.stanford.edu /~vakil/snowbird/abstracts.html (954 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		A key feature of the theory of compact Lie groups is the relationship between centers and fundamental groups; these play off against one another, at least in the semisimple case, in that the center of the simply connected form is the fundamental group of the adjoint form.
		There are explicit ways to compute the center or fundamental group of a compact Lie group in terms of the normalizer of the maximal torus.
		If $p=2$, then up to factors which do not contribute to $\pi_1X$, the normalizer of the torus in $X$ is derived by $\Ftwo$-completion from the normalizer $\NT_G$ of a maximal torus $T_G$ in a connected compact Lie group~$G$.
	math.wesleyan.edu /~mhovey/archive/LATEST (1427 words)

	AMCA: Examples of affine maximal torus fibrations of a compact Lie group by Marcos Salvai
		is a maximal torus of G. Equivalently, it is a maximal connected totally geodesic flat submanifold of G, provided that the group is endowed with a bi-invariant Riemannian metric.
		S. Given an affine Weyl chamber C, there exists a unique affine maximal torus S such that C is contained in TS.
		Fix a maximal torus T. The set T of all W-oriented affine maximal tori of G may be identified in a natural way with (G/T) ×(G/T).
	at.yorku.ca /c/a/d/q/81.htm (728 words)

	Algebraic torus (Site not responding. Last check: 2007-11-01)
		In mathematics an algebraic torus is a particular kind of algebraic group that becomes of the simple form
		They were so named by analogy the theory of tori in Lie group theory (see maximal torus).
		Each torus is dual to a Galois module its algebraic group homomorphisms to GL
	www.freeglossary.com /Algebraic_torus (321 words)

	Symplectic geometry seminar (Site not responding. Last check: 2007-11-01)
		In equivariant symplectic geometry, it is often easier to work with actions of a torus rather than a compact Lie group.
		In many cases, after understanding the torus case, there is an additional step to pass from a compact Lie group to its maximal torus.
		In this talk we show how to compute the K-theory of the symplectic reduction with respect to a Lie group G in terms of the reduction with respect to its maximal torus T. This is the K-theoretic version of results of Shaun Martin in the symplectic case, or Michel Brion in the algebraic geometry setting.
	www.math.toronto.edu /symplec/fall05/sem102805.html (160 words)

	Characteristic classes (Site not responding. Last check: 2007-11-01)
		The characteristicclass approach to curvature invariants was a particular reason to make a theory,to prove a general Gauss-Bonnet theorem.
		When the theory was put on an organised basis around 1950 (with the definitions reduced to homotopy theory) it became clearthat the most fundamental characteristic classes known at that time (the Stiefel-Whitney class,the Chern class, and the Pontryagin classes) were reflections of the classical linear groups and their maximal torus structure.
		What is more, the Chern class itself was not so new, having been reflected in the Schubert calculus on Grassmannians, and the work of the Italian school of algebraicgeometry.
	www.therfcc.org /characteristic-classes-206528.html (421 words)

	Torus - TheBestLinks.com - Conic section, Complex number, Compact space, Circle, ... (Site not responding. Last check: 2007-11-01)
		Torus - TheBestLinks.com - Conic section, Complex number, Compact space, Circle,...
		Torus, Conic section, Complex number, Compact space, Circle, Diameter, Ellipse...
		You can add this article to your own "watchlist" and receive e-mail notification about all changes in this page.
	www.thebestlinks.com /Torus.html (379 words)

	Doris Schattschneider Abstract (Site not responding. Last check: 2007-11-01)
		A key factor in the proofs of the theorems they obtained was the Galois group G(K/k) (K a splitting field for a maximal torus of G).
		These conditions are helpful in the classification of maximal k-trivial tori of the classical groups, and a partial list of possible maximal k-trivial tori of the classical groups appears in this paper at the end of section IV.
		Also, by considering the case of an arbitrary subtorus of a semi-simple algebraic group G, without any reference to the ground field of definition we obtain theorems which have as corollaries some of the results of Satake and Tits.
	www.agnesscott.edu /lriddle/women/abstracts/schattschneider_abstract.htm (151 words)

	Fachbereichskolloquium (Site not responding. Last check: 2007-11-01)
		For example a p-compact group has a maximal torus, a maximal torus normalizer and a Weyl group.
		For odd primes p, we give a classification of p-compact groups by proving that they are determined by their maximal torus normalizers.
		A number of corollaries follow easily from this classification, for example we give an affir-mative answer to the maximal torus conjecture for finite loop spaces up to Z[1/2]-localization.
	www.mathematik.uni-kassel.de /vortraege/20020507.html (146 words)

	week178
		First, every complex simple Lie group G has a bunch of maximal compact subgroups, all of which are isomorphic via conjugation inside G. People often pick one, call it "the" maximal compact subgroup, and denote it by K. But don't be fooled: there are lots!
		Third, G always has a bunch of maximal solvable subgroups, which again are all isomorphic by conjugation inside G. In case you forgot: a group B is "solvable" if when you take the subgroup B1 generated by commutators
		A maximal solvable subgroup of G is also called a "Borel" subgroup, and it's denoted B. When G = SL(n,C), an obvious choice for B is the group of upper triangular matrices with determinant 1:
	math.ucr.edu /home/baez/week178.html (3470 words)

	Lie algebra
		An abelian subalgebra of a Lie algebra is often called a torus.
		A maximal torus is also called a Cartan subalgebra.
		A maximal solvable subalgebra is called a Borel subalgebra.
	www.fastload.org /li/Lie_algebra.html (1014 words)

	LECTURES ON LIE GROUPS (Site not responding. Last check: 2007-11-01)
		There is a proper balance between, and a natural combination of, the algebraic and geometric aspects of Lie theory, not only in technical proofs but also in conceptual viewpoints.
		For example, the orbital geometry of adjoint action, is regarded as the geometric organization of the totality of non-commutativity of a given compact connected Lie group, while the maximal tori theorem of É.
		Cartan and the Weyl reduction of the adjoint action on G to the Weyl group action on a chosen maximal torus are presented as the key results that provide a clear-cut understanding of the orbital geometry.
	www.worldscibooks.com /mathematics/3835.html (159 words)

	Passman's Abstracts
		If S is a Sylow p-subgroup of G and if Fv is the unique line in V stabilized by S, then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus T on Fv.
		We also consider simple Artinian rings with involution, in characteristic not 2, and we determine those bounded Z-filtrations that are maximal subject to being stable under the action of the involution.
		Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras.
	www.math.wisc.edu /~passman/abstracts.html (3706 words)

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