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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 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.
		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.
	en.wikipedia.org /wiki/Torus (687 words)

	Algebraic torus - Wikipedia, the free encyclopedia
		In mathematics, an algebraic torus is a particular kind of algebraic group, that becomes of the simple form
		Over a field K that is not algebraically closed there exist algebraic tori that are not isomorphic over K to such a product (called a split torus).
		Each algebraic torus is dual to a Galois module, its algebraic group homomorphisms to GL
	en.wikipedia.org /wiki/Algebraic_torus (169 words)

	Abelian variety - Wikipedia, the free encyclopedia
		A complex torus of dimension g is a torus of real dimension 2 g that carries the structure of a complex manifold.
		A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure.
		Every algebraic curve C of genus g â¥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.
	en.wikipedia.org /wiki/Abelian_variety (1494 words)

	Torus [Definition]
		The torus can also be described as a quotient In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.
		The first homology group In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group).
		The k -th homology group In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group).
	www.wikimirror.com /Torus (3492 words)

	Algebraic groups (Site not responding. Last check: 2007-10-14)
		For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i.e.
		In case n is 1 this notion is the same as that of elliptic curve, and every complex torus gives rise to such a curve, for n > 1 it has been known since Bernhard Riemann that the algebraic variety condition imposes extra constraints on a complex torus.
		An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods, equivalently, it is a function in the function field of an abelian variety.
	read-and-go.hopto.org /Algebraic-groups (436 words)

	Torus - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-14)
		In geometry, a torus is a solid of revolution generated by revolving a circle about an axis coplanar with the circle.
		According to the broadest definition, the generator of a torus need not be a circle but could also be an ellipse or any other conic section.
		In nuclear physics a torus is a large fusion reactor which is very roughly the shape of an elliptical torus.
	www.indexlistus.de /keyword/Torus.php (233 words)

	Foundations of Algebraic Topology
		Algebraic methods are used to solve for the transform of the solution to the differential equation.
		For the torus the equivalence classes correspond to the number of times the loop passes through the hole of the torus in a designated direction.
		Clearly the torus is not topologically equivalent to the sphere because their first homotopy groups are different.
	www.sjsu.edu /faculty/watkins/algtop.htm (1167 words)

	PlanetMath:
		algebraically independent (in algebraically dependent) owned by mathcam
		algebraic dependence (in algebraically dependent) owned by mathcam
		algebraic numbers may be set in a sequence (= algebraic numbers are countable) owned by pahio
	planetmath.org /encyclopedia/A (1968 words)

	John Greenlees's Preprint Archive
		R.Bruner and J.P.C.Greenlees ``The algebraic Bredon-Loffler conjecture.'' 13pp
		A complete algebraic model is given for the category of rational S^1 spectra is given in Part I. In Part II the algebraic counterparts of various change of groups functors are described.
		We study connected graded algebra over a field which satisfy a local cohomology theorem (such as the cohomology ring of a group which is a discrete or profinite virtual duality group or a compact Lie group).
	www.shef.ac.uk /~pm1jg/preprints.html (2758 words)

	Taras E. Panov (English)
		Algebraic and differential topology (cobordism theories and genera, torus actions, homotopy theory), algebraic geometry (toric varieties), combinatorics and combinatorial geometry (polytopes, complexes, arrangements).
		Torus actions and combinatorics of polytopes (with Victor M. Buchstaber, in Russian), Trudy Matematicheskogo Instituta imeni V.A.Steklova 225 (1999), 96--131; English translation in: Proceedings of the Steklov Institute of Mathematics 225 (1999), 87--120; arXiv: math.AT/9909166.
		Torus actions and combinatorics of polytopes (in Russian), Trudy Matematicheskogo Instituta imeni V.A.Steklova 225 (1999), 96-131; English translation in: Proceedings of the Steklov Institute of Mathematics 225 (1999), 87-120; arXiv: math.AT/9909166.
	higeom.math.msu.su /people/taras/english.html (1515 words)

	Algebraic curves (Site not responding. Last check: 2007-10-14)
		The genus of a knot of a knot (mathematics) K is defined as the minimal genus of all Seifert surfaces for K. The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded Disk (mathematics) without rendering the resultant manifold disconnected.
		There is a definition of genus of any algebraic curve C. When the field of definition for C is the complex numbers, and C has no tangent space, then that definition coincides with the topological definition applied to the Riemann surface of C (its manifold of complex points).
		The definition of elliptic curve from algebraic geometry is non-singular curve of genus 1.
	read-and-go.hopto.org /Algebraic-curves (248 words)

	Elliptic Curves and Elliptic Functions
		For every algebraic function, it is possible to construct a specific surface such that the function is "single-valued" on the surface as a domain of definition.
		One kind is a type of algebraic variety with the technical property of being "complete", called an abelian variety, since the group operation (it turns out) must be commutative.
		The other kind is a linear algebraic group, which is (isomorphic to) an algebraic subgroup of a general linear group - i.
	cgd.best.vwh.net /home/flt/flt03.htm (3548 words)

	Citebase - Algebraic Quantization on the Torus and Modular Invariance (Site not responding. Last check: 2007-10-14)
		Citebase - Algebraic Quantization on the Torus and Modular Invariance
		Algebraic Quantization on the Torus and Modular Invariance
		are described in the framework of an algebraic quantization procedure on a group.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9702004 (624 words)

	[No title]
		The main mathematical input is the model for SO(2)-equivariant theories* * given in Part I of [4], and the analysis of O(2)-equivariant cohomology theories in [3],* * together with special cases of results on rational Mackey functors from [2].
		We may choose a maximal torus T = SO(2) consisting of rotations around the z-* axis, and hence its normalizer N = O(2), which includes all half turns with axis in the x *y-plane.
		We proved in 6.1 that the algebraic model of (G; c)-spectra is a full subcate* *gory of the al- gebraic model of (N; c)-spectra, given in [3].
	hopf.math.purdue.edu /Greenlees/so3q.txt (8061 words)

	Analytic Jacobians of Hyperelliptic Curves (Site not responding. Last check: 2007-10-14)
		The existence of a Riemann form on a torus is a necessary and sufficient condition for the torus to be embeddable into projective space.
		The analytic Jacobian as a torus equals C^g/Lambda, where Lambda is the Z-lattice spanned by the columns of the period matrix.
		Returns the endomorphism ring, as a matrix algebra, of the given analytic Jacobian A. Suppose the analytic Jacobian has big period matrix P. The second return value is a list of alpha-matrices such that alpha P = P M, for each generator, M, of the matrix algebra.
	mad.epfl.ch /magma/text1223.htm (3554 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		aperiodic dense partition of torus by the 1-dimensional subsets.
		Still the "algebraic picture" is surprisingly tractible: the coordinate ring of the leaf space is represented by an AF C*-algebra whose K-group is an ordered free abelian group of rank 2.
		In general, given non-commutative AF algebra one can take its Bratteli diagram, whose set of infinite paths can be identified with the leaf space of a lamination on some surface.
	www.math.ucalgary.ca /~nikolaev/statement.html (2591 words)

	Torus
		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.
		In topology, a torus or n -torus refers to a product of n circles.
		The torus discussed above is the 2-torus -- the product of just two circles.
	www.brainyencyclopedia.com /encyclopedia/t/to/torus.html (396 words)

	Letter to Nicolaas Kuiper
		The easiest way to see this is to consider a (3,2) knot, and observe that it projects stereographically to a (3,2) torus knot from (0,0,0,1) and to a (2,3) torus knot from (0,1,0,0).
		When we do so, we find that in general the strip is immersed except at a number of points where "twisting" occurs, as well as a finite number of places where the planar projection of the original curve has a double point.
		In the case of the torus knots above, the planar projection of the original curve will be locally convex, so the self- linking number of the image of the curve in 3-space can be read off as the algebraic number of crossing points.
	www.geom.uiuc.edu /~banchoff/self-linking/NK.html (983 words)

	Simple Groups of Finite Morley rank
		In algebraic groups the 2-elements are unipotent if the characteristic is 2, and semisimple otherwise; so in characteristic 2 one expects T=1, and otherwise one expects U=1.
		A minimal counterexample to the Algebraicity Conjecture would be a K-group; conversely, a K-group is either a K-group or a minimal counterexample to the algebraicity conjecture.
		In the odd type case, any simple K-group of finite Morley rank is either algebraic*, or is minimal connected simple with Prüfer 2-rank at most two.
	www.rci.rutgers.edu /~cherlin/FMR (556 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		The purpose of this course is to build a dictionary between concrete convex objects resembling simplices and modern algebraic geometry.
		A toric variety is generally speaking a normal variety X containing a torus T as an open subset.
		The torus T must act on X extending the action T x T --> T. Toric varieties are truly amazing objects where real and non-trivial computations are possible just using elementary convex geometry.
	home.imf.au.dk /niels/toric.html (163 words)

	Algebraic Topology: Two-dimensional manifolds
		The basic examples are the sphere, let us call it M(0), the torus, let us call it M(1), and the real projective plane, let us call it N(1).
		Now the sphere M(0) and the torus M(1) are orientable, but the real projective plane N(1) is not.
		In the above we saw that nonhomeomorphic compact 2-manifolds are distinguished by their fundamental group.
	homepages.cwi.nl /~aeb/at/algtop-4.html (990 words)

	Torus - free-definition (Site not responding. Last check: 2007-10-14)
		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.
		Topologically, a torus is defined as product of two circles : S
		The torus can also be desribed as a quotient of the Euclidean plane under the identifications
	www.free-definition.com /Torus.html (526 words)

	SIMMER March 1998 Presentation Topic
		The torus is a doughnut-shaped surface formed by taking a cylinder and joining the two circular ends together.
		For example, the characteristic of a sphere is 2, while that of a torus is 0.
		Problem 6: Show that the two-holed torus and the connected sum of the one-holed torus and the projective plane are not topologically equivalent.
	www.math.toronto.edu /mathnet/simmer/topic.mar98.html (683 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		It should be possible to present algebraic topology from this point of view, but none of the books seem to do it.
		The generators for the cohomology of the torus are two "picket fences" which look like --->^ / / / /-->^ / / / /-->^ / / / / Wrap one of these each way around the torus.
		To quote Bott and Tu (Differential Forms in Algebraic Topology, page 4), "One of the hallmarks of a topologist is a sound intuition for the coboundary operator." The introduction to their book, if you can understand it, is fantastic.
	www-swiss.ai.mit.edu /users/dae/notes/cohomology (1390 words)

	Math7800: Toric geometry (Site not responding. Last check: 2007-10-14)
		This course is aimed at students of algebraic geometry, but geometers of any flavour may find the subject interesting, particularly given the applications to mirror symmetry.
		Lectures 6 and 7: The torus action on a toric variety.
		A toric variety is decomposed into torus orbits, the E-polynomial computes Hodge-Deligne numbers and the orbit closures are constructed.
	www.math.utah.edu /~craw/math7800 (366 words)

	Allen Hatcher's Homepage (Site not responding. Last check: 2007-10-14)
		This is the first in a series of three textbooks in algebraic topology having the goal of covering all the basics while remaining readable by newcomers seeing the subject for the first time.
		This is intended to be a readable introduction to spectral sequences, with emphasis on their applications to algebraic topology.
		Further in the algebraic direction one could talk about quadratic extensions of Q, and maybe even elliptic curves.
	www.math.cornell.edu /~hatcher (1123 words)

	torus - OneLook Dictionary Search
		Phrases that include torus : torus fracture, mandibular torus, maximal torus, supraorbital torus, torus tubarius, more...
		Words similar to torus : tore, tori, toroid, more...
		This is a OneLook Word of the Day, which means it might be in the news.
	www.onelook.com /?w=torus&loc=wotd (450 words)

	POV-Ray: Documentation: 2.7.7.2 Internal Functions
		An algebraic cylinder is what you get if you take any 2d curve and plot it in 3d.
		The Ovals of Cassini are a generalization of the torus shape.
		The "Torus Gumdrop" surface is something like a torus with a couple of gumdrops hanging off the end.
	www.povray.org /documentation/view/3.6.1/448 (3721 words)

	Toric Varieties and Algebraic Monoids (ResearchIndex)
		Abstract: Introduction Let K be a field and K = Gm the multiplictive group of K viewed as an algebraic group over K.
		A K -torus is an algebraic K -group isomorphic to a direct product of such groups.
		[5] and also [4], [8]) one studies algebraic varieties which occur as closures of torus orbits for algebraic actions of a torus on an algebraic variety.
	citeseer.ist.psu.edu /neeb92toric.html (303 words)

	Monomial - Enpsychlopedia (Site not responding. Last check: 2007-10-14)
		This can be phrased in the language of algebraic groups, in terms of the existence of a group action of an algebraic torus (equivalently by a multiplicative group of diagonal matrices).
		This area is studied under the name of torus embeddings.
		In group representation theory, a monomial representation is a particular kind of induced representation.
	www.grohol.com /psypsych/Monomial (342 words)

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