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Topic: Exterior algebra

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Clifford algebra

Tensor algebra

	PlanetMath: exterior algebra (Site not responding. Last check: 2007-10-10)
		The exterior algebra is also known as the Grassmann algebra after its inventor Hermann Grassmann who created it in order to give algebraic treatment of linear geometry.
		Exterior algebra is also an essential prerequisite to understanding de Rham's theory of differential cohomology.
		This is version 26 of exterior algebra, born on 2002-04-07, modified 2006-09-17.
	planetmath.org /encyclopedia/ExteriorAlgebra.html (1001 words)

	Encyclopedia :: encyclopedia : Exterior algebra (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-10)
		In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace.
		Thus the exterior algebra forms a graded algebra where the grade is given by k.
		Exterior powers find their main application in differential geometry, where they are used to define differential forms.
	www.hallencyclopedia.com.cob-web.org:8888 /topic/Exterior_algebra.html (1242 words)

	C++ page (Site not responding. Last check: 2007-10-10)
		Exterior Algebra also known as Grassmann Algebra is an algebra of vectors.
		The Grassmann algebra is the direct sum of these multivectors up to multivectors of the dimension of the vector space.
		It may be shown that the dimension the Grassmann algebra is two to the nth power where n is the dimension of the vector space.
	www.physics.uci.edu /~rader/project/overview.html (411 words)

	PlanetMath: geometric algebra
		Geometric algebra is a Clifford algebra which has been used with great success in the modeling of a wide variety of physical phenomena.
		The underlying philosophical justification for this is the interpretation that the unit imaginary has geometric significance which naturally arises from the properties of the algebra and the interaction of its various subspaces.
		This is version 12 of geometric algebra, born on 2002-12-17, modified 2006-06-10.
	planetmath.org /encyclopedia/GeometricAlgebra.html (493 words)

	Sets, Clifford Groups and Algebras, and McKay Correspondence
		Since it is not in the same /\27 exterior algebra as the other 3 independent fundamental representations, it should not be a vertex in the same plane as the triangle formed by them, but should be a 4th vertex outside that plane, with all 4 vertices forming a tetrahedron.
		Since it is not in the same /\248 exterior algebra as the 5 pentagon-vertex fundamental representations of E8, it should not be a vertex in the same plane as the pentagon.
		Also, since it is not in the same /\3,875 exterior algebra as the two bipyramid-peak-vertex fundamental representations of E8, it should not be a vertex on the same line as the pentagonal bipyramid axis.
	www.valdostamuseum.org /hamsmith/DCLG-McKay.html (4093 words)

	Algebra -- from Wolfram MathWorld (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-10)
		This is the meaning mathematicians associate with the word "algebra." When there is the possibility of confusion, this field of mathematics is often referred to as abstract algebra.
		An algebra is sometimes implicitly assumed to be associative or to possess a multiplicative identity.
		(Note that linear algebra, which is the study of linear sets of equations and their transformation properties, is not an algebra in the formal sense of the word.) Other more exotic algebras that have been investigated and found to be of interest are usually named after one or more of their investigators.
	mathworld.wolfram.com.cob-web.org:8888 /Algebra.html (457 words)

	Exterior Algebra -- from Wolfram MathWorld
		Exterior algebra is the algebra of the exterior product, also called an alternating algebra or Grassmann algebra.
		linear combinations of the monomials are the elements of an exterior algebra.
		The differential k-forms in modern geometry are an exterior algebra, and play a role in multivariable calculus.
	mathworld.wolfram.com /ExteriorAlgebra.html (243 words)

	CSDC : Cartan's Calculus: The exterior differential (Site not responding. Last check: 2007-10-10)
		This concept is a rule of differentiation that transports an element of a lower dimensional exterior algebra subspace to the next higher dimensional exterior algebra subspace.
		For example, the exterior derivative of an p = N - 1 form carries it into the a p = N form.
		In the Cartan exterior calculus, the operations are the same and are represented by the same exterior derivative, but the operation operates on different algebraic subspaces.
	www22.pair.com /csdc/ed3/ed3fre4.htm (321 words)

	Cartan's Calculus: The exterior product (Site not responding. Last check: 2007-10-10)
		He devised an algebra based upon the concept of exterior multiplication, an idea to be found in the earlier works of Grassman.
		Every element of the Algebra is composed of primitive basis elements of an N dimensional linear vector space, and the 1 dimensional space of functions.
		The next thing that Cartan employed is the concept of the exterior derivative, which is a rule of differentiation that transports an element of a lower dimensional exterior algebra subspace to the next higher dimensional subspace.
	www.uh.edu /~rkiehn/ed3/ed3fre3.htm (386 words)

	Exterior Paints -- Recommendations and Resources (Site not responding. Last check: 2007-10-10)
		In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree.
		The exterior derivative of a differential form of degree ''k'' is a differential form of degree ''k'' + 1.
		In topology, the exterior of a subset ''S'' of a topological space ''X'' is the union of all open sets of ''X'' which do not meet ''S''.
	www.becomingapediatrician.com /health/48/exterior-paints.html (877 words)

	Algebras of electromagnetics
		Discusses tensors, Clifford algebras (spinors are elements of minimal left or right ideals of Clifford algebras, which explains why the word spinor appears so often in Clifford algebra literature) and applications.
		Kot, G. James: "Clifford algebra in electromagnetics", Proceedings of the International Symposium on Electromagnetic Theory, URSI International Union of Radio Science, Aristotle University of Thessaloniki, 25-28 May 1998, Thessaloniki, Greece.
		Algebra of forms with its de Rham operator is a standard example of cohomology in practice.
	users.tkk.fi /~ppuska/elmag_alg.html (2561 words)

	Hypercubes, Hexagons, Halayudha, Hermes, and Plato
		The 8 of 27=8+12+6+1 is the dimension of one half-spinor representation of the D4 Lie algebra Spin(0,8).
		The D4, D5, and E6 Lie algebras are used to construct the D4-D5-E6 physics model.
		The 28 of the center column is the dimension of the D4 Lie algebra Spin(0,8).
	www.valdostamuseum.org /hamsmith/cube.html (2015 words)

	[No title]
		When A is the full Steenrod algebra, this is difficult alrea* *dy for the case E = E(2) = (F2[2; 3;.
		We prove the theorem in the case where A is the mod 2 Steenrod algebra; the proof easily generalizes to any sub-Hopf algebra.
		Assume that z is "not detected" by any exterior algebra E A (i.e., the restriction *E(z) = 0 for all E).
	www.math.purdue.edu /research/atopology/Hopkins-Palmieri/palmieri.txt (1506 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Praslov, Problems and Theorems in Linear Algebra (has a long chapter on multilinear algebra and is more readable) If you are ready for multilinear algebra over a module, try D.
		Newsgroups: sci.math.research Subject: Re: multilinear algebra sources Date: Mon, 02 Mar 1998 23:26:57 +0200 ortiz wrote: > While mathematical litterature on linear algebra abounds in books, few > > references are devoted to multilinear algebra (i mean topics like > tensor > products, tensor algebra and especially symmetric and exterior > algebras).
		The topics such as tensors, spinors and exterior forms etc are dealt, along with stuff that is not to be found in any other book (such as directed integration theory or method of mobiles).
	www.math.niu.edu /~rusin/known-math/98/multilin_alg (358 words)

	CSDC : Cartan's Calculus: the interior product. (Site not responding. Last check: 2007-10-10)
		The exterior operations are defined on a variety for which the constraints of metric have not been defined.
		Another operation in the Cartan exterior algebra is useful.
		Currents are N-1 form structures on the N dimensional exterior algebra dual to the 1-form.
	www22.pair.com /csdc/ed3/ed3fre5.htm (460 words)

	Description (Site not responding. Last check: 2007-10-10)
		XIDEAL extends the method to exterior algebras using algorithms from [2].
		The basis 1-forms for the exterior algebra are automatically extracted from the input.
		The exterior algebra can be based on either an abstract vector space or the cotangent space at some fixed point on a manifold.
	www.uni-koeln.de /REDUCE/xideal/node1.html (408 words)

	Havlicek/Multilinear Algebra (Site not responding. Last check: 2007-10-10)
		Covariant, contravariant and mixed tensors, Classical definition and notation of a tensor in terms of coordinates, Structure tensor of an algebra, Mixed tensor algebra, Universal property of the tensor algebra.
		Exterior powers and p-vectors, Grassmann coordinates of subspaces, Alternation operator, Exterior powers of linear mappings, Exterior algebra, Duality and p-forms, Exterior algebra, Decomposable p-vectors.
		Quadratic forms, Clifford mappings, Cifford mappings and Exterior Algebra, Clifford Algebras, Dimension of a Clifford Algebra, Examples of Clifford Algebras, Structures on a Clifford Algebra.
	www.geometrie.tuwien.ac.at /havlicek/multilineare.html (179 words)

	Algebraic Programming and Differential Forms -- from Mathematica Information Center
		This article describes a general method for linear algebra and exterior algebra computations.
		The article concludes with an application to exterior algebras with differential forms.
		Included is the package DifferentialForms.m, a package for symbolic computation with exterior differential forms in R^n.
	library.wolfram.com /infocenter/Articles/2324 (163 words)

	[No title]
		The exterior derivative of (1/2)(x d[y] - y d[x]) is computed \ d[ (1/2)(x d[y] - y d[x]) ].
		That is d[expr] \ calculates the exterior derivative of expr provided the head of expr is not Symbol.
		The procedure works for any change of coordinates." Orientation::usage= "Orientation[metric] calculates a unit length generator of the top \ dimensional exterior algebra on the variables contained in metric.
	www.willamette.edu /~zizza/Software/DifferentialForms.m (2111 words)

	Sorin Popescu's HomePage
		Algebra and Geometry of Points in Projective Space, Napoli, February 9-12, 2000.
		Exterior algebra methods and other new directions in Algebraic Geometry, Commutative Algebra and Combinatorics, 8-15 September 2001, Ettore Majorana Centre, Erice, Sicily, Italy.
		Algebraic Geometry: conference in honour of Joseph Le Potier and Christian Peskine, Paris, June 15-18, 2004
	www.math.sunysb.edu /~sorin (733 words)

	expression math variable
		Does not assume an extensive knowledge of algebra, however, so could also be used in computer science and engineering curricula.
		Idempotent analysis is a new branch of mathematical analysis concerned with functional spaces and their mappings when the algebraic structure is generated by an idempotent operation.
		The subject area is generally referred to as multilinear algebra.
	www.algebra-online.com /expression-math-variable.html (435 words)

	Exterior algebra methods for the minimal resolution conjecture, David Eisenbud, Sorin Popescu, Frank-Olaf Schreyer, ...
		Exterior algebra methods for the minimal resolution conjecture, David Eisenbud, Sorin Popescu, Frank-Olaf Schreyer, Charles Walter
		Exterior algebra methods for the minimal resolution conjecture
		Our proof begins like a variation of that of Eisenbud and Popescu, but uses exterior algebra methods as explained by Eisenbud, G.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1087575156 (642 words)

	Math 600 (Abstract Algebra I) (Site not responding. Last check: 2007-10-10)
		Rather, the prerequisite is to make sure you have enough facility with algebraic proofs to keep up with the pace, which will be much faster than in undergraduate courses.
		This course is a basic introduction to abstract algebra.
		To prepare mathematics graduate students for the written qualifying exam in algebra.
	www.math.umd.edu /~jmr/600 (323 words)

	Year long programme on Hilbert functions (Site not responding. Last check: 2007-10-10)
		There is a deep connection between graded modules over a polynomial ring and graded modules over a "Koszul dual" exterior algebra, usually expressed by saying that they have the same derived category.
		In the last two decades Eisenbud published papers on many aspects of Algebraic Geometry and Commutative Algebra and became interested in Combinatorics and Statistics.
		He is an articulate, original expositor in both writting and speech, able to describe eloquently the nature of mathematics, the links among branches of mathematics, the ties of mathematics with science and engineering, and the role of mathematics in applications.
	www.math.iitb.ac.in /~hilbert/eisen.html (243 words)

	Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2002)
		The dimension of the space of generalized c-eigenvalues (i.e., of the nilspace of T-cI) is usually called the algebraic multiplicity of c (since it's the multiplicity of c as a root of the characteristic polynomial of T), to distinguish it from the ``geometric multiplicity'' which is the dimension of ker(T-cI).
		The ``top exterior power'' is a subspace of the ``exterior algebra'' of V, which is the quotient of the tensor algebra by the ideal generated by {v*v: v in V}.
		We'll still have to construct the sign homomorphism from the symmetry group of order dim(V) to {1,-1} to make sure that this exterior algebra is as large as we expect it to be, and that in particular that the (dim(V))-th exterior power has dimension 1 rather than zero.
	www.math.harvard.edu /~elkies/M55a.05/index.html (2742 words)

	Re: exterior and geometric calculus (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-10)
		Toby Bartels wrote: > Of course, now you might argue that maybe "exterior algebra" > should be used for exterior algebra together with a Hodge operator.
		> I'd argue against that on the grounds that the algebraic operation > in exterior algebra is the exterior product -- hence the name -- > and that you should alter the name if you add another operation.
		The word "exterior" indicates that the main ingredient is a Grassmann algebra and "geometry" makes sure we have a Hodge duality on that.
	www.lns.cornell.edu.cob-web.org:8888 /spr/2001-12/msg0037573.html (282 words)

	Exterior algebra - Education - Information - Educational Resources - Encyclopedia - Music (via CobWeb/3.1 ... (Site not responding. Last check: 2007-10-10)
		Exterior algebra - Education - Information - Educational Resources - Encyclopedia - Music (via CobWeb/3.1 planet03.csc.ncsu.edu)
		When dealing with differentiable manifolds, we define a "differential k-form" to be a function that assigns to every point x of the manifold an element of the k-th exterior power of the cotangent space at x.
		In the de Rham and Alexander-Spanier cohomology theories it is used to define a multiplication on the associated cohomology ring, in an analogous way to the cup product in singular cohomology.
	education.music.us.cob-web.org:8888 /E/Exterior-algebra.htm (1321 words)

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