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Topic: Clifford algebra

Related Topics

Exterior algebra

Representations of Clifford algebras

Tensor algebra

Algebra over a field

Geometric algebra

Multilinear algebra

Quaternion

Galois theory

William Clifford

Math

Functional analysis

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Dirac equation

Dirac monopole

	PlanetMath: geometric algebra (Site not responding. Last check: 2007-10-07)
		Geometric algebra is a Clifford algebra which has been used with great success in the modeling of a wide variety of physical phenomena.
		Clifford algebra is considered a more general algebraic framework than geometric algebra.
		This is version 9 of geometric algebra, born on 2002-12-17, modified 2004-11-29.
	planetmath.org /encyclopedia/GeometricAlgebra.html (383 words)

	What ARE Clifford Algebras and Spinors?
		If the algebra A is also generated as a ring by the copies of R and X or, equivalently, as a real algebra by {1} and X, then A is said to be a (real) Clifford algebra for X (Clifford's term...
		Since odd dimensional Clifford algebras are the sum of two matrix algebras, spinors for odd dimensional Clifford algebras are rows (or columns) of one of the matrix algebras, the matrix subalgebras of the even-grade elements of the graded Clifford algebra.
		The non-isomorphism of the octonions O with the Clifford algebra Cl(3) is due to the nonassociativity of the 7-dimensional vector cross-product.
	www.tony5m17h.net /clfpq.html (5336 words)

	Section I. Geometric Algebra
		Abstract: The claim that Clifford algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry.
		Clifford algebra is used to develop a coordinate-free algebraic formulation of projective geometry.
		Relations among Clifford algebras of different dimensions are interpreted geometrically as "projective and conformal splits." The conformal split is employed to simplify and elucidate the pin and spin representations of the conformal group for arbitrary dimension and signature.
	modelingnts.la.asu.edu /html/GeoAlg.html (717 words)

	Clifford algebra - Term Explanation on IndexSuche.Com
		The Clifford algebra exists and can be constructed as follows: take the tensor_algebra T(V) and mod out by the ideal generated by :''v'' tensor ''v'' - ''q''(''v'') 1.
		Using the anticommutation relation we can always express an element of the Clifford algebra as a linear combination of monomials of type :e_{i_1} e_{i_2} e_{i_3} \cdots e_{i_n}, i_1 which gives an explicit isomorphism with the exterior algebra.
		Physicists usually consider the Clifford algebra to be spanned by matrices γ1,...,γ''n'' which have the property that :γ''i'' γ''j'' + γ''j'' γ''i'' = 2 η''i'',''j'' where η is the matrix of a quadratic form of type p,q with respect to an orthonormal basis e1,..., e''n''.
	www.indexsuche.com /Clifford_algebra.html (467 words)

	GluCat: Generic library of universal Clifford algebra templates (Site not responding. Last check: 2007-10-07)
		Paul Leopardi, "A generalized FFT for Clifford algebras", Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 11, Number 5, pp.
		Arvind Raja, "Object-oriented implementations of Clifford algebras in C++: a prototype", in Clifford Algebras with Numeric and Symbolic Computations, Rafal Ablamowicz, P. Lounesto, Josep M. Parra (Editors), 1996.
		Wene, "The Idempotent stucture of an infinite dimensional Clifford algebra", pp.
	glucat.sourceforge.net (2306 words)

	PlanetMath: Clifford algebra (Site not responding. Last check: 2007-10-07)
		In this case, the algebra is generated by
		This is version 5 of Clifford algebra, born on 2002-12-21, modified 2003-06-18.
		Exterior algebra notation by lars_h on 2005-07-24 14:50:29
	www.planetmath.org /encyclopedia/CliffordAlgebra2.html (165 words)

	Clifford Algebra (Site not responding. Last check: 2007-10-07)
		I will demonstrate the use of the unipodal algebra to solve for the roots of the quadratic in a manner similar to that which I first did by solving for the roots to the cubic equation in 1985.
		Clifford Algebra allows for any field to act as the set of scalars of the algebra, whereas Geometric Algebra, by convention, allows the use of only real numbers as the set of scalars.
		I recently talked to him about this subtle distinction between "geometric" and Clifford algebra to be sure that my characterization of the difference between them is accurately represented according to his viewpoint.
	www.ajnpx.com /html/CliffordAlgebra.html (796 words)

	Axiom Portal - clifford
		Clifford algebra appears to be a kind of Lie algebra and is also non-commutative.
		Clifford algebras are currently used in non commutative geometry via the term spectral triple, where a Dirac operator is included.
		Clifford algebras are usually defined using quadratic forms, the concept of a Clifford algebra can be generalized by methods developed by Claude Chevalley to bilinear forms.
	page.axiom-developer.org /zope/Plone/wiki/Clifford (1428 words)

	Clifford algebra - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-07)
		The theory of Clifford algebras is intimately connected with the theory of quadratic forms and
		In characteristic not 2 the algebra Cℓ(V,Q) inherits a Z-grading from the canonical isomorphism with the exterior algebra Λ(V).
		classification of Clifford algebras is isomorphic to the algebra of 4 by 4 complex matrices.
	evil-wire.luvfeed.org /cache/3771 (3294 words)

	CLIFHISTORY
		Educated at Trinity College of Cambridge University, Clifford, at age 26, was appointed Professor of Appled Mathematics and Mechanics at University College, London, on the recommendation of the great William Clerk Maxwell (1831-79).
		Clifford observed that Grassmann developed the idea of directed number from the quantitative point of view, while Hamilton emphasized the operational interpretation.
		Clifford wrote two papers setting forth his ideas, but died prematurely of tuberculosis, leaving a young wife and two tiny girls.
	members.fortunecity.com /jonhays/clifhistory.htm (2455 words)

	Geometric algebra (Clifford algebra)
		GABLE (Geometric AlgeBra Learning Environment, by Leo Dorst, Steve Mann and Tim Bouma) is a tutorial written in Matlab; lots of visualization, and a (hopefully) generally accessible tutorial text to accompany it.
		A paper Honing geometric algebra for its use in the computer sciences (Leo Dorst, 2001) published in the book Geometric Computing with Clifford Algebras, ed.
		Clifford people, a useful index to practitioners in Clifford algebra and geometric algebra.
	www.wins.uva.nl /~leo/clifford (645 words)

	Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 38(1), 33 - 72 (1997) (Site not responding. Last check: 2007-10-07)
		Abstract: The discriminant algebra of a nonsingular quadratic form $q$ over a commutative ring is usually defined as the centralizer of the even Clifford algebra in the full Clifford algebra.
		We also show that the discriminant algebra admits a natural homomorphism into the centralizer of the even Clifford algebra in the full Clifford algebra which is an isomorphism in the nonsingular or semiregular case.
		Similarly to the discriminant algebra, there is a natural homomorphism into the Clifford algebra, and its image is contained in a Jordan subalgebra $J$ which we call the {\it Jordan center \/} since it is defined by suitable centralizer conditions.
	www.emis.de /journals/BAG/vol.38/no.1/4.html (351 words)

	CLICAL for Clifford Algebra Calculations
		Oriented volume elementes, or segments of subspaces, are represented by simple multivectors, which are homogeneous and decomposable elements in the exterior algebra.
		Clifford algebras are used to handle rotations and oriented subspaces.
		CLICAL is intended for researchers and teachers of Clifford algebras and spinors.
	www.hut.fi /~ppuska/mirror/Lounesto/CLICAL.htm (568 words)

	Directory - Science: Math: Algebra: Geometric Algebra (Site not responding. Last check: 2007-10-07)
		Geometric Algebra and its Applications in Mathematical Physics · cached · C.J.L. Doran's thesis on applications of Clifford algebras.
		Advances in Applied Clifford Algebras · cached · This journal publishes research papers and notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops in the area of Clifford Algebras and their applications to other branches of mathematics and physics, and in certain cognate areas.
		Clifford · cached · A Maple package for computations in Clifford algebras of an arbitrary bilinear form.
	www.incywincy.com /default?p=794524 (288 words)

	Clifford Algebra to Geometric Calculus : A Unified Language for Mathematics and Physics (Fundamental Theories of ...
		Clifford Algebra to Geometric Calculus : A Unified Language for Mathematics and Physics (Fundamental Theories of Physics), Springer, D. Hestenes, Garret Sobczyk
		The authors show how the traditional languages of complex numbers, quaternions, matrices, vectors, tensors, spinors and differential forms are all subsumed by the elegant language of Clifford algebra, and their calculi by Clifford analysis.
		Quite apart from the pleasure that the clarity of Clifford algebra/analysis affords, its value also lies in making it easier to understand what has already been discovered, and thus extending the mathematical grasp of the human mind.
	allentech.net /bookstore/item_9027725616.html (529 words)

	CEAC Clifford Algebra Introduction
		There is now on a separate page to introduce Clifford algebra and provide a link to the introductory material which is available elsewhere.
		Clifford Numbers and their inverses calculated using the matrix representation.
		The application of Clifford algebra to calculations of multicomponent chemical composition.
	www.ceac.aston.ac.uk /research/staff/jpf/clifford (671 words)

	APPLICATIONS OF THE CLIFFORD-GRASSMANN ALGEBRA TO THE PLANE GEOMETRY (Site not responding. Last check: 2007-10-07)
		The applications of Clifford algebra to plane geometry are shown in two different but complementary cases: the Euclidean and pseudo-Euclidean planes.
		The geometric (Clifford) product is exactly equivalent to the matrix product of the matrix representations of vectors and complex numbers (real 2×2 matrices).
		Due to its intrinsically geometric nature, the geometric algebra allows to solve geometric equations (which should be a main objective) without direct reference to coordinates, although the translation to these is always possible.
	www.terra.es /personal/rgonzal1/poster/poster.htm (1013 words)

	Clifford Algebra to Geometric Calculus : A Unified Language for Mathematics and Physics (Fundamental Theories of ... (Site not responding. Last check: 2007-10-07)
		Clifford Algebra to Geometric Calculus : A Unified Language for Mathematics and Physics (Fundamental Theories of Physics)
		Clifford Algebra to Geometric Calculus : A Unified Language for Mathematics and Physics (Fundamental Theories of Physics) Review: i have been working a few years in geometric calculus and i believe this book should be in every house of every geometrist and every person that is intersted in geometric concepts with physics applications
		Clifford Algebra to Geometric Calculus : A Unified Language for Mathematics and Physics (Fundamental Theories of Physics) Review: In 1990, I came across Hestenes' book New Foundations for Classical Mechanics in a university bookstore, and immediately purchased it.
	www.textkit.com /0_9027725616.html (597 words)

	Havlicek/Multilinear Algebra (Site not responding. Last check: 2007-10-07)
		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.
		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.
		Clifford Groups and Spin Groups, Isometries, The Centre of a Clifford Algebra, Semisimple Modules and Representation of Algebras, Spaces of Spinors.
	www.geometrie.tuwien.ac.at /havlicek/multilineare.html (179 words)

	geometric algebra (Site not responding. Last check: 2007-10-07)
		Geometric algebra (based on Clifford algebra) Geometric algebra is a very convenient representational and computational system for geometry.
		Clifford algebra is considered a more general algebraic framework than...
		In mathematics, geometric algebra is a term applied to the theory of Clifford algebras and related...
	learning-gd.com /articles/289/geometric-algebra.html (234 words)

	David Lamb's Clifford Algebra Primer
		A Clifford algebra, like A, is going to be constructed in layers.
		By now, you should be seeing a pattern to how the different layers of the Clifford algebra are being built.
		The construction is complete, and we have our Clifford algebra, A, with a full multiplication for all elements.
	www.oakland.edu /~latcha/spatial/clifford3.html (1196 words)

	Mind Matter Unification/The Foundations of Quantum Mechanics
		I consider [Clifford Space] to be similar to a quaternionic space, with several graded orders of infinite dimensions, rather in the manner of the graded Clifford algebra.
		For Cl(-+++) = Cl(1,3), the full spinor space (minimal ideal of the Clifford algebra) is 1x2 quaternionic column of the 2x2 quaternionic matrix and each half-spinor space is 1x1 quaternion column, or, in other words, a fermion is a quaternion.
		In summary, since there is extant literature on quaternionic manifolds and spaces, and since the quaternions are one of the Clifford algebras, it does not seem to me to be much of a stretch to generate a Clifford space or a Clifford manifold.
	www.rialian.com /rnboyd/mind-matter-unification.htm (1545 words)

	Cambridge University GA Research Group
		Our group works on applications of geometric algebra in physics, computer science and engineering.
		A study of the Dirac equation in a fl hole background produces the first calculations of the bound state spectrum.
		Conformal geometric algebra provides simple algorithms for blending between circles.
	www.mrao.cam.ac.uk /~clifford (117 words)

	GCI Generated Page: Abstracts (Site not responding. Last check: 2007-10-07)
		The numerical evaluation of Clifford algebra equations becomes more and more important as Clifford algebra becomes more prominent in applied fields like computer vision, computer graphics and robotics.
		CLU implements the basic Clifford algebra operations and also contains functions to solve multivector equations and to automatically analyze multivectors with regard to their geometric meaning.
		This ranges from Fourier-Rotor transforms over pose estimation and solving for versors to the teaching of basic Clifford algebra concepts in Euclidean, projective and conformal space.
	clifford.physik.uni-konstanz.de /cgi-BF/mysql_abs.cgi?&id=103 (336 words)

	Multi-Boundary Algebra
		Multiboundary algebra is a deviation from the standard formalism of abstract algebra, which has multiple operators but only one type of parenthese.
		The somewhat greater complexity of Ons algebra is compensated by the increased richness of the structures obtained.
		Multi-boundary algebra envisions a universe in which various simple and composite entities coexist and interpenetrate within various spaces, and in which the results of coexistence and interpenetration depend on the composition of the entities involved, and the space in which the entities exist.
	www.goertzel.org /papers/Multi.html (2854 words)

	Clifford algebra bei eLexi - das Onlinelexikon (Site not responding. Last check: 2007-10-07)
		C(q) defined by the following universal property: for every associative algebra A over k with a linear map j : V
		The Clifford algebra exists and can be constructed as follows: take the tensor algebra T(V), and construct its quotient ring by the ideal generated by the elements
		Physicists usually consider the Clifford algebra to be spanned by matrices γ
	www.elexi.de /en/c/cl/clifford_algebra.html (523 words)

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