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Topic: Biquaternion

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Quaternion

	[No title] (Site not responding. Last check: 2007-11-06)
		The biquaternions and their algebra represent the formalism of the usual quantum mechanics: the statements of the usual quantum mechanics are statements of the biquaternions and their algebra; - the next step signed the very innovative turning-point of our research.
		In particular, the important result to have used LHBT with UU+# 1 and UU+ = I, I being a new generalized unity, is that all the basic physical laws of quantum mechanics are generalized respect to the particular level of physical reality that is representedby the functional l = l(....).
		When we speak about quantum mechanics as it was formulated in 1927, we have always to remember that it is not the general theory of the microphysical world; it is the theory that was originally conceived for the structure of the atoms, and for the electromagnetic interactions at large.
	amasci.com /weird/vortex/fusion.doc (1225 words)

	Biquaternion Information
		In mathematics, a biquaternion (or complex quaternion) is an element of the quaternion algebra over the complex numbers.
		The algebra of biquaternions can be consider as a tensor product C⊗H where C is the field of complex numbers and H is the algebra of real quaternions.
		The actual exhibition of individual Lorentz transformations involves extensions of inner automorphisms of the group of units of biquaternions to the singular elements through inversive ring geometry.
	www.bookrags.com /wiki/Biquaternion (525 words)

	Science Fair Projects - Biquaternion
		In mathematics, a biquaternion is a numeric and geometric concept developed by William Kingdon Clifford, William Rowan Hamilton, and Alexander MacAuley in the nineteenth century.
		The collection of all biquaternions forms a vector space of four complex dimensions or eight real dimensions.
		Then {a + b ι j : a, b ∈ R } is a subring of biquaternions isomorphic to the split-complex number ring.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Biquaternion (452 words)

	Hypercomplex number - Wikipedia, the free encyclopedia
		While for Cayley-Dickson constructs, split-complex algebra, and Clifford algebra all non-real bases are anti-commutative, use of a commutative imaginary base leads to four dimensional Tessarines, eight dimensional biquaternions, and 16 dimensional conic sedenions.
		Tessarines offer a commutative and associative multiplication, biquaternions are associative but not commutative, and conic sedenions are not associative and not commutative.
		Biquaternions contain nilpotents, conic sedenions are also not power associative.
	en.wikipedia.org /wiki/Hypercomplex_number (758 words)

	Reference.com/Encyclopedia/Biquaternion
		In mathematics, a biquaternion (or complex quaternion) is an element of the (unique) quaternion algebra over the complex numbers.
		The algebra of biquaternions can be consider as a tensor product C⊗H (taken over the reals) where C is the field of complex numbers and H is the algebra of real quaternions.
		The actual exhibition of individual Lorentz transformations involves extensions of inner automorphisms of the group of units of biquaternions to the singular elements through inversive ring geometry.
	www.reference.com /browse/wiki/Biquaternion (618 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		As we have shown in (Conte, 1994c), the biquaternion quantum mechanics uses Lin ear Homogeneous Biquaternion Transformations, LHBT (.,.,.), to generalize the s tatements of the usual quantum mechanics.
		The way to gene ralize Schrödinger's equation by biquaternions implies necessarily to consider that the new quantization, instead of Planck's constant, becomes a dynamical va riable in the induced quantum theory.
		As consequence,this new theory, generaliz ed with respect to the usual quantum mechanics, is finalized to analyze physica l systems where it is of importance to examine the consequences of quantum fluc tuations I in the quantum of action h.
	www.eskimo.com /~bilb/freenrg/cofu.wri (2010 words)

	Clifford biquaternion Information
		A Clifford biquaternion is a concept from geometric algebra.
		The idea is to replace the complex numbers used in an ordinary (Hamilton) biquaternion with split-complex numbers.Thus q = w + x i + y j + z k, with w, x, y, z ∈ D is a Clifford biquaternion.
		The collection of all Clifford biquaternions forms a Clifford algebra of dimension 8 over the real line R.
	www.bookrags.com /Clifford_biquaternion (143 words)

	Biquaternion - Definitions from Dictionary.com
		Did you mean Bi quaternion (in dictionary) or Biquaternion (in encyclopedia)?
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	dictionary.reference.com /browse/Biquaternion (43 words)

	The group SK_2 of a biquaternion algebra, by Baptiste Calmes
		The group SK_2 of a biquaternion algebra, by Baptiste Calmes
		In this paper, we build an exact sequence relating the group SK_2 of a biquaternion algebra - kernel of the reduced norm - over a field F with Galois cohomology groups of F. This exact sequence is an analogue for K_2 of an exact sequence from M. Rost for K_1.
		This paper provides the proofs of the results announced in preprint
	www.math.uiuc.edu /K-theory/0663 (137 words)

	Quaternions - LoveToKnow 1911 (Site not responding. Last check: 2007-11-06)
		To fix a weighted point and a weighted plane in Euclidean space we require 8 scalars, and not the 12 scalars of a tri-quaternion.
		We should expect some species of biquaternion to suffice.
		Let n, w be two quasi-scalars such that r t e =n, con = w, nw =w 2 = o.
	www.1911encyclopedia.org /Quaternions (5768 words)

	The Work of W.K. Clifford
		This geometry was an extension of his algebra of biquaternions.
		Both systems, the new geometry and biquaternions, made use of Clifford's concept of geometric parallelism whereby parallel lines need not exist in the same plane.
		Clifford could not have known that his new geometry was far too advanced for a scientific community that was just beginning to cope with the repercussions pursuant to the discovery of ordinary non-Euclidean geometries.
	members.aol.com /jebco1st/Paraphysics/twist1.htm (5016 words)

	[No title]
		Our particular 4-dimensional representation of the biquaternions naturally g* ives rise to 4-dimensional representations of important low dimensional Lie groups a *nd algebras.
		There is another representation of the biquaternions in which the traceless * ma- trices are given by ` ' A~T F = 0~A x(iA~) EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 11 These matrices differ from the previous set in Example 4_by_changing the -i to *+i.
		Similarly, the iden* tity component of the Lorentz group intersects I + S in a subgroup which is isomorph *ic to the other by compex conjugation.
	www.math.purdue.edu /research/atopology/Gottlieb/eigbndl.txt (5948 words)

	Acta Physica Slovaca, volume 53, June 2003, no.3
		In the present article, after defining biquaternions, the general properties of biquaternion's algebra are introduced.
		The matrix representations of biquaternions are presented, as well.
		A new biquaternionic definition of the Dirac equation and its solution are given by the use of biquaternion's basis.
	www.acta.sav.sk /acta03/no3 (1373 words)

	Talk:Clifford biquaternion - Wikipedia, the free encyclopedia
		W.K.Clifford wrote the following footnote to justify use of the term biquaternion:
		Hamilton's biquaternion is quaternion with complex coefficients; but it is convenient (as Prof.
		Pierce remarks) to suppose from the beginning that all scalars may be complex.
	en.wikipedia.org /wiki/Talk:Clifford_biquaternion (102 words)

	Polarization - Biocrawler (Site not responding. Last check: 2007-11-06)
		The effects in that space correspond to a Lorentz boost of velocity parameter 2β along the given direction, or a rotation of angle 2φ about the given axis.
		These transformations may also be written as biquaternions (quaternions with complex elements), where the elements are related to the Jones matrix in the same way that the Stokes parameters are related to the coherency matrix.
		They may then be applied in pre- and post-multiplication to the quaternion representation of the coherency matrix, with the usual exploitation of the quaternion exponential for performing rotations and boosts taking a form equivalent to the matrix exponential equations above (See: Quaternion rotation).
	www.biocrawler.com /encyclopedia/Polarization (2810 words)

	The revealed secrets of classical electrodynamics
		(Bi)quaternion numbers and quaternion calculus were discovered by Rowan Hamilton and is very suitable for expressing 4-vectors or 8-vectors.
		Keep in mind that one biquaternion equations is in fact a compact notation of two scalar equations (real scalar equals real scalar and imaginary scalar equals imaginary scalar) and two vector equations (real vector equals real vector and imaginary vector equals imaginary vector).
		Maxwell's original theory was formulated in quaternion form as well, however, his theory did not include the scalar field S as defined in equation (3).
	home.wanadoo.nl /raccoon (6697 words)

	Amazon.com: Biquaternion (Site not responding. Last check: 2007-11-06)
		"Preliminary sketch of biquaternions," Proceedings of the London Mathematical...
		(or as geometrically interpreted), to biquaternions considered as imaginary (or as...
		quaternions was Hamilton's discovery of biquaternions in 1844.
	www.amazon.com /s?ie=UTF8&keywords=Biquaternion&tag=lexico&index=blended&link_code=qs&page=1 (1030 words)

	Quaternions are not biquaternions (Site not responding. Last check: 2007-11-06)
		>the Theory of Functions of a Biquaternion Variable.
		need to make this point: quaternions are not biquaternions (which
		Biquaternions not only have the imaginary basis vectors I, J, and K, but they allow t, x, y, and z to take the complex values of the form
	world.std.com /~sweetser/quaternions/spr/notbiquat.html (455 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Chapter 2:The quantization at the origin of BQM,Introduction,Biquaternion Proof of Quantization,The harmonic Oscillator in BQM,Biquaternion Analysis of the Orbital Angular Momentum,The hydrogen Atom using Biquaternions.
		Chapetr 5:On Some Advanced Applications of BQM,Introduction,quarks as biquaternions,A BQM Study of internal Isospin symmetry,Are quarks composite obiects?,The BQM and the basic foundations of Living Matter,Biquaternion Quantum Mechanical formulation of coherent quantum states in physical and Biological systems,Magnetic monopoles and final considerations.
		Biquaternions are considered in a new manner for the first time.
	www.clifford.org /anonftp/clf-alg/books/conte.txt (315 words)

	CAT.INIST
		Triality, biquaternion and vector representation of the Dirac equation
		The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion.
		A bilinear law of composition for biquaternion is defined by means of Levi-civita symbol and Lorentzian metric only.
	cat.inist.fr /?aModele=afficheN&cpsidt=15938577 (162 words)

	The revealed secrets of classical electrodynamics
		The theory of electrodynamics with scalar field, expressed in biquaternion equations
		Cornelius Lanczos and Andre Gsponer wrote several papers about applying biquaternion in physics, see for instance the paper of Andre Gsponer: THE PHYSICAL HERITAGE OF SIR W.R., for further references.
		The most important implication of the biquaternion form is the natural appearance of the scalar field in the equations.
	www.truth.myweb.nl (6697 words)

	Le groupe SK_2 d'une algebre de biquaternions, by Baptiste Calmes (Site not responding. Last check: 2007-11-06)
		Le groupe SK_2 d'une algebre de biquaternions, by Baptiste Calmes
		In this note, I give the main ideas of the proof of the exactness of a sequence relating the group SK_2 of a biquaternion algebra and the Galois cohomology of its center.
		It is the analogue of Rost's exact sequence for SK_1 of a biquaternion algebra.
	www.math.uiuc.edu /K-theory/0616 (66 words)

	CiteULike: Algebrodynamics in complex space-time and the complex-quaternionic origin of Minkowski geometry (Site not responding. Last check: 2007-11-06)
		We present a scheme of biquaternionic algebrodymamics based on a nonlinear generalization of the Cauchy-Riemann holomorphy conditions considered therein as fundamental field equations.
		The automorphism group SO(3,C) of the biquaternion algebra acts as a proper Lorentz group on a real space whose coordinates are bilinear in the complex coordinates of biquaternionic vector space.
		@misc{citeulike:518798, abstract = {We present a scheme of biquaternionic algebrodymamics based on a nonlinear generalization of the Cauchy-Riemann holomorphy conditions considered therein as fundamental field equations.
	www.citeulike.org /user/jrw/article/518798 (299 words)

	Uni Göttingen, Mathematisches Institut: Homepage Ina Kersten
		Karim Johannes Becher: On the Number of Square Classes of a Field of Finite Level, Documenta Math.
		d'une algèbre de biquaternions et cohomologie galoisienne, These.
		T.Y.Lam, D.Leep, J-P.Tignol: Biquaternion algebras and quartic extensions, IHES Publ.
	www.uni-math.gwdg.de /kersten/papers.html (209 words)

	IngentaConnect The group SK2 of a biquaternion algebra
		IngentaConnect The group SK2 of a biquaternion algebra
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		of a biquaternion algebra" title="post to CiteUlike">Post to CiteUlike
	www.ingentaconnect.com /content/els/1631073x/2003/00000337/00000001/art00261 (52 words)

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