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Topic: Orthogonal group

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	NationMaster - Encyclopedia: Special orthogonal group (Site not responding. Last check: 2007-11-07)
		In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication.
		O(n,R) is isomorphic to the group of isometries of R
		O(n,R) is a subgroup of the Euclidean group E(n), the group of isometries of R As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.
	www.nationmaster.com /encyclopedia/Special-orthogonal-group (615 words)

	Orthogonal group - Definition, explanation
		In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.
		For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.
		As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.
	www.calsky.com /lexikon/en/txt/o/or/orthogonal_group.php (1279 words)

	PlanetMath: examples of groups (Site not responding. Last check: 2007-11-07)
		This is the automorphism group of the given object and captures its “internal symmetries”.
		In Galois theory, the symmetry groups of field extensions (or equivalently: the symmetry groups of solutions to polynomial equations) are the central object of study; they are called Galois groups.
		All these matrix groups are Lie groups: groups which are differentiable manifolds such that the group operations are smooth maps.
	planetmath.org /encyclopedia/ExamplesOfGroups.html (1002 words)

	PlanetMath: dimension of the special orthogonal group (Site not responding. Last check: 2007-11-07)
		The theorem on decomposing orthogonal operators as rotations and reflections suggests that all elements of
		"dimension of the special orthogonal group" is owned by stevecheng.
		This is version 5 of dimension of the special orthogonal group, born on 2005-07-19, modified 2006-01-04.
	www.planetmath.org /encyclopedia/DimensionOfTheSpecialOrthogonalGroup.html (515 words)

	20: Group Theory and Generalizations
		Groups acting on vector spaces are subgroups of the matrix groups studied in Linear Algebra.
		Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology.
		Nielsen's theorem: subgroups of free groups are free.
	www.math.niu.edu /~rusin/known-math/index/20-XX.html (2774 words)

	Service Modeling Language Working Group Charter
		The mission of the Service Modeling Language (SML) Working Group, part of the XML Activity, is to produce W3C Recommendations for Service Modeling Language by refining the “Service Modeling Language” (SML) Member Submission, addressing implementation experience and feedback for the specifications.
		Therefore, this Working Group shall be schedule-driven and the W3C Recommendation for SML shall remain compatible to the extent possible with the existing SML models.
		The SML Working Group may adjust the timing and duration of meetings to address the workload and assure that the goals and schedule of this charter are achieved.
	www.w3.org /2007/03/SML_Charter.html (1953 words)

	PlanetMath: orthogonal group (Site not responding. Last check: 2007-11-07)
		is called the orthogonal group with respect to
		Cross-references: transpose, inverse, invertible linear transformations, group, equivalent, inner product, positive definite, general linear group, subgroup, vectors, preserve, linear transformation, vector space, real, symmetric bilinear form, non-degenerate
		This is version 3 of orthogonal group, born on 2002-02-22, modified 2006-04-05.
	www.planetmath.org /encyclopedia/OrthogonalGroup.html (109 words)

	Lie group Summary
		The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2.
		The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
		The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the standard model, whose dimension corresponds to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
	www.bookrags.com /Lie_group (4005 words)

	Generalized orthogonal group (Site not responding. Last check: 2007-11-07)
		In mathematics, the generalized orthogonal group,O(p, q) is the Lie group of all linear transformations of a p + q = ndimensional real vector space which leave invariant a nondegenerate,symmetric, bilinear form of signature (p, q).
		The generalized special orthogonal group, SO(p, q) is the subgroup of O(p, q) having determinant 1.
		For complex spaces, all groups O(p,q; C) are isomorphic with the usual orthogonal group O(p + q; C).
	www.therfcc.org /generalized-orthogonal-group-239828.html (186 words)

	physics - Generalized orthogonal group
		In mathematics, the generalized orthogonal group, O(p, q) is the Lie group of all linear transformations of a p + q = n dimensional real vector space which leave invariant a nondegenerate, symmetric, bilinear form of signature (p, q).
		The generalized special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1.
		For complex spaces, all groups O(p, q; C) are isomorphic with the usual orthogonal group O(p + q; C).
	www.physicsdaily.com /physics/Generalized_orthogonal_group (291 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: 2007-11-07)
		A (real) Lie group is a mathematical group which is also a finite-dimensional real smooth manifold, and in which the group operations of multiplication and inversion are smooth maps.
		The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2.
		The group of smooth maps from a manifold to a finite dimensional group is called a gauge group, and is used in quantum field theory and Donaldson theory.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Lie_group (3298 words)

	600-cell
		This is closely related to a presentation for the rotation group of the icosahedron.
		The rank of a Coxeter group is an important invariant; it is merely the number of generating reflections of the group.
		The rank-2 Coxeter groups are the set of all dihedral groups, the groups acting on regular polygons in 2-dimensional space.
	homepages.wmich.edu /~drichter/600cells.htm (1004 words)

	Standard Groups
		Construct the orthogonal group Omega(n, K) (which is the kernel of the spinor norm map on SO(n, K)) corresponding to the n-dimensional vector space V over the field K = GF(q), where n is odd, n >= 3 and q is a prime power.
		Construct the orthogonal group Omega^ + (n, K) (which is the kernel of the spinor norm map on SO^ + (n, K)) corresponding to the n-dimensional vector space V over the field K = GF(q), where n is even, n >= 2 and q is a prime power.
		Given a matrix group G and a permutation group H, construct action of the wreath product on the tensor power of G by H, which is the (image of) the wreath product in its action on the tensor power (of the space that G acts on).
	www.math.wisc.edu /help/magma/text317.html (1638 words)

	Wikinfo \| Orthogonal group
		Over the field C of complex numbers, O(n,C) and SO(n,C) are complex Lie groups of dimension n(n-1)/2 over C (which means real dimension is twice that).
		For n > 2 the fundamental group of SO(n,C) is Z/2Z whereas of the fundamental group of SO(2,C) is Z.
		The Lie algebra associated to O(n,C) and SO(n,C) consists of the skew-symmetric complex n-by-n matrices, with the Lie bracket given by the commutator.
	www.wikinfo.org /wiki.php?title=Orthogonal_group (619 words)

	Orthogonality and Least Squares (Site not responding. Last check: 2007-11-07)
		The idea is, of course, that orthogonal bases are much easier to work with and lend themselves nicely to performing various computations such as finding vector coordinates with respect to this basis and projecting vectors on various subspaces of a vector space.
		Orthogonal matrices are incredibly important in geometry and physics.
		One of the reasons why, is that orthogonal 2 x 2 matrices with positive determinant represent rigid motions of the plane that keep the origin fixed.
	math.ucsd.edu /~math20f/Spring/Lab8S/Lab8S.html (2350 words)

	Search Results for Orthogonal (Site not responding. Last check: 2007-11-07)
		In addition to his work on potential theory and orthogonal set of functions he published a short paper on the problem of the maximum value an of a positive integer belonging to a set of n positive integers whose reciprocals add to 1.
		A further important text on this topic was The unitary and rotation groups (1962) which concentrated on representations of unitary and orthogonal groups.
		Christoffel not only contributed to all these fields, but his interests extended to orthogonal polynomials and continued fractions, and the applications of his work to the foundations of tensor analysis, to geodetical science, to the theory of shock waves, to the dispersion of light.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Orthogonal&CONTEXT=1 (2309 words)

	Maths - Group Theory - Martin Baker
		If a given law is symmetric, or invariant, with respect to a set of actions that form a lie group, then noethers theorem tells us there is a conserved physical quantity associated with each generator of the lie group.
		The Orthogonal group in 3 dimensions is denoted by O(3).
		Groups are often categorised in a way that is independant of the number of dimentions.
	www.euclideanspace.com /maths/algebra/groups/lie/index.htm (485 words)

	Orthogonal group - Glasgledius (Site not responding. Last check: 2007-11-07)
		Both O(n,F) and SO(n,F) are algebraic groups, because the condition that a matrix have its own transpose as inverse can be expressed as a set of polynomial equations in the entries of the matrix.
		Over the field R of real numbers, the orthogonal group O(n,R) and the special orthogonal group SO(n,R) form real compact Lie groups of dimension n(n-1)/2.
		Over the field C of complex numbers, O(n,C) and SO(n,C) are complex Lie groups of dimension n(n-1)/2 over C.
	www.glasglow.com /E2/or/Orthogonal_group.html (293 words)

	Table of contents for Library of Congress control number 2005050093
		Representation of the symmetric group by permutation matrices 17 6.
		Formulae giving those characters of the symmetric group on m letters which are attached to two and three element parti- tions of m in terms of the class numbers (a).
		The analysis of the representations of the real orthogonal group which are furnished, by the principle of selection, by the irreducible representations of the full linear group.
	www.loc.gov /catdir/toc/fy0604/2005050093.html (645 words)

	3.2 The Symmetry Group of Friezes
		By choosing possible orthogonal groups for the frieze groups, respecting the condition that it must leave the lattice invariant (Theorem 3.1) we will demonstrate that there are 7 non-isomorphic frieze groups.
		The orthogonal groups of mm and mg are isomorphic to D
		For completing the classification of frieze groups, we need to show that among the 7 groups derived there are no mutually isomorphic ones.
	members.tripod.com /vismath1/ana/ana5.htm (464 words)

	Games of Soldiers - WEAVE
		GROUP - A diagonal (orthogonal) group is a set of stones of the same color connected by diagonal (orthogonal) adjacency.
		So, notice that a group that connects opposite edges only by using mixed connections (both diagonal and orthogonal) does not achieve victory.
		In a smaller board, he is an example of an orthogonal victory for White.
	www.di.fc.ul.pt /~jpn/gv/weave.htm (230 words)

	3.3 The Wallpaper Groups
		Their classification will be done by exhausting all possible cases: for every combination of a lattice and a orthogonal group we will construct the corresponding group, or show that such a group does not exist.
		In the presented analysis we distinguished 17 different cases for the wallpaper groups: 2 for the oblique, 6 for rectangular, 7 for centered, 11 for square, and 11 for hexagonal lattice.
		Finally, the group pmm is not isomorphic with pmg or cmm, because every glide reflection from pmm can be decomposed as the product of translations and reflections belonging to that group (Theorem 3.2, Corollary 2), which not holds for pmg or cmm.
	www.mi.sanu.ac.yu /vismath/ana/ana6.htm (1290 words)

	Permutation Representations of Linear Groups
		Integers n and q corresponding to the degree and the field GF(q) of M (GF(q^2) in the case of the unitary groups).
		Construct the projective general linear group G = PGL(n, q), i.e., the group corresponding to the action of GL(n, q) on the projective points of the n-dimensional vector space V over K = GF(q), where n >= 2 and q is a prime power.
		Construct the group G = PSigmaSp(n, q) of the projective symplectic group PSp(n, q) extended by field automorphisms of K = GF(q), where V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 4.
	www.math.lsu.edu /magma/text287.htm (2706 words)

	EDPSY 604 Contrast Coding
		The beta for D1 represents the mean difference between small hospitals and large hospitals (group 1 vs. 3) and the beta for D2 represents the mean difference between medium hospitals and large hospitals (group 2 vs. group 3).
		If a control group is one of the groups in the categorical variable, it should be coded as the last group with 0's on all vectors.
		When group g has been assigned 0's arbitrarily, it is usually appropriate to run an additional regression analysis in which group g is now assigned 1's in a new dummy vector to obtain the relevant statistics and tests for group g.
	www.uoregon.edu /~stevensj/MRA/contrasts.html (2534 words)

	No Title
		As examples I discuss a complex orthogonal group and a complex symplectic group.
		Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure.
		All the results of this paper are formulated for one concrete group, the Hilbert-Schmidt complex orthogonal group, though my methods can be applied in more general situations.
	www.math.mcmaster.ca /gordina/list_of_pub/list_of_pub.html (413 words)

	Maths - Orthogonal Properties of Quaternions - Martin Baker (Site not responding. Last check: 2007-11-07)
		The special orthogonal group, SO(n) - a square matrix where each element of the matrix is a real number.
		The special unitary group, SU(n) - a square matrix where each element of the matrix is a complex number.
		The symplectic group, Sp(n) - a square matrix where each element of the matrix is a quaternion.
	www.euclideanspace.com /maths/algebra/realNormedAlgebra/quaternions/orthogonal/index.htm (1061 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-07)
		One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form
		is the commutator subgroup of (index 2 in) the orthogonal group associated with a symmetric bilinear form
		As a consequence, its tangent mapping at a fixed point belongs to the symplectic group of the tangent space.
	eom.springer.de /s/s091820.htm (301 words)

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