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Topic: Special orthogonal group

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	DCE 1.1: Remote Procedure Call - Statechart Specification Language Semantics (Site not responding. Last check: 2007-10-16)
		This hierarchical decomposition can be viewed as either clustering of logical groups of states (the bottom-up approach) or as refinement (the top-down approach).
		Orthogonal AND states are represented graphically as rounded rectangles (parent state) divided by dashed lines.
		If an element is to be modified multiple times during a single step (for example, by different assignments to the same data items in concurrent transitions in orthogonal states), the system cannot resolve this non-determinism and is in an illegal state.
	www.opengroup.org /onlinepubs/9629399/chap8.htm (2568 words)

	Orthogonal group - Wikipedia, the free encyclopedia
		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 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.
	en.wikipedia.org /wiki/Orthogonal_group (1397 words)

	Spinor group - Wikipedia, the free encyclopedia
		In mathematics the spinor group or spin group Spin(n) is a particular double cover of the special orthogonal group SO(n, R).
		As a Lie group Spin(n) therefore shares its dimension, n(n−1)/2, and its Lie algebra with the special orthogonal group.
		Spin(3) corresponds to the group of unit quaternions (see also quaternions and spatial rotation) or the group of Pauli matrices.
	en.wikipedia.org /wiki/Spinor_group (131 words)

	Lie group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-16)
		In mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps.
		Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures.
		If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra g over F there is a unique (up to isomorphism) simply connected Lie group G with g as Lie algebra.
	xahlee.org /_p/wiki/Lie_group.html (1378 words)

	PlanetMath: simple and semi-simple Lie algebras (Site not responding. Last check: 2007-10-16)
		, the Lie algebra of the special orthogonal group (skew-symmetric matrices), and
		, the Lie algebra of the special unitary group (skew-hermitian matrices).
		Simple and semi-simple Lie algebras are one of the most widely studied classes of algebras for a number of reasons.
	planetmath.org /encyclopedia/Simple3.html (259 words)

	Standard Matrix 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.lsu.edu /magma/text296.htm (1648 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Special Linear Group $SL(n)$: A subgroup of $GL(n)$, it is the set of linear transformations on $R^n$ with determinant 1.
		Orthogonal Group $O(n)$: corresponds to reflections and rotations in an n-dimensional Euclidian space.
		Special Orthogonal Group $SO(n)$: corresponds to rotations in an n-dimensional Euclidean space.
	draal.physics.wisc.edu /Notes/GroupTheory/src (295 words)

	by Hans--Werner Henn and Hu`ynh Mui (ResearchIndex) (Site not responding. Last check: 2007-10-16)
		Abstract: Let G(n) denote either the symmetric group \Sigma(n), the orthogonal group O(n) or the unitary group U(n) and let SG(n) denote either the alternating group A(n), the special orthogonal group SO(n) or the special unitary group SU(n).
		107 Linear Representations of Finite Groups (context) - Serre - 1977
		2 Decomposition theorems for homology groups of symmetric grou..
	citeseer.ist.psu.edu /1153.html (399 words)

	PlanetMath: orthogonal group (Site not responding. Last check: 2007-10-16)
		is called the orthogonal group with respect to
		Cross-references: transpose, inverse, transformation, invertible linear transformations, group, equivalent, inner product, positive definite, general linear group, subgroup, vectors, linear transformation, vector space, real, symmetric bilinear form, non-degenerate
		This is version 2 of orthogonal group, born on 2002-02-22, modified 2002-02-22.
	planetmath.org /encyclopedia/OrthogonalGroup.html (110 words)

	so 8 (Site not responding. Last check: 2007-10-16)
		In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space.
		Like all orthogonal groups, SO(8) is not simply connected having a fundamental group isomorphic to C
		SO(8) is unique among the orthogonal groups in that its Dynkin diagram (shown right) possesses a high degree of symmetry.
	www.yourencyclopedia.net /so_8_.html (206 words)

	AME 698 Homework 7
		Denote it by SO(n) for special orthogonal group.
		Denote it by SE(n) for special Euclidean group.
		Prove that SE(n) is a Lie group; in particular, be sure to write a formula for the inverse of elements in SE(n) (which is necessary to prove that it is a group).
	patents.ame.nd.edu /ame698/hw7/hw7.html (235 words)

	Clifford algebra - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-16)
		If the bilinear form of V is non-degenerate and V is finite dimensional then the Clifford group maps onto the orthogonal group of V and the kernel consists of the nonzero elements of the field K.
		Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group.
		In the common case when V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V has dimension at least 3.
	evil-wire.luvfeed.org /cache/3771 (3294 words)

	ABSTRACTS OF PAPERS ERICH W. ELLERS
		ERICH W. Hermitian presentations of Chevalley groups I. We give a presentation for a Chevalley group arising from a Hermitian Lie algebra whose roots have all the same length.
		The Lorentz group Omega(V) is bireflectional and all involutions in Omega(V) are conjugate.
		Let G be a simple and simply-connected algebraic group that is defined and quasi-split over a field K. We investigate properties of intersections of Bruhat cells of G with conjugacy classes C of G, in particular, we consider the question, when is such an intersection not empty.
	www.math.toronto.edu /ellers/abstracts.html (817 words)

	Articles - Rotation group (Site not responding. Last check: 2007-10-16)
		Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group.
		The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).
		Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.
	gaple.com /articles/Spherical_symmetry?mySession=e0c10a661c0825729d7... (1380 words)

	Group actions
		In algebra, geometry and topology we often exploit the fact that important structures arise from families of morphisms that are indexed by a group.
		For example, rotations in the plane about the origin are indexed by the unimodular group of complex numbers; we say that this group acts on the plane and the orbit of a point at distance r from the origin is the circle of radius r.
		G is a Lie group, so G has a differentiable structure with respect to which its binary operation and the taking of inverses is smooth, and X is a smooth manifold.
	www.ma.umist.ac.uk /kd/curves/node4.html (797 words)

	Ed Pegg's Math Games - Matrix Revolutions
		This group is named the order 6 cyclic group, or C
		The set of all 3D rotation matrices is named the special orthogonal group, or SO(3).
		If you have a sphere handy, this group represents all the ways the sphere can be turned without changing the center point.
	www.maa.org /editorial/mathgames/mathgames_11_10_03.html (966 words)

	Articles - Circle group (Site not responding. Last check: 2007-10-16)
		The set of all 1Ã—1 unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1.
		The circle group is a compact Lie group, and any compact Lie group G of dimension â¥ 1 has a subgroup isomorphic to it.
		The circle group has many subgroups, but its only closed subgroups consist of roots of unity: there is one such, that is cyclic of order n, for each integer n â¥ 1.
	www.lastring.com /articles/U(1) (445 words)

	Modern Algebra Lecture Notes, 11/13/02 (Site not responding. Last check: 2007-10-16)
		(The special orthogonal group is a subgroup of the orthogonal group O
		In the case n=2, the special orthogonal group SO is the group of rotations of the plane that send the origin to itself.
		Theorem: The converse of Lagrange's Theorem is false.
	www.assumption.edu /Alfano/MAT351-FA02/Notes/111302.html (380 words)

	SO(8) (Site not responding. Last check: 2007-10-16)
		It could be either a real or complex simple Lie group of rank 4 and dimension 28.
		Like all special orthogonal groups, SO(8) is not simply connected having a fundamental group isomorphic to Z
		The triality automorphism of Spin(8) is the outer automorphism group of Spin(8) which is isomorphic to the symmetric group S
	www.worldhistory.com /wiki/S/SO(8).htm (282 words)

	groups
		Examples of Finite Groups: A = {1, -1, i, -i} where * is multiplication, B = {0, 1, 2, 3) where * is addition modulo 4.
		A subgroup is a group entirely inside another: {1, -1} is a subgroup of A, {0, 2) is is a subgroup of B.
		The makers of GAP have written an analysis of Rubik's Cube from a Group Theory perspective.
	www.mathpuzzle.com /groups.html (581 words)

	Caltech Multi-Res Modeling Group - Publications
		At the same time it is much more powerful in that it is general (no special tricks are required for different types of finite elements), and applicable for some newer approximations where traditional mesh refinement concepts are not of much help, for instance on subdivision surfaces.
		Special interface hardware includes a head-tracked stereoscopic display and sensors which track the body and handheld tools, allowing the artist to share the space of the artwork.
		Surfaces are created by moving a hand, instrumented with a special glove, through space in a semi-immersive 3D display and interaction environment (the Responsive Workbench).
	multires.caltech.edu /pubs (10934 words)

	Publication (Site not responding. Last check: 2007-10-16)
		The Lie group SO(3), the special orthogonal group, is used for illustration and simulation, but the results are general and may be applied to any Lie group.
		A spline in SE(3), the special Euclidean group, which has the dictated continuity properties at the waypoints, would be such a trajectory.
		Properties such as local control and optimality are important for these applications: aircraft are often required to change their trajectories locally by interpolating different waypoints to avoid turbulence or conflicts with other aircraft, and trajectories which minimize functions of angular velocity or acceleration are optimal in terms of fuel consumption.
	robotics.eecs.berkeley.edu /~clairet/splines.html (278 words)

	Atlas: Canonical Representations of Orthogonal Groups of Line-Bundle-Valued Ternary Quadratic Bundles over Schemes with ... (Site not responding. Last check: 2007-10-16)
		Given a line-bundle-valued ternary quadratic bundle over any scheme, there is a functorial representation of its group of orthogonal similitudes in the Witt-invariant, which by definition is the degree zero part of the associated generalised Clifford algebra bundle.
		The various orthogonal groups of a quadratic bundle are canonically determined in terms of the automorphisms of its even Clifford algebra.
		The special orthogonal group is thus identified with the subgroup of automorphisms with trivial determinant.
	atlas-conferences.com /cgi-bin/abstract/caoz-56 (625 words)

	Bin. Tet. Group
		The Tetrahedral Group is the group of orientation- preserving symmetries of an equilateral tetrahedron.
		If the tetrahedron is centered at the origin in 3-space, these symmetries correspond to as rigid rotations of 3-space, and can be represented by orthogonal 3 by 3 matrices, elements of the Special Orthogonal matrix group SO(3).
		Since the tetrahedral group is a 12-element subgroup of SO(3), the SU(2) matrices which map to elements of the tetrahedral group will form a 24-element subgroup of SU(2).
	www.math.sunysb.edu /~tony/bintet/tetgp.html (660 words)

	[No title]
		We usually think of this motion as the x, y, and z coordinates of the object, but we could equally think of it as the motion itself: which translations in space are needed to bring the thing back to the origin.
		In that sense, the location of an object (given a choice of origin) can be thought of as the element of a Lie group: the group of translations in space.
		This group is known as SO(3), for the "Special Orthogonal group in 3 dimensions." But there's a very important difference between this Lie group and the Lie group of translations mentioned before: translations commute with one another, while not all rotations do.
	www.math.niu.edu /~rusin/known-math/00_incoming/phasespace (932 words)

	and Hyperboloids
		that preserve the standard cone form a group called the orthogonal group of signature (2,1), denoted
		is known as the special orthogonal group of signature (2,1), written
		We shall see how this group is a version of the group of Möbius transformations that preserve the unit disk.
	www.math.okstate.edu /~wrightd/INDRA/conics/node3.html (271 words)

	55Q: Homotopy groups
		Tables of the homotopy groups of spheres [Hatcher].
		Fundamental group of the space of all unlabeled orthogonal frames in R^3.
		Calculating the fundamental groups of (compact, connected, orientable) surfaces.
	www.math.niu.edu /~rusin/known-math/index/55QXX.html (247 words)

	Making Polyhedra with Polyhedral Groups -- from Wolfram Library Archive
		As mentioned under the Mathworld entry, Polyhedral Group, the platonic and archimedean solids all belong to one of three symmetry groups: Tetrahedral, Octahedral, or Icosahedral.
		With these groups of rotation matrices, a single vector can be used to represent the snub cube, and many other solids.
		Special orthogonal group, polyhedra, snub, rotation group, archimedean
	library.wolfram.com /infocenter/MathSource/4807 (95 words)

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