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Topic: Generalized orthogonal group

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	NationMaster - Encyclopedia: Special orthogonal group
		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.
		This is a subgroup of the general linear group GL(n,F).
		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 (632 words)

	NationMaster - Encyclopedia: 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).
		This larger group is the group of all motions of a rigid body: each of these is a combination of a rotation about an arbitrary axis and a translation along the axis, or put differently, a combination of an element of SO(3) and an arbitrary translation.
		In general, the rotation group of an object is the symmetry group within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries.
	www.nationmaster.com /encyclopedia/Rotation-group (3815 words)

	PlanetMath: examples of groups
		More generally, any (skew) field gives rise to two groups: the additive group of all field elements with 0 as identity element, and the multiplicative group of all non-zero field elements with 1 as identity element.
		This is the automorphism group of the given object and captures its “internal symmetries”.
		The fundamental group is generalized by the higher homotopy groups.
	planetmath.org /encyclopedia/ExamplesOfGroups.html (1011 words)

	Orthogonal group - Biocrawler
		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.biocrawler.com /encyclopedia/orthogonal_group (1139 words)

	PlanetMath: dihedral group properties (Site not responding. Last check: 2007-11-07)
		A group generated by two involutions is a dihedral group.
		The subgroups of a characterisitic cyclic group are necessarily characteristic.
		Proposition 5 Quotient groups of dihedral groups are dihedral, and subgroups of dihedral groups are dihedral or cyclic.
	planetmath.org /encyclopedia/DihedralGroupProperties.html (503 words)

	What IS a Lie Group?
		F4 is the automorphism group of 3x3 matrices of octonions o11 o12 o13 o21 o22 o23 o31 o32 o33 such that o11, o22, and o33 are real (have no imaginary part), and o12, o13, o23 are the octonion conjugates of o21, o31, o32 respectively.
		Are we happy with G2 as the automorphism group of the octonions, F4 as the isometry of the [octonion] projective plane, E6 (in a noncompact form) as the collineations of the same, and E7 resp.
		A Lie algebra is a logarithm of a Lie group, and a Lie group is an exponential of a Lie algebra.
	www.valdostamuseum.org /hamsmith/Lie.html (3638 words)

	Generalized orthogonal group - Wikipedia, the free encyclopedia
		In mathematics, the generalized orthogonal group, O(p,q) is the Lie group of all linear transformations of a n = p + q 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 to the usual orthogonal group O(p + q; C).
	en.wikipedia.org /wiki/Generalized_orthogonal_group (415 words)

	Orthogonal matrix Summary
		The n×n orthogonal matrices form a group, the orthogonal group denoted by O(n), which—with its subgroups—is widely used in mathematics and the physical sciences.
		The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, the special orthogonal group SO(n) of rotations.
		Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected.
	www.bookrags.com /Orthogonal_matrix (3127 words)

	Notes on Finite Geometry (Site Map)
		Research announcement (4x4 case of diamond theorem and algebraic generalization) This research announcement was the basis for an abstract (79T-A37) in the Feb. 1979 AMS Notices.
		Portrait of O A table of the octahedral group O using the 24 patterns from the 2x2 case of the diamond theorem.
		Generating the octad generator The Miracle Octad Generator (MOG) of R. Curtis -- A correspondence between the 35 partitions of an 8-set into two 4-sets and the 35 lines of PG(3,2).
	finitegeometry.org /sc/map.html (794 words)

	CONTEXTS FOR SIMPLE SPINOR ALGEBRA
		The understanding of spinors as being attached to, and constructed from isotropic vectors in Euclidean spaces strongly suggests that a physical RÂ³ model of space in a fundamental physical theory be replaced with a CÂ³ that is the analytic continuation of RÂ³.
		With tensors in the context of linear spaces, the group is usually a group of the general group of basis substitutions, or a subgroup thereof.
		The "special orthogonal group in three dimensions", SO(3) is one of the classical Lie groups, and it is therefore simultaneously: an analytic manifold, a group and a measure space, with a left invariant Haar measure, invariant under the left action of the group on itself.
	graham.main.nc.us /~bhammel/PHYS/spinor.html (5134 words)

	Geometry of the 4x4 Square
		For a description of how the 4x4 array can be coordinatized to yield this family of partitions and also a group of 322,560 permutations (rather than the measly 384 of the hypercube) that leave affine "structural relations" undisturbed, see Finite Relativity.
		This pairing is not arbitrary; it is preserved under the action of the 244,823,040 permutations of the large Mathieu group M
		For another approach to constructing the MOG that involves the symplectic generalized quadrangle known as W(2), see Picturing the Smallest Projective 3-space.
	finitegeometry.org /sc/16/geometry.html (1637 words)

	Cornell Math - Thesis Abstracts (Lie Groups)
		Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure.
		We describe on K an analog of the Bargmann-Segal "coherent state" transform, and we prove that this generalized coherent state transform maps L^2(K) isometrically onto the space of holomorphic functions in L^2(G, \mu), where G is the complexification of K and where \mu is an appropriate heat kernel measure on G.
		The generalized coherent state transform is defined in terms of the heat kernel on the compact group K, and its analytic continuation to the complex group G.
	www.math.cornell.edu /Research/Abstracts/lie_groups.html (1159 words)

	What IS a Lie Group?
		The Bn and Dn are real rotations, denoted Spin(2n+1) and Spin(2n), and are called Spin groups, the double covers of special Orthogonal groups; the An are complex generalized rotations, denoted SU(n+1), and are called special Unitary groups; and the Cn are quaternionic generalized rotations, denoted Sp(n), and are called Symplectic groups.
		F4 is the automorphism group of 3x3 matrices of octonions o11 o12 o13 o21 o22 o23 o31 o32 o33 such that o11, o22, and o33 are real (have no imaginary part), and o12, o13, o23 are the octonion conjugates of o21, o31, o32 respectively.
		A Lie algebra is a logarithm of a Lie group, and a Lie group is an exponential of a Lie algebra.
	akbar.marlboro.edu /~mahoney/groups/Lie.html (2525 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Precise structure* * of the Thom spectra of the generalized braid groups of the types C and D is obtained.
		W.-L. Chow [9] found the presentation of this group with generators: oe2; :::; oen; a2; :::; an+1; where oej is the standard generator of the braid group Brn+1 and ai* * = = oe-11:::oe-1i-2oe2i-1oei-2:::oe1; 2 i n + 1.
		Considering the generalized braid groups of the t* ype C as the subgroups of the ordinary braid groups* our pairing can be described as putt* ing k + 1 strings of the first group* instead of the zero string of the second group.
	www.math.purdue.edu /research/atopology/Vershinin/thogebr.txt (4016 words)

	[ref] 47 Group Libraries
		Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one.
		The remaining non-nilpotent groups of order at most 2000 have been determined by Hans Ulrich Besche and Bettina Eick using the coprime split extensions method for solvable groups with certain normal Hall subgroups, the Frattini extension method for solvable group in general and the well-known cyclic extension algorithm for non-solvable groups.
		All groups in the library are primitive permutation groups of the indicated degree.
	www.math.temple.edu /computing/gap/ref/CHAP047.htm (5565 words)

	The CRUNCH group, Brown University
		The method, termed as `generalized polynomial chaos' or `Wiener-Askey polynomial chaos', is an extension of the mathematical theory of Nobert Wiener (1938).
		The generalized polynomial chaos is a broader framework which includes the Wiener-Hermite polynomial chaos as a subset.
		The results of generalized polynomial chaos are examined in model problems, and exponential convergence is demonstrated when the exact solutions are known and the appropriate type of chaos is employed.
	www.cfm.brown.edu /crunch/theses/db.html (295 words)

	Orthogonal Arrays (Site not responding. Last check: 2007-11-07)
		The columns of the orthogonal array correspond to the different variables whose effects are being analyzed.
		While there are nowadays other applications of orthogonal arrays in statistics (for example in computer experiments and survey sampling), the principal application is in the selection of level combinations for fractional factorial experiments.
		Since the rows of an orthogonal array represent runs (or tests or samples) - which require money, time, and other resources - there are always practical constraints on the number of rows that can be used in an experiment.
	www.research.att.com /~njas/doc/OA.html (2437 words)

	Orthogonal group - Definition, explanation
		Topics: Group theoretical methods, operator theory, and non-orthogonal expansions; Time-frequency methods for pseudo-differential operators; Non-orthogonal expansions and greedy algorithms; Noncommutative computational harmonic analysis...
		Topics: Group theoretical methods, operator theory, and non-orthogonal expansions; Time-frequency methods for pseudo-differential...
		Integral representations of finite groups and lattices, orthogonal representations of finite groups and group rings over p-adic integers.
	www.calsky.com /lexikon/en/txt/o/or/orthogonal_group.php (1279 words)

	Groups (Site not responding. Last check: 2007-11-07)
		All Abelian groups are isomorphic to a cyclic group or a direct product of cyclic groups (of prime powers).
		The finite Heisenberg groups are such that a,b,c are integers modulo n.
		The infinite groups are such that a,b,c are complex.
	www.cs.unm.edu /~dstrain/algebra/group_list.html (231 words)

	A group theory book(Allan Adler, Eric Lucas, David J Heisterberg)
		Furthermore, the distinctions between the group > theory and the representation theory is not emphasized.
		Galois invented the > concept of a group as part of his study of the solvability > of equations by radicals.
		Another is the use of groups in > topology, such as the fundamental group.
	yarchive.net /chem/group_theory_book.html (1374 words)

	PARG: Program Analysis Reading Group
		The Program Analysis Reading Group is a group of students and faculty that meets once a week to discuss papers drawn from a broad spectrum of research into program analysis, design, implementation, software engineering, and theory.
		Each group member is expected to read the paper, understand it to the best of your ability, and come to the meeting with questions and topics for discussion.
		Generally, BEDs are useful in applications, for example tautology checking, where the end-result as a reduced ordered BDD is small.
	pag.csail.mit.edu /reading-group (10419 words)

	Report of the CSM Polar Climate Working Group Meeting
		The CSM Polar Climate Working Group (PCWG) met on Wednesday, 23 June 1999, in Breckenridge, Colorado, as part of the Fourth Annual CSM Workshop.
		Briegleb presented results from CSM simulations showing oversimulation of the equatorward transport of sea ice on the poleward side of the Antarctic Circumpolar Current (ACC) and a general unrealistic pattern of ice transport throughout the Arctic Ocean and on the polar margins of the North Atlantic.
		The problem near the ACC could be related to the ocean model, the atmosphere model, or a combination of both, not necessarily the sea ice model.
	www.cgd.ucar.edu /csm/working_groups/Polar/Polarwg.html (1586 words)

	Bounded Complex Domains
		The 168-element group PSL(2,7) of the Klein Configuration is isomorphic to the symmetry group SL(3,2) of the Fano plane that describes Octonion multiplication.
		The automorphism group of the octonions is 14-dimensional G2.
		The quasihomogeneous bimodal singularities are associated with the 6 quadrilaterals and the 14 triangles on the Lobachevskii plane (in the latter case one must consider automorphic functions with automorphy factors corresponding to 2-, 3-, or 5- sheeted coverings)...".
	valdostamuseum.org /hamsmith/cdomain.html (8754 words)

	f4 - Article and Reference from OnPedia.com
		F4 is a popular boy band from Taiwan.
		The group started out in the television series Meteor Garden.
		F4 consists of Jerry Yan, Vic Zhou, Vanness Wu and Ken Zhu.
	www.onpedia.com /encyclopedia/f4 (68 words)

	groupname1 (Site not responding. Last check: 2007-11-07)
		Sp This symbol is a function with one argument, which should be a vector space or a module V equipped with a nondegenerate symplectic form.
		When applied to V it represents the group of all symplectic transformations of V.
		This is the group with three generators a, b, and c and relations c = a^2 = b^n, ca = ac, bc = cb, and c^2 = 1.
	www.win.tue.nl /~amc/oz/om/cds/groupname2.html (150 words)

	GAP Manual: 37.12 Irreducible Maximal Finite Integral Matrix Groups
		Any integral matrix group G of dimension n is a subgroup of GL_n(Z) as well as of GL_n(Q) and hence lies in some conjugacy class of integral matrix groups under GL_n(Z) and also in some conjugacy class of rational matrix groups under GL_n(Q).
		For any such group G, say, there is at least one class C in Q_1(n) such that G is conjugate under Q to a proper subgroup of some group H in C.
		In fact, the class C is uniquely determined for any group G occurring in the library (though there seems to be no reason to assume that this property should hold in general).
	www-groups.dcs.st-and.ac.uk /~gap/Gap3/Manual3/C037S012.htm (2344 words)

	Nathan D George \| U.C. Berkeley Mathematics
		An introduction to the linear algebra of GCS's and their descriptions as a complex Dirac structure and maximal isotropic subspaces of the direct sum of a vector space with its dual.
		Introduced complex Lie algebroids (of which generalized complex structures are a special case) and recalled the definition of (real, holomorphic and complex) Lie algebroids, briefly explaining how real and holomorphic Lie algebroids are the infinitesimal objects associated to smooth and holomorphic groupoids.
		Introduced a Riemannian metric on the generalized tangent bundle and defined generalized Kähler structures, which can be seen as a further reduction of the structure group of the generalized tangent bundle to U(n) × U(n).
	math.berkeley.edu /~natedawg/Seminars/GenGeom.html (624 words)

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