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Topic: Generalized special orthogonal group

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	Orthogonal Polynomials - Wolfram Mathematica
		"Special Functions" discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
		Generalized Laguerre polynomials LaguerreL[n, a, x] are related to hydrogen atom wave functions in quantum mechanics.
		Jacobi polynomials JacobiP[n, a, b, x] occur in studies of the rotation group, particularly in quantum mechanics.
	reference.wolfram.com /mathematica/tutorial/OrthogonalPolynomials.html (359 words)

	PlanetMath: examples of groups (Site not responding. Last check: 2007-10-12)
		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)

	The world's top orthogonal group websites (Site not responding. Last check: 2007-10-12)
		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).
		For n > 2 the fundamental group of SO(n,C) is Z/2Z whereas of the fundamental group of SO(2,C) is Z.
	www.websbiggest.com /wiki-article-tab.cfm/orthogonal_group (616 words)

	YourArt.com >> Encyclopedia >> spinor (Site not responding. Last check: 2007-10-12)
		In particular, the space of spinors is a projective representation of the orthogonal group.
		If one confines attention to the group of special orthogonal transformations, then it is possible unambiguously to take the square root of this form and obtain an identification of spinors with their duals.
		P and Q correspond to the generalized "position" and "momentum" for the Weyl quantization, although this physical fact is not important for the abstract construction of the spinors.
	www.yourart.com /research/encyclopedia.cgi?subject=/spinor (4192 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)

	!GROUP THEORY!
		Steiner attempted to generate allof the properties of a geometry from the use of its "fundamental forms"(Steiner).
		Given a finite group (G,)-where G isthe set of the group* and the * is the binary operation; and give a andb being elements of the set G; then ab=ba.
		Whena free group is given a number of operations then the most general groupthat can be built from them using the powers of these elements and theinverses of the powers.
	www.geocities.com /CapeCanaveral/Hangar/9302/group.html (3637 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-12)
		A generalized cohomology theory determined by spectra of Thom spaces and related to various structures in the stable tangent or normal bundle to a manifold.
		The general theorem that the (co)bordism group of
		Formal group for the notion of logarithm of a formal group law).
	eom.springer.de /c/c022780.htm (1949 words)

	[No title]
		The group G is obtained from G" by forming a quotient G"=A, where A is a finite subgroup of ZG", and the center of G is isomorphic to ZG"=A. This isomorphism indicates that in order to understand ZG it is enough to ____________ Date: January 21, 2000.
		Technically, a p-compact group is a pair (X; BX), where X is a p-complete space with finite mod p homology and B X is a classify- ing space for X (that is, the loop space BX is equivalent to X).
		Reflection groups and their centers In this section we recall some basic properties of finite reflection groups over fields of characteristic zero, and describe how the center of an irreducible reflection group can be determined by inspecting a finite collection of integers called the set of degrees of the reflection group.
	hopf.math.purdue.edu /Dwyer-Wilkerson/center-calc/center-calc.txt (9062 words)

	Groups for Dummies
		The portion of the theory of groups that is presented here is the part that is of use to molecular spectroscopists, usually chemists, in the classification and analysis of the complex infrared and optical spectra of molecules.
		A symmetry group is a group of spatial transformations that leaves an object unchanged.
		The proof of the orthogonality relations depended on the fact that any nonzero matrix that commutes with all the matrices of an irreducible representation is a multiple of the identity--that is, a diagonal matrix with all of its elements equal.
	www.du.edu /~jcalvert/math/groupdum.htm (7838 words)

	What IS a Lie Group? (Site not responding. Last check: 2007-10-12)
		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)

	[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)

	Clyde Davenport's Hypercomplex Special Relativity Page (Site not responding. Last check: 2007-10-12)
		We are motivated to attempt a formulation of special relativity in terms of commutative hypercomplex mathematics by one highly suggestive aspect of the mathematics, alone (refer to the Hypercomplex Math page for details on the following mathematical notation).
		We have produced a direct, concise formulation of special relativity with an arbitrary relative motion vector, and have shown that it reduces properly for the restricted case of relative velocity along the +x-axis.
		The group of relativistic transformations T corresponds to the group of Lorentz transformations of the traditional treatments, but the two cannot be related on a fundamental level because the Lorentz group is noncommutative and the T group is commutative.
	home.usit.net /~cmdaven/special.htm (4046 words)

	Mohammad Reza Pournaki
		Symmetry classes of tensors associated with certain groups and characters.
		Characterization of some finite non-abelian simple groups by the set of the order of their elements.
		On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group
	www.ipm.ac.ir /IPM/people/personalinfo.jsp?PeopleCode=IP0000008 (221 words)

	Computer, Telephony and Electronics Glossary and Dictionary
		Aside from its technical meaning, off-line is used frequently in a more general sense to describe events that occur outside of a standard procedure.
		Generally regarded as being attached and operating on a network or the Internet.
		Generalized to mean independent of, separate from, or irrelevant to.
	www.csgnetwork.com /glossaryo.html (3288 words)

	Delson Group
		U.S. delivers special reports on some emerging technology, strategy and business in wireless and mobile communications.
		"Our group is working on the integration of 3G system and the WiFi system for the global markets.
		Generic Adaptive Scheduler For High Speed Downlink Packet Access (HSDPA)
	www.delson.org /tmp/4G_Design_CD.htm (6765 words)

	Wellesley Science Chemistry
		In this setting I study properties which are invariant under the special linear group, rather than the orthogonal group.
		Geometry is a subject with a long, rich history, and a subject used in such diverse areas as astronomy (the shape of the universe), machinery design, and the study of DNA coiling.
		Riemannian geometry is the study of the shapes of manifolds (generalized surfaces) endowed with metrics (distance functions).
	www.wellesley.edu /Chemistry/WellesleyScience/wsmath.html (821 words)

	Reading Class: Hecke Algebras and Orthogonal Polynomials (Site not responding. Last check: 2007-10-12)
		generalized classical orthogonal polynomials on the ball and simplex, etc.
		Generalized Kostant convexity theorems, the constant term map on spherical Hecke algebras and branching to Levi subgroups
		Gus on Jack and Macdonald polynomials as orthogonal polys
	www.math.ucdavis.edu /~vazirani/W05/290.html (474 words)

	Linear Algebra and its Applications. (Site not responding. Last check: 2007-10-12)
		Ghazal A. Ghazal, Heinz Neudecker, On second-order and fourth-order moments of jointly distributed random matrices: a survey, Linear Algebra and its Applications 321 (1-3) (2000) pp.
		Claude Dellacherie, Servet Martínez, Jaime San Martín, Description of the sub-Markov kernel associated to generalized ultrametric matrices.
		Chuanqing Gu, Generalized inverse matrix Padé approximation on the basis of scalar products, Linear Algebra and its Applications 322 (1-3) (2001) pp.
	www1.elsevier.com /cdweb/journals/00243795/viewer.htt?viewtype=keywords (584 words)

	Orthogonal Polynomials
		Section 3.2.10 discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
		occur in studies of the rotation group, particularly in quantum mechanics.
		Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials.
	documents.wolfram.com /v4/MainBook/3.2.9.html (240 words)

	Hans Cuypers (Site not responding. Last check: 2007-10-12)
		A characterization of the symplectic and unitary 3-transposition groups, Comm.
		(with Anja Steinbach) Linear transvection groups, 1998, in the proceedings of the conference `Algebraic Combinatorics', A. Munemassa (ed), RIMS, Kyoto, 1998.
		(with A. Cohen and H. Sterk) Linear groups generated by reflection tori, Canadian J. of Mathematics 56 (1999), 1149-1174.
	www.win.tue.nl /~hansc/papers.html (766 words)

	Medical Reps (Site not responding. Last check: 2007-10-12)
		The various specialized branches of thescience of medicine correspond to equally specialized medical professions dealing with particular organs or diseases.
		The science of medicine is the body of knowledge about body systems and diseases, whilethe profession of medicine refers to the social structure of the group ofpeople formally trained to apply that knowledge to treat disease.
		There are traditional and schools of healing which are usually not considered to be part of (Western) medicine in a strictsense (see health science for an overview).
	www.relativeaccess.com /File/2956-Medical.Reps.Html (544 words)

	I P M - Homepage (Site not responding. Last check: 2007-10-12)
		Symmetry classes of tensors associated with certain groups and characters, Generalized matrix functions, Prime submodules, The structure of indecomposable injective modules, Characterization of some finite non-abelian simple groups by the set of the order of their elements, Probabilistic problems in group theory.
		On the Orthogonal Basis of the Symmetry Classes of Tensors
		non-abelian simple groups by the set of the order of their elements and
	www.ipm.ac.ir /ipm/activities/ViewProgramInfo.jsp?PTID=204 (112 words)

	UK Mathematics - Numerical Analysis, Scientific Computation
		Ph.D., 2005, Generalizations of an Inverse Free Krylov Subspace Method for the Symmetric Generalized Eigenvalue Problem, Ye.
		Solvers and Design Validation Group, UGS, 10824 Hope Street, Cypress, California, 90630, USA.
		MGNet, the Internet special interest area for multigrid, multilevel, multiscale, and domain decomposition methods.
	www.ms.uky.edu /~rcli/NASC.html (319 words)

	List of Lie group topics - Wikipedia, the free encyclopedia
		See Table of Lie groups for a list
		The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions.It is the group of hyperbolic motions of the Poincare disk model of the hyperbolic plane.
		See also: List of harmonic analysis and representation theory topics
	en.wikipedia.org /wiki/List_of_Lie_group_topics (100 words)

	Soeren Krausshar's webplace (Site not responding. Last check: 2007-10-12)
		Function spaces with reproducing kernel functions in higher dimensional symmetric domains
		Automorphic forms and functions for arithmetic subgroups of the orthogonal group in several variables
		Generalized CM-lattices in higher dimensional Euclidean spaces and related algebraic number fields
	cage.rug.ac.be /~krauss (159 words)

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