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Topic: Orbit group theory

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	PlanetMath: group
		Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions.
		See Also: subgroup, cyclic group, simple group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group, Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group,
		This is version 16 of group, born on 2001-08-29, modified 2005-03-14.
	planetmath.org /encyclopedia/Group.html (277 words)

	MA3131 Group Theory
		One of the major goals of the module is to develop enough theory to be able to discuss the classification of the finite simple groups, at least in broad general terms.
		This course aims to present the fundamental ideas of group theory by studying the structure theorems and decomposition concepts that arise in attempts to understand groups in terms of less complicated groups.
		Rotman, An Introduction to the Theory of Groups, Springer-Verlag.
	www.mcs.le.ac.uk /Modules/MA-02-03/MA3131.html (691 words)

	Scientific method - Wikipedia, the free encyclopedia
		The observed difference for Mercury's precession, between Newtonian theory and relativistic theory (approximately 42 arc-seconds per century), was one of the things that occurred to Einstein as a possible early test of his theory of General Relativity.
		Einstein's theory of General Relativity makes several specific predictions about the observable structure of space-time, such as a prediction that light bends in a gravitational field and that the amount of bending depends in a precise way on the strength of that gravitational field.
		Thomas Kuhn denied that it is ever possible to isolate the theory being tested from the influence of the theory in which the observations are grounded.
	en.wikipedia.org /wiki/Scientific_method (5174 words)

	List of group theory topics - Wikipedia, the free encyclopedia
		See also: List of abstract algebra topics, List of category theory topics, list of Lie group topics.
		Mathematical objects which have (or make use of) a group operation
		Other mathematical disciplines which make great use of groups
	en.wikipedia.org /wiki/List_of_group_theory_topics (166 words)

	Predrag Cvitanović, research overview
		The field theory that I have used in my research is quite different from the textbook field theory, so I wrote my own textbook.
		The studies of the QCD gauge invariance led to investigations of its group theoretical structure, and to the development of new methods for evaluating group-theoretic weights.
		I plan to work towards a periodic orbits theory of spatiotemporal "turbulence" of infinite dimensional dynamical systems such as the Kuramoto-Sivashinsky equation, and deterministic theory of far-from-equilibrium processes in settings such as transport in the Lorentz gas and other idealized gases.
	www.nbi.dk /~predrag/papers/Overview.html (1059 words)

	PlanetMath: orbit
		This is version 2 of orbit, born on 2002-01-21, modified 2003-11-05.
		Object id is 1517, canonical name is Orbit.
		(Group theory and generalizations :: Semigroups :: Representation of semigroups; actions of semigroups on sets)
	planetmath.org /encyclopedia/Orbit.html (34 words)

	MTH-3D15 : Theory of Finite Groups
		Group theory is a large topic which interconnects with many branches of pure and applied mathematics.
		Overview: Group Theory has two main roots, one in geometry where groups of geometrical transformations were studied, the other in algebra and the theory of equations where groups of substitutions of variables (i.e.
		Abstract groups began to emerge with Jordan's seminal Traité des substitutions et des equations algébriques (1870) while the definition of abstract groups in general appears to be due to Weber (1882).
	www.mth.uea.ac.uk /maths/syllabuses/0506/3D1505.html (693 words)

	SwRI Spring 1999 Technology Today Article (Site not responding. Last check: 2007-10-11)
		The third theory suggested that the moon formed as an independent planetary body that was later "captured" by the Earth during a close pass.
		For example, it was difficult in both the capture and co-formation models to account for the lack of a large lunar iron core, because both predicted that the moon formed from the same mix of materials as the terrestrial planets, which typically contain a more substantial abundance of iron.
		The giant impact theory emerged from this conference with nearly consensus support, enhanced by new models of planet formation that suggested large impacts might indeed be common events in the end stages of terrestrial planet formation.
	www.swri.edu /3pubs/ttoday/spring99/moon.htm (2538 words)

	Imagine the Universe! Dictionary
		A theory of cosmology in which the expansion of the universe is presumed to have begun with a primeval explosion (referred to as the "Big Bang").
		The geometric theory of gravitation developed by Albert Einstein, incorporating and extending the theory of special relativity to accelerated frames of reference and introducing the principle that gravitational and inertial forces are equivalent.
		The inclination of a planet's orbit is the angle between the plane of its orbit and the ecliptic; the inclination of a moon's orbit is the angle between the plane of its orbit and the plane of its primary's equator.
	imagine.gsfc.nasa.gov /docs/dictionary.html (10412 words)

	Orbit (mathematics)
		In mathematics, an orbit is a concept in group theory.
		The orbit of an element x of X is the set of elements of X to which x can be moved by the elements of G; it is denoted by Gx.
		The orbits of a group action are the equivalence classes of the equivalence relation on X defined by x ~ y iff there exists g in G with x = g.
	www.fastload.org /or/Orbit_(mathematics).html (234 words)

	Group Theory & Rubik's Cube
		Group theory is the study of the algebra of transformations and symmetry.
		Given an element x of a group G, the orbit of x is the set of all elements of G which are generated by x, i.e.
		A representation of a group G is a set of matrices M which are homomorphic to the group.
	akbar.marlboro.edu /~mahoney/courses/Spr00/rubik.html (3602 words)

	Mathematics and Statistics - MATH225 Group Theory (Site not responding. Last check: 2007-10-11)
		The aim of this course is to introduce students to the basics of the theory of groups.
		Emphasis will be given to finite groups, and for much of the first half of the module groups will be thought of in terms of their Cayley tables, although this approach will be superseded by a more abstract treatment in the second half.
		We also look at maps between groups which 'preserve structure'; this gives a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same'.
	www.maths.lancs.ac.uk /department/study/years/second/modules/math225 (548 words)

	Normal basis - TheBestLinks.com - Cryptography, Characteristic, Elliptic curve cryptography, Finite field, ... (Site not responding. Last check: 2007-10-11)
		In mathematics, a normal basis in field theory is a special kind basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group.
		In algebraic number theory the study of the more refined question of the existence of a normal integral basis is part of Galois module theory.
		In the case of finite fields, this means that each of the basis elements is related to any one of them by applying the p-th power mapping repeatedly, where p is the characteristic of the field.
	www.thebestlinks.com /Normal_basis.html (543 words)

	Theory Group Home Page (Site not responding. Last check: 2007-10-11)
		Secondly, the presence of a protoplanet alters the gravitational potential and induces a transfer of angular momentum from the inner region of the disk to the outer region.
		Consequently, gas in the vicinity of the planet tends to move away radially from the planetary orbit.
		The figure has been obtained after an evolutionary time of 2000 orbits of the planet, which is now located at a radial distance of about 0.82.
	www.mpia-hd.mpg.de /THEORY/gallery/00-01/00-01.html (366 words)

	Orbit (disambiguation) - Wikipedia, the free encyclopedia
		The word orbit can mean more than one thing:
		Orbit (anatomy) - the socket in the skull which accommodates an eye
		ORBit - an object request broker (ORB) for CORBA
	en.wikipedia.org /wiki/Orbit_(disambiguation) (103 words)

	Topology & Group Theory Seminar
		Abstract: The rich theory of Coxeter groups can be used to provide algebraic constructions of finite volume geometric (particularly hyperbolic) n-manifolds.
		Combinatorial and arithmetic properties of these groups are used to establish basic properties of the resulting manifolds, such as their volume, size of automorphism group, orientability and so on.
		The cobordism theory of smooth high-dimensional sphere knots was completely solved (or at least turned into an algebraic problem) in the 1960s through the work of Kervaire, who showed that all even-dimensional knots are cobordant, and J. Levine, who showed that the cobordism type of an odd-dimensional knot is determined by its Seifert matrix.
	math.vanderbilt.edu /~hughescb/TopGTSemF03.html (811 words)

	[No title]
		In this case the group is virtually free [S] and we may take 0to be a free group of finite index.
		It is well-known that this group has no subgroups of order p2,and furthermore the number of conjugacy classes of elements of order p in is equal tothe class number of p, Cl(p).
		The centralize* r of any such element flwill be isomorphic to the group* of units U in Z[i], where ii* *s a primitive p-th root of unity.
	hopf.math.purdue.edu /Adem/otk.abstract (2023 words)

	[No title]
		The other extreme would be size of group n!, size of orbit n, number of generators 2, and in that case I think it really would be very inefficient to loop over all group elements.
		But there is another problem with trying to loop over all g in G. Groups are virtually always input as generating sets, which in problems of this type are likely to be permutations or matrices.
		Or if you had a generator of order r and some point in the orbit fixed by that generator, then you could stop when orblen was G/r.
	www.math.niu.edu /~rusin/known-math/00_incoming/orbits (406 words)

	Ergodic Theory at UEA
		Problems, examples, and methods in ergodic theory come from many branches of mathematics, including number theory, combinatorics, harmonic analysis, coding theory and group theory.
		Research activity at UEA in ergodic theory is focused on higher dimensional Markov shifts (dynamical systems in which the acting group is a lattice) and connections between arithmetic and ergodic theory.
		`Ergodic Theory' by I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Springer-Verlag (1981).
	www.mth.uea.ac.uk /~h720/research (534 words)

	MC341 Group Theory
		This module builds upon MC242 (Introduction to Groups) and gives an overview of the main ideas of group theory.
		Conjugates and conjugacy classes of elements and of subsets; normal subgroups; quotient groups; centralisers and normalisers; the centre of a group; normal closures; homomorphisms and isomorphisms; kernels and images; the isomorphism theorems; the alternating groups A
		J.J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag.
	www.mcs.le.ac.uk /Modules/Year3/MC341.html (658 words)

	MA30110 - GROUP THEORY
		The concept of a group occurs naturally in situations involving symmetry or in which some quantity is being preserved; for example, various letters such as A, S and I possess different numbers of symmetries and rigid motions preserve distance.
		The principal structure theorems for finite groups will be described and applied in a variety of group theoretic contexts.
		To provide a deeper understanding of the concepts and techniques of abstract algebra, introduced in module MA20310, by focusing on the group concept, starting with an axiomatic development of group theory, establishing a structure theory?mainly in the context of finite groups?and giving brief illustrations of a selection of applications of group theory.
	www.aber.ac.uk /modules/2004/MA30110.html (361 words)

	Diamond Theory: Symmetry in Binary Spaces (Site not responding. Last check: 2007-10-11)
		For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4x4 array.
		In the 4x4 case, D is a four-diamond figure (left, below) and G is a group of 322,560 permutations generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants.
		A table of the octahedral group O using the 24 patterns from the 2x2 case of the diamond theorem.
	m759.freeservers.com (1917 words)

	Theory Group Home Page (Site not responding. Last check: 2007-10-11)
		The existence of such multiple stars in forming systems suggests that the evolution of circumstellar material in orbit around or infalling onto those stars will be significantly altered from the evolution in a single star+disk system.
		Measured from the beginning of the simulation, the system has evolved for nearly five binary orbit periods at the time of this snapshot.
		The trajectory of each component is counterclockwise and periapse occurs when the stars (at the center of each disk) reach the y=0 axis and are 35~AU apart.
	www.phys.lsu.edu /~andy/Images/02-00.html (442 words)

	Vignettes on automorphic and modular forms, representations, L-functions, and number theory
		We want to prove that the singular homology of quotients X/Gamma is the group homology of Gamma, under some mild conditions on X (such as that X be a ball).
		Recollection of some basic facts on discrete series of real reductive groups, with table showing which classical groups do and don't have discrete series, holomorphic discrete series, and quaternionic discrete series.
		Standard basic features of representation theory of p-adic reductive groups: exactness of Jacquet module functors, Jacquet's lemmas, admissibility and finite-generation of Jacquet modules of admissible finitely-generated smooth representations.
	www.math.umn.edu /~garrett/m/v (1093 words)

	Theory Group : Hyperfine interaction
		The measurement of the nuclear spin-relaxation in heterojunctions is a challenging experimental problem, since the number of nuclear spins interacting with the two-dimensional electrons is negligibly small compared to their total number in a sample.
		These measurements shows close similarity between the magnetic field dependence of the nuclear spin-relaxation rate and the magnetoresistance in quantum Hall effect, as it was predicted in [2] thus demonstrating clearly the importance of the coupling of nuclear spin to the conduction electron spins in the nuclear relaxation in these systems.
		Ingenious transport measurements by McEuen group have demonstrated the possibility of studying the edge states by using the hyperfine interaction between the nuclear and electron spins.
	www.magniel.com /group/r_hyper.html (1540 words)

	FREE GROUPS
		These are problems about free groups, their automorphisms and related issues.
		(F14) Let F be a non-cyclic free group of finite rank, and G a finitely generated residually finite group.
		Suppose that A, B, H, K are free groups of finite ranks.
	zebra.sci.ccny.cuny.edu /web/nygtc/problems/probfree.html (1478 words)

	Tenure-track Faculty (Site not responding. Last check: 2007-10-11)
		He is interested in representation theory for reductive Lie groups; a typical example of a reductive group is the group of all invertible matrices (of fixed size).
		In other cases the coefficients could arise from the values of group characters; from counting solutions to a set of equations modulo the primes; from the values of characters of representations; or as the eigenvalues of the Laplacian operator on a suitable space.
		Twistor theory is a geometric approach to mathematical physics, which, among other things, provides a way of using complex geometry to solve physically interesting differential equations.
	www.math.okstate.edu /undergrad/temp/node76.html.xxx (1624 words)

	Bifurcation Theory Reading Group
		Equivariant Bifurcation Theory The systematic study of equivariant bifurcation problems originated in the 1970 and is still in progress.
		The theory focussed primarily on bifurction from equilibria in equivariant vector fields.
		Twisted equivariance In the study of bifurcations from periodic solutions with spatiotemporal symmetry, it has turned out to be useful to recognize the property of twisted equivariance and twisted reversibility for a first-hit/pull-back map that captures the local dynamics near the periodic solution.
	www.ma.ic.ac.uk /~jswlamb/BifReadingGroup2002.html (684 words)

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