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Topic: Poincar group

	Poincare biography
		Poincaré was a scientist preoccupied by many aspects of mathematics, physics and philosophy, and he is often described as the last universalist in mathematics.
		Poincaré realised that indeed he had made an error and Mittag-Leffler made strenuous efforts to prevent the publication of the incorrect version of the memoir.
		Poincaré was absolutely correct, however, in his criticism that those like Russell who wished to axiomatise mathematics; they were doomed to failure.
	www-groups.dcs.st-and.ac.uk /history/Biographies/Poincare.html (3230 words)

	Encyclopedia: PoincarÃ© group (Site not responding. Last check: 2007-10-21)
		In mathematics, an abelian group is a commutative group, i.
		In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written:.
		In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful.
	www.nationmaster.com /encyclopedia/Poincar%E9-group (942 words)

	HENRI POINCARE
		Poincaré was a pioneer in hyperbolic geometry, which in the 1970's and 1980's became important in the study of 3-manifolds.
		Poincaré made his reputation in the theory of automorphic functions, which are functions invariant under linear fractional transformations.
		Poincaré's essay won the prize in 1889, even though he only partially solved the problem; what Poincaré found was that (to use modern terminology) mathematical chaos was lurking in Newton's equations for three or more bodies.
	www.usna.edu /Users/math/meh/poincare.html (514 words)

	DC MetaData for: On Poincarè Transformations and the Modular Group of the Algebra Associated with a Wedge (Site not responding. Last check: 2007-10-21)
		DC MetaData for: On Poincarè Transformations and the Modular Group of the Algebra Associated with a Wedge
		On Poincarè Transformations and the Modular Group of the Algebra Associated with a Wedge
		Poincar\'e covariant, fulfils wedge duality and moreover the
	www.esi.ac.at /Preprint-shadows/esi499.html (140 words)

	Group Theory (Site not responding. Last check: 2007-10-21)
		The most familiar group in physics is the rotation group governing rotations in three-dimensional space.
		Later, in the last two chapters of the book, some applications of this oscillator formalism to hadronic phenomenology are given, for example, to give an explanation of the mass spectra of hadrons and the study the deformation properties of relativistic hadrons.
		The idea is to show the relationship between the little groups of massive and massless particles (isomorphic to O(3) and E(2), respectively).
	www2.physics.umd.edu /~yskim/home/groth.html (593 words)

	[No title]
		Denote the group of invertible elements of $A$ by $A^$, its group* of automorphisms by $\operatorname{Aut}A$, and the center of $A$ by $\Cal Z$.
		Weyl groups of type $D_n$ \endsubhead To describe the Weyl groups of type $D_n$, write the elements of $B_n$ in the form $(c_1,\dots,c_n)\sigma$ with $c_i \in C_2, \sigma \in S_n$; then $D_n$ is the subgroup consisting of all elements in which an even number of the $c_i$ are equal to the unit element of $C_2$.
		The group consisting of all permutations of the entries in the sequences together with all operations which replace a single entry by its complement is the full group of permutations of all $2^n$ of the sequences of $0$'s and $1$'s.
	www.cs.biu.ac.il /~mschaps/wreath8.tex (14136 words)

	Jules Henri Poincaré (Site not responding. Last check: 2007-10-21)
		Poincaré can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables.
		He was able to show that any 2-dimensional surface having the same fundamental group as the two-dimensional surface of a sphere is topologically equivalent to a sphere.
		Poincaré was also first to consider the possibility of chaos in a deterministic system, in his work on planetary orbits.
	www.stetson.edu /~efriedma/periodictable/html/Pr.html (360 words)

	Poincare
		Birman, J. "Poincaré's Conjecture and the Homeotopy Group of a Closed, Orientable 2-Manifold." J. Austral.
		Papakyriakopoulos, C. "A Reduction of the Poincaré Conjecture to Group Theoretic Conjectures." Ann.
		Zeeman, E. "The Poincaré Conjecture for." In Topology of 3-Manifolds and Related Topics, Proceedings of the University of Georgia Institute, 1961.
	library.thinkquest.org /C006364/ENGLISH/problem/Poincare.htm (367 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		For example, the first homology group of a space (using integer coefficients) is the abelianization (G/[G,G]) of the fundamental group.
		Poincar\'e 's original "poincare conjecture" asked if a 3-dimensional manifold with trivial homology had to be a sphere; he himself gave a counterexample by finding a quotient of S^3 by a perfect group (G=its own commutator subgroup).
		You can do it with homotopy groups too (in which case you ought even to ask about non-abelian groups for \pi_1) and the answer is still "yes"; the groups are certain CW complexes called Eilenberg-MacLane spaces.
	www.math.niu.edu /~rusin/papers/known-math/94/detect (797 words)

	Read about Poincaré group at WorldVillage Encyclopedia. Research Poincaré group and learn about Poincaré group here! (Site not responding. Last check: 2007-10-21)
		In physics and mathematics, the Poincaré group is the
		group extension of the Lorentz group by a vector
		Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as an
	encyclopedia.worldvillage.com /s/b/Poincar%E9_group (118 words)

	Abstract (Site not responding. Last check: 2007-10-21)
		On Poincar\'e transformations and the modular group of the algebra associated with a wedge
		It will be shown that in a theory of local observables the modular group of the algebra of any wedge domain acts local iff the theory is Poincar\'e covariant, fulfils wedge duality and moreover the observables fulfil some reality condition with respect to the representation of the Lorentz group.
		If in addition to this representation of the Poincar\'e group the theory happens to be covariant with respect to a second representation of the Poincar\'e group then both representations differ only by a representation of the Lorentz group, which is a gauge transformation, i.e.
	www.lqp.uni-goettingen.de /lqp/papers/98/03/98031000.html (133 words)

	[No title]
		The topological groups G that have the finiteness property that K(n)(G) is finite in each degree will be called K(n)-locally stably duali zable groups, and among these we can single out the K(n)-compact groups* as those whose classifying space BG is a K(n)-local space.
		For finite groups G this follows as in [Kl01], but the natural generality for t* he the- ory appears to be to allow topological Galois groups* G that are E-locally stably dualizable, as considered here.
		The Poincar'e duality equivalence and the inverse Poincar'e equivalence * *pro- vide a chain of equivalences S[G] ^ S-adG ^ SadG ' DG+ ^ S-adG ' S[G], which is equivariant with respect to the standard left action on both copies of S[G], the trivial action on S-adG and the standard left action on SadG.
	hopf.math.purdue.edu /Rognes/dualizable.txt (5687 words)

	[No title]
		Introduction}}\vspace{5mm} A new family of unitary representations of the Poincar\'{e} group in four space-time dimensions is given, which describe two boson systems with interaction.
		The difficulty in passing from 3-d to 4-d comes from the highter dimension of the Poincar\'{e} group (passing from 7 to 10 dimensions) and from the non-integrability of some kernels.
		Thus from theorem 5, $c$ is the interaction kernel in the centre-of-mass frame of a continuous unitary representation of the Poincar\'{e} group.
	mpej.unige.ch /mp_arc/papers/98-305 (5398 words)

	Abstract (Site not responding. Last check: 2007-10-21)
		On Poincar\'e transformations and the modular group of the algebra associated with a wedge
		Borchers, H.-J. It will be shown that in a theory of local observables the modular group of the algebra of any wedge domain acts local iff the theory is Poincar\'e covariant, fulfils wedge duality and moreover the observables fulfil some reality condition with respect to the representation of the Lorentz group.
		If in addition to this representation of the Poincar\'e group the theory happens to be covariant with respect to a second representation of the Poincar\'e group then both representations differ only by a representation of the Lorentz group, which is a gauge transformation, i.e.
	www.lqp.uni-goe.de /papers/98/03/98031000.html (135 words)

	[No title]
		The group generated by the $\tau_i$, $i \in I$, is denoted by $\Ts$ and forms a subgroup of the transformations on the index set $I$.
		Because of the requirement that the admissible family be mapped onto itself by the isometry group of the space-time, an admissible family $\Ws$ in the case of Minkowski space must contain the orbit of each of its elements under the action of the Poincar\'e group.
		In Section 4.1 we consider the elements of the transformation group $\Ts$ associated with any such state and establish a considerable extension of the Alexandrov-Zeeman-Borchers-Hegerfeldt theorems by showing that these maps are induced by point transformations which form a subgroup $\Gs$ of the Poincar\'e group.
	www.ma.utexas.edu /mp_arc/papers/98-402 (11807 words)

	[No title]
		Group theory provides a precise language with which to describe the possible symmetries of a physical system.
		In the case of crystallographic groups, the successful classification of relevant symmetry groups in the nineteenth century made possible many predictions of what may actually be observed in nature.
		The first chapter provides a gentle introduction to groups (both finite and compact) and group actions, with excursions into the role of groups in crystallography, the classification of finite subgroups of $O(3)$, and the interplay of the icosahedral group and Euler's formula with fullerenes (``buckyballs'').
	www.ams.org /bull/pre-1996-data/199510/199510011.tex.html (1309 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Title: "Group Theoretical Foundations of Quantum Mechanics" Author: Ronald Mirman Abstract: The laws of physics are greatly constrained, perhaps even fully determined, by the geometry of space, expressed through its trans- formation groups and related algebras.
		Author: Ronald Mirman Title: "Massless Representations of the Poincare Group: electromagnetism, gravitation, quantum mechanics, geometry" Abstract: The Poincar group is the transformation group of the geometry of our space (not the symmetry group, although interestingly it is that also).
		And the group requires a quantum theory of gravity, which is, of course, general relativity.
	www.clifford.org /anonftp/pub/books/mirman.txt (726 words)

	symmetries
		Poincaré realized that the symmetry group of Maxwell's equations was (at least) the Poincaré group.
		The group consisting of the Poincaré group and dilations is sometimes called the "Weyl group".
		The representation theory of the Poincar´ group dominates relativistic physics, while the representation theory of the Galilei group dominates nonrelativistic physics.
	math.ucr.edu /home/baez/symmetries.html (2452 words)

	[No title]
		In a group of order $40$ the number of Sylow 5-subgroups must be 1 by Sylow's theorem, so it cannot be simple.
		In a simple group of order 56 there must be 8 Sylow 7-subgroups and so 48 elements of order 7.
		The remaining $56-48 = 8$ elements of the group must comprise the unique Sylow 2-subgroup, which must therefore be normal which is absurd.} \item Show that there is no non-abelian finite simple group of order greater than 60 but less that 168.
	www.bath.ac.uk /~masgcs/math0038/t8s.txt (1106 words)

	Path groups in gauge theory and gravity (Site not responding. Last check: 2007-10-21)
		Path Group (PG, consisting of curves in Minkowski space, MS), provides the group-theoretical interpretation of covariant derivatives.
		Generalized Poincarand#1081; Group (GPG, the semidirect product of PG by Lorentz group) applies to gravity, MS being a standard tangent space to a point of the real (curved) space-time.
		Generalized path groups (with paths in a group manifold) apply to topology.
	www.ccr.jussieu.fr /group24/PlenarySpeakers/mensky.htm (125 words)

	E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws
		It is the group of all continuous coordinate transformations with continuous derivatives, often called the group of general coordinate transformations.
		Such pseudotensors are covariant with respect to the linear transformations of the Poincar group and may be used in asymptotic spacetime regions far from gravitating sources to derive a principle of energy conservation.
		For theories whose symmetry group is an infinite continuous group, the main results of theorem II are that there are certain identities, or "dependencies" as she called them [1], between Lagrange functions of the theory and their derivatives.
	www.physics.ucla.edu /~cwp/articles/noether.asg/noether.html (5320 words)

	Lorentz covariance and kinetic charge, by P. Basarab-Horwath, R. F. Streater and J. Wright; superselection sectors. (Site not responding. Last check: 2007-10-21)
		There is a one-to-one correspondence between inequivalent covariant displaced Fock representations of the free relativistic field and the 1-cohomology of the Poincaré group with coefficients in the one-particle space.
		The 1-cohomology groups of the restricted Poincaré group are calculated.
		We recover P-invariance in a direct integral representation possessing a gauge group, and a superselection structure labelled by the velocities of the condensed states of matter which are the cocycles determining each irreducible component of the representation.
	www.mth.kcl.ac.uk /~streater/kinetic.html (165 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		For example, since Poincar\'e, it is known how to associate the group $\pi_1(X,x_0)$ to a topological space $X$ and to one of its points $x_0$; this group is called the {\em Poincar\'e group} or the {\em first homotopy group} of the space $X$ based on $x_0$.
		This group is null if and only if the space $X$ is {\em simply connected at $x_0$}; in another case, the group measures the lack of simple connectivity.
		Since the domain group is a free group, it is enough to define the operators on the generators of these groups; to avoid any confusion, we denote $\gamma(x)$ the generator of $\Omega_n$ which comes from the element $x$ of $X_{n+1}$.
	www-fourier.ujf-grenoble.fr /~sergerar/Papers/mega.tex (5338 words)

	The Irreducible Unitary Representations of the Extended Poincaré Group in (1+1) Dimensions (ResearchIndex)
		The Irreducible Unitary Representations of the Extended Poincaré Group in (1+1) Dimensions (2002)
		1 The representations of the oscillator group (context) - Streater - 1967
		1 Gravity and the Poincare group (context) - Grignani, Nardelli - 1992
	citeseer.ist.psu.edu /547259.html (704 words)

	TMPh V. I. Lagno, V. I. Fushchich - Reduction of self-dual Yang--Mills equations with respect to subgroups of the ... (Site not responding. Last check: 2007-10-21)
		TMPh V. Lagno, V. Fushchich - Reduction of self-dual Yang--Mills equations with respect to subgroups of the extended Poincar\'e group
		Lagno, V. Fushchich – Reduction of self-dual Yang--Mills equations with respect to subgroups of the extended Poincar\'e group
		For the vector potential of the Yang–Mills field in the Minkowski space $R(1,3)$, we construct the ansatze that are invariant under three-parameter subgroups of the extended Poincaré group $\widetilde P(1,3)$.
	math.ras.ru /tmph/scripts/tmph2.cgi?paperinfo=147 (97 words)

	GCI Generated Page: Abstracts (Site not responding. Last check: 2007-10-21)
		A geometric approach to the standard model in terms of the Clifford algebra $C\!\ell_{7}$ is advanced.
		These rotations comprise additional exterior transformations that commute with the Poincar\'{e} group and form the group $SU(2)_{L}$, interior ones that constitute $SU(3)_{C}$, and a unique group of coupled double-sided rotations with $U(1)_{Y}$ symmetry.
		The spinor mediates a physical coupling of Poincar\'{e} and isotopic symmetries within the restrictions of the Coleman-Mandula theorem.
	clifford.physik.uni-konstanz.de /cgi-BF/mysql_abs.cgi?&id=70 (186 words)

	[No title]
		New representations of the Poincar\'{e} group are given, which describe two bosons with interaction in four space-time dimensions.
		We add to the free Hamiltonian and the free Lorentz generators new interaction terms, without changing the Poincar\'{e} algebra commutation rules, and such that the algebra representation can be integrated to a unitary representation of the group on $L^2(I\!\!R^6,\sigma_2)$.
		The present paper is an extended and improved version of a previous one entitled "Relativistic quantum models for two bosons with interaction in the Schr\"{o}dinger picture", available at mp-arc 96-545.
	www.ma.utexas.edu /mp_arc/a/98-305 (195 words)

	Exact solutions of Dirac equation and induced representations of the Poincar'e group on the lattice (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Fundamental Physics Group, Oviedo Fundamental Physics Group, Oviedo...
		Abstract: We deduce the structure of Dirac field on the lattice from the discrete version of differential geometry and from the representation of the integral Lorentz transformations.
		The Analysis of the induced representations of the Poincar'e group on the lattice reveals that they are reducible a result that can be considered a group theoretical approach to the problem of fermion doubling.
	citeseer.ist.psu.edu /155049.html (239 words)

	On the invariance properties of the Klein--Gordon equation with
		Here we attempt to find the nature of the external electromagnetic field such that the KG equation with external electromagnetic field is invariant.
		Lie's extended group method is applied to obtain the class of external electromagnetic field which admits the invariance of the KG equation.
		Klein--Gordon equation; Lie analysis; prolongation group; Poincar\'e group.
	www.ias.ac.in /pramana/v61/p483/abs.htm (104 words)

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