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Topic: Discrete Heisenberg group

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Crocodile exoskeleton

Abelia

	Is String Theory in Knots?
		The symmetric group is the symmetry of fermions and bosons, while the braid group from knot theory plays the same role for anyons.
		From the Symmetric Group to the Braid Group
		This means that the braid group is also a candidate for part of the universal symmetry according to the principle of event-symmetric space-time.
	www.weburbia.com /pg/knots.htm (2891 words)

	Nobel Prize in Physics 1932 and 1933 - Presentation Speech (Site not responding. Last check: )
		Heisenberg now considered the combination of all the oscillations of such a spectrum as one system, for the mathematical handling of which, he set out certain symbolical rules of calculation.
		Heisenberg's quantum mechanics has been applied by himself and others to the study of the properties of the spectra of atoms and molecules, and has yielded results which agree with experimental research.
		Heisenberg has shown that according to quantum mechanics it is inconceivable to determine, at a given instant of time, both the position taken up by a particle and its velocity.
	nobelprize.org /nobel_prizes/physics/laureates/1933/press.html (2628 words)

	Lie group Summary
		The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
		The Heisenberg group is a Lie group of dimension 3, used in quantum mechanics.
		The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the standard model, whose dimension corresponds to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
	www.bookrags.com /Lie_group (4005 words)

	Knowledge and Reality - Torah & Science (Site not responding. Last check: )
		Heisenberg took this one step further: He challenged the notion of simple causality in nature, that every determinate cause in nature is followed by the resulting effect.
		Heisenberg even used this basis to reject the theorems of the older Ernst Schrodinger, claiming they assumed the existence of entities that could not be verified, and were therefore metaphysical.
		Heisenberg reasoned that just as Einstein had rejected the notion of absolute time and absolute space since these were no more than metaphysical concepts, so he and his colleagues can reject Schrodinger's wave mechanics on the same grounds.
	www.chabad.org /library/article.asp?AID=2778 (5904 words)

	[No title]
		For instance for a discrete group with torsion EG can never have a finite-dimensional model, whereas this is possible for EFIN (G) and the minimal dimension is related to the notion of virtual cohomological dimension.
		In particular we conclude for a discrete group G that a G-CW -complex X is the same as a CW -complex X with G-action such that for each open cell e X and each g 2 G with ge \ e 6= ; left multiplication with g induces the identity on e.
		Denote by Out(Fn) the group of outer automorphisms of Fn, i.e.
	www.math.purdue.edu /research/atopology/Lueck/lueck_classifyingspaces1203.txt (13567 words)

	index
		Volume 1 gives an introduction to harmonic analysis on the simplest symmetric spaces - Euclidean space, the sphere, and the Poincaré upper half plane H and fundamental domains for discrete groups of isometries such as SL(2,Z) in the case of H. The emphasis is on examples, applications, history.
		Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner that is both accessible to the beginner and suitable for graduate research.
		The second part emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices.
	math.ucsd.edu /~aterras (1506 words)

	Interview with Werner Heisenberg - F. David Peat (Site not responding. Last check: )
		In recent years Heisenberg adopted the unpopular position of criticizing research in elementary particle physics and proposing that symmetries and not elementary particles form the fundamental starting-point for a description of the world.
		We began by asking Heisenberg to recall the early days of quantum theory but it became apparent that great men have no desire to live in the past and he was just as eager to talk about the future of physics.
		So there are a number of groups which are fundamental in the sense that in describing the smallest particles we refer to their behaviour and transformations.
	www.fdavidpeat.com /interviews/heisenberg.htm (5887 words)

	[No title] (Site not responding. Last check: )
		Subject: asymptotics of random walks on finitely generated groups Date: 7 May 1999 09:30:08 -0500 Newsgroups: sci.math.research Suppose G is an infinite group generated by {g_1,...
		For example, the discrete Heisenberg group of order 3 consisting of 3x3 upper-triangular matrices with 1's on the main diagonal and any three integers above the main diagonal has a polynomial growth rate of order four (cf.
		For the discrete Heisenberg group of order 3, some Monte Carlo computations I did around 1989 suggest that the asymptotics of the DU_i's are O(x^{-3}) for non-zero x.
	www.math.niu.edu /~rusin/known-math/99/randwalk_grps (427 words)

	Andreas Strömbergsson
		An associated dynamical Dirichlet series is found to have a convenient closed rational form and analytic properties of the Dirichlet series are related to orbit growth asymptotics; depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.
		In such a quantization the dimension N of the state space play a number theoretically important role and I will discuss the large differences between the case when N is square free and when it is not.
		In 1989, B.Kitchens and K. Schmidt introduced an algebraic approach to the study of abelian actions of automorphisms of compact groups.
	www.math.uu.se /~astrombe/DNA/DNA.html (1774 words)

	Heisenberg group
		The Heisenberg group is a connected, simply-connected Lie group whose Lie algebra consists of matrices
		This group occurs not only in quantum mechanics but in the theory of theta functions; it is also used in Fourier analysis.
		The application that led Hermann Weyl to an explicit introduction of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent.
	www.omniknow.com /common/wiki.php?in=en&term=Discrete_Heisenberg_group (2476 words)

	Heisenberg's Physics and Philosophy
		The idea that energy could be emitted or absorbed only in discrete energy quanta was so new that it could not be fitted into the traditional framework of physics.
		If the atom can change its energy only by discrete energy quanta, this must mean that the atom can exist only in discrete stationary states, the lowest of which is the normal state of the atom.
		It did explain qualitatively the chemical behaviour of the atoms and their line spectra; the existence of the discrete stationary states was verified by the experiments of Franck and Hertz, Stern and Gerlach.
	www.marxists.org /reference/subject/philosophy/works/ge/heisenb2.htm (4007 words)

	CCR AND THE m DIMENSIONAL HEISENBERG ALGEBRAS
		For a kinematic Heisenberg algebra H(m) of QM in m [3] dimensional space, usually given as (generally unbounded) operators CCR acting on a specifically constructed Hilbert space, there is, axiomatically, and more abstractly, a nilpotent Lie algebra of 2m+1 dimensions given by the defining Commutation Relations (CRs).
		In the case of the Heisenberg algebra, there is an interesting cross relationship within a q-p pair familiar from classical canonical mechanics stated as, momentum is the generator of spatial translations and position is the generator of momentum translations in the context of phase space.
		The fundamental Heisenberg algebra H(1) is a Lie algebra of dimension 3, and it makes sense then to be able to see where it fits in the structure theory of all Lie algebra structures of dimension 3.
	graham.main.nc.us /~bhammel/PHYS/heisalg.html (4966 words)

	Corran Webster's Website : Research
		We show that the Gromov boundary is a quotient of the metric boundary and the quotient map is continuous, and that therefore a word-hyperbolic group has an amenable action on the metric boundary of its Cayley graph.
		We provide a geometric condition which determines whether or not every point on the metric boundary of a graph with the standard path metric is a Busemann point, that is it is the limit point of a geodesic ray.
		We show that groups such as the braid group and the discrete Heisenberg group have boundary points of the Cayley graph which are not Busemann points when equipped with their usual generators.
	www.nevada.edu /~cwebster/Research/papers.html (514 words)

	[No title]
		We will prove an analog of this result for hyperbolic groups, as well as a partial generalization of this result for the Heisenberg group: a word metric on the Heisenberg group lies within bounded GH distance from its asymptotic cone.
		In the case of the abelian group $\mathbb Z^n$ the asymptotic cone is $\mathbb R^n$ and it lies within a finite Gromov-Hausdorff distance from $\mathbb Z^n$.
		In the case of hyperbolic groups the result does not depend on the distance to the asymptotic cone because the asymptotic cone is not the Gromov-Hausdorff limit of the group with corresponding metrics.
	www.mpim-bonn.mpg.de /era-mirror/2001-01-011/2001-01-011.tex.html (3488 words)

	Wigner biography
		In fact Wigner's book on the applications of group theory to quantum mechanics was not the first to appear, since Weyl published his a little before Wigner.
		She was a physics student there but the happiness was soon repaced by much pain for she fell ill with cancer and died in 1937 less than a year after the marriage.
		While in Wisconsin Wigner showed the role of the special unitary group SU(4) in considering nuclear forces and he constructed a class of irreducible unitary representations of the Lorentz group.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Wigner.html (2309 words)

	Introduction and statement of results
		The group of holomorphic isometries of [tex2html_wrap_inline673] is the Lie group [tex2html_wrap_inline675].
		As a consequence of this criterion, it follows that Fuchsian groups, Schottky groups, groups of Schottky type and certain non-degenerate B-groups are all quasiconformally stable [3].
		Thus, for the result to hold, the space of Schottky groups modulo conjugation by isometries would have to reduce to a point, which of course is not the case.
	www.geom.uiuc.edu /~rminer/talks/cecm/webeq/node1.html (867 words)

	Elements of Chemistry: Atoms - Physical Science lesson plan (grades 9-12) - DiscoverySchool.com (Site not responding. Last check: )
		Then divide students into groups of three or four; each one to focus on one scientist and his contribution to the understanding of quantum mechanics.
		Werner Heisenberg: In 1927 he proposed that it is impossible to know the position and velocity of an electron at the same time; this concept is called the uncertainty principle.
		Once the timeline is complete, ask each group to present a report about the scientist, identifying his contribution and how his work borrowed from that of other scientists.
	school.discovery.com /lessonplans/programs/ec_atoms (1132 words)

	HJM, Vol. 29, No. 2, 2003
		We show that group C-algebras of some connected Lie groups* have stable rank one, connected stable rank one and general stable rank one.
		Noncommutative Geometry of the Discrete Heisenberg Group, pp.
		Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry of the group C-algebra of the three dimensional discrete* Heisenberg group.
	www.math.uh.edu /~hjm/Vol29-2.html (1685 words)

	Andrzej Zuk (Site not responding. Last check: )
		On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator - Journal of Geometry and Physics, 21, p.
		The lamplighter group as a group generated by a 2-state automaton and its spectrum - Geometriae Dedicata, 87, p.
		On property (T) for discrete groups - Proceedings of the conference Rigidity in Dynamics and Geometry, Cambridge 2000, Springer 2002, ed.
	www.math.jussieu.fr /~zuk/Papers.html (391 words)

	Representation Theory of Lie Groups - IMS
		An invariant eigendistribution on a real reductive group is a distribution which is in- variant under the conjugation by elements of the group, and is an eigendistribution for the commutative algebra of left and right invariant differential operators on the group.
		It is a classical theorem for compact Lie groups that the bilinear form is symmetric or skew-symmetric depending on the action of an element in the centre of the group of order less than or equal to 2.
		Bernstein center of a reductive $p$-adic group is an analogue of the center of the enveloping algebra of a Lie algebra.
	www.ims.nus.edu.sg /Programs/liegroups/abstracts.htm (6923 words)

	Max Planck Society - Junior Research Groups
		Since 1969 the Max Planck Society has been supporting gifted, young scientists and researchers through its Independent Junior Research Groups, which run for a limited period of time.
		The Max Planck Society invites applications for new positions as Junior Research Group Leaders.
		Especially accomplished young scientists and researchers can pursue their own research program in an Independent Junior Research Group that runs for a limited period of time.
	www.mpg.de /english/institutesProjectsFacilities/juniorResearchGroups (104 words)

	[quant-ph/0507130] Extremal quantum cloning machines
		We investigate the problem of cloning a set of states that is invariant under the action of an irreducible group representation.
		We then characterize the cloners that are "extremal" in the convex set of group covariant cloning machines, among which one can restrict the search for optimal cloners.
		For a set of states that is invariant under the discrete Weyl-Heisenberg group, we show that all extremal cloners can be unitarily realized using the so-called "double-Bell states", whence providing a general proof of the popular ansatz used in the literature for finding optimal cloners in a variety of settings.
	www.arxiv.org /abs/quant-ph/0507130 (167 words)

	Christian Borgs
		I am a senior researcher, co-founder and co-head of the Theory Group at Microsoft Research, as well as an affiliate Professor of Mathematics at the University of Washington.
		Among the honors I received are the 1993 Karl-Scheel Prize of the German Physical Society for my work in finite-size scaling, and a Heisenberg Fellowship of the German Research Council.
		Among the boards and councils on which I have served are the Council of the University of Leipzig, the Editorial Board of the Journal of Statistical Physics, and the Board of Trustees of the Institute for Pure and Applied Mathematics (IPAM).
	research.microsoft.com /~borgs (1251 words)

	Homepage of Christian de Ronde
		When the dimension N of a finite dimensional Hilbert space is a prime power, we can associate to each basis states of the Hilbert space an element of a finite or Galois field, and construct a finite group of unitary transformations, the generalised Pauli group or discrete Heisenberg-Weyl group (1).
		Besides, the elements of the group are in one to one correspondence with generalised Bell states (3) that play a crucial role in many applications of quantum information.
		(C) This solution is also in one to one correspondence with a discrete version of the Wigner distribution (5,6) which in turn is directly connected to the so-called epistemic interpretation of quantum mechanics (7).
	www.vub.ac.be /CLEA/people/deronde/Thomas2005.html (556 words)

	Yale Math Calendar
		Actions of the integral Heisenberg group on compact abelian groups.
		To every countable amenable group G and every element f of its integral group ring ZG, there is associated via Pontryagin duality for ZG/ZGf a very interesting action of G by automorphisms of the compact abelian dual of ZG/ZGf.
		For abelian groups G, natural dynamical questions such as expansiveness and entropy have been more or less completely worked out, involving ideas such as varieties, amoebas, and the Mahler measure of polynomials.
	www.math.yale.edu /calendar/day.php?LocationID=17&Date=2005-11-01 (168 words)

	[No title] (Site not responding. Last check: )
		Kachurovskii A. Convergence of averages in the ergodic theorem for groups $\Bbb Z^d$.......
		Kokhas, Suvorov Spectral estimates for Laplace operator on a discrete Heisenberg group.......
		Alekseev M. A., Glebskii L. Yu., and Gordon E. On approximations of the groups, group actions and Hopy algebras.......
	www.pdmi.ras.ru /znsl/1999/v256.html (171 words)

	CJM - Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group (Site not responding. Last check: )
		are formed using the action of the 3-dimensional discrete Heisenberg group G on a set S, and the operators will act on functions on S.
		Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.
		discrete Heisenberg group, unitary representation, local solvability, difference operator
	journals.cms.math.ca /cgi-bin/vault/view/kornelson2200 (173 words)

	Keri's cv
		Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group, Canadian J. Math., vol.
		This included organization of the orientation program for new graduate students.
		Co-Founded a group for women in mathematics, University of Colorado, 1998-2001.
	www.math.grin.edu /~kornelso/cv.html (977 words)

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