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Topic: Grassmannian

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	Packings in Grassmannian spaces (Site not responding. Last check: 2007-10-18)
		More generally, the Grassmannian space G(m,n) is the space of all n-dimensional subspaces of m-dimensional Euclidean space.
		A Family of Optimal Packings in Grassmannian Manifolds [postscript, pdf], P.
		A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces [postscript, pdf], A.
	www.research.att.com /~njas/grass (410 words)

	Grassmannian - Wikipedia, the free encyclopedia
		In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted G
		This shows directly that the real Grassmannians are compact (for the same result for complex Grassmannians one applies the unitary group).
		The detailed study of the Grassmannians uses a decomposition into subsets called Schubert cells, which were first applied in enumerative geometry.
	en.wikipedia.org /wiki/Grassmannian (534 words)

	Grassmannian -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		Supposing first that K is the (Any rational or irrational number) real number or (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex number field, the easiest approach to Grassmannians is probably to consider them as homogeneous spaces.
		This shows directly that the real Grassmannians are (A small cosmetics case with a mirror; to be carried in a woman's purse) compact (for the same result for complex Grassmannians one applies the (Click link for more info and facts about unitary group) unitary group).
		It follows that the equations defining the Grassmannian can be regarded as the (Click link for more info and facts about identities) identities satisfied by k × k (A league of teams that do not belong to a major league (especially baseball)) minors.
	www.absoluteastronomy.com /encyclopedia/g/gr/grassmannian.htm (705 words)

	Talk:Gauss map - Wikipedia, the free encyclopedia
		The Grassmannian page does talk about unoriented k-subspaces, so the notation here isn't yet consistent with that there.
		It seems that in the litterature, both are called grassmannian, with definition depending on context.
		On orientations: the Grassmannian page should point out that the orthogonal group acts transitively also on oriented k-subspaces, and so on: the distinction should be clarified there.
	en.wikipedia.org /wiki/Talk:Gauss_map (311 words)

	Grassmannian (Site not responding. Last check: 2007-10-18)
		Supposing first that K is the real number or complex number field, the easiest approach to Grassmannians is probably toconsider them as homogeneous spaces.
		This shows directly that the realGrassmannians are compact (for the same result for complex Grassmannians one appliesthe unitary group).
		The detailed study of the Grassmannians uses a decomposition into subsets called Schubert cells, which were firstapplied in enumerative geometry.
	www.therfcc.org /grassmannian-220052.html (398 words)

	Abstract of PhD Thesis (Zelenko Igor)-Mathematics
		We introduce two principal symplectic invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in Lagrange Grassmannian providing the curve with a natural projective structure, and a fundamental form, which is a degree four differential on the curve.
		Further, we give the estimates for the conjugate points of rank 1 curves in the Lagrangian Grassmannian of 4-dimensional symplectic space.
		Finally, we apply our general theory of curves in the Lagrange Grassmannian to the equivalence problem of rank 2 vector distributions on n-dimensional manifold and to the problem of geodesic equivalence of control systems.
	www.graduate.technion.ac.il /theses/Abstracts.asp?Id=10435 (341 words)

	Grassmannian - Definition up Erdmond.Com
		Supposing first that K is the real_number or complex_number field, the easiest approach to Grassmannians is probably to consider them as homogeneous spaces.
		There can be other approaches: for example orthogonal_groups also act transitively, so that the Grassmannians also appear as coset spaces for those groups.
		This shows directly that the real Grassmannians are compact (for the same result for complex Grassmannians one applies the unitary_group).
	www.erdmond.com /Grassmannian.html (398 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		The determinant bundle and the tau function are constructed over the Grassmannian whereas in contrast to the usual constructions regularized determinants are involved.
		For fixed elements of the isospectral set and the variation of the parameter x the second group which acts on the Grassmannian is derived.
		It is then shown how a unified construction for both group acting on the Grassmannian has to be carried out by comparing it to the Fock bundle construction of 3 + 1 dimensional Dirac-Yang-Mills theories.
	www.elsevier.com /cdweb/journals/03930440/articles/18/3/039304409500011.abstract.en (187 words)

	Math 665: Schubert Calculus (Site not responding. Last check: 2007-10-18)
		Prove that a Schubert variety of codimension 1 (with respect to an arbitrary flag) is the intersection of the Grassmannian with a hyperplane in the Plücker embedding.
		Prove that the Grothendieck ring of a Grassmannian is isomorphic to its cohomology ring as an abstract ring.
		Prove that the number of reduced words for a Grassmannian permutation is equal to the number of standard Young tableaux of the corresponding shape.
	www.math.lsa.umich.edu /~fomin/665w02.html (970 words)

	Grassmannian Structures on Manifolds (ResearchIndex) (Site not responding. Last check: 2007-10-18)
		Abstract: Grassmannian structures on manifolds are introduced as subbundles of the second order framebundle.
		The structure group is the isotropy group of a Grassmannian.
		A canonical normal connection is constructed from a Cartan connection on the bundle and a Grassmannian curvature tensor for the structure is derived.
	citeseer.ist.psu.edu /419279.html (332 words)

	Citations: packings in Grassmannian space - Conway, Hardin, Sloane, lines, etc (ResearchIndex) (Site not responding. Last check: 2007-10-18)
		One task of the present paper is to de ne a notion of f code in Grassmannian spaces, which reduces to A code Date: April 9, 2002.
		Bounds for antipodal codes, or packings of lines, have been previously considered in [3, 7, 11] In the next section we describe the linear programming bound of [9] and a variant that is valid for antipodal codes.
		A separate proof of convergence for Algorithm 1 0 can be based on a result of Campi, Haas, and Weil [6] This says that for any zonoid and finite set of directions, there is a zonotope that has the same support function values as the zonoid in those directions.
	sherry.ifi.unizh.ch /context/15500/0 (2360 words)

	Grassmannian (Site not responding. Last check: 2007-10-18)
		Esto muestra directamente que el Grassmannians verdadero es compacto (el mismo resultado para Grassmannians complejo uno solicita a grupo unitario).
		Entonces Grassmannians se puede mostrar para ser variedades descriptivas.
		Para un ejemplo del uso de Grassmannians en geometría diferenciada, ver el mapa del gauss y en geometría descriptiva, ven que Plücker coordina.
	www.yotor.net /wiki/es/gr/Grassmannian.htm (394 words)

	AMS Research Seminars (Site not responding. Last check: 2007-10-18)
		Grassmannian Frames, the Heisenberg group, and wireless communications
		Especially attractive are so-called Grassmannian frames, which are frames of fixed redundancy that minimize the mutual correlation between frame elements.
		Furthermore I will demonstrate that the very same Grassmannian frames also yield good codes with low peak-to-average ratio for OFDM modulation.
	www.ams.ucsc.edu /seminars/may06_03.html (166 words)

	Department of Mathematics: Alex Kasman's Homepage
		The geometry of Grassmannian manifolds and associated functions which satisfy nonlinear partial differential equations of mathematical physics
		Grassmannians, Nonlinear Waves and Generalized Schur Functions was published in Contemporary Mathematics 246 (1999) pp.
		Tau-Functions, Grassmannians and Rank One Conditions with Michael Gekhtman was published as The Journal of Computational and Applied Mathematics 202 (2007) 80--87.
	math.cofc.edu /faculty/kasman (1255 words)

	Citebase - Crystals via the affine Grassmannian
		It was proved by Ginzburg and Mirkovic-Vilonen that the G(O)-equivariant perverse sheaves on the affine grassmannian of a connected reductive group G form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group G
		Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian.
		This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:math/9909077 (1015 words)

	Informal Seminar Series (Site not responding. Last check: 2007-10-18)
		Present orthogonalization methods based on Davey's algorithm are shown to have a different differential-geometric interpretation: restriction of the ODE to a Grassmannian manifold.
		Using properties of Grassmannian manifolds and their tangent spaces, a new Grassmannian integrator can be introduced which generalizes Davey's algorithm.
		Furthermore it is shown that the compound-matrix method is a dual Grassmannian integrator: it uses Pluecker coordinates for integrating on a Grassmannian manifold, and this characterization suggests a new algorithm for constructing the induced ODE on the Grassmannian manifold.
	www.maths.surrey.ac.uk /announce/Informal_seminar/Abstracts/bridges_99.html (235 words)

	Enumerative Real Algebraic Geometry: The manifold of partial flags
		in R for which the intersection of Grassmannian Schubert varieties (5.12) is transverse with all points real and there exist such choices of the t
		Many instances of these same enumerative problems have been computed, and in each instance of (1) the intersection (5.12) is transverse with all points real.
		Conjecture 5.12 has nothing to say when the Schubert data are not Grassmannian.
	www.math.univ-rennes1.fr /geomreel/raag01/surveys/ERAG/S5/3.2.html (458 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $\langle f_k,f_l \rangle$ among all frames $\frame \in {\cal F}$.
		Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames.
		We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory.
	www.math.ucdavis.edu /~strohmer/papers/2002/grass.html (174 words)

	Packings in complex Grassmannian space and their use as multiple-antenna signal constellations
		Written by: Dakshi Agrawal, Thomas J Richardson, and Ruediger L Urbanke.
		In this paper we show that the problem of designing efficient signal-constellations for multiple transmit antennas can be related to a packing problem in complex Grassmannian space.
		We describe a numerical optimization procedure for encoding good packings in complex Grassmannian space and report the best packings found by this procedure.
	www.research.ibm.com /people/a/agrawal/mac.shtml (105 words)

	IRMA Strasbourg - Publication 2000 (Site not responding. Last check: 2007-10-18)
		In this article we study differential geometric properties of the most basic infinite dimensional manifolds arising from fermionic $(1+1)$-dimensional quantum field theory: the restricted Grass- mannian and the group of based loops in a compact simple Lie group.
		We determine the Riemann curvature tensor and the (linear- ly) divergent expression corresponding to the Ricci curvature of the restricted Grassmannian after proving that the latter mani- fold is an isotropy irreducible Hermitian symmetric space.
		Using the Gauss equation of the embedding of a based loop group into the restricted Grassmannian we show that the (conditional) Ricci curvature of a based loop group is proportional to its metric.
	www-irma.u-strasbg.fr /irma/publications/2000/00010.shtml (189 words)

	Citebase - The Verlinde Algebra And The Cohomology Of The Grassmannian
		Citebase - The Verlinde Algebra And The Cohomology Of The Grassmannian
		The Verlinde Algebra And The Cohomology Of The Grassmannian
		It has long been known that the resulting low energy effective action describes a theory with a mass gap; the novelty here is that this theory in fact is equivalent at long distances to a gauged WZW model of U(k)/U(k), and hence is related to the Verlinde algebra.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-th/9312104 (1620 words)

	Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems (Site not responding. Last check: 2007-10-18)
		In this correspondence, a quantized maximum signal-to-noise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter.
		Using the distribution of the optimal beamforming vector in independent identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing.
		Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss.
	www.ece.utexas.edu /%7Erheath/papers/2002/grassbeam/index.htm (196 words)

	A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces - Calderbank, Hardin, Rains, Shor, ...
		A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces A.
		and [6] where optimal packing in a real Grassmannian space is pursued using the chordal distance as the distance metric.
		Calderbank, R. Hardin, E. Rains, P. Shor and N. Sloane, "A group-theoretic framework for the construction of packings in Grassmannian spaces," J. Algebraic Combinatorics, 1997 (submitted).
	citeseer.ist.psu.edu /calderbank97grouptheoretic.html (648 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		It turns out that the study of symplectic invariants of curves in the Lagrange Grassmanian gives the way to construct feedback or gauge invariants of a wide class of smooth control systems and geometric structures.
		In the present talk we would like to describe how to construct symplectic invariants of the curve in the Lagrange Grassmannian, using the notion of cross-ratio of four Lagrangian subspaces.
		We will discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in Lagrange Grassamnnian providing the curve with a natural projective structure, and a fundamental form, which is a well-defined fourth degree differential on the curve.
	www.math.technion.ac.il /~techm/20020203143020020203zel (206 words)

	Grassmannian Frames with Applications to Coding and Communications (Site not responding. Last check: 2007-10-18)
		For a given class F of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation
		We then introduce infinite-dimensional Grassmannian frames and analyze their connection to uniform tight frames for frames which are generated by group-like unitary systems.
		Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.
	www.ece.utexas.edu /~rheath/papers/2002/grassframes (180 words)

	Abstract: "A sagbi basis for the quantum Grassmannian" (Site not responding. Last check: 2007-10-18)
		The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree np in the Grassmannian of p-planes in (m+p)-space.
		The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new "Gröbner basis style" proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus.
		The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and Koszul, and the ideal of quantum Plücker relations has a quadratic Gröbner basis.
	www.math.tamu.edu /~sottile/abstracts/sagbi.html (180 words)

	Grassmannian -- find the ideal of a Grassmannian (Site not responding. Last check: 2007-10-18)
		variables, the routine finds the ideal of the Grassmannian of projective k-planes in P^r, using the first
		For example, the Grassmannian of projective lines in P^3:
		Caveat: currently, this ideal is constructed using relations on minors of a generic matrix.
	www.math.temple.edu /computing/Macaulay2/0974.html (92 words)

	Grassmannian - TheBestLinks.com - Complex number, Field (mathematics), Group action, Linear algebra, ... (Site not responding. Last check: 2007-10-18)
		Grassmannian - TheBestLinks.com - Complex number, Field (mathematics), Group action, Linear algebra,...
		Grassmannian, Complex number, Field (mathematics), Group action, Linear algebra...
		)/H. This then provides a topology on the Grassmannian, and a smooth structure.
	www.thebestlinks.com /Grassmannian.html (466 words)

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