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	Flag manifold - Definition, explanation
		In mathematics, a flag manifold (or flag variety) is the set of all flags in a finite-dimensional vector space V.
		Flag manifolds are called complete or partial according to whether one considers complete or partial flags.
		The standard flag associated with this basis is the one where the ith subspace is spanned by the first i vectors of the basis.
	www.calsky.com /lexikon/en/txt/f/fl/flag_manifold.php (677 words)

	Science Fair Projects - Flag manifold
		In mathematics, a flag manifold (or flag variety) is the set of all flags in a finite-dimensional vector space V.
		The standard flag associated with this basis is the one where the ith subspace is spanned by the first i vectors of the basis.
		Relative to this basis, the stabilizer of the standard flag is the the group of nonsingular upper triangular matricies, which we denote by B
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Flag_manifold (782 words)

	Flag manifold - Education - Information - Educational Resources - Encyclopedia - Music
		In mathematics, a flag is an increasing sequence of subspaces of a vector space.
		According to basic results of linear algebra, any two flags of a finite-dimensional vector space V, in which the dimensions are the same, are no different from each other from a geometric point of view.
		In an infinite-dimensional space V, as used in functional analysis, the flag idea generalises to a subspace nest, namely a collection of subspaces of V that is a total order for inclusion and which further is closed under arbitrary intersections and closed linear spans.
	www.music.us /education/F/Flag-manifold.htm (809 words)

	Whitley Strieber's Unknown Country
		Mathematician Barbara Shipman has discovered that the movements of the dancing bees can be predicted by a mathematical formula called a "flag manifold," which expresses movement in the world of the tiny particles known as quarks.
		In mathematical terms, a manifold is a basic shape.
		She made this discovery when she projected the six dimensions of a flag manifold onto a two dimensional piece of paper.
	www.unknowncountry.com /news/?id=5457 (527 words)

	Quantum Honeybees \| Mind & Brain \| DISCOVER Magazine (Site not responding. Last check: )
		The flag manifold (which got its name because some imaginative mathematician thought it had a shape like a flag on a pole) happens to have six dimensions, which means mathematicians can’t visualize all the two-dimensional objects that can live there.
		The flag manifold, she notes, in addition to providing mathematicians with pure joy, also happens to be useful to physicists in solving some of the mathematical problems that arise in dealing with quarks, tiny particles that are the building blocks of protons and neutrons.
		And she does not believe the manifold’s presence both in the mathematics of quarks and in the dance of honeybees is a coincidence.
	discovermagazine.com /1997/nov/quantumhoneybees1263 (3534 words)

	Vexillology and Science (Site not responding. Last check: )
		We are as impartial as possible about the flags themselves (although we may baulk at the aesthetics of some designs), while we may still report on the emotion felt by certain flags.
		For both the Mississippi and Georgia state flag debates important information has been acquired and reported by the media as to the disposition of a flag or the way the flag is perceived within the jurisdiction.
		Flags are cultural artifacts, and as such their study can inform the development of theory in other academic disciplines, just like the study of ancient potsherds and 18th century gardening.
	www.crwflags.com /fotw/flags/vxt-sci.html (3995 words)

	Flag manifold - Wikipedia, the free encyclopedia
		In mathematics, a flag manifold (or flag variety) is the set of all flags in a finite-dimensional vector space V.
		The standard flag associated with this basis is the one where the ith subspace is spanned by the first i vectors of the basis.
		Relative to this basis, the stabilizer of the standard flag is the group of nonsingular upper triangular matrices, which we denote by B
	en.wikipedia.org /wiki/Flag_manifold (644 words)

	blog.myspace.com/opening_the_mouth (Site not responding. Last check: )
		The flag manifold (which got its name because some imaginative mathematician thought it had a shape like a flag on a pole) happens to have six dimensions, which means mathematicians cant visualize all the two-dimensional objects that can live there.
		The flag manifold, she notes, in addition to providing mathematicians with pure joy, also happens to be useful to physicists in solving some of the mathematical problems that arise in dealing with quarks, tiny particles that are the building blocks of protons and neutrons.
		And she does not believe the manifolds presence both in the mathematics of quarks and in the dance of honeybees is a coincidence.
	blog.myspace.com /opening_the_mouth (4417 words)

	Flag (linear algebra) - Wikipedia, the free encyclopedia
		In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a vector space V.
		A partial flag can be obtained from a complete flag by deleting some of the subspaces.
		In an infinite-dimensional space V, as used in functional analysis, the flag idea generalises to a subspace nest, namely a collection of subspaces of V that is a total order for inclusion and which further is closed under arbitrary intersections and closed linear spans.
	en.wikipedia.org /wiki/Flag_(linear_algebra) (251 words)

	Science Box
		The flag manifold (which got its name because some imaginative mathematician thought it had a "shape" like a flag on a pole) happens to have six dimensions, which means mathematicians can't visualize all the two-dimensional objects that can live there.
		The flag manifold, she notes, in addition to providing mathematicians with pure joy, also happens to be useful to physicists in solving some of the mathematical problems that arise in dealing with quarks, tiny particles that are the building blocks of protons and neutrons.
		And she does not believe the manifold's presence both in the mathematics of quarks and in the dance of honeybees is a coincidence.
	science.box.sk /newsread.php?newsid=6321 (3587 words)

	Springer Online Reference Works (Site not responding. Last check: )
		The existence of global (everywhere-defined) flag structures on a manifold imposes fairly-strong restrictions on its topological structure.
		For example, there is a line field, that is, a flag structure of type, on a simply-connected compact manifold if and only if its Euler characteristic vanishes.
		There is a complete flag structure on a simply-connected manifold if and only if it is completely parallelizable, that is, if its tangent bundle is trivial.
	eom.springer.de /f/f040570.htm (338 words)

	PlanetMath: flag variety
		associated to this data is the set of all flags
		In particular, the complete flag variety is isomorphic to
		This is version 2 of flag variety, born on 2003-02-13, modified 2003-08-21.
	planetmath.org /encyclopedia/FlagVariety.html (104 words)

	UR Math newsletter, Spring '98: Shipman's quantum honeybees
		In the geometry of a six dimensional flag manifold, she saw the dance of the honeybees.
		Physicists use these same flag manifolds to understand some of the phenomena associated to quarks, fundamental particles two of which are the building blocks of ordinary matter.
		According to flag manifold geometry, a dancing honeybee is interacting with the quantum world without disturbing it and using this information to organize its dance.
	www.math.rochester.edu /about/newsletters/spring98/bees.html (798 words)

	Aust. Math. Soc. Gazette Vol 24 No 4
		This article gives a simple introduction to flag manifolds and a taste of some of their connections to these diverse areas of mathematics.
		Flag manifolds have many useful properties: they are compact, complex, homogeneous manifolds, and the complex and algebraic structures are closely related.
		Both the conjugacy class of a parabolic subgroup and the flag manifold it defines are determined by the sizes of the GL_i blocks of the standard parabolic.
	www.austms.org.au /Publ/Gazette/1997/Nov97/flags.html (1526 words)

	Journal of Lie Theory, 8(1), 111-128 (1998)
		In \cite{smtt} it was proved that a necessary condition for $L$ to be $S$-admissible is that its action in $% B\left(S\right) $ is minimal and contractive where $B\left(S\right) $ is the flag manifold associated with $S$, as in \cite{smt}.
		A subgroup with a finite number of connected components is admissible if and only if its component of the identity is admissible, and if $L$ is a connected admissible group then $L$ is reductive and its semi-simple component $E$ is also admissible.
		Moreover, $E$ is transitive in $B\left(S\right) $ which turns out to be a flag manifold of $E$.
	www.univie.ac.at /EMIS/journals/JLT/vol.8_no.1/7.html (170 words)

	week181
		For each dot, the space of all figures of the corresponding type is called a "Grassmannian", and it's a manifold of the form G/P, where P is a "maximal parabolic" subgroup of G. More generally, any subset of dots in the Dynkin diagram corresponds to a type of "flag".
		Since G and P are complex Lie groups, the flag manifold G/P is a complex manifold.
		That's because their Dynkin diagrams are almost the same: for reasons I may someday explain, dimensions of flag manifolds don't care which way the little arrows on the Dynkin diagrams point, since they depend only on the reflection group associated to this diagram (see "week62").
	math.ucr.edu /home/baez/week181.html (3175 words)

	CWI Tract
		In the last part of the Tract the representation theory of the standard quantization of a generalized flag manifold is studied in detail.
		An important aspect of this theory is the algebraic description of the quantized flag manifold in terms of quantum analogues of Plücker coordinates.
		It is shown that the representations of the quantized flag manifold naturally relates to the symplectic foliation of the underlying Poisson manifold.
	www.cwi.nl /publications/Abstracts_tracts/tr-132.html (191 words)

	Custom Units of Measure (Site not responding. Last check: )
		Manifold uses units of measure in a variety of places such as in the units box used with most projections.
		The definitions of units available to Manifold are compiled into the product.
		We find it annoying that English units like feet are derived from the physical dimensions of an English king's body parts, so we will create new units based on our own body parts.
	www.manifold.net /doc/7x/custom_units_of_measure.htm (337 words)

	Vassar Math Department Colloquium Spring 07
		Abstract: The flag manifold is a geometric object that generalizes the notion of the tangents at a point to a curve.
		The geometry of the flag manifold encodes the combinatorics of the permutation group on n letters, as well as the algebra associated to the n x n invertible matrices.
		I will discuss problems about the geometry of the flag manifold that arise in computer science, and show some combinatorial approaches to solve them.
	math.vassar.edu /colloquium.html (1265 words)

	Unexplained Mysteries :: Those amazing quantum honey bees
		By watching the dance of a scout bee, other bees, called “recruits” get an exact idea of the direction and distance of where new food can be found.Even though bees were not her field of study, Shipman could never get the mystery of bee dances out of her mind.
		A manifold is a geometric shape described by certain complex math equations.
		Also, it is unlikely that the two-dimensional pattern of the bee dance is a perfect shadow for a six-dimensional flag manifold unless there is a connection.
	www.unexplained-mysteries.com /viewnews.php?id=73651 (1024 words)

	Quantum Mechanics and Honeybee Communication
		The two dimensional representation of a six dimensional flag manifold³ turns out to be a hexagon (like the comb in a beehive) which is interestingly coincidental, however, Ms.
		Shipman, upon further study, found curves that were an exact duplicate of the curves in the honeybees recruitment dance (the dance made by bees to recruit other bees to a source).
		When a honeybee dances to communicate the location of a source, and the distance from the hive to the source reaches a certain point, the bee will change from a waggle dance to a circle dance, which employs short straight line segments on the sides of the loops to define the shorter distance.
	www.n8ture.com /scienceqm.html (533 words)

	[No title]
		These manifolds were originally constructed by Bott and Samelson in [4] (without reference to flags or U-structures), but were introduced into complex cobordism theory by the second author in [24].
		We note that the isomorphism maps the geometric involution given by interchanging the factors of the normal bundle to the homotopy theoretic involution given by interchanging the fac- tors of the DU spectrum, and that it may, with further care, be naturally extended to the coordinate-free setting.
		Bounded flag manifolds In this section we introduce our family of bounded flag manifolds, and discuss their topology in terms of a cellular calculus which is intimately related to the Schubert calculus for classic flag manifolds.
	www.math.purdue.edu /research/atopology/Buchstaber-Ray/dcfmqd.txt (12405 words)

	Lie Group Seminar
		Abstract Permutation patterns have been useful in the study of the classical flag manifold (type A) and its Schubert subvarieties; the most notable result being that a Schubert variety is nonsingular if and only if it avoids the patterns 3412 and 4231.
		Delzant conjectured in 1990 that multiplicity free manifolds are uniquely determined by two data: the generic isotropy group and the image of the moment.
		Abstract Schubert calculus is the study of the cohomology algebra of the flag manifold, and one famous open problem is to find a combinatorial description of the structure constants of this algebra with respect to the Schubert basis.
	www.math.rutgers.edu /~seminars/old/Fall2001/LieGroup.html (1605 words)

	[No title]
		From: adler@pulsar.wku.edu (Allen Adler) Newsgroups: sci.math Subject: Bees and 6-dim flag manifolds Date: 05 Dec 1997 23:59:11 -0600 On NPR this evening, they interviewed someone named Barbara Shipman (I am not sure of the spelling and can't find her name in the Combined Membership List of the AMS under any of the variants I tried).
		Apparently, she was considering projections of a 6 dimensional flag manifold (call it F) into 2 space and got pictures that reminded her of the dances that bees do to communicate with the hive about food.
		We derive expressions for these integrals in terms of Pluecker coordinates on the flag manifolds in the case that all eigenvalues are zero and compare the geometry of the base locus of their level set varieties with the corresponding geometry for distinct eigenvalues.
	www.math.niu.edu /~rusin/known-math/97/bees (662 words)

	UTA Mathematics - Abstracts (Site not responding. Last check: )
		To see more of the geometry of the system, the solutions are completed by embedding them into a flag manifold (which will be introduced and explained in the talk).
		A nice feature of this is that in the flag manifold, the solutions generate a group action, where the form of the group depends on the multiplicities of the eigenvalues.
		Another nice feature is that a lot of the geometry can be seen in the structure of the moment polytope of the flag manifold (which will also be introduced in the talk).
	www.uta.edu /math/pages/main/abstracts/shipman_11_01_02.html (196 words)

	Connection of the dance of honeybee with the color group SU(3)
		In particular, almost vacuum spacetime sheets define orbits of the flag manifold so that simplest representation for the thought is as a curve in flag manifold.
		The flag manifold orbit is in principle determined by the absolute minimization of the Kähler action and the work of Barbara Shipman suggests that this dynamics reduces to the dynamics of a completely integrable system known as full Kostant-Toda lattice.
		For instance, geodesic motion in flag manifold would have SU(3) as symmetries and this would imply that Cartan algebra would define constants of motion and the momentum map would map the orbits to the points of plane.
	www.saunalahti.fi /~jawap/colors/string/honey.html (5288 words)

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