Definitions
from The Century Dictionary.
 noun The formation of fibers, or fibrous construction of a part or organ; fibrillation: as, the fibration of the white tissue of the brain; the fibration of minerals.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun algebraic topology a continuous mapping satisfying the
homotopy lifting property with respect to any space.
Etymologies
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Examples

The metric measure of a the Fubini space which gives the fibration over a projective Hilbert space is
Quantum Hyperion Sean 2008

But since the wave is complex valued, and there is this fibration given by the unitary operator over the projective Hilbert space what we measure in real variables has this range of possible outcomes.
Boltzmann’s Universe Sean 2008

As a digression this has some interesting properties, for this is the metric of the FubiniStudy space and the fibration over the projective Hilbert space.
Boltzmann’s Universe Sean 2008

These projective spaces, or null congurences can then have a fibration over then from which connection terms and curvatures are computed.
Does Space Expand? Sean 2008

Confocal polarizing microscopy and computer simulations reveal their equilibrium internal structures, manifesting both skyrmionlike and Hopf fibration features.
Naturejobs  All Jobs Ivan I. Smalyukh 2010

Confocal polarizing microscopy and computer simulations reveal their equilibrium internal structures, manifesting both skyrmionlike and Hopf fibration features.

He used the Hopf fibration, but did not consider whether, in an experiment, light could actually be forced to follow the circular paths.
Science News / Features, Blog Entries, Column Entries, Issues, News Items and Book Reviews 2008

VIDEO  The Hopf fibration fills all of space with circles., who is unaffiliated with the research.
Science News / Features, Blog Entries, Column Entries, Issues, News Items and Book Reviews 2008

António Fernandez Rañada of Madrid used the Hopf fibration to create a new solution to Maxwell's equations, and thus an example of how electromagnetism can work.
Science News / Features, Blog Entries, Column Entries, Issues, News Items and Book Reviews 2008

These littleknown solutions, constructed by Rañada, are based on the Hopf fibration.
Next Big Future 2008
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