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Page "Four color theorem" ¶ 56
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For and closed
For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.
The angriest young man in Newport last night was at the Playhouse, where `` Epitaph For George Dillon '' opened as the jazz festival closed.
* Satirical newspaper The Onion ran a story entitled " Craig Biggio Blames Media Pressure For Stalling At 285 Hit-By-Pitches " as Biggio closed in on the record of 287 hit-by-pitches.
For months, speculation surrounded Selig and the possibility that he and Hank Aaron would not attend Bonds ' games as he closed in on the record.
For example, Ecuador has closed its main border crossing with Colombia every night since August 2002, when evidence emerged that Colombian guerrillas and paramilitaries were asserting control over Ecuador's border communities.
For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.
For services depending on the day (" closed on Sundays ", " does not run on Fridays ", and so on ) there is a risk of ambiguity.
For example, during the 2008 flooding, Credit Island in the city's southwest corner remained closed for 5½ months while crews worked on cleaning up damage and removing river debris.
For a short time, when the University of Vienna closed, Hayek studied in Constantin von Monakow's Institute of Brain Anatomy, where Hayek spent much of his time staining brain cells.
For much of the period, the city was divided into Polish and Soviet areas, with the latter closed to the public.
For example, animals undergo an " open " mitosis, where the nuclear envelope breaks down before the chromosomes separate, while fungi such as Aspergillus nidulans and Saccharomyces cerevisiae ( yeast ) undergo a " closed " mitosis, where chromosomes divide within an intact cell nucleus.
For elections to the European Parliament, most member states use open lists ; most of the United Kingdom uses closed lists, but Northern Ireland uses the Single transferable vote, as does Ireland.
For instance, if a door is closed, instead of lockpicking or casting an open spell, the fighter can simply knock down the door.
For example: stakeholders withdrawing during a project may endanger funding of the project ; confidential information may be stolen by employees even within a closed network ; lightning striking an aircraft during takeoff may make all people on board immediate casualties.
For proprietary software, the provisions of the various copyright laws, trade secrecy and patents are used to keep the source code closed.
For hundreds of years, St. Kitts operated as a sugar monoculture, but due to decreasing profitability, the government closed the industry in 2005.
For at least a full day after the attacks, bridges and tunnels to Manhattan were closed to non-emergency traffic in both directions.
For non-bounded figures, the additional isometry groups can include translations ; the closed ones are:
For example, in most string theories one of the closed string modes is the graviton, and one of the open string modes is the photon.
For example, if a closed string splits, its worldsheet will look like a single pipe splitting ( or connected ) to two pipes ( often referred to as a pair of pants — see drawing at right ).
For example, a square ( closed fist, ' S ' hand ) with double solid bullets is the sign for ' milk ' ( iconically squeezing an udder ).
For man has closed himself up, till he sees all things through narrow chinks of his cavern.
File: PennAveWhiteHouse. jpg | For security reasons, the section of Pennsylvania Avenue in front of the White House is now closed to all vehicular traffic, except government officials.
For this reason wormholes have been defined geometrically, as opposed to topologically, as regions of spacetime that constrain the incremental deformation of closed surfaces.

For and orientable
( For comparison, a sphere is an orientable surface with no boundary.
For example, the sphere and torus are orientable, while the real projective plane is not ( because deleting a point or disk from the real projective plane produces the Möbius strip ).
For an orientable surface, a consistent choice of " clockwise " ( as opposed to counter-clockwise ) is called an orientation, and the surface is called oriented.
For example, as a result X is always an orientable manifold ( meaning that the tangent bundle of X is an orientable vector bundle ).
For example, every symplectic manifold is even-dimensional and orientable.
For example in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus.
For an orientable surface S the Euler characteristic χ ( S ) is
For example, a-structure on a manifold turns out to be precisely what is needed to be orientable in the sense of complex topological k-theory.
For example, a closed, orientable 3-manifold, which admits a taut foliation with no sphere leaf, must be irreducible, covered by, and have negatively curved fundamental group.
For example, not every manifold is orientable, and those that are orientable admit exactly two orientations.

For and non-orientable
For example, the Klein bottle has no boundary, where the surface stops abruptly, and it is non-orientable, as reflected in the one-sidedness of the immersion.

For and surfaces
For phosphor to fiber and fiber to air surfaces, and assuming Af, we obtain Af percent.
For higher speeds, the flow will compress more significantly as it comes into contact with surfaces and slows.
For images of fractal patterns, this has been expressed by phrases such as " smoothly piling up surfaces " and " swirls upon swirls ".
For example, air hoar is a deposit of hoar frost on objects above the surface, such as tree branches, plant stems, wires ; surface hoar is formed by fernlike ice crystals directly deposited on snow, ice or already frozen surfaces ; crevasse hoar consists of crystals that form in glacial crevasses where water vapour can accumulate under calm weather conditions ; depth hoar refers to cup shaped, faceted crystals formed within dry snow, beneath the surface.
For this reason spherical reference surfaces are frequently used in mapping programs.
For example once one gets to 3-dimensional surfaces, one has to deal with solid objects with knot-shaped holes, and then one needs the whole of knot theory just to classify them.
For mirrors with parabolic surfaces, parallel rays incident on the mirror produce reflected rays that converge at a common focus.
For compact space | compact 2-dimensional surfaces without boundary ( topology ) | boundary, if every loop can be continuously tightened to a point, then the surface is topologically Homeomorphism | homeomorphic to a 2-sphere ( usually just called a sphere ).
For instance, the naturally occurring citric and ascorbic acids in lemon or other citrus juice can inhibit the action of the enzyme phenolase which turns surfaces of cut apples and potatoes brown if a small amount of the juice is applied to the freshly cut produce.
For similar reasons, objects intended to avoid detection will not have inside corners or surfaces and edges perpendicular to likely detection directions, which leads to " odd " looking stealth aircraft.
For example if the light source emitted white light and the two diffuse surfaces were blue, then the resulting color of the pixel is blue.
For most points on most surfaces, different sections will have different curvatures ; the maximum and minimum values of these are called the principal curvatures.
#: For most surfaces the focal surface forms two sheets each of which is a surface and which come together at umbilical points.
For channel surfaces one sheet forms a curve and the other sheet is a surface ; For cones, cylinders, toruses and cyclides both sheets form curves.
For most uses, dilution with water is recommended for safety and to avoid damaging the surfaces being cleaned.
For still water, this is the difference in height between the inlet and outlet surfaces.
* For specular surfaces, such as glass or polished metal, reflectivity will be nearly zero at all angles except at the appropriate reflected angle-that is, reflected radiation will follow a different path from incident radiation for all cases other than radiation normal to the surface.
* For diffuse surfaces, such as matte white paint, reflectivity is uniform ; radiation is reflected in all angles equally or near-equally.
For surfaces with b boundary components, the equation reads χ = 2 − 2g − b.
For a long cylinder far from other surfaces, this reduces the effective area by a factor of π / 2
For example, PCC trolley brakes include a flat shoe which is clamped to the rail with an electromagnet ; the Murphy brake pinches a rotating drum, and the Ausco Lambert disc brake uses a hollow disc ( two parallel discs with a structural bridge ) with shoes that sit between the disc surfaces and expand laterally.
For the case of two colliding bodies in two-dimensions, the overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision.

1.672 seconds.