Demon algorithm - Revision history

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-16T14:45:54Z

References: Fix CS1 deprecated date parameter errors

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Кeeр portion sizeѕ in order to~~ [~~http://www.Dict.cc/?s=protect protect~~] ~~yourself from gaining weight and returning to oldеr ƿractices.<br><br>Consume smаll food more often. Haѵing meals that have small the size of portіons, bսt which~~ can be ~~spacеd several hours separate will assist keep the bodyweight off as well as assist wіth digestion. Tiny, repeаted foods are linked to weigɦt loss and աeiցht maintenance аs well~~, which ~~may fend off high blood pressure, diabetes as well as otheг ailments~~. ~~Regular hаving reduces your desire for food~~, lowering the drive to excessive.<br><br>To experience optimɑl ɦealth, it's сrucial that you noսrisɦ the body proper. Use these suggestions to improve yoսг health. For the way you take in now, this will get both a lot ߋr а little effort. Аnyоne can іmprove their diet for better healtɦ insurancе and power.		[[Image:Parallelogramoid.svg\|thumb\|right\|Levi-Civita's parallelogramoid]]
			In the [[mathematics\|mathematical]] field of [[differential geometry]], the '''Levi-Civita parallelogramoid''' is a [[quadrilateral]] in a [[curved space]] whose construction generalizes that of a [[parallelogram]] in the [[Euclidean plane]]. It is named for its discoverer, [[Tullio Levi-Civita]]. Like a parallelogram, two opposite sides ''AA''′ and ''BB''′ of a parallelogramoid are parallel (via [[parallel transport]] {{Not a typo\|along }} side ''AB'') and the same length as each other, but the fourth side ''A''′''B''′ will not in general be parallel to or the same length as the side ''AB,'' although it will be straight (a [[geodesic]]).

			==Construction==
			A parallelogram in [[Euclidean geometry]] can be constructed as follows:
			* Start with a straight line segment ''AB'' and another straight line segment ''AA''′.
			* Slide the segment ''AA''′ along ''AB'' to the endpoint ''B'', keeping the angle with ''AB'' constant, and remaining in the same plane as the points ''A'', ''A''′, and ''B''.
			* Label the endpoint of the resulting segment ''B''′ so that the segment is ''BB''′.
			* Draw a straight line ''A''′''B''′.

			In a curved space, such as a [[Riemannian manifold]] or more generally any manifold equipped with an [[affine connection]], the notion of "straight line" generalizes to that of a [[geodesic]]. In a suitable [[neighborhood (mathematics)\|neighborhood]] (such as a ball in a [[normal coordinates\|normal coordinate system]]), any two points can be joined by a geodesic. The idea of sliding the one straight line along the other gives way to the more general notion of [[parallel transport]]. Thus, assuming either that the manifold is [[complete space\|complete]], or that the construction is taking place in a suitable neighborhood, the steps to producing a Levi-Civita parallelogram are:
			* Start with a geodesic ''AB'' and another geodesic ''AA''′. These geodesics are assumed to be parameterized by their [[arclength]] in the case of a Riemannian manifold, or to carry a choice of [[affine parameter]] in the general case of an affine connection.
			* "Slide" ([[parallel transport]]) the [[tangent vector]] of ''AA''′ from ''A'' to ''B''.
			* The resulting tangent vector at ''B'' generates a geodesic via the [[exponential map]]. Label the endpoint of this geodesic by ''B''′, and the geodesic itself ''BB''′.
			* Connect the points ''A''′ and ''B''′ by the geodesic ''A''′''B''′.

			==Quantifying the difference from a parallelogram==
			The length of this last geodesic constructed connecting the remaining points ''A''′''B''′ may in general be different than the length of the base ''AB''. This difference is measured by the [[Riemann curvature tensor]]. To state the relationship precisely, let ''AA''′ be the exponential of a tangent vector ''X'' at ''A'', and ''AB'' the exponential of a tangent vector ''Y'' at ''A''. Then

			:<math>\|A'B'\|^2 = \|AB\|^2 + \frac{8}{3}\langle R(X,Y)X,Y\rangle + \text{higher order terms}</math>

			where terms of higher order in the length of the sides of the parallelogram have been suppressed.

			==Discrete approximation==
			[[Image:Schild's ladder step 4.svg\|thumb\|right\|Two rungs of [[Schild's ladder]]. The segments ''A''<sub>1</sub>''X''<sub>1</sub> and ''A''<sub>2</sub>''X''<sub>2</sub> are an approximation to first order of the [[parallel transport]] of ''A''<sub>0</sub>''X''<sub>0</sub> along the curve.]]
			[[Parallel transport]] can be discretely approximated by [[Schild's ladder]], which approximates Levi-Civita parallelogramoids by approximate parallelograms.

			==References==
			*{{citation \| first = Élie \| last = Cartan \| authorlink=Élie Cartan\|title = Geometry of Riemannian Spaces\| publisher = Math Sci Press, Massachusetts \| year = 1983}}

			[[Category:Curvature (mathematics)]]
			[[Category:Differential geometry]]
			[[Category:Quadrilaterals]]

en>MaxwellDecoherence: /* The demon algorithm */ equilibriate > equilibrate

2012-07-30T20:18:56Z

The demon algorithm: equilibriate > equilibrate

New page

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