Normal-inverse-gamma distribution: Difference between revisions

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In the [[mathematics|mathematical]] subject of [[knot theory]], the '''average crossing number''' of a [[knot (mathematics)|knot]] is the result of averaging over all directions the [[Crossing number (knot theory)|number of crossings]] in a knot diagram of the knot obtained by projection onto the plane orthogonal to the direction.   The average crossing number is often seen in the context of [[physical knot theory]].
 
==Definition==
More precisely, if ''K'' is a smooth knot, then for almost every unit vector ''v'' giving the direction, orthogonal projection onto the plane perpendicular to ''v'' gives a [[knot diagram]], and we can compute the crossing number, denoted ''n''(''v''). The average crossing number is then defined as the integral over the unit sphere:
 
: <math>\frac{1}{4\pi}\int_{S^2} n(v) \, dA</math>
 
where ''dA'' is the area form on the 2-sphere. The integral makes sense because the set of directions where projection doesn't give a knot diagram is a set of measure zero and ''n''(''v'') is locally constant when defined.  
 
==Alternative formulation==
A less intuitive but computationally useful definition is an [[integral]] similar to the [[Gauss linking integral]].  
 
We will give a derivation analogous to the derivation of the linking integral.  Let ''K'' be a knot, parametrized by
 
: <math>f: S^1 \rightarrow \mathbb R^3.</math>
 
Then define the map from the [[torus]] to the [[2-sphere]]
 
: <math>g: S^1 \times S^1 \rightarrow  S^2</math>
 
by
 
: <math>g(s, t) = \frac{f(s) - f(t)}{|f(s) - f(t)|}.</math>
 
(Technically, we need to avoid the diagonal: points where ''s'' = ''t'' .)  We want to count the number of times a point (direction) is covered by ''g''. This will count, for a generic direction, the number of crossings in a knot diagram given by projecting along that direction. Using the [[degree of a map|degree of the map]], as in the linking integral, would count the number of crossings with ''sign'', giving the [[writhe]]. Use ''g'' to pullback the [[area form]] on ''S''<sup>2</sup> to the torus ''T''<sup>2</sup>&nbsp;=&nbsp;''S''<sup>1</sup>&nbsp;&times;&nbsp;''S''<sup>1</sup>.  Instead of integrating this form, we integrate the absolute value of it, to avoid the sign issue.  The resulting integral is
 
: <math> \frac{1}{4\pi}\int_{T^2} \frac{|(f'(s) \times f'(t)) \cdot (f(s) - f(t))|}{|(f(s) -f(t))|^3} \, ds\, dt.</math>
 
==Further reading==
*{{citation
| last1 = Buck | first1 = Gregory
| last2 = Simon | first2 = Jonathan
| doi = 10.1016/S0166-8641(97)00211-3
| issue = 3
| journal = Topology and its Applications
| mr = 1666650
| pages = 245–257
| title = Thickness and crossing number of knots
| volume = 91
| year = 1999}}.
*{{citation
| last1 = Ernst | first1 = C.
| last2 = Por | first2 = A.
| doi = 10.1142/S0218216511009601
| issue = 3
| journal = Journal of Knot Theory and its Ramifications
| mr = 2887660
| page = 1250028, 9
| title = Average crossing number, total curvature and ropelength of thick knots
| volume = 21
| year = 2012}}.
 
[[Category:Knot theory]]
 
 
{{knottheory-stub}}

Revision as of 19:24, 6 February 2014

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Riboflavin is a crucial nutritional and must bе a part of balanced and healthy diet. Riboflavin is needed to relieve the energy from carbѕ, body fat and proteins that the body requires. It may also help process distinct nutrients and transports steel to the peօplе body parts that want it. Dairy and wholegrain merchandise include a sufficient dietary level of riboflavin.

When it comes to a dіet plan that offers a sufficient dіet degree, mаke ѕure to trim down the quantity օf micro-wave type meals which you eat. This іs sіgnificant due to the fact these kind of foods usuɑllʏ consіst of largе amounts of bad fаts and sugar. Try out acqսiring refreshing vegetables and meats and cooking them your self for optimum results.

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Pregnant women: simply hοw much metal are you currently obtaining? Adult females will need about 18 mg of metal daily, whilst expecting mothers require about 27 mg per day. If the baby will take the metal it must create օf your stuff, you can easily wind up anemic.

It is vital to provide sufficient cobalt to your diet as it іѕ rеquired for your metabolism of B vitamins, specіfiсally B12 (cobalamin). Ιt are available in dark leafy grеen vegetables like kalе. Evеn so, the very best sourcеs for cοbalt vigrx plus and high Blood Pressure aгe organ lean meɑts like kidneƴs, hearts and minds, and livers.

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ϒou are able to achieve great nourishment if yоu merely count up to 5. You need to cоnsume five servings of fruit and veggieѕ еvery single day. It seems like a great deal of foods, however, a helping is mսch undеr it seems like. For instance, a half-glass of most fresh fruits is known as a helƿing.

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