formulasearchengine - User contributions [en]

Theta solvent

2014-12-07T00:58:04Z

165.124.128.186:

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Square-integrable function

2014-11-07T21:00:35Z

165.124.129.186: /* Properties */ Clarified reference to earlier example

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Directional statistics

2014-01-23T21:24:25Z

165.124.128.146: /* Software */

In [[mathematics]], especially [[functional analysis]], '''Bessel's inequality''' is a statement about the coefficients of an element <math>x</math> in a [[Hilbert space]] with respect to an [[orthonormal]] [[sequence]].

Let <math>H</math> be a Hilbert space, and suppose that <math>e_1, e_2, ...</math> is an orthonormal sequence in <math>H</math>. Then, for any <math>x</math> in <math>H</math> one has
:<math>\sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2 </math>

where 〈•,•〉 denotes the [[inner product space|inner product]] in the Hilbert space <math>H</math>. If we define the infinite sum
:<math>x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k, </math>
consisting of 'infinite sum' of [[vector resolute]] <math>x</math> in direction <math>e_k</math>, Bessel's [[inequality (mathematics)|inequality]] tells us that this [[series (mathematics)|series]] [[Limit of a sequence|converges]]. One can think of it that there exists <math>x' \in H</math> which can be described in terms of potential basis <math>e_1, e_2, ...</math>.

For a complete orthonormal sequence (that is, for an orthonormal sequence which is a [[Orthonormal basis|basis]]), we have [[Parseval's identity]], which replaces the inequality with an equality (and consequently <math> x'</math> with <math> x</math>).

Bessel's inequality follows from the identity:
:<math>0 \le \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^2 = \|x\|^2 - 2 \sum_{k=1}^n |\langle x, e_k \rangle |^2 + \sum_{k=1}^n | \langle x, e_k \rangle |^2 = \|x\|^2 - \sum_{k=1}^n | \langle x, e_k \rangle |^2,</math>
which holds for any natural ''n''.

==See also==
* [[Cauchy–Schwarz inequality]]

==External links==
* {{springer|title=Bessel inequality|id=p/b015850}}
* [http://mathworld.wolfram.com/BesselsInequality.html Bessel's Inequality] the article on Bessel's Inequality on MathWorld.

{{PlanetMath attribution|title=Bessel inequality|id=3089}}

{{Functional Analysis}}

[[Category:Hilbert space]]
[[Category:Inequalities]]

Jacobi form

2013-08-11T17:30:36Z

165.124.167.212: /* Definition */

The '''effective diffusion coefficient''' (also referred to as the apparent diffusion coefficient) of a diffusant in [[atomic diffusion]] of solid [[polycrystalline]] materials like [[metal alloy]]s is often represented as a [[weighted average]] of the [[grain boundary diffusion coefficient]] and the [[lattice diffusion coefficient]].<ref name=Heitjans>P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.</ref> Diffusion along both the grain boundary and in the lattice may be modeled with an [[Arrhenius equation]]. The ratio of the grain boundary diffusion activation energy over the lattice diffusion activation energy is usually 0.4 - 0.6, so as temperature is lowered, the grain boundary diffusion component increases.<ref name=Heitjans /> Increasing temperature often allows for increased grain size, and the lattice diffusion component increases with increasing temperature, so often at 0.8Tmelt (of an alloy), the grain boundary component can be neglected.

==Modeling==
The effective diffusion coefficient can be modeled using Hart's equation when only grain boundary and lattice diffusion are dominant:

:<math> {D^{\mathrm{eff}}} = f</math> Dgb <math> + (1-f) </math>Dl.

where
:<math> {D^{\mathrm{eff}}} = </math> effective diffusion coefficient.
:Dgb = grain boundary diffusion coefficient.
:Dl = lattice diffusion coefficient.
:<math> f = \tfrac{q}{d} </math>.δ
:<math> q = </math> value based on grain shape, 1 for parallel grains, 3 for square grains.
:<math> d = </math> average grain size.
:δ <math> = </math> grain boundary width,often assumed to be 0.5 nm.

Grain boundary diffusion is significant in [[face centered cubic]] metals below about 0.8 Tmelt (Absolute). Line dislocations and other [[Crystallographic defect|crystalline defects]] can become significant below ~0.4 Tmelt in FCC metals.

== References ==
{{reflist}}
{{refbegin}}
{{refend}}

== See also ==
* [[Kirkendall effect]]
* [[Phase transformations in solids]]
* [[Mass diffusivity]]

[[Category:Diffusion]]

Wendel's theorem

2013-05-25T00:45:07Z

165.124.40.223:

In mathematics, the '''uncertainty exponent''' is a method of measuring the [[fractal dimension]] of a [[basin boundary]]. In a [[chaotic scattering]] system, the
[[invariant (mathematics)#Invariant set|invariant set]] of the system is usually not directly accessible because it is non-attracting and typically of [[measure (mathematics)|measure]] zero. Therefore, the only way to infer the presence of members
and to measure the properties of the invariant set is through the [[basin of attraction|basins of attraction]]. Note that in a scattering system, basins of attraction are not limit cycles therefore do not constitute members of the invariant set.

Suppose we start with a random trajectory and perturb it by a small amount,
<math>\epsilon</math>, in a random direction. If the new trajectory ends up
in a different basin from the old one, then it is called ''epsilon uncertain''.
If we take a large number of such trajectories,
then the fraction of them that are epsilon uncertain is the ''uncertainty fraction'',
<math>f(\epsilon)</math>, and we expect it to scale exponentially
with <math>\varepsilon</math>:

:<math>
f(\varepsilon) \sim \varepsilon^\gamma \,
</math>

Thus the uncertainty exponent, <math>\gamma</math>, is defined as follows:

:<math>
\gamma = \lim_{\varepsilon\to 0} \frac {\ln f(\varepsilon)} {\ln \varepsilon}
</math>

The uncertainty exponent can be shown to approximate the [[box-counting dimension]]
as follows:

:<math>
D_0 = N - \gamma \,
</math>

where ''N'' is the embedding dimension. Please refer to the article on [[chaotic mixing]] for an example of numerical computation of the ''uncertainty dimension''
compared with that of a box-counting dimension.

==References==
*C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, ''Final state sensitivity: An obstruction to predictability'', [[Phys. Letters]] 99A: 415-418 (1983).

* {{Cite book
| author = Edward Ott
| title = Chaos in Dynamical Systems
| publisher = Cambridge University Press
| year = 1993
}}

[[Category:Chaos theory]]
[[Category:Fractals]]

Order of reaction

2012-08-01T16:24:32Z

165.124.213.80: /* First order */

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2012-05-23T08:19:49Z

165.124.208.22:

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