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Hahn–Banach theorem

2014-02-19T15:18:41Z

117.211.88.35:

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Time series

2014-01-31T10:11:57Z

117.211.27.60: /* Regression analysis */

{{about|the Airy special function|the Airy stress function employed in solid mechanics|Stress functions}}

In the physical sciences, the '''Airy function''' '''Ai(''x'')''' is a [[special function]] named after the British astronomer [[George Biddell Airy]] (1801–92). The function Ai(''x'') and the related function '''Bi(''x'')''', which is also called the '''Airy function''', but sometimes referred to as the Bairy function, are solutions to the [[differential equation]]

:<math>\frac{d^2y}{dx^2} - xy = 0 , \,\!</math>

known as the '''Airy equation''' or the '''Stokes equation'''. This is the simplest second-order [[linear differential equation]] with a turning point (a point where the character of the solutions changes from oscillatory to exponential).

The Airy function is the solution to [[Schrödinger's equation]] for a particle confined within a triangular [[potential well]] and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the [[WKB method]], when the potential may be locally approximated by a linear function of position. The triangular potential well solution is directly relevant for the understanding of many semiconductor devices.

The Airy function also underlies the form of the intensity near an optical directional [[caustic (optics)|caustic]], such as that of the [[rainbow]]. Historically, this was the mathematical problem that led Airy to develop this special function. The Airy function is also important in [[microscopy]] and [[astronomy]]; it describes the pattern, due to [[diffraction]] and [[interference]], produced by a [[point source]] of light (one which is smaller than the resolution limit of a [[microscope]] or [[telescope]]).

==Definitions==
[[Image:Airy Functions.svg|right|thumb|400px|Plot of Ai(''x'') in red and Bi(''x'') in blue]]
For real values of ''x'', the Airy function of the first kind can be defined by the [[improper Riemann integral]]:
:<math>\mathrm{Ai}(x) = \frac{1}{\pi}\int_0^\infty\cos\left(\tfrac{t^3}{3} + xt\right)\, dt\equiv \frac{1}{\pi}\lim_{b\to\infty} \int_0^b \cos\left(\tfrac{t^3}{3} + xt\right)\, dt,</math>
which converges because the positive and negative parts of the [[Riemann-Lebesgue lemma|rapid oscillations tend to cancel one another out]] (as can be checked by [[integration by parts]]).

''y'' = Ai(''x'') satisfies the Airy equation
:<math>y'' - xy = 0.</math>
This equation has two [[linear independence|linearly independent]] solutions.
Up to scalar multiplication, Ai(''x'') is the solution subject to the condition ''y'' → 0 as ''x'' → ∞.
The standard choice for the other solution is the Airy function of the second kind, denoted Bi(''x''). It is defined as the solution with the same amplitude of oscillation as Ai(''x'') as ''x'' → −∞ which differs in phase by π/2:

:<math>\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[\exp\left(-\tfrac{t^3}{3} + xt\right) + \sin\left(\tfrac{t^3}{3} + xt\right)\,\right]dt.</math>

==Properties==
The values of Ai(''x'') and Bi(''x'') and their derivatives at ''x'' = 0 are given by
:<math>\begin{align}
\mathrm{Ai}(0) &{}= \frac{1}{3^{\frac{2}{3}}\Gamma(\tfrac23)}, & \quad \mathrm{Ai}'(0) &{}= -\frac{1}{3^{\frac{1}{3}}\Gamma(\tfrac13)}, \\
\mathrm{Bi}(0) &{}= \frac{1}{3^{\frac{1}{6}}\Gamma(\tfrac23)}, & \quad \mathrm{Bi}'(0) &{}= \frac{3^{\frac{1}{6}}}{\Gamma(\tfrac13)}.
\end{align}</math>
Here, Γ denotes the [[Gamma function]]. It follows that the [[Wronskian]] of Ai(''x'') and Bi(''x'') is 1/π.

When ''x'' is positive, Ai(''x'') is positive, [[convex function|convex]], and decreasing exponentially to zero, while Bi(''x'') is positive, convex, and increasing exponentially. When ''x'' is negative, Ai(''x'') and Bi(''x'') oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.

The Airy functions are orthogonal<ref>David E. Aspnes, Physical Review, '''147''', 554 (1966)</ref> in the sense that
:<math> \int_{-\infty}^\infty \mathrm{Ai}(t+x) \mathrm{Ai}(t+y) dt = \delta(x-y)</math>
again using an improper Riemann integral.

==Asymptotic formulae==
As explained below, the Airy functions can be extended to the complex plane, giving [[entire function]]s. The asymptotic behaviour of the Airy functions as ''|z|'' goes to infinity at a constant value of arg(''z'') depends on arg(''z''). For |arg(''z'')| < π we have the following [[asymptotic formula]] for Ai(''z''):<ref>{{harvtxt|Abramowitz|Stegun|1970|p=[http://people.math.sfu.ca/~cbm/aands/page_448.htm 448]}}, Eqns 10.4.59 and 10.4.63</ref>

:<math>\mathrm{Ai}(z)\sim \frac{e^{-\frac{2}{3}z^{\frac{3}{2}}}}{2\sqrt\pi\,z^{\frac{1}{4}}}</math>

and a similar one for Bi(''z''), but only applicable when |arg(''z'')| < π/3:

:<math>\mathrm{Bi}(z)\sim \frac{e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,z^{\frac{1}{4}}}.</math>

A more accurate formula for Ai(''z'') and a formula for Bi(''z'') when π/3 < |arg(''z'')| < π or, equivalently, for Ai(−''z'') and Bi(−''z'') when |arg(''z'')| < 2π/3 but not zero, are:<ref>{{harvtxt|Abramowitz|Stegun|1970|p=[http://people.math.sfu.ca/~cbm/aands/page_448.htm 448]}}, Eqns 10.4.60 and 10.4.64</ref>
:<math>\begin{align}
\mathrm{Ai}(-z) &{}\sim \frac{\sin \left(\frac23z^{\frac{3}{2}}+\frac{\pi}{4} \right)}{\sqrt\pi\,z^{\frac{1}{4}}} \\[6pt]
\mathrm{Bi}(-z) &{}\sim \frac{\cos \left(\frac23z^{\frac{3}{2}}+\frac{\pi}{4} \right)}{\sqrt\pi\,z^{\frac{1}{4}}}.
\end{align}</math>
When |arg(''z'')| = 0 these are good approximations but are not asymptotic because the ratio between Ai(−''z'') or Bi(−''z'') and the above approximation goes to infinity whenever the sine or cosine goes to zero.
[[Asymptotic analysis|Asymptotic expansions]] for these limits are also available. These are listed in (Abramowitz and Stegun, 1954) and (Olver, 1974).

==Complex arguments==
We can extend the definition of the Airy function to the complex plane by
:<math>\mathrm{Ai}(z) = \frac{1}{2\pi i} \int_{C} \exp\left(\tfrac{t^3}{3} - zt\right)\, dt,</math>
where the integral is over a path ''C'' starting at the point at infinity with argument −π/2 and ending at the point at infinity with argument π/2. Alternatively, we can use the differential equation ''y''′′ − ''xy'' = 0 to extend Ai(''x'') and Bi(''x'') to [[entire function]]s on the complex plane.

The asymptotic formula for Ai(''x'') is still valid in the complex plane if the principal value of ''x''2/3 is taken and ''x'' is bounded away from the negative real axis. The formula for Bi(''x'') is valid provided ''x'' is in the sector {''x'' ∈ '''C''' : |arg(''x'')| < (π/3)−δ} for some positive δ. Finally, the formulae for Ai(−''x'') and Bi(−''x'') are valid if ''x'' is in the sector {''x'' ∈ '''C''' : |arg(''x'')| < (2π/3)−δ}.

It follows from the asymptotic behaviour of the Airy functions that both Ai(''x'') and Bi(''x'') have an infinity of zeros on the negative real axis. The function Ai(''x'') has no other zeros in the complex plane, while the function Bi(''x'') also has infinitely many zeros in the sector {''z'' ∈ '''C''' : π/3 < |arg(''z'')| < π/2}.

===Plots===

{| style="text-align:center" align=center
! <math>\Re \left[ \mathrm{Ai} ( x + iy) \right] </math>
! <math>\Im \left[ \mathrm{Ai} ( x + iy) \right] </math>
! <math>| \mathrm{Ai} ( x + iy) | \, </math>
! <math>\mathrm{arg} \left[ \mathrm{Ai} ( x + iy) \right] \, </math>
|-
|[[Image:AiryAi Real Surface.png|200px]]
|[[Image:AiryAi Imag Surface.png|200px]]
|[[Image:AiryAi Abs Surface.png|200px]]
|[[Image:AiryAi Arg Surface.png|200px]]
|-
|[[Image:AiryAi Real Contour.svg|200px]]
|[[Image:AiryAi Imag Contour.svg|200px]]
|[[Image:AiryAi Abs Contour.svg|200px]]
|[[Image:AiryAi Arg Contour.svg|200px]]
|}

{| style="text-align:center" align=center
! <math>\Re \left[ \mathrm{Bi} ( x + iy) \right] </math>
! <math>\Im \left[ \mathrm{Bi} ( x + iy) \right] </math>
! <math>| \mathrm{Bi} ( x + iy) | \, </math>
! <math>\mathrm{arg} \left[ \mathrm{Bi} ( x + iy) \right] \, </math>
|-
|[[Image:AiryBi Real Surface.png|200px]]
|[[Image:AiryBi Imag Surface.png|200px]]
|[[Image:AiryBi Abs Surface.png|200px]]
|[[Image:AiryBi Arg Surface.png|200px]]
|-
|[[Image:AiryBi Real Contour.svg|200px]]
|[[Image:AiryBi Imag Contour.svg|200px]]
|[[Image:AiryBi Abs Contour.svg|200px]]
|[[Image:AiryBi Arg Contour.svg|200px]]
|}

==Relation to other special functions==
For positive arguments, the Airy functions are related to the [[Bessel function#Modified Bessel functions|modified Bessel functions]]:
:<math>\begin{align}
\mathrm{Ai}(x) &{}= \frac1\pi \sqrt{\frac{x}{3}} \, K_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right), \\
\mathrm{Bi}(x) &{}= \sqrt{\frac{x}{3}} \left(I_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) + I_{-\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right)\right).
\end{align}</math>
Here, ''I''±1/3 and ''K''1/3 are solutions of

:<math>x^2y'' + xy' - \left (x^2 + \tfrac{1}{9} \right )y = 0.</math>

The first derivative of Airy function is
:<math> \mathrm{Ai'}(x) = - \frac{x} {\pi \sqrt{3}} \, K_{\frac{2}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) .</math>

Functions ''K''1/3 and ''K''2/3 can be represented in terms of rapidly converged integrals<ref>M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690-707 \ (2004).</ref> (see also [[Bessel function#Modified Bessel functions|modified Bessel functions]] )

For negative arguments, the Airy function are related to the [[Bessel function]]s:
:<math>\begin{align}
\mathrm{Ai}(-x) &{}= \sqrt{\frac{x}{9}} \left(J_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) + J_{-\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right)\right), \\
\mathrm{Bi}(-x) &{}= \sqrt{\frac{x}{3}} \left(J_{-\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right) - J_{\frac{1}{3}}\left(\tfrac23 x^{\frac{3}{2}}\right)\right).
\end{align}</math>
Here, ''J''±1/3 are solutions of

:<math>x^2y'' + xy' + \left (x^2 - \tfrac{1}{9} \right )y = 0.</math>

The [[Scorer's function]]s solve the equation ''y''′′ − ''xy'' = 1/π. They can also be expressed in terms of the Airy functions:
:<math>\begin{align}
\mathrm{Gi}(x) &{}= \mathrm{Bi}(x) \int_x^\infty \mathrm{Ai}(t) \, dt + \mathrm{Ai}(x) \int_0^x \mathrm{Bi}(t) \, dt, \\
\mathrm{Hi}(x) &{}= \mathrm{Bi}(x) \int_{-\infty}^x \mathrm{Ai}(t) \, dt - \mathrm{Ai}(x) \int_{-\infty}^x \mathrm{Bi}(t) \, dt.
\end{align}</math>

==Fourier transform==
Using the definition of the Airy function Ai(''x''), it is straightforward to show its [[Fourier transform]] is given by
:<math>\mathcal{F}(\mathrm{Ai})(k) := \int_{-\infty}^{\infty} \mathrm{Ai}(x)\ e^{- 2\pi i k x}\,dx = e^{\frac{i}{3}(2\pi k)^3}.</math>

==Fabry–Pérot interferometer Airy Function==
The transmittance function of a [[Fabry–Pérot interferometer]] is also referred to as the ''Airy Function'':<ref>{{cite book | first=Eugene|last=Hecht|year=1987|title=Optics|edition=2nd ed.|publisher=Addison Wesley|isbn=0-201-11609-X}} Sect. 9.6</ref>
:<math>T_e = \frac{1}{1+F\sin^2(\frac{\delta}{2})},</math>

where both surfaces have reflectance ''R'' and

:<math> F = \frac{4R}{{(1-R)^2}}</math>

is the ''coefficient of finesse''.

==History==
The Airy function is named after the [[British (people)|British]] [[astronomer]] and [[physicist]] [[George Biddell Airy]] (1801–1892), who encountered it in his early study of [[optics]] in physics (Airy 1838). The notation Ai(''x'') was introduced by [[Harold Jeffreys]]. Airy had become the British [[Astronomer Royal]] in 1835, and he held that post until his retirement in 1881.

==See also==

*The proof of [[Witten's conjecture]] used a matrix-valued generalization of the Airy function.
*[[Airy zeta function]]

==Notes==
<references />

==References==
*{{AS ref|10|446}}
* {{citation|last=Airy |year=1838|title= On the intensity of light in the neighbourhood of a caustic|journal=Transactions of the Cambridge Philosophical Society|volume=6|pages= 379–402|url=http://books.google.com/?id=-yI8AAAAMAAJ&dq=Transactions+of+the+Cambridge+Philosophical+Society+1838|publisher=University Press}}
* Olver (1974). ''Asymptotics and Special Functions,'' Chapter 11. Academic Press, New York.
* {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.6.3. Airy Functions | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=289}}
* {{Citation | last1=Vallée | first1=Olivier | last2=Soares | first2=Manuel | title=Airy functions and applications to physics | url=http://www.worldscibooks.com/physics/p345.html | publisher=Imperial College Press | location=London | isbn=978-1-86094-478-9 | mr=2114198 | year=2004}}

==External links==
* {{springer|title=Airy functions|id=p/a011210}}
* {{MathWorld | urlname=AiryFunctions | title=Airy Functions}}
* Wolfram function pages for [http://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/ Ai] and [http://functions.wolfram.com/Bessel-TypeFunctions/AiryBi/ Bi] functions. Includes formulas, function evaluator, and plotting calculator.
* {{dlmf|title= Airy and related functions |id=9|first=F. W. J.|last= Olver}}

[[Category:Special functions]]
[[Category:Special hypergeometric functions]]
[[Category:Ordinary differential equations]]

Strassen algorithm

2014-01-07T10:50:45Z

117.211.86.74: /* History */

In [[mathematics]], a '''Hurwitz quaternion''' (or '''Hurwitz integer''') is a [[quaternion]] whose components are ''either'' all [[integer]]s ''or'' all [[half-integer]]s (halves of an odd integer; a mixture of integers and half-integers is not allowed). The set of all Hurwitz quaternions is

:<math>H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.</math>

It can be confirmed that ''H'' is closed under quaternion multiplication and addition, which makes it a [[subring]] of the [[ring (mathematics)|ring]] of all quaternions '''H'''.

A '''Lipschitz quaternion''' (or '''Lipschitz integer''') is a quaternion whose components are all [[integer]]s. The set of all Lipschitz quaternions

:<math>L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}</math>

forms a subring of the Hurwitz quaternions ''H''.

As a [[group (mathematics)|group]], ''H'' is [[free abelian group|free abelian]] with generators {(1 + ''i'' + ''j'' + ''k'')/2, ''i'', ''j'', ''k''}. It therefore forms a [[lattice (group)|lattice]] in '''R'''4. This lattice is known as the [[F4 lattice|''F''4 lattice]] since it is the [[root lattice]] of the [[semisimple Lie algebra]] [[F4 (mathematics)|''F''4]]. The Lipschitz quaternions ''L'' form an index 2 sublattice of ''H''.

The [[group of units]] in ''L'' is the order 8 [[quaternion group]] ''Q'' = {±1, ±''i'', ±''j'', ±''k''}. The [[group of units]] in ''H'' is a nonabelian group of order 24 known as the [[binary tetrahedral group]]. The elements of this group include the 8 elements of ''Q'' along with the 16 quaternions {(±1 ± ''i'' ± ''j'' ± ''k'')/2} where signs may be taken in any combination. The quaternion group is a [[normal subgroup]] of the binary tetrahedral group ''U''(''H''). The elements of ''U''(''H''), which all have norm 1, form the vertices of the [[24-cell]] inscribed in the [[3-sphere]].

The Hurwitz quaternions form an [[order (ring theory)|order]] (in the sense of [[ring theory]]) in the [[division ring]] of quaternions with [[rational number|rational]] components. It is in fact a [[maximal order]]; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an ''integral quaternion'', also form an order. However, this latter order is not a maximal one, and therefore (as it turns out) less suitable for developing a theory of [[left ideal]]s comparable to that of [[algebraic number theory]]. What [[Adolf Hurwitz]] realised, therefore, was that this definition of Hurwitz integral quaternion is the better one to operate with. This was one major step in the theory of maximal orders, the other being the remark that they will not, for a non-commutative ring such as '''H''', be unique. One therefore needs to fix a maximal order to work with, in carrying over the concept of an [[algebraic integer]].

The [[field norm|(arithmetic, or field) norm]] of a Hurwitz quaternion, given by <math>a^2+b^2+c^2+d^2</math>, is always an integer. By a [[Lagrange's four-square theorem|theorem of Lagrange]] every nonnegative integer can be written as a sum of at most four [[square (algebra)|squares]]. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. A Hurwitz integer is a [[prime element]] if and only if its norm is a [[prime number]].

==See also==
* [[Gaussian integer]]
* [[Eisenstein integer]]
* The Lie group [[F4 (mathematics)|F4]]
* The [[E8 lattice|E8 lattice]]

==References==
*John Horton Conway, Derek Alan Smith (2003), [http://books.google.co.uk/books?id=E_HCwwxMbfMC On quaternions and octonions: their geometry, arithmetic, and symmetry], A K Peters Ltd., ISBN 978-1-56881-134-5

[[Category:Quaternions]]

Law of supply

2013-12-08T20:53:28Z

117.211.84.18: /* See also */

In [[abstract algebra]], a branch of pure [[mathematics]], the [[algebraic structure]] '''group with operators''' or Ω-'''group''' can be viewed as a [[group (mathematics)|group]] with a [[Set (mathematics)|set]] Ω which operates on the elements of the group in a special way.

Groups with operators were extensively studied by [[Emmy Noether]] and her school in the 1920s. She employed the concept in her original formulation of the three [[Noether isomorphism theorem]]s.

{{Algebraic structures |Module}}

== Definition ==

A '''group with operators''' (''G'', <math>\Omega</math>) can be defined{{sfn|Bourbaki|1974|p=31}} as a group ''G'' together with an action of a set <math>\Omega</math> on ''G'' :
:<math>\ \Omega \times G \rightarrow G : (\omega , g) \mapsto g^{\omega}</math>
which is distributive relatively to the group law :
:<math>\ (gh)^{\omega} = g^{\omega} h^{\omega}.</math>
For each <math>\omega \in \Omega </math>, the application
:<math>\ g \mapsto g^{\omega}</math>
is then an endomorphism of ''G''. From this, it results that a Ω-group can also be viewed as a group ''G'' with an [[indexed family]] <math>(u_{\omega})_{\omega \in \Omega}</math> of endomorphisms of ''G''.

<math>\Omega</math> is called the '''operator domain'''. The associate [[endomorphisms]]{{sfn|Bourbaki|1974|pp=30–31}} are called the '''homotheties''' of ''G''.

Given two groups ''G'', ''H'' with same operator domain <math>\Omega</math>, a '''homomorphism''' of groups with operators is a group homomorphism ''f'':''G''<math>\to</math>''H'' satisfying
:<math>\forall \omega \in \Omega, \forall g \in G : f(g^\omega)=(f(g))^\omega.</math>

A subgroup ''S'' of ''G'' is called a '''stable subgroup''', '''<math>\omega</math>-subgroup''' or '''<math>\Omega</math>-invariant subgroup''' if it respects the homotheties, that is
:<math>\forall s \in S, \forall \omega \in \Omega : s^\omega \in S.</math>

== Category-theoretic remarks ==
In [[category theory]], a '''group with operators''' can be defined{{sfn|Mac Lane|1998|p=41}} as an object of a [[functor category]] '''Grp'''''M'' where ''M'' is a monoid (''i.e.'', a category with one object) and '''Grp''' denotes the [[category of groups]]. This definition is equivalent to the previous one, provided <math>\Omega</math> is a monoid (otherwise we may expand it to include the identity and all compositions).

A [[morphism]] in this category is a [[natural transformation]] between two functors (''i.e.'' two groups with operators sharing same operator domain ''M''). Again we recover the definition above of a homomorphism of groups with operators (with ''f'' the [[Natural_Transformation#Definition|component]] of the natural transformation).

A group with operators is also a mapping
:<math>\Omega\rightarrow\operatorname{End}_{\mathbf{Grp}}(G),</math>

where <math>\operatorname{End}_{\mathbf{Grp}}(G)</math> is the set of group [[endomorphism]]s of ''G''.

== Examples ==
* Given any group ''G'', (''G'', ∅) is trivially a group with operators
* Given an ''R''-[[module (mathematics)|module]] ''M'', ''R'' acts by [[scalar multiplication]] on the underlying Abelian group ''M'', so (''M'', ''R'') is a group with operators.
* As a special case of the above, every [[vector space]] over ''k'' is a group with operators (''V'', ''k'').

==Applications==

The [[Jordan–Hölder theorem]] also holds in the context of operator groups. The requirement that a group have a [[composition series]] is analogous to that of [[compact space|compactness]] in [[topology]], and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set", i.e. talk about [[composition series]] where each (normal) subgroup is an operator-subgroup relative to the operator set ''X'', of the group in question.

==See also==
*[[Group action]]

==Notes==
{{reflist}}

== References ==
*{{cite book | ref=harv | last=Bourbaki | first=Nicolas | title=Elements of Mathematics : Algebra I Chapters 1–3 | publisher=Hermann | year=1974 | isbn=2-7056-5675-8}}
*{{cite book | ref=harv | last=Bourbaki | first=Nicolas | title=Elements of Mathematics : Algebra I Chapters 1–3 | publisher=Springer-Verlag | year=1998 | isbn=3-540-64243-9}}
*{{cite book | ref=harv | last=Mac Lane | first=Saunders | title=Categories for the Working Mathematician | publisher=Springer-Verlag | year=1998 | isbn=0-387-98403-8}}

[[Category:Group actions]]
[[Category:Universal algebra]]

Dunkerley's method

2013-05-04T17:29:24Z

117.211.84.18:

<math>fgf</math>[[Image:Oceanic Whitetip Shark.png|left|200px]]
The '''oceanic whitetip shark''', ''Carcharhinus longimanus'', is a large [[pelagic zone|pelagic]] [[shark]] of tropical and warm temperate seas. It is a stocky shark, most notable for its long, white tipped rounded fins.

This aggressive but slow-moving fish dominates feeding frenzies, and has attacked more humans than all other shark species combined — it is a notable danger to survivors of oceanic ship wrecks and downed aircraft. Recent studies have shown that its numbers are in steep decline — its large fins are highly valued as the chief ingredient of [[shark-fin soup]] and, as with other shark species, the oceanic whitetip faces mounting pressure from [[fishery|fishing]] throughout its range.

The oceanic whitetip shark was first described by naturalist [[Rene Primevere Lesson]] in his account of observations made during [[Louis Isidore Duperrey|Louis Duperrey]]'s 1822–1825 circumnavigation of the world on the corvette ''[[Coquille]]''. Lesson described two specimens found in the [[Tuamotu Archipelago]] in [[French Polynesia]], and [[binomial nomenclature|named]] the shark ''Squalus maou'' after a Polynesian word for "shark". However, Lesson's description and name were forgotten.

'''More on the [[oceanic whitetip shark]]'''

Barsoum elements

2013-04-12T07:59:10Z

117.211.86.74:

{{Infobox enzyme
| Name = Unsaturated rhamnogalacturonyl hydrolase
| EC_number = 3.2.1.172
| CAS_number =
| IUBMB_EC_number = 3/2/1/172
| GO_code =
| image =
| width =
| caption =
}}
'''Unsaturated rhamnogalacturonyl hydrolase''' ({{EC number|3.2.1.172}}, ''YteR'', ''YesR'') is an [[enzyme]] with system name ''2-O-(4-deoxy-beta-L-threo-hex-4-enopyranuronosyl)-alpha-L-rhamnopyranose hydrolase''.<ref>{{cite journal | title = A novel glycoside hydrolase family 105: the structure of family 105 unsaturated rhamnogalacturonyl hydrolase complexed with a disaccharide in comparison with family 88 enzyme complexed with the disaccharide |author = Itoh, T., Ochiai, A., Mikami, B., Hashimoto, W. and Murata, K. |journal = J. Mol. Biol. |year = 2006 |volume = 360 |pages = 573–585 |pmid = 16781735}}</ref><ref>{{cite journal | title = 1.6 Å crystal structure of YteR protein from ''Bacillus subtilis'', a predicted lyase |author = Zhang, R., Minh, T., Lezondra, L., Korolev, S., Moy, S.F., Collart, F. and Joachimiak, A. |journal = Proteins |year = 2005 |volume = 60 |pages = 561–565 |pmid = 15906318}}</ref><ref>{{cite journal | title = Structure of unsaturated rhamnogalacturonyl hydrolase complexed with substrate |author = Itoh, T., Ochiai, A., Mikami, B., Hashimoto, W. and Murata, K. |journal = Biochem. Biophys. Res. Commun. |year = 2006 |volume = 347 |pages = 1021–1029 |pmid = 16870154}}</ref> This enzyme [[catalysis|catalyses]] the following [[chemical reaction]]

: 2-O-(4-deoxy-beta-L-threo-hex-4-enopyranuronosyl)-alpha-L-rhamnopyranose + H2O <math>\rightleftharpoons</math> 5-dehydro-4-deoxy-D-glucuronate + L-[[rhamnopyranose]]

The enzyme is part of the degradation system for rhamnogalacturonan I in ''[[Bacillus subtilis]]'' strain 168.

== References ==
{{reflist}}

== External links ==
* {{MeshName|Unsaturated+rhamnogalacturonyl+hydrolase}}

[[Category:EC 3.2.1]]

Pumping lemma for context-free languages

2012-08-31T05:22:41Z

117.211.42.174: /* Usage of the lemma */

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Kruskal's algorithm

2012-08-24T10:58:07Z

117.211.84.18: /* Description */

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Pay attention to species of fish being a major dinner rather than red meat. Some little known vitamins and minerals that provide important nourishment for your personal skin area are selenium, L-Carnitine and zinc. L-Carnitine normally comes in a supplement kind. Zinc is something you can find in numerous nuts and whole grain products, as well as mushrooms and chicken eggs. Selenium is normally based in the garden soil that will grow vegetables. However, hydroponic fresh vegetables generally usually do not. When you eat an eating plan rich in by natural means cultivated food items as well as L-carnitine, you will definately get very good healthy balance. When eating, your must focus on fruits and vegetables, then proteins. Carbs must be your cheapest top priority. Even though sugars are important, most people have a tendency to eat too much. Consuming vegetables, fresh fruits and proteins at the start will give you less yearnings for big numbers of carbs. 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Good diet is all about picking more healthy foods typically and ingesting unhealthy foods only as being a treat. If this information has demonstrated you nearly anything, it must be that there are plenty of effortless things that you can be doing in order to get much better nutrients to you.

Carnot cycle

2012-08-02T13:13:37Z

117.211.123.49: /* Stages of the Carnot Cycle */

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Generalized Fourier series

2012-02-25T19:26:40Z

117.211.91.130:

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