formulasearchengine - User contributions [en]

Airy function

2013-11-09T20:22:08Z

46.115.90.113: /* See also */ Airy zeta function

[[Image:Dawson+ Function.svg|thumb|300px|right|The Dawson function, <math>F(x) = D_+(x)</math>, around the origin]]
[[Image:Dawson- Function.svg|thumb|300px|right|The Dawson function, <math>D_-(x)</math>, around the origin]]

In [[mathematics]], the '''Dawson function''' or '''Dawson integral''' (named for [[John M. Dawson]]) is either
:<math>F(x) = D_+(x) = e^{-x^2} \int_0^x e^{t^2}\,dt</math>,
also denoted as ''F''(''x'') or ''D''(''x''), or alternatively
:<math>D_-(x) = e^{x^2} \int_0^x e^{-t^2}\,dt\!</math>.

The Dawson function is the one-sided Fourier-Laplace sine transform of the Gaussian function,
:<math>D_+(x) = \frac12 \int_0^\infty e^{-t^2/4}\,\sin{(xt)}\,dt.</math>
It is closely related to the [[error function]] erf, as
:<math> D_+(x) = {\sqrt{\pi} \over 2} e^{-x^2} \mathrm{erfi} (x) = - {i \sqrt{\pi} \over 2} e^{-x^2} \mathrm{erf} (ix) </math>
where erfi is the imaginary error function, {{nowrap|1=erfi(''x'') = −''i'' erf(''ix'').}} Similarly,
:<math>D_-(x) = \frac{\sqrt{\pi}}{2} e^{x^2} \mathrm{erf}(x)</math>
in terms of the real error function, erf.

In terms of either erfi or the [[Faddeeva function]] ''w''(''z''), the Dawson function can be extended to the entire [[complex plane]]:<ref>Mofreh R. Zaghloul and Ahmed N. Ali, "[http://dx.doi.org/10.1145/2049673.2049679 Algorithm 916: Computing the Faddeyeva and Voigt Functions]," ''ACM Trans. Math. Soft.'' '''38''' (2), 15 (2011). Preprint available at [http://arxiv.org/abs/1106.0151 arXiv:1106.0151].</ref>
:<math>F(z) = {\sqrt{\pi} \over 2} e^{-z^2} \mathrm{erfi} (z) = \frac{i\sqrt{\pi}}{2} \left[ e^{-z^2} - w(z) \right]</math>,
which simplifies to
:<math>D_+(x) = F(x) = \frac{\sqrt{\pi}}{2} \operatorname{Im}[ w(x) ]</math>
:<math>D_-(x) = i F(-ix) = -\frac{\sqrt{\pi}}{2} \left[ e^{x^2} - w(-ix) \right]</math>
for real ''x''.

For |''x''| near zero, {{nowrap|1=''F''(''x'') ≈ ''x'',}}
and for |''x''| large, {{nowrap|1=''F''(''x'') ≈ 1/(2''x'').}}
More specifically, near the origin it has the series expansion

:<math> F(x) = \sum_{k=0}^{\infty} \frac{(-1)^k \, 2^k}{(2k+1)!!} \, x^{2k+1}
= x - \frac{2}{3} x^3 + \frac{4}{15} x^5 - \cdots</math>

''F''(''x'') satisfies the differential equation

:<math> \frac{dF}{dx} + 2xF=1\,\!</math>

with the initial condition ''F''(0) = 0.

==References==
*{{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 6.9. Dawson's Integral | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=302}}
*{{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=N. M. |last=Temme}}

<references />

== External links ==
* [http://www.gnu.org/software/gsl/manual/html_node/Dawson-Function.html gsl_sf_dawson] in the [[GNU Scientific Library]]
* [http://www.moshier.net/#Cephes Cephes] – C and C++ language special functions math library
* [http://ab-initio.mit.edu/Faddeeva Faddeeva Package] – C++ code for the Dawson function of both real and complex arguments, via the [[Faddeeva function]]
* [http://mathworld.wolfram.com/DawsonsIntegral.html Dawson's Integral] ''(at Mathworld)''
* [http://nlpc.stanford.edu/nleht/Science/reference/errorfun.pdf Error functions]

[[Category:Special functions]]
[[Category:Gaussian function]]

{{mathapplied-stub}}

Airy zeta function

2013-11-09T20:18:32Z

46.115.90.113: MathWorld

In [[modular arithmetic]], '''Barrett reduction''' is a reduction [[algorithm]] introduced in 1986 by P.D. Barrett.<ref>{{cite doi|10.1007/3-540-47721-7_24}}</ref> A naive way of computing

:<math>c = a \,\bmod\, n. \, </math>

would be to use a fast [[division algorithm]]. Barrett reduction is an algorithm designed to optimize this operation assuming <math>n</math> is constant, and <math>a<n^2</math>, replacing divisions by multiplications.

== Naive idea ==

Let <math>m=1/n</math> be the inverse of <math>n</math> as a [[floating point]] number. Then

:<math>a \,\bmod\, n = a-\lfloor a m\rfloor n </math> where <math>\lfloor x \rfloor</math> denotes the [[floor function]],
assuming m was computed with sufficient accuracy.

== Barrett algorithm ==

Barrett algorithm is a [[fixed-point]] analog which expresses everything in terms of integers.
Let ''k'' be the smallest integer such that <math>2^k>n</math>. Think of ''n'' as representing the fixed-point number <math> n 2^{-k} </math>.
We precompute m such that <math> m = \lfloor 4^k/n \rfloor</math>. Then ''m'' represents the fixed-point number <math> m 2^{-k} \approx (n 2^{-k})^{-1} </math>.

Let <math>q = \left\lfloor \frac{m a}{4^k} \right\rfloor </math> and <math>r = a - q n</math>.

Clearly <math>a \equiv r \pmod{n}</math>, and if <math>a<n^2</math> then <math>r<2 n </math>.
So
:<math>a \,\bmod\, n = \begin{cases} r & \text{if } r<n \\ r-n & \text{otherwise} \end{cases}</math>

== Barrett algorithm for polynomials ==

It is also possible to use Barrett algorithm for polynomial division, by reversing polynomials
and using X-adic arithmetic.{{clarify|date=January 2014}}

== See also ==

[[Montgomery reduction]] is another similar algorithm.

==References==

{{Reflist}}

== Sources ==
*Chapter 14 of [[Alfred J. Menezes]], Paul C. van Oorschot, and [[Scott A. Vanstone]]. [http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography]. CRC Press, 1996. ISBN 0-8493-8523-7.
*Bosselaers, ''et al.'', "Comparison of Three Modular Reduction Functions," Advances in Cryptology-Crypto'93, 1993. [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.3779]
*W. Hasenplaugh, G. Gaubatz, V. Gopal, [http://www.acsel-lab.com/arithmetic/html/Hasenplaugh_W.html "Fast Modular Reduction"], ARITH 18

[[Category:Computer arithmetic]]
[[Category:Cryptographic algorithms]]
[[Category:Modular arithmetic]]

Z function

2013-11-09T20:05:21Z

46.115.90.113: MathWorld

{{refimprove|date=November 2009}}
In [[geometry]], the '''tangent cone''' is a generalization of the notion of the [[tangent space]] to a [[manifold]] to the case of certain spaces with singularities.

== Definition in convex geometry ==

Let ''K'' be a [[closed set|closed]] [[convex subset]] of a real [[vector space]] ''V'' and ∂''K'' be the [[boundary (topology)|boundary]] of ''K''. The '''solid tangent cone''' to ''K'' at a point ''x'' ∈ ∂''K'' is the [[closure (mathematics)|closure]] of the cone formed by all half-lines (or rays) emanating from ''x'' and intersecting ''K'' in at least one point ''y'' distinct from ''x''. It is a [[convex cone]] in ''V'' and can also be defined as the intersection of the closed [[Half-space (geometry)|half-space]]s of ''V'' containing ''K'' and bounded by the [[supporting hyperplane]]s of ''K'' at ''x''. The boundary ''T''''K'' of the solid tangent cone is the '''tangent cone''' to ''K'' and ∂''K'' at ''x''. If this is an [[affine subspace]] of ''V'' then the point ''x'' is called a '''smooth point''' of ∂''K'' and ∂''K'' is said to be '''differentiable''' at ''x'' and ''T''''K'' is the ordinary [[tangent space]] to ∂''K'' at ''x''.


== Definition in algebraic geometry ==

[[File:Node (algebraic geometry).png|thumb|right|200px|y2 = x3 + x2 (red) with tangent cone (blue)]]

Let ''X'' be an [[affine algebraic variety]] embedded into the affine space ''k''''n'', with the defining ideal ''I'' ⊂ ''k''[''x''1,…,''x''''n'']. For any polynomial ''f'', let in(''f'') be the homogeneous component of ''f'' of the lowest degree, the ''initial term'' of ''f'', and let in(''I'') ⊂ ''k''[''x''1,…,''x''''n''] be the homogeneous ideal which is formed by the initial terms in(''f'') for all ''f'' ∈ ''I'', the ''initial ideal'' of ''I''. The '''tangent cone''' to ''X'' at the origin is the Zariski closed subset of ''k''''n'' defined by the ideal in(''I''). By shifting the coordinate system, this definition extends to an arbitrary point of ''k''''n'' in place of the origin. The tangent cone serves as the extension of the notion of the tangent space to ''X'' at a regular point, where ''X'' most closely resembles a [[differentiable manifold]], to all of ''X''. (The tangent cone at a point of ''k''''n'' that is not contained in ''X'' is empty.)

For example, the nodal curve

: <math>C: y^2=x^3+x^2 </math>

is singular at the origin, because both [[partial derivative]]s of ''f''(''x'', ''y'') = ''y''2 − ''x''3 − ''x''2 vanish at (0, 0). Thus the [[Zariski tangent space]] to ''C'' at the origin is the whole plane, and has higher dimension than the curve itself (two versus one). On the other hand, the tangent cone is the union of the tangent lines to the two branches of ''C'' at the origin,

: <math> x=y,\quad x=-y. </math>

Its defining ideal is the principal ideal of ''k''[''x''] generated by the initial term of ''f'', namely ''y''2 − ''x''2 = 0.

The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general [[Noetherian]] [[scheme (mathematics)|schemes]]. Let ''X'' be an [[algebraic variety]], ''x'' a point of ''X'', and (''O''''X'',''x'', ''m'') be the [[local ring]] of ''X'' at ''x''. Then the '''tangent cone''' to ''X'' at ''x'' is the [[spectrum of a ring|spectrum]] of the [[associated graded ring]] of ''O''''X'',''x'' with respect to the [[Completion (ring theory)#I-adic topology|''m''-adic filtration]]:

:<math>\operatorname{gr}_m O_{X,x}=\bigoplus_{i\geq 0} m^i / m^{i+1}.</math>

== See also ==
* [[Cone]]
* [[Monge cone]]
* [[Convex cone#Examples of convex cones|Normal cone]]

== References ==

* {{Springer|title=Tangent cone|id=T/t092120|author=M. I. Voitsekhovskii}}

{{DEFAULTSORT:Tangent Cone}}
[[Category:Convex geometry]]
[[Category:Algebraic geometry]]