List of integrals of Gaussian functions - Revision history

76.168.199.32: /* Indefinite integrals */

2014-08-20T19:12:04Z

Indefinite integrals

← Older revision		Revision as of 21:12, 20 August 2014
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en>Olfau: I corrected formulae 2 and 3 which were incorrect: the first one could take negative values. The sum of the two for certain values of a and b should be given by formula 12 but was not.

2014-02-10T21:08:03Z

I corrected formulae 2 and 3 which were incorrect: the first one could take negative values. The sum of the two for certain values of a and b should be given by formula 12 but was not.

Show changes

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q635136

2013-03-21T22:48:56Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q635136

← Older revision		Revision as of 00:48, 22 March 2013
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	~~Hello~~ and ~~welcome. My name~~ is ~~Hiedi Patillo~~ and ~~Adore it~~. ~~Some~~ [~~http~~:/~~/Www~~.~~Squidoo~~.~~com~~/~~search~~/~~results?q~~=~~time+ago time ago~~] I ~~thought he would live~~ in ~~Ohio~~. ~~Debt collecting~~ is ~~just how I generate income~~. ~~To fix computers~~ is ~~something I will never give~~ -up. ~~Her husband~~ and ~~her maintain web page~~. ~~You might want~~ to ~~verify it out~~: ~~http~~://~~www~~.~~jordans4sale~~.~~com~~/~~308~~-~~nike~~-~~air~~-~~jordan~~-~~womens~~<br><br>~~Feel free~~ to ~~surf~~ to ~~my web blog [~~http://~~www~~.~~jordans4sale~~.com/~~308~~-~~nike~~-~~air~~-~~jordan~~-~~womens retro jordans for sale~~]		In [[mathematics]] and [[optimization]], a '''pseudo-Boolean function''' is a [[function (mathematics)\|function]] of the form
			:<math>f:\mathbf{B}^n \rightarrow \mathbb{R}</math>,
			where '''B''' = {0, 1} is a ''[[Boolean domain]]'' and ''n'' is a nonnegative integer called the [[arity]] of the function. Any pseudo-Boolean function can be written uniquely as a [[multi-linear]] polynomial: {{Citation needed\|date=July 2013}}
			:<math>f(\boldsymbol{x}) = a + \sum_i a_ix_i + \sum_{i<j}a_{ij}x_ix_j + \sum_{i<j<k}a_{ijk}x_ix_jx_k + \ldots</math>
			An important class of pseudo-Boolean functions are the [[supermodular function\|submodular functions]], because polynomial-time algorithms exists for minimizing them. The '''degree''' of the pseudo-Boolean function is simply the degree of the polynomial.

			In many settings (e.g., in Fourier analysis of pseudo-Boolean functions), a pseudo-Boolean function is viewed as a function <math>f</math> that maps <math>\{-1,1\}^n</math> to <math>\mathbb{R}</math>. Again in this case we can uniquely write <math>f</math> as a multi-linear polynomial:
			<math> f(x)= \sum_{I\subseteq [n]}\hat{f}(I)\prod_{i\in I}x_i, </math> where <math> \hat{f}(I) </math> are Fourier coefficients of <math>f</math> and <math>[n]=\{1,...,n\}</math>. For a nice and simple introduction to Fourier analysis of pseudo-Boolean functions, see.<ref name="odon">O'Donnell, 2008</ref>

			==Optimization==
			Minimizing (or, equivalently, maximizing) a pseudo-Boolean function is [[NP-Hard]]. This can easily be seen by formulating, for example, the [[maximum cut]] problem as maximizing a pseudo-Boolean function.{{Citation needed\|date=July 2013}}

			===Submodularity===
			A pseudo-Boolean function ''f'' is said to be '''submodular''' if
			:<math> f(\boldsymbol{x}) + f(\boldsymbol{y}) \ge f(\boldsymbol{x} \wedge \boldsymbol{y}) + f(\boldsymbol{x} \vee \boldsymbol{y}) </math>
			for every '''''x''''' and '''''y'''''. This is a very important concept, because a submodular pseudo-boolean function can be minimized in polynomial time.{{Citation needed\|date=August 2011}}

			===Roof Duality===
			If ''f'' is a quadratic polynomial, a concept called ''roof duality'' can be used to obtain a lower bound for its minimum value.<ref name="boroshammer">Boros and Hammer, 2002</ref> Roof duality may also provide a partial assignment of the variables, indicating some of the values of a minimizer to the polynomial. Several different methods of obtaining lower bounds were developed only to later be shown to be equivalent to what is now called roof duality.<ref name="boroshammer" />

			===Reductions===
			If the degree of ''f'' is greater than 2, one can always employ ''reductions'' to obtain an equivalent quadratic problem with additional variables.<ref name="ishikawa2011">Ishikawa, 2011</ref> One possible reduction is
			:<math>\displaystyle -x_1x_2x_3=\min_{z\in\mathbf{B}}z(2-x_1-x_2-x_3)</math>
			There are other possibilities, for example,
			:<math> \displaystyle -x_1x_2x_3=\min_{z\in\mathbf{B}}z(-x_1+x_2+x_3)-x_1x_2-x_1x_3+x_1. </math>
			Different reductions lead to different results. Take for example the following cubic polynomial:<ref name="kahlstrandmark">Kahl and Strandmark, 2011</ref>
			:<math> \displaystyle f(\boldsymbol{x})=-2x_1+x_2-x_3+4x_1x_2+4x_1x_3-2x_2x_3-2x_1x_2x_3. </math>
			Using the first reduction followed by roof duality, we obtain a lower bound of -3 and no indication on how to assign the three variables. Using the second reduction, we obtain the (tight) lower bound of -2 and the optimal assignment of every variable (which is <math> {(0,1,1)}</math>).

			===Polynomial Compression Algorithms===
			Consider a pseudo-Boolean function <math> f </math> as a mapping from <math>\{-1,1\}^n</math> to <math>\mathbb{R}</math>. Then <math> f(x)= \sum_{I\subseteq [n]}\hat{f}(I)\prod_{i\in I}x_i. </math> Assume that each coefficient <math>\hat{f}(I)</math> is integral.
			Then for an integer <math> k </math> the problem P of deciding whether <math> f(x) </math> is more or equal to <math> k </math> is NP-complete. It is proved in <ref name="crow">Crowston et al., 2011</ref> that
			in polynomial time we can either solve P or reduce the number of variables to <math> O(k^2\log k) </math>.
			Let <math> r </math> be the degree of the above multi-linear polynomial for <math> f </math>. Then <ref name="crow">Crowston et al., 2011</ref> proved that in polynomial time we can either solve P or reduce the number of variables to <math> r(k-1) </math>.

			==See also==
			*[[Boolean function]]

			==References==
			* {{cite journal\|last=Boros\|coauthors=Hammer\|title=Pseudo-Boolean Optimization\|journal=Discrete Applied Mathematics\|year=2002\|volume=123\|doi=10.1016/S0166-218X(01)00341-9}}
			* {{cite journal\|last=Crowston\|coauthors=Fellows, Gutin, Jones, Rosamond, Thomasse, Yeo\|title=Simultaneously Satisfying Linear Equations Over GF(2): MaxLin2 and Max-r-Lin2 Parameterized Above Average.\|journal=Proc. of FSTTCS 2011\|year=2011\|url=http://arxiv.org/abs/1104.1135}}
			* {{cite journal\|last=Ishikawa\|title=Transformation of general binary MRF minimization to the first order case\|journal=IEEE Trans. Pattern Analysis and Machine Intelligence\|year=2011\|volume=33\|number=6\|pages=1234–1249}}
			* {{cite journal\|last=Rother\|coauthors=Kolmogorov, Lempitsky, Szummer\|title=Optimizing Binary MRFs via Extended Roof Duality\|journal=International Conference on Computer Vision and Pattern Recognition\|year=2007\|url=http://research.microsoft.com/pubs/67978/cvpr07-QPBOpi.pdf}}
			* {{cite journal\|last=Kahl\|coauthors=Strandmark\|title=Generalized Roof Duality for Pseudo-Boolean Optimization\|journal=International Conference on Computer Vision \|year=2011\|url=http://www.maths.lth.se/vision/publdb/reports/pdf/kahl-strandmark-iccv-11.pdf}}
			* {{cite journal\|last=O'Donnell\|first=Ryan\|title=Some topics in analysis of Boolean functions\|journal={{ECCC\|2008\|08\|055}}\|year=2008\|url=http://www.eccc.uni-trier.de/eccc-reports/2008/TR08-055/}}

			==Notes==
			<references />

			[[Category:Mathematical optimization]]

en>Daniel Strobusch: /* References */

2012-08-21T07:50:01Z

References

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