List of integrals of Gaussian functions: Difference between revisions

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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.
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In [[mathematics]] and [[optimization]], a '''pseudo-Boolean function''' is a [[function (mathematics)|function]] of the form
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:<math>f:\mathbf{B}^n \rightarrow \mathbb{R}</math>,
where '''B'''&nbsp;=&nbsp;{0,&nbsp;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]]

Revision as of 23:08, 10 February 2014



Oscillation is described as high speed vibration, generally as rapid as 22,000 oscillations per minute. This 11 amp recipro saw is strikingly powerful and also strikingly lightweight (at only 7 lbs) winning this Sawzall the very best power to weight ratio in its class. If you liked this article and Finest Angle Grinder For Steel also you would like to receive more info pertaining to Ideal Correct Angle Die Grinder; you can look here, i implore you to visit our own web-page. The saw has a super rapid - 2800 Drillmaster Angle Grinder Reviews SPM variable speed trigger, and fantastic feature, so the sucker definitely moves quickly, but since it only has a �" stroke, it doesn't have the exact same length of motion as Makita's JR3070CT. For high-efficiency, light-duty use Bosch's RS15 reciprocating saw is a brilliant machine.

Look at your requirements and quantity of usage your saw will see prior to investing a fantastic deal of cash into something that will commit 75% of its life on a shelf or in the tool box of your truck. Even though there are countless models to select from, this guide is based off business requirements and overall buyer critiques of each and every listed model. Beneath you will come across reciprocating saws reviews of the major models readily available.

What I genuinely like about the DeWalt DC385K 18-volt Ni-Cad cordless reciprocating saw kit is that the blade has four positions for cutting, so even if the saw is facing handle up you can turn the blade about to face the project at a number of angles, guaranteeing you get the most effective cut feasible with out damaging the blade clamp or the shoe. I can try to remember a time when you required an Allen wrench to get rid of the blade from a reciprocating saw.

If you are working on a project exactly where there is no other way to pull out a nail for the reason that of its difficult hold, you can use a reciprocating saw to reduce the head of the nail and not bother pulling it up. You can also use this with pins in the same way. The DeWalt also had a wonderful set up with a multi-position that enables the saw to go from pistol grip to entirely strait. This saw is lots for cutting 2" to five" limbs and smaller shrubs out.

I never like to let the branches grow over the roof and with the chain saw the job of cutting them is quickly completed. I took my cordless chain saw into the woods in back of my house and reduce a few sections of an oak log about 21/two feet extended. The hand saw was able to fit amongst the ground and the bottom of the fence. Whether or not you use a corded or cordless energy saw, each do the similar job.

When the ideal mixture of pivot head, saw and material are made use of, it should result in very few challenges when it comes to this. This allows the saw to reduce various sorts of supplies such as woods, PVC pipes, shingles, tiles and so on. Additionally, considering that this type of saw does not require much space, you can use it in places exactly where a frequent saw would not attain. The main issue of this saw is the absence of a saw blade.

If allowed, it's also an exceptional idea to hold and play around a bit with the reciprocating saw prior to you obtain. Being in a position to essentially hold the saw ahead of producing a monetary commitment guarantees you are fully pleased with the instrument and its bodyweight, stability, and so forth. I initially utilized my silky katanaboy to cut the fronds and then my reciprocating saw to do the angled cuts....was way easier than using a had saw for the angled cuts. I cut a range of materials with this saw.

This Dewalt�Bare Tool DC�385B�Cordless Reciprocating Saw operates on 18 volts and received four.five stars from additional than 185 Amazon reviewers. This Dewalt�DW304PK�Reciprocating Saw has a four position blade clamp that allows you versatility and flush cutting. The saw has an ergonomic style and a soft grip that permits you to manage it for a lengthy period of time with out fatigue. This saw expenses just under�$one hundred online and received four.5 stars from Amazon reviewers. This to me looks like a wonderful cost for this saw.

The reciprocating saw, in some cases known as a recipro saw or saws-all, is able to cut via thin and thick material alike. The highly effective, feature-rich SKIL 9205 Reciprocating Saw supplies rapidly, effective cutting in each and every application and unmatched energy for its class. This saw is equipped with advanced capabilities to get the job accomplished. The 9205 Reciprocating Saw is the perfect tool for any homeowner needing to trim trees or complete little projects around the house.

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