List of integrals of Gaussian functions: Difference between revisions

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In [[mathematics]] and [[optimization]], a '''pseudo-Boolean function''' is a [[function (mathematics)|function]] of the form
ireport.cnn.com - [http://ireport.cnn.com/docs/DOC-1169363 http://ireport.cnn.com/docs/DOC-1169363]. Deodorize! Nothing turns off a potential buyer than the usual "funny" or unpleasant detect. You've heard for the bake bread or cookies in the oven stunt.it's a lot in order to just use plug in deodorizers.<br><br>You always be barking within the wrong tree-When the average person decides to obtain involved in Dean Graziosi, where do they turn? Typically, they leave trying to see a cheap houses. After all, foreclosures are increasingly popular these instances. That said, you may find yourself purchasing one particular foreclosure lists and trying to find that perfect, undervalued residential. And that's not to say they aren't out there-they are. But let us assure you that stripping the home, redesigning it, and renting it out will be no cakewalk. And atmosphere difficult, yet it's not while the real finances are. When you obtain real estate mentoring, you'll find that you may be looking for gold out of all wrong companies. There are better alternatives in the market.<br><br>1031 Exchange - This money-saving tip can save you tens of thousands of tax dollars over precisely of your real estate investing full-time job. When you sell a property for a profit you would normally get hit along with a substantial capital gains fiscal. This technique a person to to defer your tax penalty indefinitely - until death in like. Make use of the 1031 exchange to defer taxes you'd otherwise have to pay. While it's true that "you can't take it with you", why a person take the taxes with you?<br><br>When putting your home for sale, you needs to keep in mind that leads will obtain it unappealing if there is rust in the sink, or leaks your plumbing. Check pipes, faucets and toilets to certain each is working properly and is updated.<br><br>Set office hours.You'll add to your enjoyment in order to set office hours. You'll find that you focus better and accomplish more when you have a finite long to work. If you gave yourself seven days a week to work, you'd probably spread out whatever work had to be able to accomplished over those week. Whether you're a part-time or full-time investor, set your office hours and look after them.<br><br>Right the market is ideal for doing Sub2 assignments. Many homes turn out to be underwater, meaning the seller owes more on the mortgage than a home is worth. Really are a few sellers who no longer afford the repayments on their mortgage and also are either struggling to get the payments monthly or are behind in their payments usually are facing real estate. In Find and Assign, I have a matrix that shows concerning options a vendor has on getting associated with their property, along i'm able to costs each and every. If you are able display a seller how they will walk far from their property and making the mortgage payments without affecting their credit, you possess a motivated seller, and person that would be receptive towards your offer.<br><br>The nice thing about buying a home foreclosure may lender is willing to offer the property at a great lower price in order to recover the unpaid loan amount, which can be 30% to 50% under market value. The thing is, banks don't really want to hold on to real estate properties, especially ones in which foreclosures and consequently they are just present doing nothing and not making some cash for loan company. Banks are in the business of lending money to make money. They're not in the business of selling real estate. They can be in small business of owning real estate sometimes. Only if they're making money it!<br><br>Ion television airs old movies, reruns of old sitcoms like Alice or Mama's Family, and broadcast paid shows. On average, I've counted 10-12 hours on weekends of nothing but 30 or 60 minute advertisements. Use bad, wishing to get of individuals offenders are MSNBC and CNBC. Both air 10-14 infomercial a lot. Supposedly they're a news which includes a business news channel correspondingly. Seems to me, they should be rechristened thusly, MarketingSNBC and CommericalsNBC.
:<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]]

Latest revision as of 20:12, 20 August 2014

ireport.cnn.com - http://ireport.cnn.com/docs/DOC-1169363. Deodorize! Nothing turns off a potential buyer than the usual "funny" or unpleasant detect. You've heard for the bake bread or cookies in the oven stunt.it's a lot in order to just use plug in deodorizers.

You always be barking within the wrong tree-When the average person decides to obtain involved in Dean Graziosi, where do they turn? Typically, they leave trying to see a cheap houses. After all, foreclosures are increasingly popular these instances. That said, you may find yourself purchasing one particular foreclosure lists and trying to find that perfect, undervalued residential. And that's not to say they aren't out there-they are. But let us assure you that stripping the home, redesigning it, and renting it out will be no cakewalk. And atmosphere difficult, yet it's not while the real finances are. When you obtain real estate mentoring, you'll find that you may be looking for gold out of all wrong companies. There are better alternatives in the market.

1031 Exchange - This money-saving tip can save you tens of thousands of tax dollars over precisely of your real estate investing full-time job. When you sell a property for a profit you would normally get hit along with a substantial capital gains fiscal. This technique a person to to defer your tax penalty indefinitely - until death in like. Make use of the 1031 exchange to defer taxes you'd otherwise have to pay. While it's true that "you can't take it with you", why a person take the taxes with you?

When putting your home for sale, you needs to keep in mind that leads will obtain it unappealing if there is rust in the sink, or leaks your plumbing. Check pipes, faucets and toilets to certain each is working properly and is updated.

Set office hours.You'll add to your enjoyment in order to set office hours. You'll find that you focus better and accomplish more when you have a finite long to work. If you gave yourself seven days a week to work, you'd probably spread out whatever work had to be able to accomplished over those week. Whether you're a part-time or full-time investor, set your office hours and look after them.

Right the market is ideal for doing Sub2 assignments. Many homes turn out to be underwater, meaning the seller owes more on the mortgage than a home is worth. Really are a few sellers who no longer afford the repayments on their mortgage and also are either struggling to get the payments monthly or are behind in their payments usually are facing real estate. In Find and Assign, I have a matrix that shows concerning options a vendor has on getting associated with their property, along i'm able to costs each and every. If you are able display a seller how they will walk far from their property and making the mortgage payments without affecting their credit, you possess a motivated seller, and person that would be receptive towards your offer.

The nice thing about buying a home foreclosure may lender is willing to offer the property at a great lower price in order to recover the unpaid loan amount, which can be 30% to 50% under market value. The thing is, banks don't really want to hold on to real estate properties, especially ones in which foreclosures and consequently they are just present doing nothing and not making some cash for loan company. Banks are in the business of lending money to make money. They're not in the business of selling real estate. They can be in small business of owning real estate sometimes. Only if they're making money it!

Ion television airs old movies, reruns of old sitcoms like Alice or Mama's Family, and broadcast paid shows. On average, I've counted 10-12 hours on weekends of nothing but 30 or 60 minute advertisements. Use bad, wishing to get of individuals offenders are MSNBC and CNBC. Both air 10-14 infomercial a lot. Supposedly they're a news which includes a business news channel correspondingly. Seems to me, they should be rechristened thusly, MarketingSNBC and CommericalsNBC.

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