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| A '''Lévy flight''' is a [[random walk]] in which the step-lengths have a [[probability distribution]] that is [[heavy-tailed distribution|heavy-tailed]]. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. The "Lévy" in "Lévy flight" is a reference to the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]].
| | Have we been thinking "how do I speed up my computer" lately? Well chances are in the event you are reading this article; then we may be experiencing one of various computer issues that thousands of people discover that they face regularly.<br><br>Install an anti-virus software. If you absolutely have that on you computer then carry out a full program scan. If it finds any viruses found on the computer, delete those. Viruses invade the computer plus make it slower. To protect the computer from many viruses, it is better to keep the anti-virus software running whenever we use the internet. You may also fix the safety settings of your internet browser. It usually block unknown plus dangerous sites plus also block off any spyware or malware struggling to receive into the computer.<br><br>So, this advanced dual scan is not only among the greater, but it really is additionally freeware. And as of all of this that several regard CCleaner among the better registry products in the marketplace today. I would add that I personally prefer Regcure for the simple reason that it has a better interface plus I recognize for a truth which it is ad-ware without charge.<br><br>Fixing tcpip.sys blue screen is simple to do with registry repair software.Trying to fix windows blue screen error on your own can be challenging considering if you remove or damage the registry it can cause serious damage to a computer. The registry should be cleaned plus all erroneous plus incomplete information removed to stop blue screen errors from occurring.The advantage of registry repair software is not limited to simply getting rid of the blue screen on business.We may be amazed at the greater plus more improved speed plus performance of your computer system after registry cleaning is completed. Registry cleaning can definitely develop a computer's functioning abilities, incredibly whenever we choose a certain registry repair software that is extremely powerful.<br><br>Use a [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014]. This might search your Windows registry for three types of keys which can definitely hurt PC performance. These are: duplicate, missing, plus corrupted.<br><br>Reinstall Windows 7 - If nothing seems to work, reinstall Windows 7 with all the installation disc which came with the pack. Kindly backup or restore all your information to a flash drive or another difficult drive/CD etc. before operating the reinstallation.<br><br>Across the top of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Every of these tabs will show we the results of that area. The Junk Files are mainly temporary files such as web data, photos, internet pages... And they are just taking up storage.<br><br>Ally Wood is a pro software reviewer and has worked inside CNET. Then she is functioning for her own review software company to provide suggestions to the software creator and has completed deep test inside registry cleaner software. After reviewing the top registry cleaner, she has written complete review on a review site for you which is accessed for free. |
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| The term "Lévy flight" was coined by [[Benoît Mandelbrot]],<ref name=D1>Mandelbrot (1982, p.289)</ref> who used this for one specific definition of the distribution of step sizes. He used the term '''Cauchy flight''' for the case where the distribution of step sizes is a [[Cauchy distribution]],<ref>Mandelbrot (1982, p.290)</ref> and '''Rayleigh flight''' for when the distribution is a [[normal distribution]]<ref>Mandelbrot (1982, p.288)</ref> (which is not an example of a heavy-tailed probability distribution).
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| Later researchers have extended the use of the term "Lévy flight" to include cases where the random walk takes place on a discrete grid rather than on a continuous space.<ref name=Kleinberg/><ref name=Li/>
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| A Lévy flight is a [[random walk]] in which the steps are defined in terms of the step-lengths, which have a certain [[probability distribution]], with the directions of the steps being isotropic and random.
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| The particular case for which Mandelbrot used the term "Lévy flight"<ref name=D1/> is defined by the [[survivor function]] (commonly known as the survival function) of the distribution of step-sizes, ''U'', being<ref>Mandelbrot (1982, p.294)</ref>
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| :<math>\operatorname{Pr}(U>u) = \begin{cases}
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| 1 &:\ u < 1,\\
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| u^{-D} &:\ u \ge 1.
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| \end{cases}</math>
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| Here ''D'' is a parameter related to the [[fractal dimension]] and the distribution is a particular case of the [[Pareto distribution]]. Later researchers allow the distribution of step sizes to be any distribution for which the [[survival function]] has a power-like tail{{citation needed|date=August 2011}}
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| :<math>\operatorname{Pr}(U>u) = O(u^{-k}),</math>
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| for some ''k'' satisfying 1 < k < 3. (Here the notation ''O'' is the [[Big O notation]].) Such distributions have an infinite [[variance]]. Typical examples are the symmetric [[stable distribution]]s.
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| ==Properties==
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| Lévy flights are, by construction, [[Markov property|Markov processes]]. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a [[stable distribution]].{{citation needed|date=December 2010}}
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| The exponential scaling of the step lengths gives Lévy flights a [[scale invariance|scale invariant]] property,{{citation needed|date=December 2010}} and they are used to model data that exhibits clustering.{{citation needed|date=December 2010}} | |
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| [[Image:LevyFlight.svg|thumb|left|256px|Figure 1. An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at [0,0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. [[stable distribution|stable]]) distribution with ''α'' = 1 and ''β'' = 0 which is a [[Cauchy distribution]]. Note the presence of large jumps in location compared to the Brownian motion illustrated in Figure 2.]]
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| [[Image:BrownianMotion.svg|thumb|right|256px|Figure 2. An example of 1000 steps of an approximation to a Brownian motion type of Lévy flight in two dimensions. The origin of the motion is at [0, 0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. [[stable distribution|stable]]) distribution with ''α'' = 2 and ''β'' = 0 (''i.e.,'' a [[normal distribution]]).]]
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| <br style="clear:both;" />
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| ==Applications==
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| The definition of a Lévy flight stems from the mathematics related to [[chaos theory]] and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomena. Examples include [[earthquake]] data analysis, [[financial mathematics]], [[cryptography]], signals analysis as well as many applications in [[astronomy]], [[biology]], and [[physics]].
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| Another application is the [[Lévy flight foraging hypothesis]]. When sharks and other ocean predators can’t find food, they abandon [[Brownian motion]], the random motion seen in swirling gas molecules, for Lévy flight — a mix of long trajectories and short, random movements found in turbulent fluids. Researchers analyzed over 12 million movements recorded over 5,700 days in 55 radio-tagged animals from 14 ocean predator species in the Atlantic and Pacific Oceans, including [[silky shark]]s, [[yellowfin tuna]], blue marlin and swordfish. The data showed that Lévy flights interspersed with Brownian motion can describe the animals' hunting patterns.<ref>{{cite web |last=Witze |first=Alexandra |title=Sharks Have Math Skills |url=http://news.discovery.com/animals/sharks/sharks-math-hunt.htm |work=discovery.com |accessdate=22 February 2013}}</ref><ref>{{cite web |last=Dacey |first=James |title=Sharks hunt via Lévy flights |url=http://physicsworld.com/cws/article/news/2010/jun/11/sharks-hunt-via-levy-flights |work=physicsworld.com |accessdate=22 February 2013}}</ref> Birds and other animals<ref>{{cite journal |title=Optimizing the success of random searches |doi=10.1038/44831 |journal=Nature |year=1999 |last1=Viswanathan |first1=G. M. |last2=Buldyrev |first2=S. V. |last3=Havlin |first3=Shlomo |authorlink3=Shlomo Havlin |last4=da Luz |first4=M. G. E. |last5=Raposo |first5=E. P. |last6=Stanley |first6=H. E. |authorlink6=H. Eugene Stanley |volume=401 |issue=6756 |page=911|bibcode = 1999Natur.401..911V }}</ref> (including humans)<ref>{{cite web|url=http://well.blogs.nytimes.com/2014/01/01/navigating-our-world-like-birds-and-bees/ |date=January 1, 2014 |title=Navigating Our World Like Birds and Bees |first=Gretchen |last=Reynolds |work=[[The New York Times]]}}</ref> follow Lévy paths when searching for food.
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| Efficient routing in a network can be performed by links having a Levy flight length distribution with specific values of alpha.<ref name=Kleinberg/><ref name=Li/>
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| ==See also==
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| *[[Fat-tailed distribution]]
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| *[[Heavy-tailed distribution]]
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| *[[Lévy process]]
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| *[[Lévy alpha-stable distribution]]
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| *[[Lévy flight foraging hypothesis]]
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| == Notes ==
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| {{reflist|refs=
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| <ref name=Kleinberg>{{cite journal|title=Navigation in a small world|doi=10.1038/35022643 |journal=Nature|year=2000|volume=406|pages=845|issue=6798|pmid=10972276|author=J. M. Kleinberg|bibcode = 2000Natur.406..845K }}</ref>
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| <ref name=Li>{{cite journal|title=Towards Design Principles for Optimal Transport Networks|journal=PRL|year=2010|volume=104|pages=018701| | |
| url=http://havlin.biu.ac.il/Publications.php?keyword=Towards+Design+Principles+for+Optimal+Transport+Networks&year=*&match=all|author=G. Li, S. D. S. Reis, A. A. Moreira, [[Shlomo Havlin|S. Havlin]], [[H. Eugene Stanley|H. E. Stanley]], and J. S. Andrade, Jr|bibcode=2010PhRvL.104a8701L|last2=Reis|last3=Moreira|last4=Havlin|last5=Stanley|last6=Andrade|doi=10.1103/PhysRevLett.104.018701}}</ref>
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| }}
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| ==References==
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| *{{cite book |last=Mandelbrot |first=Benoit B. |authorlink=Benoît Mandelbrot |title=The Fractal Geometry of Nature |year=1982 |publisher=W. H. Freeman |location=New York |isbn=0-7167-1186-9 |edition=Updated and augm. |oclc=7876824}}
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| ==Further reading==
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| *{{cite doi|10.1016/S0378-4371(02)01157-3}}
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| *{{cite doi|10.1016/S0378-4371(00)00071-6}}
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| *{{cite doi|10.1063/1.527644}}
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| *{{Cite journal |last1=Shlesinger |first1=Michael F. |last2=Klafter |first2=Joseph |last3=Zumofen |first3=Gert |title=Above, below and beyond Brownian motion |doi=10.1119/1.19112 |journal=American Journal of Physics |volume=67 |issue=12 |pages=1253–1259 |date=December 1999 |url=http://caos.fs.usb.ve/~srojas/Teaching/USB/MC_Intro/MC_readings_a/MC_a4_brownian_1.pdf|bibcode = 1999AmJPh..67.1253S }}
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| ==External links==
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| *[http://plus.maths.org/issue11/features/physics_world/ A comparison of the paintings of Jackson Pollock to a Lévy flight model]
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| {{Fractals}}
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| {{DEFAULTSORT:Levy Flight}}
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| [[Category:Fractals]]
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| [[Category:Stochastic processes]]
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Have we been thinking "how do I speed up my computer" lately? Well chances are in the event you are reading this article; then we may be experiencing one of various computer issues that thousands of people discover that they face regularly.
Install an anti-virus software. If you absolutely have that on you computer then carry out a full program scan. If it finds any viruses found on the computer, delete those. Viruses invade the computer plus make it slower. To protect the computer from many viruses, it is better to keep the anti-virus software running whenever we use the internet. You may also fix the safety settings of your internet browser. It usually block unknown plus dangerous sites plus also block off any spyware or malware struggling to receive into the computer.
So, this advanced dual scan is not only among the greater, but it really is additionally freeware. And as of all of this that several regard CCleaner among the better registry products in the marketplace today. I would add that I personally prefer Regcure for the simple reason that it has a better interface plus I recognize for a truth which it is ad-ware without charge.
Fixing tcpip.sys blue screen is simple to do with registry repair software.Trying to fix windows blue screen error on your own can be challenging considering if you remove or damage the registry it can cause serious damage to a computer. The registry should be cleaned plus all erroneous plus incomplete information removed to stop blue screen errors from occurring.The advantage of registry repair software is not limited to simply getting rid of the blue screen on business.We may be amazed at the greater plus more improved speed plus performance of your computer system after registry cleaning is completed. Registry cleaning can definitely develop a computer's functioning abilities, incredibly whenever we choose a certain registry repair software that is extremely powerful.
Use a tuneup utilities 2014. This might search your Windows registry for three types of keys which can definitely hurt PC performance. These are: duplicate, missing, plus corrupted.
Reinstall Windows 7 - If nothing seems to work, reinstall Windows 7 with all the installation disc which came with the pack. Kindly backup or restore all your information to a flash drive or another difficult drive/CD etc. before operating the reinstallation.
Across the top of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Every of these tabs will show we the results of that area. The Junk Files are mainly temporary files such as web data, photos, internet pages... And they are just taking up storage.
Ally Wood is a pro software reviewer and has worked inside CNET. Then she is functioning for her own review software company to provide suggestions to the software creator and has completed deep test inside registry cleaner software. After reviewing the top registry cleaner, she has written complete review on a review site for you which is accessed for free.