formulasearchengine - User contributions [en]

Stress functions

2014-01-16T22:35:12Z

89.70.239.182: /* Airy stress function */

{{Probability distribution
| type = density
| pdf_image =
| cdf_image =
| notation = Tukey(''λ'')
| parameters = ''λ'' ∈ '''R''' — [[shape parameter]]
| support = ''x'' ∈ [−1/''λ'', 1/''λ''] for ''λ'' > 0, ''x'' ∈ '''R''' for ''λ'' ≤ 0
| pdf = <math>(Q(p;\lambda)\,,Q'(p;\lambda)^{-1}),\, 0\leq\,p\,\leq\,1</math>
| cdf = <math>(e^{-x}+1)^{-1},\,\,\lambda\,=\,0</math>
| mean = <math>0,\,\,\lambda > -1</math>
| median = 0
| mode = 0
| variance = <math>\frac{2}{\lambda^2}\bigg(\frac{1}{1+2\lambda}-\frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\bigg),\,\,\lambda > -1/2</math> <math>\frac{ \pi^{2} }{ 3 },\,\,\lambda\,=\,0</math>
| skewness = <math>0,\,\,\lambda > -1/3</math>
| kurtosis = <math>\frac{(2\lambda+1)^2}{2(4\lambda+1)} \frac{ g_2^2\big(3g_2^2-4g_1g_3+g_4\big)}{g_4\big(g_1^2-g_2\big)^2} - 3,</math> <math> 1.2,\,\,\lambda\,=\,0,</math> where ''g''''k'' = Γ(''kλ''+1) and ''λ'' > -1/4.
| entropy = <math>h(\lambda) = \int_0^1 \log (Q'(p;\lambda))\,dp</math><ref>{{Citation |last1=Vasicek |first1=Oldrich |year=1976 |title=A Test for Normality Based on Sample Entropy |journal=Journal of the Royal Statistical Society, Series B |volume=38 |issue=1 |pages=54–59 |postscript=. }}</ref>
| mgf =
| cf = <math>\phi(t;\lambda) = \int_0^1 \exp (\,i t\,Q(p;\lambda))\,dp</math><ref>{{Citation |last1=Shaw |first1=W. T. |last2=McCabe |first2=J. |year=2009 |title=Monte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space |journal=Eprint-arXiv:0903,1592 }}</ref>
}}
Formalized by [[John Tukey]], the '''Tukey lambda distribution''' is a continuous probability distribution defined in terms of its [[quantile function]]. It is typically used to identify an appropriate distribution (see the comments below) and not used in [[statistical model]]s directly.

The Tukey lambda distribution has a single [[shape parameter]] λ. As with other probability distributions, the Tukey lambda distribution can be transformed with a [[location parameter]], μ, and a [[scale parameter]], σ. Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function.

==Quantile function==

For the standard form of the Tukey lambda distribution, the quantile function, Q(p), (i.e. the inverse of the [[cumulative distribution function]]) and the quantile density function (i.e. the derivative of the quantile function) are

:<math>
Q\left(p;\lambda\right) =
\begin{cases}
\frac{ 1 }{ \lambda } \left[p^\lambda - (1 - p)^\lambda\right], & \mbox{if } \lambda \ne 0 \\
\log(\frac{p}{1-p}), & \mbox{if } \lambda = 0,
\end{cases}</math>
:<math>Q'\left(p;\lambda\right) = p^{(\lambda-1)} + \left(1-p\right)^{(\lambda-1)}.</math>

The [[probability density function]] (pdf) and [[cumulative distribution function]] (cdf) are both computed numerically, as the Tukey lambda distribution does not have a simple, closed form for any values of the parameters except ''λ'' = 0 (see [[logistic distribution]]). However, the pdf can be expressed in parametric form, for all values of ''λ'', in terms of the quantile function and the reciprocal of the quantile density function.

==Moments==
The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution is equal to zero. The variance exists for {{nowrap|''λ'' > −½}} and is given by the formula (except when ''λ'' = 0)
: <math>
\operatorname{Var}[X] = \frac{2}{\lambda^2}\bigg(\frac{1}{1+2\lambda} - \frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\bigg).
</math>

More generally, the ''n''-th order moment is finite when {{nowrap|''λ'' > −1/''n''}} and is expressed in terms of the [[beta function]] ''Β''(''x'',''y'') (except when ''λ'' = 0) :
: <math>
\mu_n = \operatorname{E}[X^n] = \frac{1}{\lambda^n} \sum_{k=0}^n (-1)^k {n \choose k}\, \Beta(\lambda k+1,\, \lambda(n-k)+1 ).
</math>

Note that due to symmetry of the density function, all moments of odd orders are equal to zero.

==Comments==
The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,
{| class="wikitable"
|-
| ''λ'' = −1
| approx. [[Cauchy distribution|Cauchy]] ''C''(0,''π'')
|-
| ''λ'' = 0
| exactly [[logistic distribution|logistic]]
|-
| ''λ'' = 0.14
| approx. [[normal distribution|normal]] ''N''(0, 2.142)
|-
| ''λ'' = 0.5
| strictly [[concave function|concave]] (<math>\cap</math>-shaped)
|-
| ''λ'' = 1
| exactly [[continuous uniform distribution|uniform]] ''U''(−1, 1)
|-
| ''λ'' = 2
| exactly [[continuous uniform distribution|uniform]] ''U''(−½, ½)
|}

The most common use of this distribution is to generate a Tukey lambda [[PPCC plot]] of a [[data set]]. Based on the PPCC plot, an appropriate [[statistical model|model]] for the data is suggested. For example, if the maximum [[correlation]] occurs for a value of ''λ'' at or near 0.14, then the data can be modeled with a normal distribution. Values of ''λ'' less than this imply a heavy-tailed distribution (with −1 approximating a Cauchy). That is, as the optimal value of lambda goes from 0.14 to −1, increasingly heavy tails are implied. Similarly, as the optimal value of ''λ'' becomes greater than 0.14, shorter tails are implied.

Since the Tukey lambda distribution is a [[reflection symmetry|symmetric]] distribution, the use of the Tukey lambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributions. A [[histogram]] of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.<ref>{{Citation| title=Some Properties of the Range in Samples from Tukey's Symmetric Lambda Distributions| first1=Brian L. |last1=Joiner|first2=Joan R. |last2=Rosenblatt| journal=Journal of the American Statistical Association| volume=66 |issue=334 |year=1971| pages=394–399| doi=10.2307/2283943| jstor=2283943}}</ref>

==References==
{{Reflist}}

==External links==
*[http://www.itl.nist.gov/div898/handbook/eda/section3/eda366f.htm Tukey-Lambda Distribution]

{{NIST-PD}}
{{ProbDistributions|continuous-variable}}

[[Category:Continuous distributions]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Probability distributions]]

Knaster–Kuratowski fan

2013-09-26T09:27:33Z

89.70.126.226:

In [[functional analysis]], a branch of [[mathematics]], the '''Favard operators''' are defined by:

:<math>[\mathcal{F}_n(f)](x) = \frac{\sqrt{n}}{n\sqrt{c\pi}} \sum_{k=-\infty}^\infty {\exp{\left({\frac{-n}{c} {\left({\frac{k}{n}-x}\right)}^2 }\right)} f\left(\frac{k}{n}\right)}</math>

where <math>x\in\mathbb{R}</math>, <math>n\in\mathbb{N}</math>, and <math>c\in\mathbb{R^{+}}</math>.<ref name="Nowak">{{cite journal| last=Nowak | first=Grzegorz | coauthors=Aneta Sikorska-Nowak | date=14 November 2007| title=On the generalized Favard–Kantorovich and Favard–Durrmeyer operators in exponential function spaces | journal=Journal of Inequalities and Applications | volume=2007 | url=http://www.hindawi.com/journals/jia/raa.75142.html | doi=10.1155/2007/75142 | pages=1| format={{dead link|date=June 2010}}}}</ref> They are named after [[Jean Favard]].

==Generalizations==
A common generalization is:
:<math>[\mathcal{F}_n(f)](x) = \frac{1}{n\gamma_n\sqrt{2\pi}} \sum_{k=-\infty}^\infty {\exp{\left({\frac{-1}{2\gamma_n^2} {\left({\frac{k}{n}-x}\right)}^2 }\right)} f\left(\frac{k}{n}\right)}</math>

where <math>(\gamma_n)_{n=1}^\infty</math> is a positive sequence that [[Limit of a sequence|converges]] to 0.<ref name="Nowak"/> This reduces to the classical Favard operators when <math>\gamma_n^2=c/(2n)</math>.

==References==
*{{cite journal| last=Favard | first=Jean | authorlink=Jean Favard | year=1944 | title=Sur les multiplicateurs d'interpolation | journal=Journal de Mathematiques Pures et Appliquees | volume=23 | issue=9 | pages=219–247}} {{fr icon}} This paper also discussed [[Szász–Mirakyan operator]]s, which is why Favard is sometimes credited with their development (e.g. Favard–Szász operators).

===Footnotes===
<references/>

{{DEFAULTSORT:Favard Operator}}
[[Category:Approximation theory]]

{{mathanalysis-stub}}

Proxmap sort

2012-07-31T17:49:19Z

89.70.157.68: /* Pseudocode */ add indentation

The tear drop rear of the knife is perfect for nestling into your palm, sitting slightly below the muscle that controls your thumb. With no sharp or point edge, you get loads of again up from the form of your hand, particularly in piercing cuts where grip is extra essential. In slicing and roll cuts the nice [http://www.thebestpocketknifereviews.com/ best pocket knife Reviews] rounded bolster helps out as nicely. When a knife sample is that this previous, you need to figure that it stuck around for a reason. The nice grip is one purpose why the Barlow is still around. This class describes use around the house, the occasional tenting trip, or an occasional carry knife. That is essentially the most versatile class, with the widest vary of knives to fulfill this kind of use. You don’t have to fret a lot about things like weight or dimension, as a result of you will solely carry it as you want it. I discovered the Case XX Again Pocket knife to be a nice choice for everyday carry. It does not lock open, however there are occasions when I desire a conventional pocket folder over more fashionable designs. This is a great EDC knife. It seems to be good, is made up of quality supplies, and is crafted with sensible purposeful features. It is dear sufficient that Spyderco can afford to invest in making an enduring product, however it's not so costly that I'm afraid to make use of the factor. Private Creations is a one-stop shop for unique and personalized [http://abcnews.go.com/blogs/politics/2013/03/tsa-to-allow-pocket-knives-on-planes/ flick knives] gifts, from simple to crazy to sentimental. Personal Creations affords birthday gifts, wedding ceremony gifts and anniversary items for celebrating those important, romantic days. These merchandise are customizable, to present a considerate present that comes straight from the guts. I acquired a customized pocket knife and seashore towel for this assessment. Whats up each body when you find out. Victorinox Swiss Military Traditional SD Pocket Knife (Blue),And want to purchase a superb one you meet it in here “Victorinox Swiss Army Basic SD Pocket Knife (Blue)”, We've good price for merchandise, In Our Site You possibly can read reviews from actual prospects and You possibly [http://www.foxnews.com/us/2013/09/18/student-suspended-for-10-days-for-accidentally-bringing-pocket-knife-to/ how to make knives] can examine Costs Comparision before you purchase with simple you won’t miss it. Victorinox Swiss Army Traditional SD Pocket Knife (Blue) Characteristic Here are 5 folding knives which can be the very best in the market. Every is considered to be the best folding knife within the business by knife aficionados and blade fanatics in its own respect. The Native isn't one thing I take advantage of as a beater knife. I like to hold it, however even for medium obligation tasks I am going to normally grab the Delica. For heavy responsibility tasks I will grab a heavier folder just like the Tenacious and at some point a set blade knife. The Native is extra likely to be used to filet a bit of chicken for the grill than break down a storage filled with packing containers. At simply 1.eight ounces, it’s lighter than it seems, and it carries like a much smaller knife. Although it is too small for serious slicing jobs, the Shoki has worked fantastic as an everyday blade. One of the first stuff you’ll realize whenever you maintain the Sibert is that it is heavy! Weighing 12 ounces, it's a tank and, given its military design, is becoming. This knife is for tremendous heavy obligation users and military personnel solely. The weight makes it too heavy to be an EDC in my ebook. The Delica4, the fourth installment of the Delica collection, is one in every of Spyderco’s most popular and finest selling knife. The knife is ultra light weight at 2.5oz and ideal for EDC. Other, more heavy responsibility knives, can handle harder tasks however the Delica4′s VG-10 steel allows it to [http://Www.Malgefragt.net/mwiki/index.php?title=Best_Pocket_Knife withstand harsh] work if needed. If you need a nice bigger folding knife that is designed for a bit extra rugged use try my review of the Gerber Bear Grylls Scout knife or the Gerber Bear Grylls Folding Sheath knife and the Schrade First Response folder is another great knife to consider. For a smaller fastened blade knife, test our Gerber Bear Grylls Compact Fastened blade assessment and the other other knife opinions under the Video tab of our YouTube Channel or search this web site. To your convenience I’ve included hyperlinks to the Gerber Bear Grylls Compact Scout knife and the entire other knives and instruments that I’ve talked about, all through this text. Typically referred to as a toolbox on your pocket, the 3-half of in. SwissChamp packs a whopping 33 implements and is the daddy of Swiss Military Knives. Despite it is immense variety of implements, the Swiss Champ still suits in your pocket to be used anyplace from the nice open air to workbench at residence. The implements are constructed of sturdy stainless steel, carry a lifetime guarantee and are Swiss-Made, which means they've been constructed to last. They have a chemical inertness to acids and alkalis, which make them excellent for slicing fruits and vegetables, like tomatoes and citrus fruits, which have a excessive stage of acidity. The Gerber Fundamental is a rock stable fastened blade advantageous edge knife that feels and appears simple, however it is fairly advanced with the entire included options. The Basic incorporates a three.forty inch fantastic [http://www.thebestpocketknifereviews.com/ top Ten best pocket knives] edge blade with a serrated edge that takes up just lower than half of the knife blade. The knife size is about 6.60 inches lengthy and 6.80 inches long with the sheath included. The deal with of the knife has a groove for the index finger and options smaller grooves for the remainder of your fingers. The Gerber Basic is extraordinarily sturdy, helpful, and accessible.