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| In [[probability theory]] and [[directional statistics]], a '''wrapped probability distribution''' is a continuous [[probability distribution]] that describes data points that lie on a unit [[n-sphere|''n''-sphere]]. In one dimension, a wrapped distribution will consist of points on the [[unit circle]].
| | Regardless that NYC pizza is widespread, you're not prone to discover individuals folding up slices before they take a chew exterior of the Big Apple. While 38% opt for the fold methodology, 43% eat it flat (and using a fork and a knife continues to be a no-no, with only 19% [http://sfinktah.trac.cvsdude.com/vhddirector/ticket/154572 admitting] to [http://myfirstclasslife.com/members/darbygddm/activity/756281/ slicing] their slice before [http://luansantana.trechosertanejo.com.br/members/vernmaxted/activity/369212/ consuming] it). Nearly all of folks like to add a bit of kick to their pie, with fifty one% saying they throw on some pepper flakes, however most diners are [http://doc.froza.ru/index.php/Cheap_Automatic_Knives_For_Sale completely content] with a bit of grease, with solely 30% saying they like to blot the surplus oil off before chowing down.<br><br><br><br>Worldwide - if you can inform me the regulation concerning sending a knife and you might be keen to pay value for shipping, then contact me. You may additionally be liable to import tax your finish. The design criteria was for a practical anddependable customized made survival knife with a 4"(100mm) blade remaininglight and slim enough to be carried unobtrusively on the belt or tucked into acargo or rucksack pocket. Compact however capable. The design has been confirmed andtested to good effect over many seasons in some of the world's mostinhospitable environments. RWL powdered metallurgy steel. G10 handle. Fullslotted tang. Leather sheath. ------------------------- Gerber Prodigy<br><br>Chilly Metal Knives make the worlds strongest folding knives and pocket knives on the planet. Our Tri-Advert® Lock is untouchable relating to lock power. Whether or not you are navy, police, SWAT, regulation enforcement, or an outdoor enthusiast, Chilly Steel folder are the way to go in the event you value your fingers. The Black and Decker workmate is a marvel of a workbench for dwelling use as well as skilled. A folding workbench that is light-weight sufficient to hold with you and with a built in vise - what may very well be higher? WASHINGTON — Airline passengers will have to leave their knives at house in any case. And their bats and golf clubs.<br><br>The Tak Ri rides in a simple however efficient ballistic nylon sheath which contains a thermoplastic insert to safe the wide blade. Two hook and loop straps secure the throat & handle, & general the knife rides comfortably & fairly quietly on the belt. My most well-liked belts are 5.eleven TDU belts, due [http://www.thebestpocketknifereviews.com/best-automatic-knife-discover-the-pefect-model/ automatic pocket knife] to their relative rigidity, & means to tote sheath knives & multi instruments without stretching or sagging an excessive amount of. The scabbard is fully army-webbing compatible, & nicely-made, with loads of outer grommets for securing the package deal with the included size of paracord.<br><br>If you happen to aren't aware of the Kershaw SpeedSafe system I'll cowl it for you actual fast. The Kershaw SpeedSafe is a patented guide assisted-opening system; as you start to open the blade by the flipper or thumb studs the SpeedSafe torsion bar takes over to fully open the knife blade. It is tremendous fast and clean when deploying the blade. Another consideration is the way in which that you just want to carry it. There are a number of options including scout carry, belt carry, drop leg, MOLLE, or a tactical leg strap. In my opinion smaller knives are great to carry scout, however lager knives are nice candidates for a drop leg option.<br><br>The locking mechanism and the handle are as important because the blade for a folding knife. The blade needs to be simply deployable, with the flick of a wrist or a simple flip of your hand. The locking mechanism ensures that your blade stays in place when it's in use. Weak locking will lead to the blade folding again in on itself when it is getting used and should lead [http://abcnews.go.com/blogs/politics/2013/03/tsa-to-allow-pocket-knives-on-planes/ small knife] to harm. The Kershaw Leek has been among the best selling pocket knives in the market for quite some time. The 3” blade is crafted from Sandvik stainless steel and rests in a sophisticated stainless-steel deal with.<br><br>The blade of the Recon 1 is made of Japanese AUS 8A stainless steel. It was heat handled in a vacuum, then submersed in sub-zero temperatures, to get a hard, sharp blade. It's then coated with black Teflon. This keeps the blade from rusting, and keeps it from being reflective. The Teflon also makes the blade very fast, in a position to slide by way [http://www.thebestpocketknifereviews.com/best-automatic-knife-discover-the-pefect-model/ out the front automatic knife] of the toughest of supplies with little friction. Mainly, G10 is a time period for a high grade of laminate. To make this materials, producers weave glass material into epoxy resin. This process creates a robust materials with excessive affect resistance in addition to dimensional stability. |
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| Any [[probability density function]] (pdf) <math>p(\phi)</math> on the line can be "wrapped" around the circumference of a circle of unit radius.<ref name="Mardia99">{{cite book |title=Directional Statistics |last=Mardia |first=Kantilal |authorlink=Kantilal Mardia |coauthors=Jupp, Peter E. |year=1999|publisher=Wiley |location= |isbn=978-0-471-95333-3 |url=http://www.amazon.com/Directional-Statistics-Kanti-V-Mardia/dp/0471953334/ref=sr_1_1?s=books&ie=UTF8&qid=1311003484&sr=1-1#reader_0471953334 |accessdate=2011-07-19}}</ref> That is, the pdf of the wrapped variable
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| :<math>\theta=\phi \mod 2\pi</math> in some interval of length <math>2\pi</math> | |
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| is
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| : <math>
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| p_w(\theta)=\sum_{k=-\infty}^\infty {p(\theta+2\pi k)}.
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| </math>
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| which is a [[periodic summation|periodic sum]] of period <math>2\pi</math>. The preferred interval is generally <math>(-\pi<\theta\le\pi)</math> for which <math>\ln(e^{i\theta})=\arg(e^{i\theta})=\theta</math>
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| ==Theory==
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| In most situations, a process involving [[circular statistics]] produces angles (<math>\phi</math>) which lie in the interval from negative infinity to positive infinity, and are described by an "unwrapped" probability density function <math>p(\phi)</math>. However, a measurement will yield a "measured" angle <math>\theta</math> which lies in some interval of length <math>2\pi</math> (for example <math>[0,2\pi)</math>). In other words, a measurement cannot tell if the "true" angle <math>\phi</math> has been measured or whether a "wrapped" angle <math>\phi+2\pi a</math> has been measured where ''a'' is some unknown integer. That is:
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| :<math>\theta=\phi+2\pi a.</math>
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| If we wish to calculate the expected value of some function of the measured angle it will be:
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| :<math>\langle f(\theta)\rangle=\int_{-\infty}^\infty p(\phi)f(\phi+2\pi a)d\phi.</math> | |
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| We can express the integral as a sum of integrals over periods of <math>2\pi</math> (e.g. 0 to <math>2\pi</math>):
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| :<math>\langle f(\theta)\rangle=\sum_{k=-\infty}^\infty \int_{2\pi k}^{2\pi(k+1)} p(\phi)f(\phi+2\pi a)d\phi.</math>
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| Changing the variable of integration to <math>\theta'=\phi-2\pi k</math> and exchanging the order of integration and summation, we have
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| :<math>\langle f(\theta)\rangle= \int_0^{2\pi} p_w(\theta')f(\theta'+2\pi a')d\theta'</math>
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| where <math>p_w(\theta')</math> is the pdf of the "wrapped" distribution and ''a' '' is another unknown integer (a'=a+k). It can be seen that the unknown integer ''a' '' introduces an ambiguity into the expectation value of <math>f(\theta)</math>. A particular instance of this problem is encountered when attempting to take the [[Mean of circular quantities|mean of a set of measured angles]]. If, instead of the measured angles, we introduce the parameter <math>z=e^{i\theta}</math> it is seen that ''z'' has an unambiguous relationship to the "true" angle <math>\phi</math> since:
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| :<math>z=e^{i\theta}=e^{i\phi}.</math>
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| Calculating the expectation value of a function of ''z'' will yield unambiguous answers:
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| :<math>\langle f(z)\rangle= \int_0^{2\pi} p_w(\theta')f(e^{i\theta'})d\theta'</math>
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| and it is for this reason that the ''z'' parameter is the preferred statistical variable to use in circular statistical analysis rather than the measured angles <math>\theta</math>. This suggests, and it is shown below, that the wrapped distribution function may itself be expressed as a function of ''z'' so that: | |
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| :<math>\langle f(z)\rangle= \oint p_w(z)f(z)\,dz</math>
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| where <math>p_w(z)</math> is defined such that <math>p_w(\theta)\,d\theta=p_w(z)\,dz</math>. This concept can be extended to the multivariate context by an extension of the simple sum to a number of <math>F</math> sums that cover all dimensions in the feature space:
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| : <math>
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| p_w(\vec\theta)=\sum_{k_1=-\infty}^{\infty}{p(\vec\theta+2\pi k_1\mathbf{e}_1+\dots+2\pi k_F\mathbf{e}_F)}
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| </math> | |
| where <math>\mathbf{e}_k=(0,\dots,0,1,0,\dots,0)^{\mathsf{T}}</math> is the <math>k</math>th Euclidean basis vector.
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| ==Expression in terms of characteristic functions==
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| A fundamental wrapped distribution is the [[Dirac comb]] which is a wrapped delta function:
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| :<math>\Delta_{2\pi}(\theta)=\sum_{k=-\infty}^{\infty}{\delta(\theta+2\pi k)}.</math>
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| Using the delta function, a general wrapped distribution can be written
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| :<math>p_w(\theta)=\sum_{k= -\infty}^{\infty}\int_{-\infty}^\infty p(\theta')\delta(\theta-\theta'+2\pi k)\,d\theta'.</math>
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| Exchanging the order of summation and integration, any wrapped distribution can be written as the convolution of the "unwrapped" distribution and a Dirac comb:
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| :<math>p_w(\theta)=\int_{-\infty}^\infty p(\theta')\Delta_{2\pi}(\theta-\theta')\,d\theta'.</math>
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| The Dirac comb may also be expressed as a sum of exponentials, so we may write:
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| :<math>p_w(\theta)=\frac{1}{2\pi}\,\int_{-\infty}^\infty p(\theta')\sum_{n=-\infty}^{\infty}e^{in(\theta-\theta')}\,d\theta'</math>
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| again exchanging the order of summation and integration,
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| :<math>p_w(\theta)=\frac{1}{2\pi}\,\sum_{n=-\infty}^{\infty}\int_{-\infty}^\infty p(\theta')e^{in(\theta-\theta')}\,d\theta'</math>
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| using the definition of <math>\phi(s)</math>, the [[Characteristic function (probability theory)|characteristic function]] of <math>p(\theta)</math> yields a [[Laurent series]] about zero for the wrapped distribution in terms of the characteristic function of the unwrapped distribution:<ref name="Mardia72">{{Cite book|title=Statistics of Directional Data |last=Mardia |first=K. |year=1972 |publisher=Academic press |location=New York}}
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| </ref>
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| :<math>p_w(\theta)=\frac{1}{2\pi}\,\sum_{n=-\infty}^{\infty} \phi(-n)\,e^{in\theta} = \frac{1}{2\pi}\,\sum_{n=-\infty}^{\infty} \phi(-n)\,z^n </math>
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| or
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| :<math>p_w(z)=\frac{1}{2\pi i}\,\sum_{n=-\infty}^{\infty} \phi(-n)\,z^{n-1}. </math>
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| By analogy with linear distributions, the <math>\phi(m)</math> are referred to as the characteristic function of the wrapped distribution<ref name="Mardia72"/> (or perhaps more accurately, the characteristic [[sequence]]). This is an instance of the [[Poisson summation formula]] and it can be seen that the Fourier coefficients of the Fourier series for the wrapped distribution are just the Fourier coefficients of the Fourier transform of the unwrapped distribution at integer values.
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| ==Moments==
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| The moments of the wrapped distribution <math>p_w(z)</math> are defined as:
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| :<math>
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| \langle z^m \rangle = \oint p_w(z)z^m \, dz.
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| </math> | |
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| Expressing <math>p_w(z)</math> in terms of the characteristic function and exchanging the order of integration and summation yields:
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| :<math>
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| \langle z^m \rangle = \frac{1}{2\pi i}\sum_{n=-\infty}^\infty \phi(-n)\oint z^{m+n-1}\,dz.
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| </math>
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| From the [[Residue (complex analysis)|theory of residues]] we have
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| :<math> | |
| \oint z^{m+n-1}\,dz = 2\pi i \delta_{m+n}
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| </math>
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| where <math>\delta_k</math> is the [[Kronecker delta]] function. It follows that the moments are simply equal to the characteristic function of the unwrapped distribution for integer arguments:
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| :<math>
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| \langle z^m \rangle = \phi(m).
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| </math>
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| == Entropy ==
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| The [[Entropy (information theory)|information entropy]] of a circular distribution with probability density <math>f_w(\theta)</math> is defined as:<ref name="Mardia99"/>
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| :<math>H = -\int_\Gamma f_w(\theta)\,\ln(f_w(\theta))\,d\theta</math>
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| where <math>\Gamma</math> is any interval of length <math>2\pi</math>. If both the probability density and its logarithm can be expressed as a [[Fourier series]] (or more generally, any [[integral transform]] on the circle) then the orthogonality property may be used to obtain a series representation for the entropy which may reduce to a [[closed form expression|closed form]].
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| The moments of the distribution <math>\phi(n)</math> are the Fourier coefficents for the Fourier series expansion of the probability density: | |
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| :<math>f_w(\theta)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty \phi_n e^{-in\theta}</math>
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| If the logarithm of the probability density can also be expressed as a Fourier series:
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| :<math>\ln(f_w(\theta))=\sum_{m=-\infty}^\infty c_m e^{im\theta}</math>
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| where
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| :<math>c_m=\frac{1}{2\pi}\int_\Gamma \ln(f_w(\theta))e^{-i m \theta}\,d\theta</math>
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| Then, exchanging the order of integration and summation, the entropy may be written as:
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| :<math>H=-\frac{1}{2\pi}\sum_{m=-\infty}^\infty\sum_{n=-\infty}^\infty c_m \phi_n \int_\Gamma e^{i(m-n)\theta}\,d\theta</math>
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| Using the orthogonality of the Fourier basis, the integral may be reduced to:
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| :<math>H=-\sum_{n=-\infty}^\infty c_n \phi_n</math>
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| For the particular case when the probability density is symmetric about the mean, <math>c_{-m}=c_m</math> and the logarithm may be written:
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| :<math>\ln(f_w(\theta))= c_0 + 2\sum_{m=1}^\infty c_m \cos(m\theta)</math>
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| and
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| :<math>c_m=\frac{1}{2\pi}\int_\Gamma \ln(f_w(\theta))\cos(m\theta)\,d\theta</math>
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| and, since normalization requires that <math>\phi_0=1</math>, the entropy may be written:
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| :<math>H=-c_0-2\sum_{n=1}^\infty c_n \phi_n</math>
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| ==See also==
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| * [[Wrapped normal distribution]]
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| * [[Wrapped Cauchy distribution]]
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| {{More footnotes|date=July 2011}}
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| ==References==
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| <references/>
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| * {{Cite book|title=Statistics of Earth Science Data |last=Borradaile |first=Graham |year=2003 |publisher=Springer |isbn=978-3-540-43603-4 |url=http://books.google.com/books?id=R3GpDglVOSEC&printsec=frontcover&source=gbs_navlinks_s#v=onepage&q=&f=false |accessdate=31 December 2009}}
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| * {{Cite book|title=Statistical Analysis of Circular Data |last=Fisher |first=N. I. |year=1996 |publisher=Cambridge University Press |location= |isbn=978-0-521-56890-6 |url=http://books.google.com/books?id=IIpeevaNH88C&dq=%22circular+variance%22+fisher&source=gbs_navlinks_s |accessdate=2010-02-09}}
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| ==External links==
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| * [http://www.codeproject.com/Articles/190833/Circular-Values-Math-and-Statistics-with-Cplusplus Circular Values Math and Statistics with C++11], A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics
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| {{ProbDistributions|directional}}
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Wrapped Distribution}}
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| [[Category:Types of probability distributions]]
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| [[Category:Directional statistics]]
| |
Regardless that NYC pizza is widespread, you're not prone to discover individuals folding up slices before they take a chew exterior of the Big Apple. While 38% opt for the fold methodology, 43% eat it flat (and using a fork and a knife continues to be a no-no, with only 19% admitting to slicing their slice before consuming it). Nearly all of folks like to add a bit of kick to their pie, with fifty one% saying they throw on some pepper flakes, however most diners are completely content with a bit of grease, with solely 30% saying they like to blot the surplus oil off before chowing down.
Worldwide - if you can inform me the regulation concerning sending a knife and you might be keen to pay value for shipping, then contact me. You may additionally be liable to import tax your finish. The design criteria was for a practical anddependable customized made survival knife with a 4"(100mm) blade remaininglight and slim enough to be carried unobtrusively on the belt or tucked into acargo or rucksack pocket. Compact however capable. The design has been confirmed andtested to good effect over many seasons in some of the world's mostinhospitable environments. RWL powdered metallurgy steel. G10 handle. Fullslotted tang. Leather sheath. ------------------------- Gerber Prodigy
Chilly Metal Knives make the worlds strongest folding knives and pocket knives on the planet. Our Tri-Advert® Lock is untouchable relating to lock power. Whether or not you are navy, police, SWAT, regulation enforcement, or an outdoor enthusiast, Chilly Steel folder are the way to go in the event you value your fingers. The Black and Decker workmate is a marvel of a workbench for dwelling use as well as skilled. A folding workbench that is light-weight sufficient to hold with you and with a built in vise - what may very well be higher? WASHINGTON — Airline passengers will have to leave their knives at house in any case. And their bats and golf clubs.
The Tak Ri rides in a simple however efficient ballistic nylon sheath which contains a thermoplastic insert to safe the wide blade. Two hook and loop straps secure the throat & handle, & general the knife rides comfortably & fairly quietly on the belt. My most well-liked belts are 5.eleven TDU belts, due automatic pocket knife to their relative rigidity, & means to tote sheath knives & multi instruments without stretching or sagging an excessive amount of. The scabbard is fully army-webbing compatible, & nicely-made, with loads of outer grommets for securing the package deal with the included size of paracord.
If you happen to aren't aware of the Kershaw SpeedSafe system I'll cowl it for you actual fast. The Kershaw SpeedSafe is a patented guide assisted-opening system; as you start to open the blade by the flipper or thumb studs the SpeedSafe torsion bar takes over to fully open the knife blade. It is tremendous fast and clean when deploying the blade. Another consideration is the way in which that you just want to carry it. There are a number of options including scout carry, belt carry, drop leg, MOLLE, or a tactical leg strap. In my opinion smaller knives are great to carry scout, however lager knives are nice candidates for a drop leg option.
The locking mechanism and the handle are as important because the blade for a folding knife. The blade needs to be simply deployable, with the flick of a wrist or a simple flip of your hand. The locking mechanism ensures that your blade stays in place when it's in use. Weak locking will lead to the blade folding again in on itself when it is getting used and should lead small knife to harm. The Kershaw Leek has been among the best selling pocket knives in the market for quite some time. The 3” blade is crafted from Sandvik stainless steel and rests in a sophisticated stainless-steel deal with.
The blade of the Recon 1 is made of Japanese AUS 8A stainless steel. It was heat handled in a vacuum, then submersed in sub-zero temperatures, to get a hard, sharp blade. It's then coated with black Teflon. This keeps the blade from rusting, and keeps it from being reflective. The Teflon also makes the blade very fast, in a position to slide by way out the front automatic knife of the toughest of supplies with little friction. Mainly, G10 is a time period for a high grade of laminate. To make this materials, producers weave glass material into epoxy resin. This process creates a robust materials with excessive affect resistance in addition to dimensional stability.