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Comparison sort

2014-02-19T16:42:02Z

117.194.200.187: there are references in the article

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Template:Violence against Hindus

2014-02-13T16:01:16Z

117.194.196.170: unsourced political clash have been marked as violence against Hindus?

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1964 PRL symmetry breaking papers

2014-02-03T15:17:28Z

117.194.82.26:

In [[statistics]], '''Samuelson's inequality''', named after the economist [[Paul Samuelson]],<ref>Paul Samuelson, "How Deviant Can You Be?", ''[[Journal of the American Statistical Association]]'', volume 63, number 324 (December, 1968), pp. 1522–1525 {{JSTOR|2285901}}</ref> also called the '''Laguerre–Samuelson inequality''',<ref>Jensen, Shane Tyler (1999) [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape10/PQDD_0027/MQ50799.pdf The Laguerre–Samuelson Inequality with Extensions and Applications in Statistics and Matrix Theory] MSc Thesis. Department of Mathematics and Statistics, [[McGill University]].</ref> after the mathematician [[Edmond Laguerre]], proved that every one of any collection ''x''1, ..., ''x''''n'', is within √(''n'' − 1) sample [[standard deviation]]s of their sample mean. In other words, if we let

: <math> \overline{x} = \frac{x_1+\cdots+x_n}{n} </math>

be the sample [[mean]] and

: <math> s = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^2 } </math>

be the standard deviation of the sample, then

: <math> \overline{x} - s\sqrt{n-1} \le x_i \le \overline{x} + s\sqrt{n-1}\qquad \text{for }i = 1,\dots,n. </math><ref>''Advances in Inequalities from Probability Theory and Statistics'', by Neil S. Barnett and Sever Silvestru Dragomir, Nova Publishers, 2008, page 164</ref>

Equality holds on the left [[if and only if]] the ''n'' − 1 smallest of the ''n'' numbers are equal to each other, and on the right [[iff]] the ''n'' − 1 largest ones are equal.

Samuelson's inequality may be considered a reason why [[studentized residuals|studentization of residuals]] should be done externally.

==Relationship to polynomials==

Samuelson was not the first to describe this relationship. The first to discover this relationship was probably [[Laguerre]] in 1880 while investigating the [[Equation solving|root]]s (zeros) of [[polynomial]]s.<ref name=Jensen1999>Jensen, Shane Tyler (1999) [http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape10/PQDD_0027/MQ50799.pdf The Laguerre–Samuelson Inequality with Extensions and Applications in Statistics and Matrix Theory] MSc Thesis. Department of Mathematics and Statistics, [[McGill University]]</ref><ref name=Laguerre1880>Laguerre E. (1880) Mémoire pour obtenir par approximation les racines d'une équation algébrique qui a toutes les racines réelles. Nouv Ann Math 2e série, 19, 161-172, 193-202</ref>

Consider a polynomial

: <math> a_0x^n + a_1x^{n-1} + \ldots + a_{n-1}x + a_n = 0 </math>

Without loss of generality let <math>a_0 = 1</math> and let

: <math> t_1 = \sum x_i </math> and <math> t_2 = \sum x_i^2 </math>

Then

: <math> a_1 = - \sum x_i = -t_1 </math>

and

: <math> a_2 = \sum x_ix_j = \frac{t_1^2 - t_2}{2} \qquad \text{ where } i < j </math>

In terms of the coefficients

: <math> t_2 = a_1^2 - 2a_2 </math>

Laguerre showed that the roots of this polynomial were bounded by

: <math> -a_1 / n \pm b \sqrt{n - 1} </math>

where

: <math> b = \frac{\sqrt{nt_2 - t_1}}{n} = \frac{\sqrt{na_1^2 + a_1 - 2na_2}}{n} </math>

Inspection shows that <math>-\tfrac{a_1}{n}</math> is the [[mean]] of the roots and that ''b'' is the standard deviation of the roots.

Laguerre failed to notice this relationship with the means and standard deviations of the roots being more interested in the bounds themselves. This relationship permits a rapid estimate of the bounds of the roots and may be of use in their location.

===Note===

When the coefficients <math> a_1 </math> and <math> a_2 </math> are both zero no information can be obtained about the location of the roots.

== References ==
{{reflist}}

[[Category:Statistical inequalities]]

On Physical Lines of Force

2014-01-29T15:32:34Z

117.194.89.46:

In [[complex analysis]] a branch of mathematics, the '''residue at infinity''' is a [[Residue (complex analysis)|residue]] of a [[holomorphic function]] on an [[Annulus (mathematics)|annulus]] having an infinite external radius. The ''infinity'' <math>\infty</math> is a point added to the local space <math>\mathbb C </math> in order to render it [[compact space|compact]] (in this case it is a [[Alexandroff extension|one-point compactification]]). This space noted <math> \hat{\mathbb C} </math> is [[isomorphism|isomorphic]] to the [[Riemann sphere]].<ref>Michèle AUDIN, ''Analyse Complexe'', cursus notes of the university of Strasbourg [http://www-irma.u-strasbg.fr/~maudin/analysecomp.pdf available on the web], pp. 70–72</ref> One can use the residue at infinity to calculate some [[integral]]s.

==Definition==
Given a holomorphic function ''f'' on an [[Annulus (mathematics)|annulus]] <math> A(0, R, \infty) </math> (centered at 0, with inner radius <math>R</math> and infinite outer radius), the '''residue at infinity''' of the function ''f'' can be defined in terms of the usual [[residue (mathematics)|residue]] as follows:

:<math> \mathrm{Res}(f,\infty) = \mathrm{Res}\left( {-1\over z^2}f\left({1\over z}\right), 0 \right)</math>

Thus, one can transfer the study of <math> f(z) </math> at infinity to the study of <math> f(1/z) </math> at the origin.

Note that <math>\forall r > R</math>, we have

:<math> \mathrm{Res}(f, \infty) = {-1\over 2\pi i}\int_{C(0, r)} f(z) \, dz</math>

== See also ==
* [[Riemann sphere]]
* [[Algebraic variety]]
* [[Residue theorem]]

== References ==
{{Translation/Ref|fr|Résidu à l'infini|oldid=59523358}}
<references/>
* Murray R. Spiegel, ''Variables complexes'', Schaum, ISBN 2-7042-0020-3
* [[Henri Cartan]], ''Théorie analytique des fonctions d'une ou plusieurs varaiables complexes'', Hermann, 1961

[[Category:Complex analysis]]

Charge-coupled device

2014-01-28T10:58:17Z

117.194.86.107: /* History */

[[Image:CCD.jpg|thumb|A specially developed CCD used for [[ultraviolet]] imaging in a wire bonded package]]

A '''charge-coupled device''' ('''CCD''') is a device for the movement of electrical charge, usually from within the device to an area where the charge can be manipulated, for example conversion into a digital value. This is achieved by "shifting" the signals between stages within the device one at a time. CCDs move charge between capacitive ''bins'' in the device, with the shift allowing for the transfer of charge between bins.

The CCD is a major piece of technology in [[digital imaging]]. In a CCD [[image sensor]], [[pixel]]s are represented by [[Doping (Semiconductors)|p-doped]] [[MOSFET#Metal–oxide–semiconductor structure|MOS]] capacitors. These capacitors are biased above the threshold for inversion when image acquisition begins, allowing the conversion of incoming [[photon]]s into electron charges at the semiconductor-oxide interface; the CCD is then used to read out these charges. Although CCDs are not the only technology to allow for light detection, CCD image sensors are widely used in professional, medical, and scientific applications where high-quality image data is required. In applications with less exacting quality demands, such as consumer and professional [[digital camera]]s, [[active pixel sensor]]s (CMOS) are generally used; the large quality advantage CCDs enjoyed early on has narrowed over time.

==History==
[[File:Nobel Prize 2009-Press Conference KVA-19.jpg|thumb|240px|George E. Smith and Willard Boyle, 2009]]
The charge-coupled device was invented in 1969 at [[American Telephone & Telegraph|AT&T]] [[Bell Labs]] by [[Willard Boyle]] and [[George E. Smith]].<ref name="patent">See US3792322 [http://www.google.com./patents/US3792322] and US3796927 [http://www.google.com/patents/US3796927]</ref>
The lab was working on [[semiconductor]] [[bubble memory]] when Boyle and Smith conceived of the design of what they termed, in their notebook, "Charge 'Bubble' Devices".<ref>
{{Cite book
| title = Scientific charge-coupled devices
| author = James R. Janesick
| publisher = SPIE Press
| year = 2001
| isbn = 978-0-8194-3698-6
| page = 4
| url = http://books.google.com/?id=3GyE4SWytn4C&pg=PA3&dq=charge+bubble+device+boyle+smith&q=charge%20bubble%20device%20boyle%20smith
}}</ref>
A description of how the device could be used as a [[shift register]]. The essence of the design was the ability to transfer charge along the surface of a semiconductor from one storage capacitor to the next. The concept was similar in principle to the [[bucket-brigade device]] (BBD), which was developed at Philips Research Labs during the late 1960s. The first patent (4,085,456) on the application of CCDs to imaging was assigned to [[Michael F. Tompsett|Michael Tompsett]].<ref>{{US Patent|4,085,456}}</ref>

The initial paper describing the concept<ref>
{{Cite journal
| journal = Bell Sys. Tech. J.
| title = Charge Coupled Semiconductor Devices
| author = W. S. Boyle and G. E. Smith
| volume = 49| issue = 4
| pages = 587–593
| date = April 1970
| url =
}}</ref> listed possible uses as a memory, a delay line, and an imaging device. The first experimental device<ref>
{{Cite journal
| journal = Bell Sys. Tech. J.
| title = Experimental Verification of the Charge Coupled Device Concept
| author = G. F. Amelio, M. F. Tompsett, and G. E. Smith
| volume = 49| issue = 4
| pages = 593–600
| date = April 1970
| url =
}}</ref> demonstrating the principle was a row of closely spaced metal squares on an oxidized silicon surface electrically accessed by wire bonds.

The first working CCD made with integrated circuit technology was a simple 8-bit shift register.<ref>
{{Cite journal
| journal = Applied Physics Lettersfrom
| doi = 10.1063/1.1653327
| title = Charge Coupled 8-bit Shift Register
| author = M. F. Tompsett, G. F. Amelio, and G. E. Smith
| volume = 17
| pages = 111–115
| date = 1 August 1970
| url =
|bibcode = 1970ApPhL..17..111T }}</ref> This device had input and output circuits and was used to demonstrate its use as a shift register and as a crude eight [[pixel]] linear imaging device.
Development of the device progressed at a rapid rate. By 1971, Bell researchers lead by Michael Tompsett were able to capture images with simple linear devices.<ref>
{{Cite journal
| doi = 10.1109/T-ED.1971.17321
| journal = IEEE Transactions on Electron Devices
| title = Charge-coupled imaging devices: Experimental results
| author = Tompsett, M.F.; Amelio, G.F. ; Bertram, W.J., Jr.; Buckley, R.R.; McNamara, W.J.; Mikkelsen, J.C., Jr.; Sealer, D.A.
| volume = 18
| issue = 11
| publisher =
| pages = 992–996
| issn = 0018-9383
| date = November 1971
| url =
}}</ref>
Several companies, including [[Fairchild Semiconductor]], [[RCA]] and [[Texas Instruments]], picked up on the invention and began development programs. Fairchild's effort, led by ex-Bell researcher [[Gil Amelio]], was the first with commercial devices, and by 1974 had a linear 500-element device and a 2-D 100 x 100 pixel device. [[Steven Sasson]], an electrical engineer working for [[Kodak]], invented the first [[digital still camera]] using a Fairchild {{nowrap|100 x 100}} CCD in 1975.<ref name=ap>Dobbin, Ben. (2005-09-08) [http://www.seattlepi.com/business/article/Kodak-engineer-had-revolutionary-idea-the-first-1182624.php Kodak engineer had revolutionary idea: the first digital camera]. seattlepi.com. Retrieved on 2011-11-15.</ref> The first [[KH-11 KENNAN]] reconnaissance satellite equipped with charge-coupled device array ({{nowrap|800 x 800}} pixels)<ref>[http://www.globalsecurity.org/space/systems/kh-11.htm globalsecurity.org - KH-11 KENNAN], 2007-04-24</ref> technology for imaging was launched in December 1976.<ref>{{cite web|title=NRO review and redaction guide (2006 ed.)| url=http://www.fas.org/irp/nro/declass.pdf |publisher=National Reconnaissance Office}}</ref> Under the leadership of Kazuo Iwama, [[Sony]] also started a large development effort on CCDs involving a significant investment. Eventually, Sony managed to mass-produce CCDs for their [[camcorder]]s. Before this happened, Iwama died in August 1982; subsequently, a CCD chip was placed on his tombstone to acknowledge his contribution.<ref>{{Cite book|last=Johnstone |first=B. |title=We Were Burning: Japanese Entrepreneurs and the Forging of the Electronic Age |year=1999 |publisher=Basic Books |location=New York |isbn=0-465-09117-2 |postscript= }}</ref>

In January 2006, Boyle and Smith were awarded the [[National Academy of Engineering]] [[Charles Stark Draper Prize]],<ref>{{Cite web|url=http://www.nae.edu/NAE/awardscom.nsf/weblinks/CGOZ-6K9L6P?OpenDocument |title=Charles Stark Draper Award |postscript= }}</ref> and in 2009 they were awarded the [[Nobel Prize for Physics]],<ref>{{Cite web|url=http://nobelprize.org/nobel_prizes/physics/laureates/2009/ |title=Nobel Prize website |postscript= }}</ref> for their invention of the CCD concept.
Michael Tompsett was awarded the 2010 National Medal of Technology and Innovation for pioneering work and electronic technologies including the design and development of the first charge coupled device (CCD) imagers. He was also awarded the 2012 IEEE Edison Medal "For pioneering contributions to imaging devices including CCD Imagers, cameras and thermal imagers".

==Basics of operation==
[[File:CCD charge transfer animation.gif|thumb|250px|right|The charge packets (electrons, blue) are collected in potential wells (yellow) created by applying positive voltage at the gate electrodes (G). Applying positive voltage to the gate electrode in the correct sequence transfers the charge packets.]]

In a CCD for capturing images, there is a photoactive region (an [[epitaxy|epitaxial]] layer of silicon), and a transmission region made out of a [[shift register]] (the CCD, properly speaking).

An image is projected through a [[lens (optics)|lens]] onto the capacitor array (the photoactive region), causing each capacitor to accumulate an electric charge proportional to the [[light]] intensity at that location. A one-dimensional array, used in line-scan cameras, captures a single slice of the image, while a two-dimensional array, used in video and still cameras, captures a two-dimensional picture corresponding to the scene projected onto the focal plane of the sensor. Once the array has been exposed to the image, a control circuit causes each capacitor to transfer its contents to its neighbor (operating as a shift register). The last capacitor in the array dumps its charge into a [[charge amplifier]], which converts the charge into a [[voltage]]. By repeating this process, the controlling circuit converts the entire contents of the array in the semiconductor to a sequence of voltages. In a digital device, these voltages are then sampled, digitized, and usually stored in memory; in an analog device (such as an analog video camera), they are processed into a continuous analog signal (e.g. by feeding the output of the charge amplifier into a low-pass filter) which is then processed and fed out to other circuits for transmission, recording, or other processing.

[[Image:CCD line sensor.JPG|thumb|right|"One-dimensional" CCD [[image sensor]] from a [[fax machine]]]]

==Detailed physics of operation==

=== Charge generation ===
Before the MOS capacitors are exposed to light, they are [[biasing|biased]] into the depletion region; in n-channel CCDs, the silicon under the bias gate is slightly ''p''-doped or intrinsic. The gate is then biased at a positive potential, above the threshold for strong inversion, which will eventually result in the creation of a ''n'' channel below the gate as in a [[MOSFET]]. However, it takes time to reach this thermal equilibrium: up to hours in high-end scientific cameras cooled at low temperature.<ref>For instance, the specsheet of PI/Acton's [http://www.princetoninstruments.com/Uploads/Princeton/Documents/Datasheets/Princeton_Instruments_SPEC-10_2K_eXcelon_rev_N3_9.22.2011.pdf SPEC-10 camera] specifies a dark current of 0.3 electron per pixel per hour at -110 °C.</ref> Initially after biasing, the holes are pushed far into the substrate, and no mobile electrons are at or near the surface; the CCD thus operates in a non-equilibrium state called deep depletion.<ref name=sze>{{cite book
| last1 = Sze
| first1 = S. M.
| last2 = Ng
| first2 = Kwok K.
| authorlink = Simon Sze
| title = Physics of semiconductor devices
| publisher = [[John Wiley and Sons]]
| edition = 3
| year = 2007
| isbn = 978-0-471-14323-9
}} Chapter 13.6.</ref>
Then, when [[electron–hole pair]]s are generated in the depletion region, they are separated by the electric field, the electrons move toward the surface, and the holes move toward the substrate. Four pair-generation processes can be identified:
* photo-generation (up to 95% of [[quantum efficiency]]),
* generation in the depletion region,
* generation at the surface, and
* generation in the neutral bulk.

The last three processes are known as dark-current generation, and add noise to the image; they can limit the total usable integration time. The accumulation of electrons at or near the surface can proceed either until image integration is over and charge begins to be transferred, or thermal equilibrium is reached. In this case, the well is said to be full (corresponding typically to about 105 electrons per pixel<ref name=sze />).

=== Design and manufacturing ===
The photoactive region of a CCD is, generally, an [[epitaxial]] layer of [[silicon]]. It is lightly ''p'' doped (usually with [[boron]]) and is grown upon a [[substrate (materials science)|substrate]] material, often p++. In buried-channel devices, the type of design utilized in most modern CCDs, certain areas of the surface of the silicon are [[ion implantation|ion implanted]] with [[phosphorus]], giving them an n-doped designation. This region defines the channel in which the photogenerated charge packets will travel. [[Simon Sze]] details the advantages of a buried-channel device:<ref name=sze />
<blockquote>This thin layer (= 0.2–0.3 nm) is fully depleted and the accumulated photogenerated charge is kept away from the surface. This structure has the advantages of higher transfer efficiency and lower dark current, from reduced surface recombination. The penalty is smaller charge capacity, by a factor of 2–3 compared to the surface-channel CCD.</blockquote> The gate oxide, i.e. the [[capacitor]] [[dielectric]], is grown on top of the epitaxial layer and substrate.

Later in the process, [[polysilicon]] gates are deposited by [[chemical vapor deposition]], patterned with [[photolithography]], and etched in such a way that the separately phased gates lie perpendicular to the channels. The channels are further defined by utilization of the [[LOCOS]] process to produce the [[channel stop]] region.

Channel stops are thermally grown [[oxide]]s that serve to isolate the charge packets in one column from those in another. These channel stops are produced before the polysilicon gates are, as the LOCOS process utilizes a high-temperature step that would destroy the gate material. The channel stops are parallel to, and exclusive of, the channel, or "charge carrying", regions.

Channel stops often have a p+ doped region underlying them, providing a further barrier to the electrons in the charge packets (this discussion of the physics of CCD devices assumes an [[electron]] transfer device, though hole transfer is possible).

The clocking of the gates, alternately high and low, will forward and reverse bias the diode that is provided by the buried channel (n-doped) and the epitaxial layer (p-doped). This will cause the CCD to deplete, near the [[p-n junction]] and will collect and move the charge packets beneath the gates—and within the channels—of the device.

CCD manufacturing and operation can be optimized for different uses. The above process describes a frame transfer CCD. While CCDs may be manufactured on a heavily doped p++ wafer it is also possible to manufacture a device inside p-wells that have been placed on an n-wafer. This second method, reportedly, reduces smear, [[dark current (physics)|dark current]], and [[infrared]] and red response. This method of manufacture is used in the construction of interline-transfer devices.

Another version of CCD is called a peristaltic CCD. In a peristaltic charge-coupled device, the charge-packet transfer operation is analogous to the peristaltic contraction and dilation of the [[digestive system]]. The peristaltic CCD has an additional implant that keeps the charge away from the silicon/[[silicon dioxide]] interface and generates a large lateral electric field from one gate to the next. This provides an additional driving force to aid in transfer of the charge packets.

==Architecture==
The CCD image sensors can be implemented in several different architectures. The most common are full-frame, frame-transfer, and interline. The distinguishing characteristic of each of these architectures is their approach to the problem of shuttering.

In a full-frame device, all of the image area is active, and there is no electronic shutter. A mechanical shutter must be added to this type of sensor or the image smears as the device is clocked or read out.

With a frame-transfer CCD, half of the silicon area is covered by an opaque mask (typically aluminum). The image can be quickly transferred from the image area to the opaque area or storage region with acceptable smear of a few percent. That image can then be read out slowly from the storage region while a new image is integrating or exposing in the active area. Frame-transfer devices typically do not require a mechanical shutter and were a common architecture for early solid-state broadcast cameras. The downside to the frame-transfer architecture is that it requires twice the silicon real estate of an equivalent full-frame device; hence, it costs roughly twice as much.

The interline architecture extends this concept one step further and masks every other column of the image sensor for storage. In this device, only one pixel shift has to occur to transfer from image area to storage area; thus, shutter times can be less than a microsecond and smear is essentially eliminated. The advantage is not free, however, as the imaging area is now covered by opaque strips dropping the [[fill factor]] to approximately 50 percent and the effective [[quantum efficiency]] by an equivalent amount. Modern designs have addressed this deleterious characteristic by adding microlenses on the surface of the device to direct light away from the opaque regions and on the active area. Microlenses can bring the fill factor back up to 90 percent or more depending on pixel size and the overall system's optical design.

[[Image:ArgusCCD.jpg|thumb|right|CCD from a 2.1 [[megapixel]] [[Argus (camera company)|Argus]] digital camera]]

The choice of architecture comes down to one of utility. If the application cannot tolerate an expensive, failure-prone, power-intensive mechanical shutter, an interline device is the right choice. Consumer snap-shot cameras have used interline devices. On the other hand, for those applications that require the best possible light collection and issues of money, power and time are less important, the full-frame device is the right choice. Astronomers tend to prefer full-frame devices. The frame-transfer falls in between and was a common choice before the fill-factor issue of interline devices was addressed. Today, frame-transfer is usually chosen when an interline architecture is not available, such as in a back-illuminated device.

CCDs containing grids of [[pixel]]s are used in [[digital camera]]s, [[image scanner|optical scanners]], and video cameras as light-sensing devices. They commonly respond to 70 percent of the [[wiktionary:incident|incident]] light (meaning a quantum efficiency of about 70 percent) making them far more efficient than [[photographic film]], which captures only about 2 percent of the incident light.

[[Image:2.1 MP CCD Close Up.JPG|thumb|right|CCD from a 2.1 [[megapixel]] [[Hewlett-Packard]] digital camera]]

Most common types of CCDs are sensitive to near-infrared light, which allows [[infrared photography]], [[night-vision]] devices, and zero [[lux]] (or near zero lux) video-recording/photography. For normal silicon-based detectors, the sensitivity is limited to 1.1 μm. One other consequence of their sensitivity to infrared is that infrared from [[remote control]]s often appears on CCD-based digital cameras or camcorders if they do not have infrared blockers.

Cooling reduces the array's [[dark current (physics)|dark current]], improving the sensitivity of the CCD to low light intensities, even for ultraviolet and visible wavelengths. Professional observatories often cool their detectors with [[liquid nitrogen]] to reduce the dark current, and therefore the [[thermal noise]], to negligible levels.

==Use in astronomy==
Due to the high quantum efficiencies of CCDs, linearity of their outputs (one count for one photon of light), ease of use compared to photographic plates, and a variety of other reasons, CCDs were very rapidly adopted by astronomers for nearly all UV-to-infrared applications.

Thermal noise and [[cosmic ray]]s may alter the pixels in the CCD array. To counter such effects, astronomers take several exposures with the CCD shutter closed and opened. The average of images taken with the shutter closed is necessary to lower the random noise. Once developed, the [[dark frame subtraction|''dark frame'' average image is then subtracted]] from the open-shutter image to remove the dark current and other systematic defects ([[dead pixel]]s, hot pixels, etc.) in the CCD.

The [[Hubble Space Telescope]], in particular, has a highly developed series of steps (“data reduction pipeline”) to convert the raw CCD data to useful images.<ref name = "ESO-Hainaut-CCD image processing">
{{Cite web| url = http://www.eso.org/~ohainaut/ccd | title = Basic CCD image processing |date = December 2006| last = Hainaut | first = Oliver R. | accessdate = January 15, 2011 }} 
{{Cite web| url = http://www.eso.org/~ohainaut/ccd/sn.html | title = Signal, Noise and Detection | date = June 1, 2005 | last = Hainaut | first = Oliver R. | accessdate = October 7, 2009 }} 
{{Cite web| url = http://www.eso.org/~ohainaut/images/imageProc.html | title = Retouching of astronomical data for the production of outreach images | date = May 20, 2009 | last = Hainaut | first = Oliver R. | accessdate = October 7, 2009 }} (Hainaut is an astronomer at the [http://www.eso.org/~ohainaut/cv.html European Southern Observatory]) 
</ref>

CCD cameras used in [[astrophotography]] often require sturdy mounts to cope with vibrations from wind and other sources, along with the tremendous weight of most imaging platforms. To take long exposures of galaxies and nebulae, many astronomers use a technique known as [[autoguider|auto-guiding]]. Most autoguiders use a second CCD chip to monitor deviations during imaging. This chip can rapidly detect errors in tracking and command the mount motors to correct for them.

[[Image:SDSSFaceplate.gif|thumb|right|Array of 30 CCDs used on [[Sloan Digital Sky Survey]] telescope imaging camera, an example of "drift-scanning."]]

An interesting unusual astronomical application of CCDs, called ''drift-scanning'', uses a CCD to make a fixed telescope behave like a tracking telescope and follow the motion of the sky. The charges in the CCD are transferred and read in a direction parallel to the motion of the sky, and at the same speed. In this way, the telescope can image a larger region of the sky than its normal field of view. The [[Sloan Digital Sky Survey]] is the most famous example of this, using the technique to produce the largest uniform survey of the sky yet accomplished.

In addition to astronomy, CCDs are also used in astronomical analytical instrumentation such as [[spectrometer]]s.

==Color cameras==
[[Image:Bayer pattern on sensor.svg|thumb|A [[Bayer filter]] on a CCD]]
[[File:Webcam CCD - 640x480px Colour.jpg|thumb|CCD color sensor]]
[[File:An RGGB Bayer Colour Filter on a 1980's vintage Sony PAL Camcorder CCD.png|right|thumb|x80 microscope view of an RGGB Bayer filter on a 240 line Sony CCD PAL Camcorder CCD sensor]]

Digital color cameras generally use a [[Bayer filter|Bayer mask]] over the CCD. Each square of four pixels has one filtered red, one blue, and two green (the [[human eye]] is more sensitive to green than either red or blue). The result of this is that [[luminance]] information is collected at every pixel, but the color resolution is lower than the luminance resolution.

Better color separation can be reached by three-CCD devices ([[3CCD]]) and a [[dichroic prism|dichroic beam splitter prism]], that splits the [[image]] into [[red]], [[green]] and [[blue]] components. Each of the three CCDs is arranged to respond to a particular color. Many [[professional video camera|professional video]] camcorders, and some semi-professional camcorders, use this technique, although developments in competing CMOS technology have made CMOS sensors, both with beam-splitters and bayer filters, increasingly popular in high-end video and digital cinema cameras. Another advantage of 3CCD over a Bayer mask device is higher [[quantum efficiency]] (and therefore higher light sensitivity for a given aperture size). This is because in a 3CCD device most of the light entering the aperture is captured by a sensor, while a Bayer mask absorbs a high proportion (about 2/3) of the light falling on each CCD pixel.

For still scenes, for instance in microscopy, the resolution of a Bayer mask device can be enhanced by [[microscanning]] technology. During the process of [[co-site sampling|color co-site sampling]], several frames of the scene are produced. Between acquisitions, the sensor is moved in pixel dimensions, so that each point in the visual field is acquired consecutively by elements of the mask that are sensitive to the red, green and blue components of its color. Eventually every pixel in the image has been scanned at least once in each color and the resolution of the three channels become equivalent (the resolutions of red and blue channels are quadrupled while the green channel is doubled).

===Sensor sizes===
{{Main|Image sensor format}}
Sensors (CCD / CMOS) come in various sizes, or image sensor formats. These sizes are often referred to with an inch fraction designation such as 1/1.8″ or 2/3″ called the [[optical format]]. This measurement actually originates back in the 1950s and the time of [[video camera tube|Vidicon tubes]].

==Electron-multiplying CCD==
[[Image:EMCCD2 color en.svg|thumb|Electrons are transferred serially through the gain stages making up the multiplication register of an [[#Electron-multiplying CCD|EMCCD]]. The high voltages used in these serial transfers induce the creation of additional charge carriers through impact ionisation.]]
[[Image:Output vs input electrons.png|thumb|in an [[#Electron-multiplying CCD|EMCCD ]] there is a dispersion (variation) in the number of electrons output by the multiplication register for a given (fixed) number of input electrons (shown in the legend on the right). The probability distribution for the number of output electrons is plotted [[logarithm]]ically on the vertical axis for a simulation of a multiplication register. Also shown are results from the [[empiricism|empirical]] fit equation shown on this page.]]

An '''electron-multiplying CCD''' (EMCCD, also known as an L3Vision CCD, L3CCD or Impactron CCD) is a charge-coupled device in which a gain register is placed between the shift register and the output amplifier. The gain register is split up into a large number of stages. In each stage, the electrons are multiplied by [[impact ionization]] in a similar way to an [[avalanche diode]]. The gain probability at every stage of the register is small (''P'' < 2%), but as the number of elements is large (N > 500), the overall gain can be very high (<math>g = (1 + P)^N</math>), with single input electrons giving many thousands of output electrons. Reading a signal from a CCD gives a noise background, typically a few electrons. In an EMCCD, this noise is superimposed on many thousands of electrons rather than a single electron; the devices' primary advantage is thus their negligible readout noise.

EMCCDs show a similar sensitivity to [[charge-coupled device#Intensified charge-coupled device|Intensified CCDs]] (ICCDs). However, as with ICCDs, the gain that is applied in the gain register is stochastic and the ''exact'' gain that has been applied to a pixel's charge is impossible to know. At high gains (> 30), this uncertainty has the same effect on the [[signal-to-noise ratio]] (SNR) as halving the [[quantum efficiency]] (QE) with respect to operation with a gain of unity. However, at very low light levels (where the quantum efficiency is most important), it can be assumed that a pixel either contains an electron — or not. This removes the noise associated with the stochastic multiplication at the risk of counting multiple electrons in the same pixel as a single electron. To avoid multiple counts in one pixel due to coincident photons in this mode of operation, high frame rates are essential. The dispersion in the gain is shown in the graph on the right. For multiplication registers with many elements and large gains it is well modelled by the equation:

<math>P\left (n \right ) = \frac{\left
(n-m+1\right )^{m-1}}{\left (m-1 \right )!\left
(g-1+\frac{1}{m}\right )^{m}}\exp \left ( -
\frac{n-m+1}{g-1+\frac{1}{m}}\right )</math> if <math>n \ge m </math>

where ''P'' is the probability of getting ''n'' output electrons given ''m'' input electrons and a total mean multiplication register gain of ''g''.

Because of the lower costs and better resolution, EMCCDs are capable of replacing ICCDs in many applications. ICCDs still have the advantage that they can be gated very fast and thus are useful in applications like [[range gate|range-gated imaging]]. EMCCD cameras indispensably need a cooling system — using either [[thermoelectric cooling]] or liquid nitrogen — to cool the chip down to temperatures in the range of -65°C to -95°C. This cooling system unfortunately adds additional costs to the EMCCD imaging system and may yield condensation problems in the application. However, high-end EMCCD cameras are equipped with a permanent hermetic vacuum system confining the chip to avoid condensation issues.

The low-light capabilities of EMCCDs primarily find use in astronomy and biomedical research, among other fields. In particular, their low noise at high readout speeds makes them very useful for a variety of astronomical applications involving low light sources and transient events such as [[lucky imaging]] of faint stars, high speed [[photon counting]] photometry, [[Fabry-Pérot|Fabry-Pérot spectroscopy]] and high-resolution spectroscopy. More recently, these types of CCDs have broken into the field of biomedical research in low-light applications including [[small animal imaging]], [[single-molecule|single-molecule imaging]], [[Raman spectroscopy]], [[super resolution microscopy]] as well as a wide variety of modern [[fluorescence microscopy]] techniques thanks to greater SNR in low-light conditions in comparison with traditional CCDs and ICCDs.

In terms of noise, commercial EMCCD cameras typically have clock-induced charge (CIC) and dark current (dependent on the extent of cooling) that together lead to an effective readout noise ranging from 0.01 to 1 electrons per pixel read. However, recent improvements in EMCCD technology have led to a new generation of cameras capable of producing significantly less CIC, higher charge transfer efficiency and an EM gain 5 times higher than what was previously available.<ref>As specified in {{cite web|title=Nüvü Caméras' specsheet| url=http://www.nuvucameras.com/wp-content/uploads/2013/08/Nuvu-Cameras-HNu-512-Specsheet.pdf }}</ref> These advances in low-light detection lead to an effective total background noise of 0.001 electrons per pixel read, a noise floor unmatched by any other low-light imaging device.<ref>
{{cite journal
| last1 = Daigle | first1 = Olivier
| last2 = Djazovski | first2 = Oleg
| last3 = Laurin | first3 = Denis
| last4 = Doyon | first4 = René
| last5 = Artigau | first5 = Étienne
| title = Characterization results of EMCCDs for extreme low light imaging
| date = July 2012
| url = http://www.nuvucameras.com/wp-content/uploads/2012/07/Characterization-results-of-EMCCDs-for-extreme-low-light-imaging.pdf}}</ref>

==Frame transfer CCD==
[[Image:Vertical smear.jpg|thumb|right|300px|Vertical smear]]

The frame transfer CCD imager was the first imaging structure proposed for CCD Imaging by Michael Tompsett at Bell Laboratories. A '''frame transfer CCD''' is a specialized CCD, often used in [[astronomy]] and some [[professional video camera]]s, designed for high exposure efficiency and correctness.

The normal functioning of a CCD, astronomical or otherwise, can be divided into two phases: exposure and readout. During the first phase, the CCD passively collects incoming [[photon]]s, storing [[electron]]s in its cells. After the exposure time is passed, the cells are read out one line at a time. During the readout phase, cells are shifted down the entire area of the CCD. While they are shifted, they continue to collect light. Thus, if the shifting is not fast enough, errors can result from light that falls on a cell holding charge during the transfer. These errors are referred to as "vertical smear" and cause a strong light source to create a vertical line above and below its exact location. In addition, the CCD cannot be used to collect light while it is being read out. Unfortunately, a faster shifting requires a faster readout, and a faster readout can introduce errors in the cell charge measurement, leading to a higher noise level.

A frame transfer CCD solves both problems: it has a shielded, not light sensitive, area containing as many cells as the area exposed to light. Typically, this area is covered by a reflective material such as aluminium. When the exposure time is up, the cells are transferred very rapidly to the hidden area. Here, safe from any incoming light, cells can be read out at any speed one deems necessary to correctly measure the cells' charge. At the same time, the exposed part of the CCD is collecting light again, so no delay occurs between successive exposures.

The disadvantage of such a CCD is the higher cost: the cell area is basically doubled, and more complex control electronics are needed.

==Intensified charge-coupled device==
{{Main|Image intensifier}}
An intensified charge-coupled device (ICCD) is a CCD that is optically connected to an image intensifier that is mounted in front of the CCD.

An image intensifier includes three functional elements: a [[photocathode]], a [[micro-channel plate]] (MCP) and a [[phosphor]] screen. These three elements are mounted one close behind the other in the mentioned sequence. The photons which are coming from the light source fall onto the photocathode, thereby generating photoelectrons. The photoelectrons are accelerated towards the MCP by an electrical control voltage, applied between photocathode and MCP. The electrons are multiplied inside of the MCP and thereafter accelerated towards the phosphor screen. The phosphor screen finally converts the multiplied electrons back to photons which are guided to the CCD by a fiber optic or a lens.

An image intensifier inherently includes a [[shutter (photography)|shutter]] functionality: If the control voltage between the photocathode and the MCP is reversed, the emitted photoelectrons are not accelerated towards the MCP but return to the photocathode. Thus, no electrons are multiplied and emitted by the MCP, no electrons are going to the phosphor screen and no light is emitted from the image intensifier. In this case no light falls onto the CCD, which means that the shutter is closed. The process of reversing the control voltage at the photocathode is called [[gating]] and therefore ICCDs are also called gateable CCD cameras.

Besides the extremely high sensitivity of ICCD cameras, which enable single photon detection, the gateability is one of the major advantages of the ICCD over the [[charge-coupled device#Electron-multiplying CCD|EMCCD]] cameras. The highest performing ICCD cameras enable shutter times as short as 200 [[picosecond]]s.

ICCD cameras are in general somewhat higher in price than EMCCD cameras because they need the expensive image intensifier. On the other hand EMCCD cameras need a cooling system to cool the EMCCD chip down to temperatures around 170 [[Kelvin|K]]. This cooling system adds additional costs to the EMCCD camera and often yields heavy condensation problems in the application.

ICCDs are used in [[night vision devices]] and in a large variety of scientific applications.

== Blooming ==

When a CCD exposure is long enough, eventually the electrons that collect in the "bins" in the brightest part of the image will overflow the bin, resulting in blooming. The structure of the CCD allows the electrons to flow more easily in one direction than another, resulting in vertical streaking.<ref>
Phil Plait. [http://www.badastronomy.com/bad/misc/planetx/soho.html "The Planet X Saga: SOHO Images"]
</ref><ref>
Phil Plait. [http://blogs.discovermagazine.com/badastronomy/2009/03/13/why-king-triton-how-nice-to-see-you/ "Why, King Triton, how nice to see you!"]
</ref><ref>
Thomas J. Fellers and Michael W. Davidson. [http://learn.hamamatsu.com/articles/ccdsatandblooming.html "CCD Saturation and Blooming"]
</ref>

Some anti-blooming features that can be built into a CCD reduce its sensitivity to light by using some of the pixel area for a drain structure.<ref>
{{cite book
| title = Solid-State Imaging With Charge-Coupled Devices
| author = Albert J. P. Theuwissen
| publisher = Springer
| year = 1995
| isbn = 9780792334569
| pages = 177–180
| url = http://books.google.com/books?id=dchEKTHNCMcC&pg=PA177
}}</ref>
[[James M. Early]] developed a vertical anti-blooming drain that would not detract from the light collection area, and so did not reduce light sensitivity.

==See also==
<div style="-moz-column-count:3; column-count:3;">
* [[Photodiode]]
* [[Active pixel sensor|CMOS sensor]]
* [[Angle-sensitive pixel]]
* [[Rotating line camera]]
* [[Superconducting camera]]
* [[Wide dynamic range]]
* [[Hole Accumulation Diode]] (HAD)
* [[Andor Technology]] – Manufacturer of EMCCD cameras
* [[Roper Industries#PI/Acton|PI/Acton]] – Manufacturer of EMCCD cameras
* [[Stanford Computer Optics]] – Manufacturer of ICCD cameras
* [[Time delay and integration]] (TDI)
* [[Glossary of video terms]]
</div>

==References==
{{Reflist}}

==External links==
{{Commons category|Charge-coupled devices}}
* [http://www.jyi.org/volumes/volume3/issue1/features/peterson.html Journal Article On Basics of CCDs]
* [http://www.microscopyu.com/articles/digitalimaging/ccdintro.html Nikon microscopy introduction to CCDs]
* [http://micro.magnet.fsu.edu/primer/digitalimaging/concepts/concepts.html Concepts in Digital Imaging Technology]
* [http://zebu.uoregon.edu/ccd.html CCDs for Material Scientists]
* [http://www.ing.iac.es/~smt/LLLCCD/marcl3.htm A general L3CCD page with many links]
* [http://www.mpia.de/homes/tubbs/papers/lllccd/sern_main.html Paper discussing the performance of L3CCDs]
* [http://www.mrao.cam.ac.uk/telescopes/coast/theses/rnt/node72.html Statistical properties of multiplication registers including derivation of the equation above]
* [http://arxiv.org/abs/astro-ph/0407315 More statistical properties]
* [http://www.ast.cam.ac.uk/~optics/Lucky_Web_Site/guide_to_l3ccds.htm L3CCDs used in astronomy]
*[http://www.niscair.res.in/sciencecommunication/Popularization%20of%20Science/SciRep/scirep2k5/scirep_feb05.asp] Heart of a Camera, Science Reporter, February, 2005, Volume 42, Number 2
{{Photography}}

{{DEFAULTSORT:Charge-Coupled Device}}
[[Category:American inventions]]
[[Category:Integrated circuits]]
[[Category:Image processing]]
[[Category:Image sensors]]
[[Category:Image scanners]]
[[Category:Astronomical imaging]]

{{Link GA|de}}

Electromagnetic wave equation

2014-01-27T14:56:47Z

117.194.82.30: /* The origin of the electromagnetic wave equation */

{{howto|date=September 2010}}
'''Betting arbitrage''', '''miraclebets''', '''surebets''', '''sports arbitraging''' is a particular case of [[arbitrage]] arising on betting markets due to either [[bookmaker|bookmakers']] different opinions on event outcomes or plain errors. When conditions allow, by placing one bet per each outcome with different betting companies, the bettor can make a profit regardless of the outcome. In the bettors' slang an arbitrage is often referred to as an '''arb'''; people who use arbitrage are called '''arbers'''.

A typical arbitrage profit is around 2 percent, often less; however 4-5 percent profit is occasionally obtained, and sometimes even 20 percent.{{Citation needed|date=July 2013}} Arbitrage betting involves relatively large sums of money (stakes are bigger than in normal betting), due to large sums of money being required to generate a decent profit. It is usually detected quickly by bookmakers. Arbitrage betting is almost always insufficiently profitable due to detection, hackers, unreliable betting websites, limiting of stakes, and the use of high percentage arbitrages to trick bettors into giving security details.

Bookmakers generally disapprove of betting arbitrage, and restrict or close the accounts of those who they suspect of engaging in arbitrage betting. Although arbitrage betting has existed since the beginnings of bookmaking, the rise of the Internet, odds-comparison websites and [[betting exchanges]] have made the practice easier to perform. On the other hand, these changes also made it easier for bookmakers to keep their odds in line with the market, because arbitrage bettors are basically acting as [[market maker]]s.

The best way of generating profit via sports arbitrage, which has been established in Britain, consists of highly experienced "key men" employing others to place bets on their behalf, so as to avoid detection and increase accessibility to retail bookmakers. This allows the financiers or key arbitragers to stay at a computer to keep track of market movement.

Arbitrage is an extremely fast-paced process and its successful performance requires lots of time, experience, dedication and discipline, and especially liquidity.

== Arbitrage in theory ==

There are a number of potential arbitrage deals. Below is an explanation of some of them including formulas and risks associated with them. The table below introduces a number of variables that will be used to formalise the arbitrage models.

{| border="1"
|-
|Variable || Explanation
|-
| <math>s_1</math> || Stake in outcome 1
|-
| <math>s_2</math> || Stake in outcome 2
|-
| <math>o_1</math> || Odds for outcome 1
|-
| <math>o_2</math> || Odds for outcome 2
|-
| <math>r_1</math> || Return if outcome 1 occurs
|-
| <math>r_2</math> || Return if outcome 2 occurs
|}

=== Arbitrage using bookmakers ===
This type of arbitrage takes advantage of different odds offered by different bookmakers. For an example of an event with only two possible outcomes (e.g. a [[tennis]] match - either Federer wins or Henman wins), the two bookmakers have different ideas of who has the best chances of winning. They offer the following [[Fixed-odds betting|Fixed-odds gambling]] on the outcomes of the event:

{| border="1"
|-
| ||Bookmaker 1 || Bookmaker2
|-
| Outcome 1 || 1.25 || 1.43
|-
| Outcome 2 || 3.9 || 2.85
|}

For an individual bookmaker, the sum of the inverse of all outcomes of an event will always be greater than 1.
<math>1.25^{-1} + 3.9^{-1} = 1.056</math> and <math>1.43^{-1} + 2.85^{-1} = 1.051</math>

The bookmaker's return rate is <math>1- (1.25*3.9)/(1.25+3.9)= 5.34%</math>, which is the amount the bookmaker earns on offering bets at some event. Bookmaker 1 will in this example expect to earn 5.34% on bets on the tennis game. Usually these gaps will be in the order 8 - 12%. The idea is to find odds at different bookmakers, where the sum of the inverse of all the outcomes are below 1, meaning that the bookmakers disagree on the chances of the outcomes. This discrepancy can be used to obtain a profit.

For instance if one places a bet on outcome 1 at bookmaker 2 and outcome 2 at bookmaker 1:

<math>1.43^{-1} + 3.9^{-1} = 0.956</math>

Placing a bet of $100 on outcome 1 with bookmaker 2 and a bet of <math>$100*1.43/3.9 = 36.67</math> on outcome 2 at bookmaker 1 would ensure the bettor a profit.

In case outcome 1 comes out, one could collect <math>r_1 = $100 * 1.43 = $143</math> from bookmaker 2. In case outcome 2 comes out, one could collect <math>r_2 = $36.67 * 3.9 = $143</math> from bookmaker 1. One would have invested $136.67, but have collected $143, a profit of $6.33 (4.6%) no matter the outcome of the event.

So for 2 odds <math>o_1</math> and <math>o_2</math>, where <math>o_1^{-1} + o_2^{-1} < 1</math>. If one wishes to place stake <math>s_1</math> at outcome 1, then one should place <math>s_2 = s_1 * o_1 / o_2</math> at outcome 2, to even out the odds, and receive the same return no matter the outcome of the event.

Or in other words, if there are two outcomes, a 2/1 and a 3/1, by covering the 2/1 with $500 and the 3/1 with $333, one is guaranteed to win $1000 at a cost of $833, giving a 20% profit. More often profits exists around the 4% mark or less.

Reducing the risk of human error is vital being that the mathematical formula is sound and only external factors add "risk". Numerous online arbitrage calculator tools exist to help bettors get the math right. For example, arbitrage calculators can handle calculations for both book arbitrage ("back/back" or "lay/lay") and "back/lay" arbitrage opportunities on an intra-exchange or inter-exchange basis, and is free.

For arbitrages involving three outcomes (e.g. a game which can be won, lost or drawn) having the odds <math>o_1</math> for Outcome 1, <math>o_2</math> for outcome 2 and <math>o_3</math> for outcome 3 with their respective bids being <math>b_1</math>, <math>b_2</math> and <math>b_3</math> and sum of the bids being B.

The amount required to bet on each possibility in order to ensure profit can be calculated by

<math>b_1 = B / (1 + (o_1/o_2) + (o_1/o_3) )</math>

<math>b_2 = B / (1 + (o_2/o_1) + (o_2/o_3) )</math>

<math>b_3 = B / (1 + (o_3/o_1) + (o_3/o_2) )</math>

=== Back-lay sports arbitrage ===
[[Betting exchange]]s such as [[Smarkets]] have opened up a new range of arbitrage possibilities since on the exchanges it is possible to ''lay'' (i.e. to bet against) as well as to back an outcome. Arbitrage using only the back or lay side might occur on betting exchanges. It is in principle the same as the arbitrage using different bookmakers. Arbitrage using back and lay side is possible if a lay bet on one exchange provides shorter odds than a back bet on another exchange or bookmaker. However, the commission charged by the bookmakers and exchanges must be included into calculations.

Back-lay sports arbitrage is often called "scalping" or "trading". Scalping is not actually arbitrage, but short term trading. In the context of sports arbitrage betting a scalping trader or scalper looks to make lots of small profits, which in time can add up. In theory a trader could turn a small investment into large profits by re-investing his earlier profits into future bets so as to generate [[exponential growth]]. Scalping relies on liquidity in the markets and that the odds will fluctuate around a mean point. A key advantage to scalping on one exchange is that most exchanges charge commission only on the net winnings in a particular event, thus ensuring that even the smallest favorable difference in the odds will guarantee some profit.

=== Bonus sports arbitrage ===
Many bookmakers offer first time users a signup bonus in the range $10–200 for depositing an initial amount. They typically demand that this amount is wagered a number of times before the bonus can be withdrawn. Bonus sport arbitraging is a form of sports arbitraging where you hedge or back your bets as usual, but since you received the bonus, a small loss can be allowed on each wager (2–5%), which comes off your profit. In this way the bookmakers wagering demand can be met and the initial deposit and sign up bonus can be withdrawn with little loss.

The advantage over usual betting arbitrage is that it is a lot easier to find bets with an acceptable loss, instead of an actual profit. Since most bookmakers offer these bonuses this can potentially be exploited to harvest the sign up bonuses.

By signing up to various bookmakers, it is possible to turn these "free" bets into cash fairly quickly, and either making a small arbitrage, or in the majority of cases, making a small loss on each bet, or trade. However, it is relatively time consuming to find close matched bets or arbitrages, which is where a middleman service is useful. As many bookmakers require a certain turnover of the bonus amount, matching money from different bookmakers against each other enables the player to in effect quickly "play free" the money of the losing bookmaker and in effect transfer it to the winning bookmaker. By avoiding most of the turnover requirements in this way the player can usually expect a 70-80% return on investment.

As well as spending time physically matching odds from various bet sites to exchanges, the other draw back with bonus bagging and arbitrage trading in this sense is that often the free bets are "non-stake returned". This effectively reduces the odds, in decimal format, by 1. Therefore, in order to reduce "losses" on the free bet, it is necessary to place a bet with high odds, so that the percentage difference of the decrease in odds is minimised.

=== Shop Arbitrage(Sharbing) ===

Shop arbitrage (also known as sharbing, an almagamation of shop and arbitrage) is the process of using a highstreet bookmaker's coupons and a betting exchange to create an arbitrage position. This is made possible as whilst online prices change quickly to close these positions, high street bookmakers are slower to change the prices on their printed coupons.

== Arbitrage in practice ==
While often claimed to be "risk-free", this is only true if an arbitrage is successfully completed; in reality, there are several threats to this:

*'''Disappearance of arbitrage:''' Arbitrages in online sports markets have a median lifetime of around 15 minutes,<ref>"How quickly is temporary market inefficiency removed?" Ben R. Marshall ''The Quarterly Review of Economics and Finance'' '''49''' (2009) 917–930</ref> after which the difference in odds underpinning them vanishes through betting activity. Without rapid alerting (Surebet Monitor services or software) and action, it is possible to fail to make all the "legs" of the arbitrage before it vanishes, thus transforming it from a risk-free arbitrage into a conventional bet with the usual risks involved. High street bookmakers however, offer their odds days in advance and rarely change them once they have been set. These arbitrages can have a lifetime of several hours.

*'''Hackers:''' Due to the large number of accounts that have to be created and managed (containing personal details such as email, name, address, eWallet, credit card information and often even a copy of the bettor's ID/passport or driver's license), arbitrage traders are highly susceptible to cyber fraud, such as bank account theft. While making deposits is usually made easy and quick, making withdrawals always requires proof of identity in the form of passport/driver license, copies of which need to be shared with the bookies via fax/email or even postal mail, which causes additional identity theft risks. It is estimated that at least 50%-60% of all arbitrage bettors fall victim to some form of cyber crime. Traders are often attracted to high odds comparison sites that yield high percentage profits per stake (5-30%); this is often used by hackers to lure a high number of arbitrage bettors that then place large sums of money on these arb's, only to lose all of the profit and even entire savings in bank accounts to hackers or untrustworthy websites, which may further use the gathered data to sell personal data to criminals.

*'''Making errors:''' In the excitement of the action and due to the high number of bets placed, it is not uncommon to make a mistake (like traders on financial markets). For example the appropriate stakes may be incorrectly calculated, or be placed on the wrong "legs" of the arb, locking in a loss, or there may be inadequate funds in one of the accounts to complete the arb. Those errors might temporarily have an important impact. In the long term, the benefit will depend on the odds. For example one could actually make more money by placing the "wrong" bet where the outcome happens to be beneficial, though not justified by the arbitrage calculation. However, repetition of this stroke of luck is unlikely, assuming the bookmaker has calculated the odds so they make a profit. Websites and bet placement interfaces differ between bookies, so that arbitrage bettors need to be familiar with different web interfaces. In some sports different bookmakers deal with outcomes in different ways (they differ in their handling of - for example - player withdrawal due to injury in tennis, overtime in [[ice hockey]]), meaning that both "legs" can lose. Matching terms for all bookmakers is time consuming, requires lots of expertise and experience, while still being fairly error-prone.

*'''Detection:''' There are only very few bookies who openly tolerate arbing (such as for example pinnaclesports.com). Many bookmakers may now be using shared security servers in order to pinpoint people suspected of arbitrage betting; they can simply limit stakes to make arbing unprofitable and even close accounts without honouring a bet that was placed. Loss of deposited money into a bookmaker could occur. This usually leads to unprofitable arbing as the most successful bookmakers are so adept at identifying arbitrage bettors, without these countermeasure the gambling industry would not be able to generate a large overall profit consistently every year.

*'''Stake reviewal:''' Some bookies are known to accept only very small stakes by default, while requiring larger stakes to be manually reviewed before being accepted, which basically makes it difficult for an arbitrage better to determine if a leg was completely accepted or not, until it may be too late.

*'''Bet cancellation:''' If a bettor places bets so as to make an arbitrage and one bookmaker cancels a bet, the bettor could find himself in a bad position because he is actually betting with all the risks implied. The bettor can repeat the bet that has been cancelled so as minimize the risk, but if he cannot get the same odds he had before he may be forced to take a loss. In some cases the situation arises when there are very high potential payouts by the [[bookie]], perhaps due to an unintentional error made while quoting odds. Many jurisdictions allow bookmakers to cancel bets in the event of such a "palpable" ["obvious"] error in the quoted odds This is often loosely defined as an obvious mistake, but whether a "palp" in fact has been made is often the sole discretion of the bookmaker.

Other potential problems include:
* Bookmakers who encourage responsible gambling will close accounts where they see only large losses, unaware that the arbitrage trader has made wins at other books.
* Capital diffusion is serious; many bookmakers make it very easy to deposit funds and difficult to withdraw them (requiring lots of additional information, and documents as proof of identiy, i.e. a passport/ID copy). Making a return involves many bets spread over typically many bookmakers so keeping track is a considerable challenge, and requires excellent record-keeping and discipline.
* Responding to an available arb, may require funds to be transferred from one bookie to another, going through one or more eWallet accounts - often, withdrawals require reviews before being approved.
* While there are commercial software products and web services available to help with some of these tasks, these are usually still fairly complicated tools which also further lower the total RoR because of the initial investment required, or even the monthly subscription fees involved.
* Arbitrage bettors using software tools or web services to find arbitrages will often make an existing arbitrage even more prominent and obvious to the bookie because of the number of arbitrage bettors placing bets on the same outcome, so that the lifetime of an arbitrage found via such tools is often even much shorter than the average 15 minutes. Thus, the risk of seeing bets revoked is also often much higher for arbitrages found via such tools than for arbitrages found manually, that are not shared with other arbitrage bettors.
* In addition, arbing often involves making use of bookie bonuses; these, however, usually need to be turned over a number of times before a bettor becomes eligible for withdrawal, which may further reduce his total liquidity.
* Foreign currency movements can wipe out small percentage gains and can make quick calculation of stakes difficult.
* Transferring funds between bookmakers and eWallets may create additional costs at some point; most bookmakers and/or eWallets limit deposits to certain amounts per month.
* Withdrawals are often limited to a certain amount per month or to a certain number of free withdrawals per month
* Withdrawals are often charged for, not just on the side of the bookie, but sometimes also on the eWallet side (transfer to the bettor's bank account).
* In some countries, additional costs are imposed by government taxes, so that the final profit is further reduced by a fixed percentage of say 5% (Germany/Europe).
* Professional arbitrage betting may eat up considerable time and energy and requires lots of experience and liquidity, as well as sufficient funds to recover from inevitable losses that will happen sooner or later due to the aforementioned reasons.
* Typically, arbitrages have a profit margin of only 2-5% - many other arbitrages are regarded as "high risk" ("palps"). Accordingly, profits accumulated through 20-40 successful arbitrages can be lost on a single failed bet.
* Won arbitrages realized via betting exchanges are typically subject to a commission fee of about 4-5%. However, frequent punters may be subject to a discount of up to 60%, i.e. a minimum commission rate of 3%).
* Smaller betting exchanges may not be able to deal with consistently winning punters.<ref>{{cite web|url=https://www.bookmakersreview.com/bookmakers-rating-news/smarkets-says-they-cannot-support-consistently-winning-punters/48018 |title=Smarkets says they cannot support consistently winning punters |publisher=Bookmakers Review |date= |accessdate=2013-11-22}}</ref>

==See also==
* [[Advantage gambling]]
* [[Dutch book]]
* [[Mathematics of bookmaking]]

==External links==
*[http://www.thegamblingtimes.com/board/content/tgt-sharbing-guide-210/ Guide to Sharbing]

==References==
<references/>

[[Category:Sports betting]]
[[Category:Investment]]
[[Category:Wagering]]
[[Category:Gambling terminology]]

Graviscalar

2014-01-11T04:41:49Z

117.194.87.171:

{{Refimprove|date=July 2007}}
In [[theoretical physics]], the '''Peierls bracket''' is an equivalent description of the [[Poisson bracket]]. It directly follows from the [[action (physics)|action]] and does not require the [[canonical coordinate]]s and their [[canonical momenta]] to be defined in advance.

The bracket

:<math>[A,B]</math>

is defined as

:<math>D_A(B)-D_B(A)</math>,

as the difference between some kind of action of one quantity on the other, minus the flipped term.

In [[quantum mechanics]], the Peierls bracket becomes a [[commutator]] i.e. a [[Lie bracket of vector fields|Lie bracket]].

==References==
{{Citizendium}}
<references />

Peierls, R. "The Commutation Laws of Relativistic Field Theory,"
Proc. R. Soc. Lond. A August 21, 1952 214 1117 143-157.

[[Category:Theoretical physics]]

{{physics-stub}}

Acceptance sampling

2013-12-12T06:58:50Z

117.194.4.149:

In [[computational number theory]], '''Cipolla's algorithm''' is a technique for solving a [[Congruence relation|congruence]] of the form

:<math>x^2\equiv n \pmod{p}.</math>

where <math>x,n \in \mathbf{F}_{p}</math>, so ''n'' is the square of ''x'', and where <math>p</math> is an [[Parity (mathematics)|odd]] [[Prime number|prime]]. Here <math>\mathbf{F}_p</math> denotes the finite [[Field (mathematics)|field]] with <math>p</math> [[Element (mathematics)|elements]]; <math>\{0,1,\dots,p-1\}</math>. The [[algorithm]] is named after [[Michele Cipolla]], an [[Italy|Italian]] [[Mathematics|mathematician]] who discovered it in the year 1907.

==The algorithm==

'''Inputs:'''
*<math>p</math>, an odd prime,
*<math>n \in \mathbf{F}_p</math>, which is a square.

'''Outputs:'''
*<math>x \in \mathbf{F}_p</math>, satisfying <math> x^2= n . </math>

Step 1 is to find an <math>a \in \mathbf{F}_p</math> such that <math>a^2 - n</math> is not a square. There is no known algorithm for finding such an <math>a</math>, except the [[trial and error]] method. Simply pick an <math>a</math> and by computing the [[Legendre symbol]] <math>(a^2-n|p)</math> one can see whether <math>a</math> satisfies the condition. The chance that a random <math>a</math> will satisfy is <math>(p-1)/2p</math>. With <math>p</math> large enough this is about <math>1/2</math>.<ref>R. Crandall, C. Pomerance Prime Numbers: A Computational Perspective Springer-Verlag, (2001) p. 157</ref> Therefore, the expected number of trials before finding a suitable ''a'' is about 2.

Step 2 is to compute ''x'' by computing <math>x=\left( a + \sqrt{a^2-n} \right)^{(p+1)/2}</math> within the field <math>\mathbf{F}_{p^2} = \mathbf{F}_p(\sqrt{a^2-n})</math>. This ''x'' will be the one satisfying <math> x^2 =n .</math>

If <math>x^2 = n</math>, then <math>(-x)^2 = n</math> also holds. And since ''p'' is odd, <math> x \neq -x </math>. So whenever a solution ''x'' is found, there's always a second solution, ''-x''.

==Example==

(Note: All elements before step two are considered as an element of <math>\mathbf{F}_{13}</math> and all elements in step two are considered as elements of <math>\mathbf{F}_{13^2}</math>).

Find all ''x'' such that <math>x^2 = 10.</math>

Before applying the algorithm, it must be checked that <math>10</math> is indeed a square in <math>\mathbf{F}_{13}</math>. Therefore, the Legendre symbol <math>(10 | 13)</math> has to be equal to 1. This can be computed using [[Euler's criterion]]; <math>(10 | 13) \equiv 10^6 \equiv 1 \bmod 13.</math> This confirms 10 being a square and hence the algorithm can be applied.

* Step 1: Find an ''a'' such that <math>a^2 - n</math> is not a square. As stated, this has to be done by trial and error. Choose <math>a=2</math>. Then <math>a^2 - n</math> becomes 7. The Legendre symbol <math>(7 | 13)</math> has to be -1. Again this can be computed using Euler's criterion. <math>7^6 = 343^2 \equiv 5^2 \equiv 25 \equiv -1 \bmod 13.</math> So <math>a=2</math> is a suitable choice for ''a''.
* Step 2: Compute <math>x = \left( a + \sqrt{a^2-n} \right)^{(p+1)/2} = \left( 2 + \sqrt{-6}\right)^7 .</math>

:<math>\left(2+\sqrt{-6}\right)^2 = 4 + 4\sqrt{-6} - 6 = -2 + 4 \sqrt{-6} .</math>
:<math>\left(2+\sqrt{-6}\right)^4 = \left(-2+4\sqrt{-6}\right)^2 = -1-3\sqrt{-6} .</math>
:<math>\left(2+\sqrt{-6}\right)^6 = \left(-2 + 4\sqrt{-6}\right)\left(-1-3\sqrt{-6}\right) = 9+2\sqrt{-6} .</math>
:<math>\left(2+\sqrt{-6}\right)^7 = \left(9+2\sqrt{-6}\right)\left(2+ \sqrt{-6}\right) = 6 .</math>

So <math>x = 6 </math> is a solution, as well as <math>x = -6 = 7 .</math> Indeed, <math>6^2 = 36 = 10</math> and <math> 7^2= 49 = 10 .</math>

==Proof==

The first part of the proof is to verify that <math>\mathbf{F}_{p^2} = \mathbf{F}_p(\sqrt{a^2-n}) = \{x + y\sqrt{a^2-n} : x,y \in \mathbf{F}_p\}</math> is indeed a field. For the sake of notation simplicity, <math>\omega</math> is defined as <math>\sqrt{a^2-n}</math>. Of course, <math>a^2-n</math> is a quadratic non-residue, so there is no [[square root]] in <math>\mathbf{F}_p</math>. This <math>\omega</math> can roughly be seen as analogous to the complex number [[Imaginary unit|i]].
The field arithmetic is quite obvious. [[Addition]] is defined as
:<math>\left(x_1 + y_1 \omega \right) + \left(x_2 + y_2 \omega \right) = \left(x_1 + x_2 \right) + \left(y_1 + y_2\right) \omega</math>.
[[Multiplication]] is also defined as usual. With keeping in mind that <math>\omega^2 = a^2-n</math>, it becomes
:<math>\left(x_1 + y_1 \omega \right)\left(x_2 + y_2 \omega \right) = x_1 x_2 + x_1 y_2 \omega + y_1 x_2 \omega + y_1 y_2 \omega^2 = \left( x_1 x_2 + y_1 y_2 \left(a^2-n\right)\right) + \left(x_1 y_2 + y_1 x_2 \right) \omega</math>.
Now the field properties have to be checked.
The properties of closure under addition and multiplication, [[associativity]], [[commutativity]] and [[distributivity]] are easily seen. This is because in this case the field <math>\mathbf{F}_{p^2}</math> is somewhat equivalent to the field of [[complex number]]s (with <math>\omega</math> being the analogon of ''i''). 
The additive [[Identity element|identity]] is <math>0</math>, more formal <math>0 + 0\omega</math>: Let <math>\alpha \in \mathbf{F}_{p^2}</math>, then
:<math>\alpha + 0 = (x+y\omega) + (0 + 0\omega) = (x + 0) + (y + 0)\omega = x+y\omega = \alpha</math>.
The multiplicative identity is <math>1</math>, or more formal <math> 1 + 0\omega</math>:
:<math>\alpha \cdot 1 = (x+y\omega)(1 + 0\omega) = \left(x\cdot 1 + 0 \cdot 0 \left(n^2-a\right)\right) + (x\cdot 0 + 1 \cdot x)\omega = x+y\omega = \alpha</math>.
The only thing left for <math>\mathbf{F}_{p^2}</math> being a field is the existence of additive and multiplicative [[Inverse element|inverses]]. It is easily seen that the additive inverse of <math>x+y\omega</math> is <math>-x-y\omega</math>, which is an element of <math>\mathbf{F}_{p^2}</math>, because <math>-x,-y \in \mathbf{F}_p</math>. In fact, those are the additive inverse elements of ''x'' and ''y''. For showing that every non-zero element <math>\alpha</math> has a multiplicative inverse, write down <math>\alpha = x_1 + y_1 \omega</math> and <math>\alpha^{-1} = x_2 + y_2 \omega</math>. In other words,
:<math>(x_1 + y_1 \omega)(x_2 + y_2 \omega) = \left( x_1 x_2 + y_1 y_2 \left(n^2-a\right)\right) + \left(x_1 y_2 + y_1 x_2 \right) \omega = 1</math>.
So the two equalities <math>x_1x_2 + y_1y_2(n^2-a) = 1</math> and <math>x_1y_2 + y_1x_2 = 0</math> must hold. Working out the details gives expressions for <math>x_2</math> and <math>y_2</math>, namely
:<math>x_2 = -y_1^{-1}x_1\left(y_1\left(n^2-a\right)-x_1^2y_1^{-1}\right)^{-1}</math>,
:<math>y_2 = \left( y_1 \left(n^2-a\right) - x_1^2y_1^{-1}\right)^{-1}</math>.
The inverse elements which are shown in the expressions of <math>x_2</math> and <math>y_2</math> do exist, because these are all elements of <math>\mathbf{F}_p</math>. This completes the first part of the proof, showing that <math>\mathbf{F}_{p^2}</math> is a field.

The second and middle part of the proof is showing that for every element <math>x+y\omega \in \mathbf{F}_{p^2} : (x+y\omega)^p = x - y\omega</math>.
By definition, <math>\omega^2=a^2-n</math> is not a square in <math>\mathbf{F}_p</math>. Euler's criterion then says that
:<math>\omega^{p-1} = \left(\omega^2\right)^{\frac{p-1}{2}} = -1</math>.
Thus <math>\omega^p = -\omega</math>. This, together with [[Fermat's little theorem]] (which says that <math>x^p = x</math> for all <math>x \in \mathbf{F}_{p}</math>) and the knowledge that in fields of [[Characteristic (algebra)|characteristic]] ''p'' the equation <math>\left(a+b\right)^p = a^p + b^p</math> holds, shows the desired result
:<math>(x+y\omega)^p = x^p + y^p \omega^p = x - y\omega</math>.

The third and last part of the proof is to show that if <math>x_0=\left(a+\omega \right)^{\frac{p+1}{2}} \in \mathbf{F}_{p^2}</math>, then <math>x_0^2=n \in \mathbf{F}_p</math>. 
Compute
:<math>x_0^2 = \left(a+\omega \right)^{p+1} = (a+\omega)(a+\omega)^{p}=(a+\omega)(a-\omega)=a^2 - \omega^2 = a^2 - \left(a^2 - n \right) = n</math>.
Note that this computation took place in <math>\mathbf{F}_{p^2}</math>, so this <math>x_0 \in \mathbf{F}_{p^2}</math>. But with [[Lagrange's theorem (number theory)|Lagrange's theorem]], stating that a non-zero [[Integer polynomial|polynomial]] of degree ''n'' has at most ''n'' roots in any field ''K'', and the knowledge that <math>x^2-n</math> has 2 roots in <math>\mathbf{F}_p</math>, these roots must be all of the roots in <math>\mathbf{F}_{p^2}</math>. It was just shown that <math>x_0</math> and <math>-x_0</math> are roots of <math>x^2-n</math> in <math>\mathbf{F}_{p^2}</math>, so it must be that <math>x_0, -x_0 \in \mathbf{F}_p</math>.<ref>[http://people.math.gatech.edu/~mbaker/pdf/cipolla2011.pdf M. Baker ''Cipolla's Algorithm for finding square roots mod p'']</ref>

==Speed of the algorithm==
After finding a suitable ''a'', the number of operations required for the algorithm is <math>4m + 2k - 4</math> multiplications, <math>4m-2</math> sums, where ''m'' is the number of [[Numerical digit|digits]] in the [[Binary numeral system|binary representation]] of ''p'' and ''k'' is the number of ones in this representation. To find ''a'' by trial and error, the expected number of computations of the Legendre symbol is 2. But one can be lucky with the first try and one may need more than 2 tries. In the field <math>\mathbf{F}_{p^2}</math>, the following two equalities hold
:<math>(x+y\omega)^2 = \left(x^2 + y^2 \omega^2 \right) + \left(\left(x+y\right)^2-x^2-y^2\right)\omega,</math>
where <math>\omega^2 = a^2-n</math> is known in advance. This computation needs 4 multiplications and 4 sums.
:<math>\left(x+y\omega\right)^2\left(c + \omega \right) = \left( cd - b\left(x+d\right)\right) + \left(d^2 - by\right)\omega,</math>
where <math>d=(x+yc)</math> and <math>b=ny</math>. This operation needs 6 multiplications and 4 sums.

Assuming that <math>p \equiv 1 \pmod 4,</math> (in the case <math>p \equiv 3 \pmod 4</math>, the direct computation <math>x \equiv \pm n^{\frac{p+1}{4}}</math> is much faster) the binary expression of <math>(p+1)/2</math> has <math>m-1</math> digits, of which ''k'' are ones. So for computing a <math>(p+1)/2</math> power of <math>\left(a + \omega \right)</math>, the first formula has to be used <math>n-k-1</math> times and the second <math>k-1</math> times.

For this, Cipolla's algorithm is better than the [[Tonelli-Shanks algorithm]] if and only if <math>S(S-1) > 8m+20</math>, with <math>2^{S}</math> being the maximum power of 2 which divides <math>p-1</math>.<ref>[http://www.springerlink.com/content/xgxe68edy03la96p/fulltext.pdf Gonzalo Tornaria ''Square roots modulo p'']</ref>

==References==

<references/>
* E. Bach, J.O. Shallit ''Algorithmic Number Theory: Efficient algorithms'' MIT Press, (1996)

{{number theoretic algorithms}}

[[Category:Modular arithmetic]]
[[Category:Number theoretic algorithms]]
[[Category:Articles containing proofs]]

Breakdown voltage

2013-11-24T14:13:14Z

117.194.80.164:

In [[mathematics]], the '''cohomology operation''' concept became central to [[algebraic topology]], particularly [[homotopy theory]], from the 1950s onwards, in the shape of the simple definition that if ''F'' is a [[functor]] defining a [[cohomology theory]], then a cohomology operation should be a [[natural transformation]] from ''F'' to itself. Throughout there have been two basic points:

#the operations can be studied by combinatorial means; and
#the effect of the operations is to yield an interesting [[bicommutant]] theory.

The origin of these studies was the work of Pontryagin, Postnikov, and [[Norman Steenrod]], who first defined the [[Pontryagin square]], [[Postnikov square]], and [[Steenrod square]] operations for [[singular cohomology]], in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a [[natural diagonal]] map, at [[cochain]] level. The general theory of the [[Steenrod algebra]] of operations has been brought into close relation with that of the [[symmetric group]].

In the [[Adams spectral sequence]] the ''bicommutant'' aspect is implicit in the use of [[Ext functor]]s, the [[derived functor]]s of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a ''derived'' level. The convergence is to groups in [[stable homotopy theory]], about which information is hard to come by. This connection established the deep interest of the cohomology operations for [[homotopy theory]], and has been a research topic ever since. An [[extraordinary cohomology theory]] has its own cohomology operations, and these may exhibit a richer set on constraints.

==Formal definition==
A '''cohomology operation''' <math>\theta</math> of type

:<math>(n,q,\pi,G)\,</math>

is a [[natural transformation]] of functors

:<math>\theta:H^{n}(-,\pi)\to H^{q}(-,G)\,</math>

defined on [[CW complex]]es.

==Relation to Eilenberg-MacLane spaces==
Cohomology of CW complexes is [[representable functor|representable]] by an [[Eilenberg-MacLane space]], so by the [[Yoneda lemma]] a cohomology operation of type <math>(n,q,\pi,G)</math> is given by a [[homotopy]] class of maps <math>K(\pi,n) \to K(G,q)</math>. Using [[representable functor|representability]] once again, the cohomology operation is given by an element of <math>H^{q}(K(\pi,n),G)</math>.

Symbolically, letting <math>[A,B]</math> denote the set of homotopy classes of maps from <math>A</math> to <math>B</math>,

::<math>\displaystyle\mathrm{Nat}(H^n(-,\pi),H^q(-,G)) = \mathrm{Nat}([-,K(\pi,n)],[-,K(G,q)]) = [K(\pi,n),K(G,q)] = H^q(K(\pi,n);G)</math>

==See also==

*[[Secondary cohomology operation]]

==References==

*{{Citation | last1=Mosher | first1=Robert E. | last2=Tangora | first2=Martin C. | title=Cohomology operations and applications in homotopy theory | origyear=1968 | url=http://books.google.com/books?id=FFCaPwAACAAJ | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-46664-4 | mr=0226634 | year=2008}}
*{{Citation | last1=Steenrod | first1=N. E. | editor1-last=Epstein | editor1-first=D. B. A. | editor1-link=David B. A. Epstein | title=Cohomology operations | url=http://books.google.com/books?id=CF3bt4oYZ2oC | publisher=[[Princeton University Press]] | series= Annals of Mathematics Studies | isbn=978-0-691-07924-0 | mr=0145525 | year=1962 | volume=50}}

{{DEFAULTSORT:Cohomology Operation}}
[[Category:Algebraic topology]]

Minimum deviation

2013-11-12T16:51:50Z

117.194.87.199:

In [[differential geometry]], '''Hilbert's theorem''' (1901) states that there exists no complete [[regular surface]] <math>S</math> of constant negative [[gaussian curvature]] <math>K</math> [[immersion (mathematics)|immersed]] in <math>\mathbb{R}^{3}</math>. This theorem answers the question for the negative case of which surfaces in <math>\mathbb{R}^{3}</math> can be obtained by isometrically immersing [[complete manifold]]s with [[constant curvature]].

Hilbert's theorem was first treated by [[David Hilbert]] in, "Über Flächen von konstanter Krümmung" ([[Trans. Amer. Math. Soc.]] 2 (1901), 87-99). A different proof was given shortly after by E. Holmgren, "Sur les surfaces à courbure constante negative," (1902).

==Proof==
The [[proof (mathematics)|proof]] of Hilbert's theorem is elaborate and requires several [[lemma (mathematics)|lemma]]s. The idea is to show the nonexistence of an isometric [[immersion (mathematics)|immersion]]
:<math>\varphi = \psi \circ \exp_p: S' \longrightarrow \mathbb{R}^{3}</math>

of a plane <math>S'</math> to the real space <math>\mathbb{R}^{3}</math>. This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and [[Michael Spivak|Spivak]].

''Observations'': In order to have a more manageable treatment, but without loss of generality, the [[curvature]] may be considered equal to minus one, <math>K=-1</math>. There is no loss of generality, since it is being dealt with constant curvatures, and similarities of <math>\mathbb{R}^{3}</math> multiply <math>K</math> by a constant. The [[exponential map]] <math>\exp_p: T_p(S) \longrightarrow S</math> is a [[local diffeomorphism]] (in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an [[inner product]] in the [[tangent space]] of <math>S</math> at <math>p</math>: <math>T_p(S)</math>. Furthermore, <math>S'</math> denotes the geometric surface <math>T_p(S)</math> with this inner product. If <math>\psi:S \longrightarrow \mathbb{R}^{3}</math> is an isometric immersion, the same holds for
:<math>\varphi = \psi \circ \exp_o:S' \longrightarrow \mathbb{R}^{3}</math>.

The first lemma is independent from the other ones, and will be used at the end as the counter statement to reject the results from the other lemmas.

'''Lemma 1''': The area of <math>S'</math> is infinite. 
''Proof's Sketch:'' 
The idea of the proof is to create a [[global isometry]] between <math>H</math> and <math>S'</math>. Then, since <math>H</math> has an infinite area, <math>S'</math> will have it too. 
The fact that the [[Hyperbolic manifold|hyperbolic plane]] <math>H</math> has an infinite area comes by computing the [[surface integral]] with the corresponding [[coefficient]]s of the [[First fundamental form]]. To obtain these ones, the hyperbolic plane can be defined as the plane with the following inner product around a point <math>q\in \mathbb{R}^{2}</math> with coordinates <math>(u,v)</math> 
:<math>E = \left\langle \frac{\partial}{\partial u}, \frac{\partial}{\partial u} \right\rangle = 1 \qquad F = \left\langle \frac{\partial}{\partial u}, \frac{\partial}{\partial v} \right\rangle = \left\langle \frac{\partial}{\partial v}, \frac{\partial}{\partial u} \right\rangle = 0 \qquad G = \left\langle \frac{\partial}{\partial v}, \frac{\partial}{\partial v} \right\rangle = e^{u} </math> 

Since the hyperbolic plane is unbounded, the limits of the integral are [[Infinity|infinite]], and the area can be calculated through
:<math>\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{u} du dv = \infty</math>

Next it is needed to create a map, which will show that the global information from the hyperbolic plane can be transfer to the surface <math>S'</math>, i.e. a global isometry. <math>\varphi: H \rightarrow S'</math> will be the map, whose domain is the hyperbolic plane and image the [[2-dimensional manifold]] <math>S'</math>, which carries the inner product from the surface <math>S</math> with negative curvature. <math>\varphi</math> will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces,
:<math>\psi:T_p(H) \rightarrow T_{p'}(S')</math>.

That is
:<math>\varphi = \exp_{p'} \circ \psi \circ \exp_p^{-1}</math>,

where <math>p\in H, p' \in S'</math>. That is to say, the starting point <math>p\in H</math> goes to the tangent plane from <math>H</math> through the inverse of the exponential map. Then travels from one tangent plane to the other through the isometry <math>\psi</math>, and then down to the surface <math>S'</math> with another exponential map.

The following step involves the use of [[polar coordinates]], <math>(\rho, \theta)</math> and <math>(\rho', \theta')</math>, around <math>p</math> and <math>p'</math> respectively. The requirement will be that the axis are mapped to each other, that is <math>\theta=0</math> goes to <math>\theta'=0</math>. Then <math>\varphi</math> preserves the first fundamental form. 
In a geodesic polar system, the [[Gaussian curvature]] <math>K</math> can be expressed as
:<math>K = - \frac{(\sqrt{G})_{\rho \rho}}{\sqrt{G}}</math>.

In addition K is constant and fulfills the following differential equation
:<math>(\sqrt{G})_{\rho \rho} + K\cdot \sqrt{G} = 0</math>

Since <math>H</math> and <math>S'</math> have the same constant Gaussian curvature, then they are locally isometric ([[Minding's Theorem]]). That means that <math>\varphi</math> is a local isometry between <math>H</math> and <math>S'</math>. Furthermore, from the Hadamard's theorem it follows that <math>\varphi</math> is also a covering map. 
Since <math>S'</math> is simply connected, <math>\varphi</math> is a homeomorphism, and hence, a (global) isometry. Therefore, <math>H</math> and <math>S'</math> are globally isometric, and because <math>H</math> has an infinite area, then <math>S'=T_p(S)</math> has an infinite area, as well. <math>\square</math>

'''Lemma 2''': For each <math>p\in S'</math> exists a parametrization <math>x:U \subset \mathbb{R}^{2} \longrightarrow S', \qquad p \in x(U)</math>, such that the coordinate curves of <math>x</math> are asymptotic curves of <math> x(U) = V'</math> and form a Tchebyshef net.

'''Lemma 3''': Let <math>V' \subset S'</math> be a coordinate [[neighborhood]] of <math>S'</math> such that the coordinate curves are asymptotic curves in <math>V'</math>. Then the area A of any quadrilateral formed by the coordinate curves is smaller than <math>2\pi</math>.

The next goal is to show that <math>x</math> is a parametrization of <math>S'</math>.

'''Lemma 4''': For a fixed <math>t</math>, the curve <math>x(s,t), -\infty < s < +\infty </math>, is an asymptotic curve with <math>s</math> as arc length.

The following 2 lemmas together with lemma 8 will demonstrate the existence of a [[parametrization]] <math>x:\mathbb{R}^{2} \longrightarrow S'</math>

'''Lemma 5''': <math>x</math> is a local diffeomorphism.

'''Lemma 6''': <math>x</math> is [[surjective]].

'''Lemma 7''': On <math>S'</math> there are two differentiable linearly independent vector fields which are tangent to the [[asymptotic curve]]s of <math>S'</math>.

'''Lemma 8''': <math>x</math> is [[injective]]. 

''Proof of Hilbert's Theorem:'' 
First, it will be assumed that an isometric immersion from a [[complete surface]] with negative curvature<math>S</math> exists: <math>\psi:S \longrightarrow \mathbb{R}^{3}</math>

As stated in the observations, the tangent plane <math>T_p(S)</math> is endowed with the metric induced by the exponential map <math>\exp_p: T_p(S) \longrightarrow S</math> . Moreover, <math>\varphi = \psi \circ \exp_p:S' \longrightarrow \mathbb{R}^{3}</math> is an isometric immersion and Lemmas 5,6, and 8 show the existence of a parametrization <math>x:\mathbb{R}^{2} \longrightarrow S'</math> of the whole <math>S'</math>, such that the coordinate curves of <math>x</math> are the asymptotic curves of <math>S'</math>. This result was provided by Lemma 4. Therefore, <math>S'</math> can be covered by a union of "coordinate" quadrilaterals <math>Q_{n}</math> with <math> Q_{n} \subset Q_{n+1}</math>. By Lemma 3, the area of each quadrilateral is smaller than <math>2 \pi </math>. On the other hand, by Lemma 1, the area of <math>S'</math> is infinite, therefore has no bounds. This is a contradiction and the proof is concluded. <math>\square</math>

==References==
* {{aut|Do Carmo, Manfredo}}, ''Differential Geometry of Curves and Surfaces'', Prentice Hall, 1976.
* {{aut|[[Michael Spivak|Spivak, Michael]]}}, ''A Comprenhensive Introduction to Differential Geometry'', Publish or Perish, 1999.

{{DEFAULTSORT:Hilberts theorem}}
[[Category:Hyperbolic geometry]]
[[Category:Theorems in differential geometry]]
[[Category:Articles containing proofs]]

Supplee's paradox

2013-08-28T03:03:24Z

117.194.82.117:

In [[computer science]], an '''interval tree''' is an [[ordered tree data structure|ordered tree]] [[data structure]] to hold [[Interval (mathematics)|intervals]]. Specifically, it allows one to efficiently find all intervals that overlap with any given interval or point. It is often used for windowing queries, for instance, to find all roads on a computerized map inside a rectangular viewport, or to find all visible elements inside a three-dimensional scene. A similar data structure is the [[segment tree]].

The trivial solution is to visit each interval and test whether it intersects the given point or interval, which requires Θ(''n'') time, where ''n'' is the number of intervals in the collection. Since a query may return all intervals, for example if the query is a large interval intersecting all intervals in the collection, this is [[asymptotically optimal]]; however, we can do better by considering [[output-sensitive algorithm]]s, where the runtime is expressed in terms of ''m'', the number of intervals produced by the query. Interval trees are dynamic, i.e., they allow insertion and deletion of intervals. They obtain a query time of O(log ''n'') while the preprocessing time to construct the data structure is O(''n'' log ''n'') (but the space consumption is O(''n'')). If the endpoints of intervals are within a small integer range (e.g., in the range [1,...,O(''n'')]), faster data structures exist with preprocessing time O(''n'') and query time O(1+''m'') for reporting ''m'' intervals containing a given query point.

==Naive approach==
In a simple case, the intervals do not overlap and they can be inserted into a simple [[binary search tree]] and queried in O(log ''n'') time. However, with arbitrarily overlapping intervals, there is no way to compare two intervals for insertion into the tree since orderings sorted by the beginning points or the ending points may be different. A naive approach might be to build two parallel trees, one ordered by the beginning point, and one ordered by the ending point of each interval. This allows discarding half of each tree in O(log ''n'') time, but the results must be merged, requiring O(''n'') time. This gives us queries in O(''n'' + log ''n'') = O(''n''), which is no better than brute-force.

Interval trees solve this problem. This article describes two alternative designs for an interval tree, dubbed the ''centered interval tree'' and the ''augmented tree''.

== Centered interval tree ==

Queries require O(log ''n'' + ''m'') time, with ''n'' being the total number of intervals and ''m'' being the number of reported results. Construction requires O(''n'' log ''n'') time, and storage requires O(''n'') space.

=== Construction ===

Given a set of ''n'' intervals on the number line, we want to construct a data structure so that we can efficiently retrieve all intervals overlapping another interval or point.

We start by taking the entire range of all the intervals and dividing it in half at ''x_center'' (in practice, ''x_center'' should be picked to keep the tree relatively balanced). This gives three sets of intervals, those completely to the left of ''x_center'' which we'll call ''S_left'', those completely to the right of ''x_center'' which we'll call ''S_right'', and those overlapping ''x_center'' which we'll call ''S_center''.

The intervals in ''S_left'' and ''S_right'' are recursively divided in the same manner until there are no intervals left.

The intervals in S_center that overlap the center point are stored in a separate data structure linked to the node in the interval tree. This data structure consists of two lists, one containing all the intervals sorted by their beginning points, and another containing all the intervals sorted by their ending points.

The result is a ternary tree with each node storing:
* A center point
* A pointer to another node containing all intervals completely to the left of the center point
* A pointer to another node containing all intervals completely to the right of the center point
* All intervals overlapping the center point sorted by their beginning point
* All intervals overlapping the center point sorted by their ending point

=== Intersecting ===

Given the data structure constructed above, we receive queries consisting of ranges or points, and return all the ranges in the original set overlapping this input.

==== With an Interval ====

First, we can reduce the case where an interval ''R'' is given as input to the simpler case where a single point is given as input. We first find all ranges with beginning or end points inside the input interval ''R'' using a separately constructed tree. In the one-dimensional case, we can use a simple tree containing all the beginning and ending points in the interval set, each with a pointer to its corresponding interval.

A binary search in O(log ''n'') time for the beginning and end of R reveals the minimum and maximum points to consider. Each point within this range references an interval that overlaps our range and is added to the result list. Care must be taken to avoid duplicates, since an interval might both begin and end within ''R''. This can be done using a binary flag on each interval to mark whether or not it has been added to the result set.

The only intervals not yet considered are those overlapping ''R'' that do not have an endpoint inside ''R'', in other words, intervals that enclose it. To find these, we pick any point inside ''R'' and use the algorithm below to find all intervals intersecting that point (again, being careful to remove duplicates).

==== With a Point ====

The task is to find all intervals in the tree that overlap a given point ''x''. The tree is walked with a similar recursive algorithm as would be used to traverse a traditional binary tree, but with extra affordance for the intervals overlapping the "center" point at each node.

For each tree node, ''x'' is compared to ''x_center'', the midpoint used in node construction above. If ''x'' is less than ''x_center'', the leftmost set of intervals, ''S_left'', is considered. If ''x'' is greater than ''x_center'', the rightmost set of intervals, ''S_right'', is considered.

As each node is processed as we traverse the tree from the root to a leaf, the ranges in its ''S_center'' are processed. If ''x'' is less than ''x_center'', we know that all intervals in ''S_center'' end after ''x'', or they could not also overlap ''x_center''. Therefore, we need only find those intervals in ''S_center'' that begin before ''x''. We can consult the lists of ''S_center'' that have already been constructed. Since we only care about the interval beginnings in this scenario, we can consult the list sorted by beginnings. Suppose we find the closest number no greater than ''x'' in this list. All ranges from the beginning of the list to that found point overlap ''x'' because they begin before ''x'' and end after ''x'' (as we know because they overlap ''x_center'' which is larger than ''x''). Thus, we can simply start enumerating intervals in the list until the endpoint value exceeds ''x''.

Likewise, if ''x'' is greater than ''x_center'', we know that all intervals in ''S_center'' must begin before ''x'', so we find those intervals that end after ''x'' using the list sorted by interval endings.

If ''x'' exactly matches ''x_center'', all intervals in ''S_center'' can be added to the results without further processing and tree traversal can be stopped.

=== Higher Dimensions ===

The interval tree data structure can be generalized to a higher dimension ''N'' with identical query and construction time and O(''n'' log ''n'') space.

First, a [[range tree]] in ''N'' dimensions is constructed that allows efficient retrieval of all intervals with beginning and end points inside the query region ''R''. Once the corresponding ranges are found, the only thing that is left are those ranges that enclose the region in some dimension. To find these overlaps, N interval trees are created, and one axis intersecting ''R'' is queried for each. For example, in two dimensions, the bottom of the square ''R'' (or any other horizontal line intersecting R) would be queried against the interval tree constructed for the horizontal axis. Likewise, the left (or any other vertical line intersecting R) would be queried against the interval tree constructed on the vertical axis.

Each interval tree also needs an addition for higher dimensions. At each node we traverse in the tree, ''x'' is compared with ''S_center'' to find overlaps. Instead of two sorted lists of points as was used in the one-dimensional case, a range tree is constructed. This allows efficient retrieval of all points in ''S_center'' that overlap region ''R''.

=== Deletion ===

If after deleting an interval from the tree, the node containing that interval contains no more intervals, that node may be deleted from the tree. This is more complex than a normal binary tree deletion operation.

An interval may overlap the center point of several nodes in the tree. Since each node stores the intervals that overlap it, with all intervals completely to the left of its center point in the left subtree, similarly for the right subtree, it follows that each interval is stored in the node closest to the root from the set of nodes whose center point it overlaps.

Normal deletion operations in a binary tree (for the case where the node being deleted has two children) involve promoting a node further from the root to the position of the node being deleted (usually the leftmost child of the right subtree, or the rightmost child of the left subtree). As a result of this promotion, some nodes that were above the promoted node will become descendents of it; it is necessary to search these nodes for intervals that also overlap the promoted node, and move those intervals into the promoted node. As a consequence, this may result in new empty nodes, which must be deleted, following the same algorithm again.

=== Balancing ===

The same issues that affect deletion also affect rotation operations; rotation must preserve the invariant that intervals are stored as close to the root as possible.

== Augmented tree ==

Another way to represent intervals is described in {{harvtxt|Cormen|Leiserson|Rivest|Stein|2001|loc=Section 14.3: Interval trees, pp. 311–317}}.

Both insertion and deletion require O(log ''n'') time, with ''n'' being the total number of intervals.

Use a simple ordered tree, for example a [[binary search tree]] or [[self-balancing binary search tree]], where the tree is ordered by the 'low' values of the intervals, and an extra annotation is added to every node recording the maximum high value of both its subtrees. It is simple to maintain this attribute in only O(''h'') steps during each addition or removal of a node, where ''h'' is the height of the node added or removed in the tree, by updating all ancestors of the node from the bottom up. Additionally, the [[tree rotation]]s used during insertion and deletion may require updating the high value of the affected nodes.

Now, it is known that two intervals ''A'' and ''B'' overlap only when both ''A''.low ≤ ''B''.high and ''A''.high ≥ ''B''.low. When searching the trees for nodes overlapping with a given interval, you can immediately skip:
* all nodes to the right of nodes whose low value is past the end of the given interval.
* all nodes that have their maximum 'high' value below the start of the given interval.

A total order can be defined on the intervals by ordering them first by their 'low' value and finally by their
'high' value. This ordering can be used to prevent duplicate intervals from being inserted into the tree in O(log ''n'') time, versus the O(''k'' + log ''n'') time required to find duplicates if ''k'' intervals overlap a new interval.

===Java Example: Adding a new interval to the tree===

The key of each node is the interval itself and the value of each node is the end point of the interval:

<source lang=java>
public void add(Interval i) {
put(i, i.getEnd());
}
</source>

===Java Example: Searching a point or an interval in the tree===

To search for an interval, you walk the tree, omitting those branches which can't contain what you're looking for. The simple case is looking for a point:

<source lang=java>
// Search for all intervals which contain "p", starting with the
// node "n" and adding matching intervals to the list "result"
public void search(IntervalNode n, Point p, List<Interval> result) {
// Don't search nodes that don't exist
if (n == null)
return;

// If p is to the right of the rightmost point of any interval
// in this node and all children, there won't be any matches.
if (p.compareTo(n.getValue()) > 0)
return;

// Search left children
if (n.getLeft() != null)
search(IntervalNode (n.getLeft()), p, result);

// Check this node
if (n.getKey().contains(p))
result.add(n.getKey());

// If p is to the left of the start of this interval,
// then it can't be in any child to the right.
if (p.compareTo(n.getKey().getStart()) < 0)
return;

// Otherwise, search right children
if (n.getRight() != null)
search(IntervalNode (n.getRight()), p, result);
}
</source>

The code to search for an interval is similar, except for the check in the middle:
<source lang=java>
// Check this node
if (n.getKey().overlapsWith(i))
result.add (n.getKey());
</source>

<tt>overlapsWith()</tt> is defined as:
<source lang=java>
public boolean overlapsWith(Interval other) {
return start.compareTo(other.getEnd()) <= 0 &&
end.compareTo(other.getStart()) >= 0;
}
</source>

===Higher dimension===

This can be extended to higher dimensions by cycling through the dimensions at each level of the tree. For example, for two dimensions, the odd levels of the tree might contain ranges for the ''x'' coordinate, while the even levels contain ranges for the ''y'' coordinate. However, it is not quite obvious how the rotation logic will have to be extended for such cases to keep the tree balanced.

A much simpler solution is to use nested interval trees. First, create a tree using the ranges for the ''y'' coordinate. Now, for each node in the tree, add another interval tree on the ''x'' ranges, for all elements whose ''y'' range intersect that node's ''y'' range.

The advantage of this solution is that it can be extended to an arbitrary amount of dimensions using the same code base.

At first, the cost for the additional trees might seem prohibitive but that is usually not the case. As with the solution above, you need one node per ''x'' coordinate, so this cost is the same in both solutions. The only difference is that you need an additional tree structure per vertical interval. This structure is typically very small (a pointer to the root node plus maybe the number of nodes and the height of the tree).

==Medial/length oriented tree==
Similar to Augmented tree, but in a symmetrical way, where the [[Binary Search Tree]] is ordered by the Medial point of intervals. And there is a Maximum-oriented [[Binary Heap]] in every node, ordered by the length of interval (or half of the length). Also we store minimum possible value of the subtree in each node, additional to maximum possible value (this is how it is symmetrical).

===Overlap test===
Using only start and end values of two intervals <math>\left( a_{i}, b_i \right)</math>, for <math>i=0,1</math>, the overlap test can be performed like:

<math>a_0 \leqslant a_1 < b_0</math>    OR   
<math>a_0 < b_1 \leqslant b_0</math>    OR   
<math>a_1 \leqslant a_0 < b_1</math>    OR   
<math>a_1 < b_0 \leqslant b_1</math>

But with defining:

<math>m_i = \frac{a_i + b_i}{2}</math>

<math>d_i = \frac{b_i - a_i}{2}</math>

The overlap test is simpler:

<math>\left| m_1 - m_0 \right| < d_0 + d_1</math>

===Adding interval===
Adding new intervals to the tree is the same as BST, just we use medial value as the key, and when we found/created the node to put the interval. We should push <math>d_i</math> to the [[Binary Heap]] associated to node. And update minimum and maximum possible values associated with all higher nodes.

===Searching for all overlapping intervals===
Let's use <math>a_q, b_q, m_q, d_q</math> for the query interval, and <math>M_n</math> for the key of a node (compared to <math>m_i</math> of intervals)

Starting with root node, in each node, first we check if it is possible that our query interval overlaps with the node subtree using minimum and maximum values of node (if it is not, we don't continue for this node).

Then we calculate <math>\min \left\{ d_i \right\}</math> for intervals inside this node (not its children) to overlap with query interval (knowing <math>m_i = M_n</math>):

<math>\min \left\{ d_i \right\} = \left| m_q - M_n \right| - d_q</math>

And perform a query on its [[binary heap]] for the <math>d_i</math>'s bigger than <math>\min \left\{ d_i \right\}</math>

Then we pass through both left and right children of node, doing the same thing.
In the worst-case, we have to scan all nodes of BST, but since [[Binary Heap]] query is optimum, there is not much worries (a 2- dimensional problem can not be optimum in both dimensions)

This algorithm is expected to be faster than traditional Interval Tree (Augmented tree) in search operation, adding is just a little bit slower (order of growth is the same).

== References ==
* [[Mark de Berg]], [[Marc van Kreveld]], [[Mark Overmars]], and [[Otfried Schwarzkopf]]. ''Computational Geometry'', Second Revised Edition. Springer-Verlag 2000. Section 10.1: Interval Trees, pp. 212–217.
* {{citation
| last=Cormen | first=Thomas H. | author-link =Thomas H. Cormen
| first2=Charles E. | last2=Leiserson | author2-link=Charles E. Leiserson
| first3=Ronald L. | last3=Rivest | author3-link=Ronald L. Rivest
| first4=Clifford | last4=Stein | author4-link=Clifford Stein
| title=[[Introduction to Algorithms]]
| edition=2nd
| publisher=MIT Press and McGraw-Hill
| year=2001
| isbn=0-262-03293-7
}}
* [[Franco P. Preparata]] and [[Michael Ian Shamos]]. ''Computational Geometry: An Introduction''. Springer-Verlag, 1985
* [[Jens M. Schmidt]]. ''Interval Stabbing Problems in Small Integer Ranges''. [http://dx.doi.org/10.1007/978-3-642-10631-6_18 DOI]. ISAAC'09, 2009

== External links ==
* [http://www.cgal.org/ CGAL : Computational Geometry Algorithms Library in C++] contains a robust implementation of Range Trees
* [http://code.google.com/p/intervaltree/ Interval Tree (an augmented self balancing avl tree implementation)]
* [https://github.com/gam3/interval-tree/ Interval Tree (a ruby implementation)]

{{CS-Trees}}

[[Category:Trees (data structures)]]

Faraday's laws of electrolysis

2012-08-29T15:41:13Z

117.194.231.5: /* Statements of the laws */

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2012-08-25T10:49:17Z

117.194.195.251: /* Component equivalents */

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