formulasearchengine - User contributions [en]

Travelling salesman problem

2015-01-11T16:33:41Z

129.97.124.179: /* appraoching the exact length */

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Pell's equation

2014-12-09T14:52:00Z

129.97.124.69: grammar

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Time–frequency analysis for music signals

2014-11-24T20:29:20Z

129.97.124.108: /* STFT example */

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Representation theory of finite groups

2014-11-20T16:52:26Z

129.97.124.227: Reverted confusing phrasing

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Abel's theorem

2014-08-13T19:13:57Z

129.97.9.98: /* Applications */

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Amenable Banach algebra

2014-07-27T21:14:24Z

129.97.125.113: /* Examples */

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Transmission coefficient

2014-03-11T17:02:56Z

129.97.59.100: /* Opticss */

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Ideal (ring theory)

2014-02-28T22:13:43Z

129.97.93.58: /* Definitions */

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Positive element

2014-02-27T00:07:30Z

129.97.93.58:

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Atkinson's theorem

2014-02-12T23:35:24Z

129.97.93.58: /* Sketch of proof */

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Stone–von Neumann theorem

2014-02-07T05:39:41Z

129.97.131.0: /* Representation issues of the commutation relations */ Clarified confusing notation

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Compact operator

2014-02-03T22:25:23Z

129.97.93.58: /* Important properties */ Changed "transpose" to "adjoint"

{{about|car handling|the topic in aviation|Sideslip angle}}
[[File:TreadDeflected1.jpg|thumb|350px|'Deflected' tread path, sideslip velocity and slip angle]] 
In [[vehicle dynamics]], '''slip angle'''<ref name="Pacejka">{{cite book
| title = Tire and Vehicle Dynamics
| last = Pacejka
| first = Hans B.
| edition = 2nd
| publisher = Society of Automotive Engineers
| isbn = 0-7680-1702-5
| pages = 3}}</ref> or '''sideslip angle'''<ref name="Cossalter">{{cite book
| title = Motorcycle Dynamics
| edition = Second
| last = Cossalter
| first = Vittore
| year = 2006
| publisher = Lulu.com
| isbn = 978-1-4303-0861-4
| pages = 47,111}}</ref> is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing (i.e., the angle of the vector sum of wheel forward velocity <math>v_x</math> and lateral velocity <math>v_y</math>).<ref name="Pacejka"/> For a free-rolling wheel this slip angle results in a force parallel to the axle and the component of the force perpendicular to the wheel's direction of travel is the [[cornering force]]. This cornering force increases approximately linearly for the first few degrees of slip angle, then increases non-linearly to a maximum before beginning to decrease.

The slip angle, <math>\alpha</math> is defined as 
<center><math>\alpha \triangleq -\arctan\left(\frac{v_y}{|v_x|}\right)</math></center>

==Causes==
A non-zero slip angle arises because of deformation in the [[tire]] carcass and tread. As the tire rotates, the friction between the contact patch and the road results in individual tread 'elements' (finite sections of tread) remaining stationary with respect to the road. If a side-slip velocity ''u'' is introduced, the [[contact patch]] will be deformed. As a tread element enters the contact patch the friction between road and tire means that the tread element remains stationary, yet the tire continues to move laterally. This means that the tread element will be ‘deflected’ sideways. In reality it is the tire/wheel that is being deflected away from the stationary tread element, but convention is for the co-ordinate system to be fixed around the wheel mid-plane.

As the tread element moves through the contact patch it will be deflected further from the wheel mid-plane. This deflection gives rise to the slip angle, and to the [[cornering force]]. The rate at which the cornering force builds up is described by the [[relaxation length]].

==Effects==
The ratios between the slip angles of the front and rear axles (a function of the slip angles of the front and rear tires respectively) will determine the vehicle's behavior in a given turn. If the ratio of front to rear slip angles is greater than 1:1, the vehicle will tend to [[understeer]], while a ratio of less than 1:1 will produce [[oversteer]].<ref name="Cossalter"/> Actual instantaneous slip angles depend on many factors, including the condition of the road surface, but a vehicle's [[suspension (vehicle)|suspension]] can be designed to promote specific dynamic characteristics. A principal means of adjusting developed slip angles is to alter the relative roll couple (the rate at which weight transfers from the inside to the outside wheel in a turn) front to rear by varying the relative amount of front and rear lateral [[load transfer]]. This can be achieved by modifying the height of the [[roll center]]s, or by adjusting [[roll stiffness]], either through suspension changes or the addition of an [[anti-roll bar]].

Because of asymmetries in the side-slip along the length of the contact patch, the resultant force of this side-slip occurs away from the geometric center of the contact patch, a distance described as the [[pneumatic trail]], and so creates a torque on the tire.

== Measurement of slip angle ==
There are two main ways to measure slip angle of a tire: on a vehicle as it moves, or on a dedicated testing device.

There are a number of devices which can be used to measure slip angle on a vehicle as it moves; some use optical methods, some use inertial methods, some [[GPS]] and some both GPS and inertial.

Various test machines have been developed to measure slip angle in a controlled environment. A [[motorcycle tire]] test machine is located at the [[University of Padua]]. That uses a 3 meter diameter disk that rotates under a tire held at a fixed steer and camber angle, up to 54 degrees. Sensors measure the force and moment generated, and a correction is made to account for the curvature of the track.<ref name="Cossalter"/> Other devices use the inner or outer surface of rotating drums, sliding planks, conveyor belts, or a trailer that presses the test tire to an actual road surface.<ref name="Pacejka"/>

==See also==
*[[Camber thrust]]
*[[Cornering force]]
*[[Slip (vehicle dynamics)]]
*[[Traction Circle]]

== References ==
{{reflist}}

[[Category:Car safety]]
[[Category:Automotive steering technologies]]
[[Category:Automotive suspension technologies]]
[[Category:Tires]]
[[Category:Motorcycle dynamics]]

Fenwick tree

2013-12-05T23:25:16Z

129.97.131.0: /* Applications */ capitalize

{{ Infobox scientist
| name = Gábor Tardos
| image = Gábor Tardos.jpg
| image_size =
| caption =
| birth_date = {{birth date and age|1964|07|11|df=y}}
| birth_place = [[Budapest]]
| death_date =
| death_place =
| nationality = Hungarian
| fields = [[Mathematics]]
| workplaces = [[Simon Fraser University]]
| alma_mater = [[Hungarian Academy of Sciences]]
| doctoral_advisor = [[László Babai]]
| doctoral_students =
| known_for =
| awards = [[Paul Erdős Prize|Erdős Prize]] (2000) [[EMS Prize]] (1992)
}}
'''Gábor Tardos''' (born 11 July 1964) is a [[Hungarian people|Hungarian]][[ mathematician]], currently a professor and [[Canada Research Chair]] at [[Simon Fraser University]]. He works mainly in [[combinatorics]] and [[computer science]]. He is the younger brother of [[Éva Tardos]].<ref>[http://blog.computationalcomplexity.org/2009/02/baseball-families-and-math-families.html Baseball Families and Math Families], William Gasarch, February 12, 2009.</ref>

==Mathematical results==
Tardos started with a result in [[universal algebra]]: he exhibited a maximal [[clone (algebra)|clone]] of monotone operations which is not finitely generated. He obtained partial results concerning the [[Hanna Neumann conjecture]]. With his student, [[Adam Marcus (mathematician)|Adam Marcus]], he proved a combinatorial conjecture of [[Zoltán Füredi]] and [[Péter Hajnal]] which was known to imply the [[Stanley–Wilf conjecture]]. With topological methods he proved that if <math>\mathcal{H}</math> is a finite set system consisting of the unions of intervals on two disjoint lines, then <math>\tau(\mathcal{H})\leq 2\nu(\mathcal{H})</math> holds, where <math>\tau(\mathcal{H})</math> is the least number of points covering all elements of <math>\mathcal{H}</math> and <math>\nu(\mathcal{H})</math> is the size of the largest disjoint subsystem of <math>\mathcal{H}</math>. Tardos worked out a method for optimal probabilistic fingerprint codes. Although the mathematical content is hard, the algorithm is easy to implement.

==Awards==
He received the European Mathematical Society prize for young researchers at the [[European Congress of Mathematics]] in 1992 and the [[Paul Erdős Prize|Erdős Prize]] from the [[Hungarian Academy of Sciences]] in 2000. He received a Lendület Grant from the Hungarian Academy of Sciences (2009).<ref>[http://www.mta.hu/index.php?id=634&no_cache=1&backPid=390&tt_news=11051&cHash=10423ab5d6 Lendületben az MTA]</ref> specifically devised to keep outstanding researchers in Hungary.

==Selected publications==
*{{Citation |first=G. |last=Tardos |authormask=3 |title=Optimal probabilistic fingerprint codes |journal=[[Journal of the ACM]] |volume=55 |year=2008 | doi =10.1145/780542.780561 }}.
*{{Citation |first=G. |last=Tardos |authormask=3 |title=Transversals of 2-intervals, a topological approach |journal=[[Combinatorica]] |volume=15 |year=1995 |pages=123–134 }}.
*{{Citation |first=G. |last=Tardos |authormask=3 |first2=S. |last2=Ben-David |first3=A. |last3=Borodin |authorlink4=Richard Karp |first4=R. |last4=Karp |authorlink5=Avi Wigderson |first5=A. |last5=Wigderson |title=On the power of randomization in on-line algorithms |journal=[[Algorithmica]] |volume=11 |year=1994 |pages=2–14 }}.
*{{Citation |first=G. |last=Tardos |authormask=3 |title=A maximal clone of monotone operations which is not finitely generated |journal=[[Order (journal)|Order]] |volume=3 |year=1986 |pages=211–218 }}.

==References==
{{Reflist}}

==External links==
*{{MathGenealogy |id=99198}}

{{Authority control|VIAF=2784220}}
{{Persondata 
| NAME = Tardos, Gabor
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Hungarian mathematician
| DATE OF BIRTH = 11 July 1964
| PLACE OF BIRTH = [[Budapest]]
| DATE OF DEATH =
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Tardos, Gabor}}
[[Category:1964 births]]
[[Category:Living people]]
[[Category:Hungarian mathematicians]]
[[Category:Hungarian computer scientists]]
[[Category:Combinatorialists]]
[[Category:20th-century mathematicians]]
[[Category:21st-century mathematicians]]
[[Category:Simon Fraser University faculty]]
[[Category:Canada Research Chairs]]
[[Category:International Mathematical Olympiad participants]]

Parallel axis theorem

2013-11-29T19:17:37Z

129.97.9.60: /* Mass moment of inertia */

In [[mathematical logic]], [[computational complexity theory]], and [[computer science]], the '''existential theory of the [[real number|reals]]''' is the set of all true sentences of the form

: <math> \exists X_1 \cdots \exists X_k \, F(X_1,\dots, X_k), \, </math>

where ''F''(''X''1, ..., ''X''''k'') is a [[quantifier-free formula]] in the language of [[ordered field]]s with coefficients in a [[real closed field]].<ref name="bpr06">{{citation|contribution=Existential theory of the reals|first1=Saugata|last1=Basu|first2=Richard|last2=Pollack|first3=Marie-Françoise|last3=Roy|title=Algorithms in Real Algebraic Geometry|publisher=Springer-Verlag|pages=505–532|doi=10.1007/3-540-33099-2_14|series=Algorithms and Computation in Mathematics|volume=10|year=2006|edition=2nd|isbn=978-3-540-33098-1}}.</ref>

The [[decision problem]] for the existential theory of the reals is the problem of finding an [[algorithm]] that decides, for each such formula, whether it is true or false. This decision problem is [[NP-hard]] and lies in [[PSPACE]]. Thus, it has significantly lower complexity than [[Alfred Tarski]]'s [[Real closed field#Decidability and quantifier elimination|quantifier elimination]] procedure for deciding statements in the first-order theory of the reals without the restriction to existential quantifiers.<ref name="bpr06"/>

==Complete problems==
Several problems in computational complexity and [[geometric graph theory]] may be classified as [[Complete (complexity)|complete]] for the existential theory of the reals. These include:
* recognition of [[intersection graph]]s of [[line segment]]s in the plane (that is, given an undirected graph, determining whether there is a set of line segments that has an isomorphic intersection graph);<ref name="s09"/><ref>{{citation
| last1 = Kratochvíl | first1 = Jan
| last2 = Matoušek | first2 = Jiří | author2-link = Jiří Matoušek (mathematician)
| doi = 10.1006/jctb.1994.1071
| issue = 2
| journal = [[Journal of Combinatorial Theory]] | series = Series B
| mr = 1305055
| pages = 289–315
| title = Intersection graphs of segments
| volume = 62
| year = 1994}}.</ref>
* recognition of [[unit disk graph]]s (again, given only the graph itself as input);<ref>{{citation|last1=Kang|first1=Ross J.|last2=Müller|first2=Tobias|year=2011|contribution=Sphere and dot product representations of graphs|title=Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry (SCG'11), June 13–15, 2011, Paris, France|pages=308–314}}.</ref>
* recognition of [[unit distance graph]]s, and testing whether the [[Dimension (graph theory)|dimension]] or Euclidean dimension of a graph is at most a given value.<ref>{{citation
| last = Schaefer | first = Marcus
| editor-last = Pach | editor-first = János | editor-link = János Pach
| contribution = Realizability of graphs and linkages
| doi = 10.1007/978-1-4614-0110-0_24
| pages = 461–482
| publisher = Springer
| title = Thirty Essays on Geometric Graph Theory
| year = 2013}}.</ref>
* recognition of intersection graphs of convex sets in the plane;<ref name="s09"/>
* stretchability of pseudolines (that is, given a family of curves in the plane, determining whether they are [[homeomorphic]] to a [[line arrangement]]);<ref name="s09"/><ref>{{citation
| last = Mnëv | first = N. E.
| contribution = The universality theorems on the classification problem of configuration varieties and convex polytopes varieties
| doi = 10.1007/BFb0082792
| location = Berlin
| mr = 970093
| pages = 527–543
| publisher = Springer
| series = Lecture Notes in Math.
| title = Topology and geometry—Rohlin Seminar
| volume = 1346
| year = 1988}}.</ref><ref>{{citation
| last = Shor | first = Peter W. | authorlink = Peter Shor
| contribution = Stretchability of pseudolines is NP-hard
| location = Providence, RI
| mr = 1116375
| pages = 531–554
| publisher = [[American Mathematical Society]]
| series = DIMACS Ser. Discrete Math. Theoret. Comput. Sci.
| title = Applied Geometry and Discrete Mathematics
| volume = 4
| year = 1991}}.</ref>
* determining the [[Crossing number (graph theory)|rectilinear crossing number]] of a graph (the minimum number of pairs of edges that cross in any drawing with edges drawn as straight line segments in the plane);<ref name="s09"/><ref>{{citation
| last = Bienstock | first = Daniel
| doi = 10.1007/BF02574701
| issue = 5
| journal = [[Discrete and Computational Geometry]]
| mr = 1115102
| pages = 443–459
| title = Some provably hard crossing number problems
| volume = 6
| year = 1991}}.</ref>
* the algorithmic [[Steinitz's theorem|Steinitz problem]] (given a [[Lattice (order)|lattice]], determine whether it is the face lattice of a [[convex polytope]]);<ref>{{citation
| last1 = Björner | first1 = Anders | author1-link = Anders Björner
| last2 = Las Vergnas | first2 = Michel | author2-link = Michel Las Vergnas
| last3 = Sturmfels | first3 = Bernd | author3-link = Bernd Sturmfels
| last4 = White | first4 = Neil
| last5 = Ziegler | first5 = Günter M. | author5-link = Günter Ziegler
| at = Corollary 9.5.10, p. 407
| isbn = 0-521-41836-4
| location = Cambridge
| mr = 1226888
| publisher = Cambridge University Press
| series = Encyclopedia of Mathematics and its Applications
| title = Oriented Matroids
| volume = 46
| year = 1993}}.</ref>
* testing whether a given graph can be drawn in the plane with straight line edges and with a given set of edge pairs as its crossings;<ref>{{citation
| last = Kynčl | first = Jan
| doi = 10.1007/s00454-010-9320-x
| issue = 3
| journal = [[Discrete and Computational Geometry]]
| mr = 2770542
| pages = 383–399
| title = Simple realizability of complete abstract topological graphs in P
| volume = 45
| year = 2011}}.</ref> and
* testing whether a [[regular graph|4-regular graph]] whose edges are [[edge coloring|colored]] with four colors has a drawing with edges as straight line segments of four slopes, with the slopes representing the colors in the coloring.<ref>{{citation
| last = Richter | first = David A.
| editor1-last = Brandes | editor1-first = Ulrik
| editor2-last = Cornelsen | editor2-first = Sabine
| contribution = How to draw a Tait-colorable graph
| doi = 10.1007/978-3-642-18469-7_32
| pages = 353–364
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 18th International Symposium on Graph Drawing (GD 2010)
| volume = 6502
| year = 2011}}.</ref>
Based on this, the [[complexity class]] <math>\exists \mathbb{R}</math> has been defined as the set of problems having a [[polynomial-time reduction|polynomial-time many-one reduction]] to the existential theory of the reals.<ref name="s09">{{citation|first=Marcus|last=Schaefer|contribution=Complexity of some geometric and topological problems|url=http://ovid.cs.depaul.edu/documents/convex.pdf|title=[[International Symposium on Graph Drawing|Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers]]|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|volume=5849|pages=334–344|doi=10.1007/978-3-642-11805-0_32|year=2010}}.</ref>

==References==
{{reflist|colwidth=30em}}

{{comp-sci-theory-stub}}

[[Category:Mathematical logic]]
[[Category:Computational complexity theory]]
[[Category:Real algebraic geometry]]

Quantum point contact

2013-11-25T17:21:07Z

129.97.58.107:

A '''parametric oscillator''' is a [[harmonic oscillator]] whose parameters oscillate in time. For example, a well known parametric oscillator is a child pumping a swing by periodically standing and squatting to increase the size of the swing's oscillations.<ref name=Case>{{cite web |title=Two ways of driving a child's swing |url=http://www.grinnell.edu/academic/physics/faculty/case/swing/ |first=William |last=Case |accessdate=27 November 2011}} Note: In real-life playgrounds, swings are predominantly driven, not parametric, oscillators.</ref><ref name=Case96>{{cite journal |last=Case |first=W. B. |year=1996 |title=The pumping of a swing from the standing position |journal=American Journal of Physics |volume=64 |pages=215–220 |doi=10.1119/1.18209}}</ref><ref name=Roura>{{cite journal |last1=Roura |first1=P. |last2=Gonzalez |first2=J.A. |year=2010 |title=Towards a more realistic description of swing pumping due to the exchange of angular momentum |journal=European Journal of Physics |volume=31 |pages=1195–1207 |doi=10.1088/0143-0807/31/5/020}} </ref> The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency <math>\omega</math> and damping <math>\beta</math>.

Parametric oscillators are used in many applications. The classical [[varactor]] parametric oscillator will oscillate when the diode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics, [[waveguide (electromagnetism)|waveguide]]/[[Yttrium aluminum garnet|YAG]] based parametric oscillators operate in the same fashion. The designer varies a parameter periodically in order to induce oscillations.

Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise is minimal, since a [[reactance]] (not a resistance) is varied. One common use is frequency conversion, e.g., conversion from audio to radio frequencies. Another important example is the [[Optical parametric oscillator]], which converts an input [[laser]] wave into two output waves of lower frequency (<math>\omega_s, \omega_i</math>).

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the [[instability]] phenomenon.

==History==
[[Michael Faraday]] (1831) was the first to notice oscillations of one frequency being excited by forces of double the frequency, in the crispations (ruffled surface waves) observed in a wine glass excited to "sing".<ref>Faraday, M. (1831) "On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces", ''Philosophical Transactions of the Royal Society (London)'', vol. 121, pages 299-318.</ref> Melde (1859) generated parametric oscillations in a string by employing a tuning fork to periodically vary the tension at twice the resonance frequency of the string.<ref>Melde, F. (1859) "Über Erregung stehender Wellen eines fadenförmigen Körpers" [On the excitation of standing waves on a string], ''Annalen der Physik und Chemie'' (Ser. 2), vol. 109, pages 193-215.</ref> Parametric oscillation was first treated as a general phenomenon by [[John Strutt, 3rd Baron Rayleigh|Rayleigh]] (1883,1887).<ref>Strutt, J.W. (Lord Rayleigh) (1883) "On maintained vibrations", ''Philosophical Magazine'', vol. 15, pages 229-235.</ref><ref>Strutt, J.W. (Lord Rayleigh) (1887) "On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with periodic structure", ''Philosophical Magazine'', vol.24, pages 145-159.</ref><ref>Strutt, J.W. (Lord Rayleigh) ''The Theory of Sound'', 2nd. ed. (N.Y., N.Y.: Dover, 1945), vol. 1, pages 81-85.</ref>

One of the first to apply the concept to electric circuits was [[George Francis FitzGerald]], who in 1892 tried to excite oscillations in an [[LC circuit]] by pumping it with a varying inductance provided by a dynamo.<ref name="Hong">{{cite book
| last = Hong
| first = Sungook Hong
| authorlink =
| coauthors =
| title = Wireless: From Marconi's Black-Box to the Audion
| publisher = MIT Press
| date = 201
| location =
| pages = 158-161
| url = http://books.google.com/books?id=UjXGQSPXvIcC&pg=PA165&lpg=PA164&dq=%22negative+resistance%22+ayrton+%22continuous+waves%22#v=onepage&q=%22negative%20resistance%22%20ayrton%20%22continuous%20waves%22&f=false
| doi =
| id =
| isbn = 0262082985}}</ref> Parametric amplifiers ('''paramps''') were first used in 1913-1915 for radio telephony from Berlin to Vienna and Moscow, and were predicted to have a useful future ([[Ernst Alexanderson]], 1916).<ref>Alexanderson, Ernst F.W. (April 1916) "A magnetic amplifier for audio telephony" ''[[Proceedings of the Institute of Radio Engineers]]'', vol. 4, pages 101-149.</ref> The early paramps varied inductances, but other methods have been developed since, e.g., the varactor diodes, [[klystron tube]]s, [[Josephson junctions]] and [[optical parametric oscillator|optical methods]].

==The mathematics==
:<math>
\frac{d^{2}x}{dt^{2}} + \beta(t) \frac{dx}{dt} + \omega^{2}(t) x = 0
</math>

This equation is linear in <math>x(t)</math>. By assumption, the parameters
<math>\omega^{2}</math> and <math>\beta</math> depend only on time and do ''not'' depend on the state of the oscillator. In general, <math>\beta(t)</math> and/or <math>\omega^{2}(t)</math> are assumed to vary periodically, with the same period <math>T</math>.

Remarkably, if the parameters vary at roughly ''twice'' the [[natural frequency]] of the oscillator (defined below), the oscillator phase-locks to the parametric variation and absorbs energy at a rate proportional to the energy it already has. Without a compensating energy-loss mechanism provided by <math>\beta</math>, the oscillation amplitude grows exponentially. (This phenomenon is called '''parametric excitation''', '''parametric resonance''' or '''parametric pumping'''.) However, if the initial amplitude is zero, it will remain so; this distinguishes it from the non-parametric resonance of driven simple [[harmonic oscillator]]s, in which the amplitude grows linearly in time regardless of the initial state.

A familiar experience of both parametric and driven oscillation is playing on a swing.<ref name=Case/><ref name=Case96/><ref name=Roura/> Rocking back and forth pumps the swing as a [[Harmonic_oscillator#Driven_harmonic_oscillators|driven harmonic oscillator]], but once moving, the swing can also be parametrically driven by alternately standing and squatting at key points in the swing arc. This changes moment of inertia of the swing and hence the resonance frequency, and children can quickly reach large amplitudes provided that they have some amplitude to start with (e.g., get a push). Standing and squatting at rest, however, leads nowhere.

===Transformation of the equation===

We begin by making a change of variables

:<math>
q(t) \ \stackrel{\mathrm{def}}{=}\ e^{D(t)} x(t)
</math>

where <math>D(t)</math> is a time integral of the damping

:<math>
D(t) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{2} \int^{t} d\tau \ \beta(\tau).
</math>

This change of variables eliminates the damping term

:<math>
\frac{d^{2}q}{dt^{2}} + \Omega^{2}(t) q = 0
</math>

where the transformed frequency is defined

:<math>
\Omega^{2}(t) = \omega^{2}(t) -
\frac{1}{2} \left( \frac{d\beta}{dt} \right) - \frac{1}{4} \beta^{2}.
</math>

In general, the variations in damping and frequency are relatively small perturbations

:<math>
\beta(t) = \omega_{0} \left[b + g(t) \right]
</math>

:<math>
\omega^{2}(t) = \omega_{0}^{2} \left[1 + h(t) \right]
</math>

where <math>\omega_{0}</math> and <math>b\omega_{0}</math> are constants, namely, the time-averaged oscillator frequency and damping, respectively. The transformed frequency can be written in a similar way:

:<math>
\Omega^{2}(t) = \omega_{n}^{2} \left[1 + f(t) \right],
</math>

where <math>\omega_{n}</math> is the [[natural frequency]] of the damped harmonic oscillator

:<math>
\omega_{n}^{2} \ \stackrel{\mathrm{def}}{=}\ \omega_{0}^{2} \left( 1 - \frac{b^{2}}{4} \right)
</math>

and

:<math>
\omega_{n}^{2} f(t) \ \stackrel{\mathrm{def}}{=}\ \omega_{0}^{2} \left\{h(t) -
\frac{1}{2\omega_{0}} \left( \frac{dg}{dt} \right)
- \frac{b}{2} g(t) - \frac{1}{4} g^{2}(t)\right\}.
</math>

Thus, our transformed equation can be written

:<math>
\frac{d^{2}q}{dt^{2}} + \omega_{n}^{2} \left[1 + f(t) \right] q = 0.
</math>

Remarkably, the independent variations <math>g(t)</math> and <math>h(t)</math> in the oscillator damping and resonance frequency, respectively, can be combined into a single pumping function <math>f(t)</math>. The converse conclusion is that any form of parametric excitation can be accomplished by varying either the resonance frequency or the damping, or both.

===Solution of the transformed equation===

Let us assume that <math>f(t)</math> is sinusoidal, specifically

:<math>
f(t) = f_{0} \sin 2\omega_{p}t
</math>

where the pumping frequency <math>2\omega_{p} \approx 2\omega_{n}</math> but need not equal <math>2\omega_{n}</math> exactly. The solution <math>q(t)</math> of our transformed equation may be written

:<math>
q(t) = A(t) \cos \omega_{p}t + B(t) \sin \omega_{p}t
</math>

where we have factored out the rapidly varying components (<math>\cos \omega_{p}t</math> and <math>\sin \omega_{p}t</math>) to isolate the slowly varying amplitudes <math>A(t)</math> and <math>B(t)</math>. This corresponds to Laplace's variation of parameters method.

Substituting this solution into the transformed equation and retaining only the terms first-order in <math>f_{0} \ll 1</math> yields two coupled equations

:<math>
2\omega_{p} \frac{dA}{dt} =
\left( \frac{f_{0}}{2} \right) \omega_{n}^{2} A -
\left( \omega_{p}^{2} - \omega_{n}^{2} \right) B
</math>

:<math>
2\omega_{p} \frac{dB}{dt} =
-\left( \frac{f_{0}}{2} \right) \omega_{n}^{2} B +
\left( \omega_{p}^{2} - \omega_{n}^{2} \right) A
</math>

We may decouple and solve these equations by making another change of variables

:<math>
A(t) \ \stackrel{\mathrm{def}}{=}\ r(t) \cos \theta(t)
</math>

:<math>
B(t) \ \stackrel{\mathrm{def}}{=}\ r(t) \sin \theta(t)
</math>

which yields the equations

:<math>
\frac{dr}{dt} = \left( \alpha_{\mathrm{max}} \cos 2\theta \right) r
</math>

:<math>
\frac{d\theta}{dt} = -\alpha_{\mathrm{max}}
\left[\sin 2\theta - \sin 2\theta_{\mathrm{eq}} \right]
</math>

where we have defined for brevity

:<math>
\alpha_{\mathrm{max}} \ \stackrel{\mathrm{def}}{=}\ \frac{f_{0} \omega_{n}^{2}}{4\omega_{p}}
</math>

:<math>
\sin 2\theta_{\mathrm{eq}} \ \stackrel{\mathrm{def}}{=}\ \left( \frac{2}{f_{0}} \right) \epsilon
</math>

and the detuning

:<math>
\epsilon \ \stackrel{\mathrm{def}}{=}\ \frac{\omega_{p}^{2} - \omega_{n}^{2}}{\omega_{n}^{2}}
</math>

The <math>\theta</math> equation does not depend on <math>r</math>, and linearization near its equilibrium position <math>\theta_{\mathrm{eq}}</math> shows that <math>\theta</math> decays exponentially to its equilibrium

:<math>
\theta(t) = \theta_{\mathrm{eq}} +
\left( \theta_{0} - \theta_{\mathrm{eq}} \right) e^{-2\alpha t}
</math>

where the decay constant

<math>\alpha \ \stackrel{\mathrm{def}}{=}\ \alpha_{\mathrm{max}} \cos 2\theta_{\mathrm{eq}}</math>.

In other words, the parametric oscillator phase-locks to the pumping signal <math>f(t)</math>.

Taking <math>\theta(t) = \theta_{\mathrm{eq}}</math> (i.e., assuming that the phase has locked), the <math>r</math> equation becomes

:<math>
\frac{dr}{dt} = \alpha r
</math>

whose solution is <math>r(t) = r_{0} e^{\alpha t}</math>; the amplitude of the <math>q(t)</math> oscillation diverges exponentially. However, the corresponding amplitude <math>R(t)</math> of the ''untransformed'' variable <math>x \ \stackrel{\mathrm{def}}{=}\ q e^{-D(t)}</math> need not diverge

:<math>
R(t) = r(t) e^{-D(t)} = r_{0} e^{\alpha t - D(t)}
</math>

The amplitude <math>R(t)</math> diverges, decays or stays constant, depending on whether <math>\alpha t</math> is greater than, less than, or equal to <math>D(t)</math>, respectively.

The maximum growth rate of the amplitude occurs when <math>\omega_{p} = \omega_{n}</math>. At that frequency, the equilibrium phase <math>\theta_{\mathrm{eq}}</math> is zero, implying that <math>\cos 2\theta_{\mathrm{eq}}=1</math> and <math>\alpha = \alpha_{\mathrm{max}}</math>. As <math>\omega_{p}</math> is varied from <math>\omega_{n}</math>, <math>\theta_{\mathrm{eq}}</math> moves away from zero and <math>\alpha < \alpha_{\mathrm{max}}</math>, i.e., the amplitude grows more slowly. For sufficiently large deviations of <math>\omega_{p}</math>, the decay constant <math>\alpha</math> can become purely imaginary since

:<math>
\alpha = \alpha_{\mathrm{max}}
\sqrt{1- \left( \frac{2}{f_{0}} \right)^{2} \epsilon^{2}}
</math>

If the detuning <math>\epsilon</math> exceeds <math>f_{0}/2</math>, <math>\alpha</math> becomes purely imaginary and <math>q(t)</math> varies sinusoidally. Using the definition of the detuning <math>\epsilon</math>, the pumping frequency <math>2\omega_{p}</math> must lie between <math>2\omega_{n} \sqrt{1 - \frac{f_{0}}{2}}</math> and <math>2\omega_{n} \sqrt{1 + \frac{f_{0}}{2}}</math> in order to achieve exponetial growth in <math>q</math>. Expanding the square roots in a binomial series shows that the spread in pumping frequencies that result in exponentially growing <math>q</math> is approximately <math>\omega_{n} f_{0}</math>.

==Intuitive derivation of parametric excitation==

The above derivation may seem like a mathematical sleight-of-hand, so it may be helpful to give an intuitive derivation. The <math>q</math> equation may be written in the form

:<math>
\frac{d^{2}q}{dt^{2}} + \omega_{n}^{2} q = -\omega_{n}^{2} f(t) q
</math>

which represents a simple harmonic oscillator (or, alternatively, a [[bandpass filter]]) being driven by a signal <math>-\omega_{n}^{2} f(t) q</math> that is proportional to its response <math>q</math>.

Assume that <math>q(t) = A \cos \omega_{p} t</math> already has an oscillation at frequency <math>\omega_{p}</math> and that the pumping <math>f(t) = f_{0} \sin 2\omega_{p}t</math> has double the frequency and a small amplitude <math>f_{0} \ll 1</math>. Applying a [[trigonometry|trigonometric identity]] for products of sinusoids, their product <math>q(t)f(t)</math> produces two driving signals,
one at frequency <math>\omega_{p}</math> and the other at frequency <math>3 \omega_{p}</math>

:<math>
f(t)q(t) = \frac{f_{0}}{2} A
\left( \sin \omega_{p} t + \sin 3\omega_{p} t \right)
</math>

Being off-resonance, the <math>3\omega_{p}</math> signal is attentuated and can be neglected initially. By contrast, the <math>\omega_{p}</math> signal is on resonance, serves to amplify <math>q</math> and is proportional to the amplitude
<math>A</math>. Hence, the amplitude of <math>q</math> grows exponentially unless it is initially zero.

Expressed in Fourier space, the multiplication <math>f(t)q(t)</math> is a convolution of their Fourier transforms <math>\tilde{F}(\omega)</math> and <math>\tilde{Q}(\omega)</math>. The positive feedback arises because the <math>+2\omega_{p}</math> component of <math>f(t)</math> converts the <math>-\omega_{p}</math> component of <math>q(t)</math> into a driving signal at
<math>+\omega_{p}</math>, and vice versa (reverse the signs). This explains why the pumping frequency must be near <math>2\omega_{n}</math>, twice the natural frequency of the oscillator. Pumping at a grossly different frequency would not couple (i.e., provide mutual positive feedback) between the <math>-\omega_{p}</math> and <math>+\omega_{p}</math> components of <math>q(t)</math>.

== Parametric resonance ==

'''Parametric resonance''' is the [[Parameter|parametric]]al [[resonance]] [[phenomenon]] of mechanical [[excitation]] and [[oscillation]] at certain [[frequency|frequenc]]ies (and the associated [[harmonic]]s). This effect is different from regular resonance because it exhibits the [[instability]] phenomenon.

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies.Parametric resonance takes place when the external excitation frequency equals twice the natural frequency of the system. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter. The classical example of parametric resonance is that of the vertically forced pendulum.

For small amplitudes and by linearising, the stability of the periodic solution is given by :

<math>\ddot{u} + (a + B \cos t)u =0 \ </math>

where <math>u</math> is some perturbation from the periodic solution. Here the <math>B\ \cos(t)</math> term acts as an ‘energy’ source and is said to parametrically excite the system. The Mathieu equation describes many other physical systems to a sinusoidal parametric excitation such as an LC Circuit where the capacitor plates move sinusoidally.

==Parametric amplifiers==

===Introduction===
A parametric amplifier is implemented as a [[frequency mixer|mixer]]. The mixer's gain shows up in the output as amplifier gain. The input weak signal is mixed with a strong local oscillator signal, and the resultant strong output is used in the ensuing receiver stages.

Parametric amplifiers also operate by changing a parameter of the amplifier.
Intuitively, this can be understood as follows, for a variable capacitor based amplifier.

Q [charge in a capacitor] = C x V 
therefore 
V [voltage across a capacitor] = Q/C

Knowing the above, if a capacitor is charged until its voltage equals the sampled voltage of an incoming weak signal, and if the capacitor's capacitance is then reduced (say, by manually moving the plates further apart), then the voltage across the capacitor will increase. In this way, the voltage of the weak signal is amplified.

If the capacitor is a [[varicap diode]], then the 'moving the plates' can be done simply by applying time-varying DC voltage to the varicap diode. This driving voltage usually comes from another oscillator — sometimes called a "pump".

The resulting output signal contains frequencies that are the sum and difference of the input signal (f1) and the pump signal (f2): (f1 + f2) and (f1 - f2).

A practical parametric oscillator needs the following connections: one for the "common" or "[[ground (electrical)|ground]]", one to feed the pump, one to retrieve the output, and maybe a fourth one for biasing. A parametric amplifier needs a fifth port to input the signal being amplified. Since a varactor diode has only two connections, it can only be a part of an LC network with four [[eigenvector]]s with nodes at the connections. This can be implemented as a [[transimpedance amplifier]], a [[Traveling wave tube amplifier|traveling wave amplifier]] or by means of a [[circulator]].

===Mathematical equation===
The parametric oscillator equation can be extended by adding an external driving force <math>E(t)</math>:

:<math>
\frac{d^{2}x}{dt^{2}} + \beta(t) \frac{dx}{dt} + \omega^{2}(t) x = E(t).
</math>

We assume that the damping <math>D</math> is sufficiently strong that, in the absence of the driving force <math>E</math>, the amplitude of the parametric oscillations does not diverge, i.e., that <math>\alpha t < D</math>. In this situation, the parametric pumping acts to lower the effective damping in the system. For illustration, let the damping be constant <math>\beta(t) = \omega_{0} b</math> and assume that the external driving force is at the mean resonance frequency <math>\omega_{0}</math>, i.e., <math>E(t) = E_{0} \sin \omega_{0} t</math>. The equation becomes

:<math>
\frac{d^{2}x}{dt^{2}} + b \omega_{0} \frac{dx}{dt} +
\omega_{0}^{2} \left[1 + h_{0} \sin 2\omega_{0} t \right] x =
E_{0} \sin \omega_{0} t
</math>

whose solution is roughly

:<math>
x(t) = \frac{2E_{0}}{\omega_{0}^{2} \left( 2b - h_{0} \right)} \cos \omega_{0} t.
</math>

As <math>h_{0}</math> approaches the threshold <math>2b</math>, the amplitude diverges. When <math>h \geq 2b</math>, the system enters parametric resonance and the amplitude begins to grow exponentially, even in the absence of a driving force <math>E(t)</math>.

===Advantages===
'''1''':It is highly sensitive

'''2''':low noise level amplifier for ultra high frequency and microwave radio signal

'''3''':The unique capability to operate as a wireless powered amplifier that doesn't require internal power source<ref>[http://onlinelibrary.wiley.com/doi/10.1002/mrm.23274/abstract Sensitivity Enhancement of Remotely Coupled NMR Detectors Using Wirelessly Powered Parametric Amplification]</ref>

===Other relevant mathematical results===

If the parameters of any second-order linear differential equation are varied periodically, [[Floquet analysis]] shows that the solutions must vary either sinusoidally or exponentially.

The <math>q</math> equation above with periodically varying <math>f(t)</math> is an example of a [[Hill differential equation|Hill equation]]. If <math>f(t)</math> is a simple sinusoid, the equation is called a [[Mathieu equation]].

==See also==
* [[Harmonic oscillator]]
* [[Optical parametric oscillator]]
* [[Optical parametric amplifier]]
* [[Mathieu equation]]

==References==

{{reflist}}

{{Refimprove|date=May 2008}}

==Further reading ==

* Kühn L. (1914) ''Elektrotech. Z.'', '''35''', 816-819.
* Mumford WW. (1960) "Some Notes on the History of Parametric Transducers", ''Proceedings of the Institute of Radio Engineers'', '''48''', 848-853.
* Pungs L. DRGM Nr. 588 822 (24 October 1913); DRP Nr. 281440 (1913); ''Elektrotech. Z.'', '''44''', 78-81 (1923?); ''Proc. IRE'', '''49''', 378 (1961).

==External articles==
* Elmer, Franz-Josef, "''[http://monet.physik.unibas.ch/~elmer/pendulum/parres.htm Parametric Resonance]''". unibas.ch, July 20, 1998.
* Cooper, Jeffery, "''[http://dx.doi.org/10.1137/S0036141098340703 Parametric Resonance in Wave Equations with a Time-Periodic Potential]''". SIAM Journal on Mathematical Analysis, Volume 31, Number 4, pp. 821–835. Society for Industrial and Applied Mathematics, 2000 .
* "''[http://bednorzmuller87.phys.cmu.edu/demonstrations/oscillationsandwaves/drivenoscillations/demo223.html Driven Pendulum: Parametric Resonance]''"{{dead link|date=November 2013}}. phys.cmu.edu (Demonstration of physical mechanics or classical mechanics. Resonance oscillations set up in a simple pendulum via periodically varying pendulum length.)

[[Category:Oscillators]]
[[Category:Amplifiers]]
[[Category:Dynamical systems]]
[[Category:Ordinary differential equations]]

Ratio estimator

2013-10-29T19:30:47Z

129.97.59.156:

{{Infobox enzyme
| Name = Pyrimidine oxygenase
| EC_number = 1.14.99.46
| CAS_number =
| IUBMB_EC_number = 1/14/99/46
| GO_code =
| image =
| width =
| caption =
}}
'''Pyrimidine oxygenase''' ({{EC number|1.14.99.46}}, ''RutA'') is an [[enzyme]] with system name ''uracil,FMNH2:oxygen oxidoreductase (uracil hydroxylating, ring-opening) ''.<ref>{{cite journal | title = Catalysis of a flavoenzyme-mediated amide hydrolysis |author = Mukherjee, T., Zhang, Y., Abdelwahed, S., Ealick, S.E. and Begley, T.P. |journal = J. Am. Chem. Soc. |year = 2010 |volume = 132 |pages = 5550–5551 |pmid = 20369853 |doi=10.1021/ja9107676 |issue=16 |pmc=2873085}}</ref><ref>{{cite journal | title = The Rut pathway for pyrimidine degradation: novel chemistry and toxicity problems |author = Kim, K.S., Pelton, J.G., Inwood, W.B., Andersen, U., Kustu, S. and Wemmer, D.E. |journal = J. Bacteriol. |year = 2010 |volume = 192 |pages = 4089–4102 |pmid = 20400551 |doi=10.1128/JB.00201-10 |issue=16 |pmc=2916427}}</ref> This enzyme [[catalysis|catalyses]] the following [[chemical reaction]]

: (1) [[uracil]] + FMNH2 + O2 <math>\rightleftharpoons</math> (Z)-3-ureidoacrylate peracid + FMN
: (2) [[thymine]] + FMNH2 + O2 <math>\rightleftharpoons</math> (Z)-2-methylureidoacrylate peracid + FMN

In vitro the product (Z)-3-ureidoacrylate peracid is spontaneously reduced to [[ureidoacrylate]].

== References ==
{{reflist}}

== External links ==
* {{MeshName|Pyrimidine+oxygenase}}

[[Category:EC 1.14.99]]

Strong antichain

2013-08-25T23:53:47Z

129.97.186.86:

{{Refimprove|date=December 2009}}

'''Operative temperature''' (<math>t_o</math>) is defined as a uniform temperature of a radiantly black enclosure in which an occupant would exchange the same amount of heat by [[radiation]] plus [[convection]] as in the actual nonuniform environment.<ref>ASHRAE Terminology, ASHRAE Handbook CD, 1999-2002</ref> Some references also use the terms 'equivalent temperature" or 'effective temperature' to describe combined effects of convective and radiant heat transfer.<ref>Nilsson, H.O., Comfort Climate Evaluation with Thermal Manikin Methods and Computer Simulation Models, National Institute for Working Life, 2004, pg. 37</ref> In design, operative temperature can be defined as the average of the [[Mean radiant temperature|mean radiant]] and [[Dry-bulb temperature|ambient air temperatures]], weighted by their respective [[heat transfer coefficient]]s.<ref>Thermal Comfort, ASHRAE Handbook, Fundamentals, Ch. 9, pg.3, 2009</ref> The instrument used for assessing environmental thermal comfort in terms of operative temperature is called a eupatheoscope and was invented by A. F. Dufton in 1929.<ref>Glossary of Meteorology, American Meteorological Society, < http://amsglossary.allenpress.com/glossary/browse?s=e&p=42>, accessed Sept 2010</ref> Mathematically, operative temperature can be shown as;

::<big><math>t_o = \frac{(h_r t_{mr} + h_c t_a)}{ h_r + h_c}</math></big>

where,

:<math>h_c</math> = convective heat transfer coefficient
:<math>h_r</math> = linear radiative heat transfer coefficient
:<math>t_a</math> = air temperature
:<math>t_{mr}</math> = mean radiant temperature

It is also acceptable to approximate this relationship for occupants engaged in near sedentary physical activity (with metabolic rates between 1.0 met and 1.3 met), not in direct sunlight, and not exposed to air velocities greater than 0.20 m/s (40 fpm). <ref>ANSI/ASHRAE Standard 55-2010, Thermal Environmental Conditions for Human Occupancy</ref>

::<big><math>t_o = \frac{(t_a + t_{mr})}{ 2 }</math></big>

where <math>t_a</math> and <math>t_{mr}</math> have the same meaning as above.

==Application==

Operative temperature is used in heat transfer and thermal comfort analysis in transportation and buildings.<ref>Dufton, A. F. The Equivalent Temperature of a room and its Measurement, Building Research Technical Paper No. 13. London, 1932</ref> Most [[Psychrometrics|psychrometric]] charts used in HVAC design only show the dry bulb temperature on the x-axis(abscissa), however, it is the operative temperature which is specified on the x-axis of the psychrometric chart illustrated in ANSI/ASHRAE Standard 55 – Thermal Environmental Conditions for Human occupancy.

==See also==
*[[HVAC]]
*[[Psychrometrics]]
*[[Underfloor heating]]
*[[ASHRAE]]

==References==
{{Reflist}}

{{DEFAULTSORT:Operative Temperature}}
[[Category:Heating, ventilating, and air conditioning]]

{{Ecology-stub}}

Species evenness

2013-06-27T17:25:51Z

129.97.39.192: added prime to H_max

'''Relative viscosity''' (<math>\eta_r</math>) (a synonym of "viscosity ratio") is the ratio of the [[viscosity]] of a [[solution]] (<math>\eta</math>) to the viscosity of the [[solvent]] used (<math>\eta_s</math>),
:<math>\eta_r = \frac{\eta}{\eta_s}</math>.

==References==
* [http://www.iupac.org/publications/compendium/ IUPAC Compendium of Chemical Terminology]

[[Category:Viscosity]]

{{physics-stub}}

Sylvester matrix

2013-05-06T20:08:14Z

129.97.93.38:

[[File:Set of curves Outer approximation.png|345px|thumb|right|Tolerance function (turquoise) and interval-valued approximation (red)'']]
'''Interval arithmetic''', '''interval mathematics''', '''interval analysis''', or '''interval computation''', is a method developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on [[rounding error]]s and [[measurement error]]s in [[numerical analysis|mathematical computation]] and thus developing [[numerical methods]] that yield reliable results. Very simply put, it represents each value as a range of possibilities. For example, instead of estimating the height of someone using standard arithmetic as 2.0 meters, using interval arithmetic we might be certain that that person is somewhere between 1.97 and 2.03 meters.

Whereas classical arithmetic defines operations on individual numbers, interval arithmetic defines a set of operations on [[interval (mathematics)|interval]]s:

:T · S = { ''x'' | there is some ''y'' in ''T'', and some ''z'' in ''S'', such that ''x'' = ''y'' · ''z'' }.

The basic operations of interval arithmetic are, for two intervals [''a'', ''b''] and [''c'', ''d''] that are subsets of the real line (-∞, ∞),

* [''a'', ''b''] + [''c'', ''d''] =  [''a'' + ''c'', ''b'' + ''d''],
* [''a'', ''b''] − [''c'', ''d''] =  [''a'' − ''d'', ''b'' − ''c''],
* [''a'', ''b''] × [''c'', ''d''] = [min (''a'' × ''c'', ''a'' × ''d'', ''b'' × ''c'', ''b'' × ''d''), max (''a'' × ''c'', ''a'' × ''d'', ''b'' × ''c'', ''b'' × ''d'')],
* [''a'', ''b''] ÷ [''c'', ''d''] = [min (''a'' ÷ ''c'', ''a'' ÷ ''d'', ''b'' ÷ ''c'', ''b'' ÷ ''d''), max (''a'' ÷ ''c'', ''a'' ÷ ''d'', ''b'' ÷ ''c'', ''b'' ÷ ''d'')] when 0 is not in [''c'', ''d''].

Division by an interval containing zero is not defined under the basic interval arithmetic. The addition and multiplication operations are [[commutative]], [[associative]] and sub-[[distributive]]: the set ''X'' ( ''Y'' + ''Z'' ) is a subset of ''XY'' + ''XZ''.

Instead of working with an uncertain [[real (number)|real]] <math>x</math> we work with the two ends of the interval <math>[a,b]</math> which contains <math>x</math>: <math>x</math> lies between <math>a</math> and <math>b</math>, or could be one of them. Similarly a function <math>f</math> when applied to <math>x</math> is also uncertain. Instead, in interval arithmetic <math>f</math> produces an interval <math>[c,d]</math> which is all the possible values for <math>f(x)</math> for all <math>x \in [a,b]</math>.

This concept is suitable for a variety of purposes. The most common use is to keep track of and handle rounding errors directly during the calculation and of uncertainties in the knowledge of the exact values of physical and technical parameters. The latter often arise from measurement errors and tolerances for components or due to limits on computational accuracy. Interval arithmetic also helps find reliable and guaranteed solutions to equations and optimization problems.

==Introduction==

The main focus in the interval arithmetic is on the simplest way to calculate upper and lower endpoints for the range of values of a function in one or more variables. These barriers are not necessarily the [[supremum]] or [[infimum]], since the precise calculation of those values can be difficult or impossible; it can be shown that that task is in general [[NP-hard]].

Treatment is typically limited to real intervals, so quantities of form
:<math>[a,b] = \{x \in \mathbb{R} \,|\, a \le x \le b\},</math>
where <math> a = {-\infty}</math> and <math> b = {\infty}</math> are allowed; with one of them infinite we would have an unbounded interval, while with both infinite we would have the extended real number line.

As with traditional calculations with real numbers, simple arithmetic operations and functions on elementary intervals must first be defined.<ref name="Kulisch">{{cite book|last=Kulisch|first=Ulrich|title=Wissenschaftliches Rechnen mit Ergebnisverifikation. Eine Einführung|year=1989|publisher=Vieweg-Verlag|location=Wiesbaden|language=German|isbn=3-528-08943-1}}</ref> More complicated functions can be calculated from these basic elements.<ref name="Kulisch"/>

===Example===
[[File:Interval BMI Simple Example.png|350px|thumb|left|[[Body Mass Index]] for a person 1.80m tall in relation to body weight ''m'' (in kilograms).]]
Take as an example the calculation of [[body mass index]] (BMI). The BMI is the body weight in kilograms divided by the square of height in metres. Measuring the mass with bathroom scales may have an accuracy of one kilogram. We will not know intermediate values - about 79.6 kg or 80.3 kg - but information rounded to the nearest whole number. It is unlikely that when the scale reads 80 kg, someone really weighs exactly 80.0 kg. In normal rounding to the nearest value, the scales showing 80 kg indicates a weight between 79.5 kg and 80.5 kg. The relevant range is that of all real numbers that are greater than or equal to 79.5, while less than or equal to 80.5, or in other words the interval [79.5,80.5].

For a man who weighs 80 kg and is 1.80 m tall, the BMI is about 24.7. With a weight of 79.5 kg and the same height the value is 24.5, while 80.5 kilograms gives almost 24.9. So the actual BMI is in the range [24.5,24.9]. The error in this case does not affect the conclusion (normal weight), but this is not always the position. For example, weight fluctuates in the course of a day so that the BMI can vary between 24 (normal weight) and 25 (overweight). Without detailed analysis it is not possible to always exclude questions as to whether an error ultimately is large enough to have significant influence.

Interval arithmetic states the range of possible outcomes explicitly. Simply put, results are no longer stated as numbers, but as intervals which represent imprecise values. The size of the intervals are similar to error bars to a metric in expressing the extent of uncertainty. Simple arithmetic operations, such as basic arithmetic and trigonometric functions, enable the calculation of outer limits of intervals.

===Simple arithmetic===
[[File:Interval BMI Example.png|260px|thumb|right|Body mass index for different weights in relation to height L (in metres).]]
Returning to the earlier BMI example, in determining the body mass index, height and body weight both affect the result. For height, measurements are usually in round centimetres: a recorded measurement of 1.80 metres actually means a height somewhere between 1.795 m and 1.805 m. This uncertainty must be combined with the fluctuation range in weight between 79.5 kg and 80.5 kg. The BMI is defined as the weight in kilograms divided by the square of height in metre. Using either 79.5 kg and 1.795 m or 80.5 kg and 1.805 m gives approximately 24.7. But the person in question may only be 1.795 m tall, with a weight of 80.5 kilograms - or 1.805 m and 79.5 kilograms: all combinations of all possible intermediate values must be considered. Using the interval arithmetic methods described below, the BMI lies in the interval
:<math>[79{.}5; 80{.}5]/([1{.}795; 1{.}805])^2 = [24{.}4; 25{.}0].</math>

An operation <math>{\langle\!\mathrm{op}\!\rangle}</math>, such as addition or multiplication, on two intervals is defined by

:<math>[x_1, x_2] {\,\langle\!\mathrm{op}\!\rangle\,} [y_1, y_2] = \{ x {\,\langle\!\mathrm{op}\!\rangle\,} y \, | \, x \in [x_1, x_2] \,\mbox{and}\, y \in [y_1, y_2] \} </math>.
For the four basic arithmetic operations this can become

:<math>\begin{align}[][x_1, x_2] \,\langle\!\mathrm{op}\!\rangle\, [y_1, y_2] & = \left[ \min(x_1 {\langle\!\mathrm{op}\!\rangle} y_1, x_1 \langle\!\mathrm{op}\!\rangle y_2, x_2 \langle\!\mathrm{op}\!\rangle y_1, x_2 \langle\!\mathrm{op}\!\rangle y_2),
\right.\\
&{}\qquad \left.
\;\max(x_1 {\langle\!\mathrm{op}\!\rangle}y_1, x_1 {\langle\!\mathrm{op}\!\rangle} y_2, x_2
{\langle\!\mathrm{op}\!\rangle} y_1, x_2 {\langle\!\mathrm{op}\!\rangle} y_2) \right]
\,\mathrm{,}
\end{align}
</math>

provided that <math>x {\,\langle\!\mathrm{op}\!\rangle\,} y</math> is allowed for all
<math>x\in [x_1, x_2]</math> and <math>y \in [y_1, y_2]</math>.

For practical applications this can be simplified further:

* [[Addition]]: <math>[x_1, x_2] + [y_1, y_2] = [x_1+y_1, x_2+y_2]</math>
* [[Subtraction]]: <math>[x_1, x_2] - [y_1, y_2] = [x_1-y_2, x_2-y_1]</math>
* [[Multiplication]]: <math>[x_1, x_2] \cdot [y_1, y_2] = [\min(x_1 y_1,x_1 y_2,x_2 y_1,x_2 y_2), \max(x_1 y_1,x_1 y_2,x_2 y_1,x_2 y_2)]</math>
* [[Division (mathematics)|Division]]: <math>[x_1, x_2] / [y_1, y_2] =
[x_1, x_2] \cdot (1/[y_1, y_2])</math>, where <math>1/[y_1, y_2] = [1/y_2, 1/y_1]</math> if <math>0 \notin [y_1, y_2]</math>.

For division by an interval including zero, first define
: <math>1/[y_1, 0] = [-\infty, 1/y_1]</math> and <math>1/[0, y_2] = [1/y_2, \infty]</math>.
For <math>y_1 < 0 < y2</math>, we get <math>1/[y_1, y_2] = [-\infty, 1/y_1] \cup [1/y_2, \infty]</math> which as a single interval gives <math>1/[y_1, y_2] = [-\infty, \infty]</math>; this loses useful information about <math>(1/y_1, 1/y_2)</math>. So typically it is common to work with <math>[-\infty, 1/y_1]</math> and <math>[1/y_2, \infty]</math> as separate intervals.

Because several such divisions may occur in an interval arithmetic calculation, it is sometimes useful to do the calculation with so-called ''multi-intervals'' of the form <math>\textstyle \bigcup_{i=1}^l [x_{i1},x_{i2}]</math>. The corresponding ''multi-interval arithmetic'' maintains a disjoint set of intervals and also provides for overlapping intervals to unite.<ref name="Dreyer">{{cite book|last=Dreyer|first=Alexander|title=Interval Analysis of Analog Circuits with Component Tolerances|year=2003|publisher=[[Shaker Verlag]]|location=Aachen, Germany|isbn=3-8322-4555-3}}</ref>{{Page needed|date=February 2011}}

Since a real number <math>r\in \mathbb{R}</math> can be interpreted as the interval <math>[r,r]</math>, intervals and real number can be freely and easily combined.

With the help of these definitions, it is already possible to calculate the range of simple functions, such as <math>f(a,b,x) = a \cdot x + b</math>.
If, for example<math>a = [1,2]</math>, <math>b = [5,7]</math> and <math>x = [2,3]</math>, it is clear

:<math>f(a,b,x) = ([1,2] \cdot [2,3]) + [5,7] = [1\cdot 2, 2\cdot 3] + [5,7] = [7,13]</math>.

Interpreting this as a function <math>f(a,b,x)</math> of the variable
<math>x</math> with interval parameters <math>a</math> and <math>b</math>, then it is possible to find the roots of this function. It is then

:<math>f([1,2],[5,7],x) = ([1,2] \cdot x) + [5,7] = 0\Leftrightarrow [1,2] \cdot x = [-7, -5]\Leftrightarrow x = [-7, -5]/[1,2],</math>
the possible zeros are in the interval <math>[-7, {-2.5}]</math>.

[[File:Interval multiplication.png|120px|right|thumb|Multiplication of positive intervals]]
As in the above example, the multiplication of intervals often only requires two multiplications. It is in fact

:<math>[x_1, x_2] \cdot [y_1, y_2] = [x_1 \cdot y_1, x_2 \cdot y_2],\text{ if }x_1, y_1 \geq 0.</math>

The multiplication can be seen as a destination area of a rectangle with varying edges. The result interval covers all levels from the smallest to the largest.

The same applies when one of the two intervals is non-positive and the other non-negative. Generally, multiplication can produce results as wide as <math>[-\infty, \infty]</math>, for example if <math>0 \cdot \infty</math> is squared. This also occurs, for example, in a division, if the numerator and denominator both contain zero.

=== Notation ===
To make the notation of intervals smaller in formulae, brackets can be used.

So we can use <math>[x] \equiv [x_1, x_2]</math> to represent an interval. For the set of all finite intervals, we can use

:<math>[\mathbb{R}] := \big\{\, [x_1, x_2] \,|\, x_1 \leq x_2 \text{ and } x_1, x_2 \in \mathbb{R} \cup \{-\infty, \infty\} \big\}</math>

as an abbreviation. For a vector of intervals <math>\big([x]_1, \ldots , [x]_n \big) \in [\mathbb{R}]^n </math> we can also use a bold font: <math>[\mathbf{x}]</math>.

Note that in such a compact notation, <math>[x]</math> should not be confused between a so-called improper or single point interval <math>[x_1, x_1]</math> and the lower and upper limit.

=== Elementary functions ===
[[File:Value domain of monotonic function.png|160px|right|thumb|Values of a monotonic function]]
Interval methods can also apply to functions which do not just use simple arithmetic, and we must also use other basic functions for redefining intervals, using already known monotonicity properties.

For [[monotonic function]]s in one variable, the range of values is also easy. If <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is monotonically rising or falling in the interval <math>[x_1, x_2]</math>, then for all values in the interval <math>y_1, y_2 \in [x_1, x_2]</math> such that <math>y_1 \leq y_2</math>, one of the following inequalities applies:
:<math>f(y_1) \leq f(y_2) </math>, or <math>f(y_1) \geq f(y_2) </math>.

The range corresponding to the interval <math>[y_1, y_2] \subseteq [x_1, x_2]</math> can be calculated by applying the function to the endpoints <math>y_1</math> and <math>y_2</math>:
:<math>f([y_1, y_2]) = \left[\min \big \{f(y_1), f(y_2) \big\}, \max \big\{ f(y_1), f(y_2) \big\}\right]</math>.

From this the following basic features for interval functions can easily be defined:
* [[Exponential function]]: <math>a^{[x_1, x_2]} = [a^{x_1},a^{x_2}]</math>, for <math>a > 1</math>,
* [[Logarithm]]: <math>\log_a\big( {[x_1, x_2]} \big) = [\log_a {x_1}, \log_a {x_2}]</math>, for positive intervals <math>[x_1, x_2]</math> and <math>a>1</math>
* Odd powers: <math>{[x_1, x_2]}^n = [{x_1}^n,{x_2}^n]</math>, for odd <math>n\in \mathbb{N}</math>.

For even powers, the range of values being considered is important, and needs to be dealt with before doing any multiplication.
For example <math>x^n</math> for <math>x \in [-1,1]</math> should produce the interval <math>[0,1]</math> when <math>n = 2, 4, 6, \ldots</math>. But if <math>[-1,1]^n</math> is taken by applying interval multiplication of form <math>[-1,1]\cdot \ldots \cdot [-1,1]</math> then the result will appear to be <math>[-1,1]</math>, wider than necessary.

Instead consider the function <math>x^n</math> as a monotonically decreasing function for <math>x < 0</math> and a monotonically increasing function for <math>x > 0</math>. So for even <math>n\in \mathbb{N}</math>:

* <math>{[x_1, x_2]}^n = [x_1^n, x_2^n]</math>, if <math>x_1 \geq 0</math>,
* <math>{[x_1, x_2]}^n = [x_2^n, x_1^n]</math>, if <math>x_2 < 0</math>,
* <math>{[x_1, x_2]}^n = [0, \max \{x_1^n, x_2^n \} ]</math>, otherwise.

More generally, one can say that for piecewise monotonic functions it is sufficient to consider the endpoints <math>x_1, x_2</math> of the interval <math>[x_1, x_2]</math>, together with the so-called ''critical points'' within the interval being those points where the monotonicity of the function changes direction.

For the [[sine]] and [[cosine]] functions, the critical points are at <math>\left( {}^1\!\!/\!{}_2 + {n}\right) \cdot \pi</math> or <math>{n} \cdot \pi</math> for all <math>n \in \mathbb{Z}</math> respectively. Only up to five points matter as the resulting interval will be <math>[-1,1]</math> if the interval includes at least two extrema. For sine and cosine, only the endpoints need full evaluation as the critical points lead to easily pre-calculated values – namely -1, 0, +1.

===Interval extensions of general functions===
In general, it may not be easy to find such a simple description of the output interval for many functions. But it may still be possible to extend functions to interval arithmetic.
If <math>f:\mathbb{R}^n \rightarrow \mathbb{R}</math> is a function from a real vector to a real number, then  <math>[f]:[\mathbb{R}]^n \rightarrow [\mathbb{R}]</math> is called an ''interval extension'' of <math>f</math> if
:<math>[f]([\mathbf{x}]) \supseteq \{f(\mathbf{y}) | \mathbf{y} \in [\mathbf{x}]\}</math>.

This definition of the interval extension does not give a precise result. For example, both <math>[f]([x_1,x_2]) =[e^{x_1}, e^{x_2}]</math> and <math>[g]([x_1,x_2]) =[{-\infty}, {\infty}]</math> are allowable extensions of the exponential function. Extensions as tight as possible are desirable, taking into the relative costs of calculation and imprecision; in this case <math>[f]</math> should be chosen as it give the tightest possible result.

The ''natural interval extension'' is achieved by combining the function rule <math>f(x_1, \cdots, x_n)</math> with the equivalents of the basic arithmetic and elementary functions.

The ''Taylor interval extension'' (of degree <math>k</math> ) is a <math>k+1</math> times differentiable function <math>f</math> defined by

:<math>[f]([\mathbf{x}]) := f(\mathbf{y}) + \sum_{i=1}^k\frac{1}{i!}\mathrm{D}^i f(\mathbf{y}) \cdot ([\mathbf{x}] - \mathbf{y})^i + [r]([\mathbf{x}], [\mathbf{x}], \mathbf{y})
</math>,
for some <math>\mathbf{y} \in [\mathbf{x}]</math>,
where <math>\mathrm{D}^i f(\mathbf{y})</math> is the <math>i</math>th order differential of <math>f</math> at the point <math>\mathbf{y}</math> and <math>[r]</math> is an interval extension of the ''Taylor remainder''

:<math>r(\mathbf{x}, \xi, \mathbf{y}) = \frac{1}{(k+1)!}\mathrm{D}^{k+1} f(\xi) \cdot (\mathbf{x}-\mathbf{y})^{k+1}. </math>

[[File:Meanvalue extension.png|220px|right|thumb|Mean value form]]
The vector <math>\xi</math> lies between <math>\mathbf{x}</math>
and <math>\mathbf{y}</math> with <math>\mathbf{x}, \mathbf{y} \in [\mathbf{x}]</math>, <math>\xi</math> is protected by <math>[\mathbf{x}]</math>.
Usually one chooses <math>\mathbf{y}</math> to be the midpoint of the interval and uses the natural interval extension to assess the remainder.

The special case of the Taylor interval extension of degree <math>k = 0</math> is also referred to as the ''mean value form''.
For an interval extension of the [[Jacobian]] <math>[J_f](\mathbf{[x]})</math>
we get

:<math>[f]([\mathbf{x}]) :=
f(\mathbf{y}) + [J_f](\mathbf{[x]}) \cdot ([\mathbf{x}] - \mathbf{y})
</math>.

A nonlinear function can be defined by linear features.

==Complex interval arithmetic==

An interval can also be defined as a locus of points at a given distance from the centre, and this definition can be extended from real numbers to [[complex number]]s.<ref>[http://books.google.com/books?id=Vtqk6WgttzcC Complex interval arithmetic and its applications], Miodrag Petkovi?, Ljiljana Petkovi?, Wiley-VCH, 1998, ISBN 978-3-527-40134-5</ref> As it is the case with computing with real numbers, computing with complex numbers involves uncertain data. So, given the fact that an interval number is a real closed interval and a complex number is an ordered pair of [[real number]]s, there is no reason to limit the application of interval arithmetic to the measure of uncertainties in computations with real numbers.<ref name="Dawood">[[Hend Dawood]] (2011). ''Theories of Interval Arithmetic: Mathematical Foundations and Applications''. Saarbrücken: LAP LAMBERT Academic Publishing. ISBN 978-3-8465-0154-2.</ref> Interval arithmetic can thus be extended, via [[complex interval numbers]], to determine regions of uncertainty in computing with complex numbers.<ref name="Dawood"/>

The basic algebraic operations for real interval numbers (real closed intervals) can be extended to complex numbers. It is therefore not surprising that complex interval arithmetic is similar to, but not the same as, ordinary complex arithmetic.<ref name="Dawood"/> It can be shown that, as it is the case with real interval arithmetic, there is no distributivity between addition and multiplication of complex interval numbers except for certain special cases, and inverse elements do not always exist for complex interval numbers.<ref name="Dawood"/> Two other useful properties of ordinary complex arithmetic fail to hold in complex interval arithmetic: the additive and multiplicative properties, of ordinary complex conjugates, do not hold for complex interval conjugates.<ref name="Dawood"/>

Interval arithmetic can be extended, in an analogous manner, to other multidimensional [[number systems]] such as [[quaternion]]s and [[octonion]]s, but with the expense that we have to sacrifice other useful properties of ordinary arithmetic.<ref name="Dawood"/>

== Interval methods ==
The methods of classical numerical analysis can not be transferred one-to-one into interval-valued algorithms, as dependencies between numerical values are usually not taken into account.

=== Rounded interval arithmetic ===
[[File:Illustration of outward rounding.png|200px|left|thumb|Outer bounds at different level of rounding]]
In order to work effectively in a real-life implementation, intervals must be compatible with floating point computing. The earlier operations were based on exact arithmetic, but in general fast numerical solution methods may not be available. The range of values of the function <math>f(x, y) = x + y</math>
for <math>x \in [0.1, 0.8]</math> and <math>y \in [0.06, 0.08]</math> are for example <math>[0.16, 0.88]</math>. Where the same calculation is done with single digit precision, the result would normally be <math>[0.2, 0.9]</math>. But <math>[0.2, 0.9] \not\supseteq [0.16, 0.88]</math>,
so this approach would contradict the basic principles of interval arithmetic, as a part of the domain of <math>f([0.1, 0.8], [0.06, 0.08])</math> would be lost.
Instead, it is the outward rounded solution <math>[0.1, 0.9]</math> which is used.

The standard [[IEEE 754]] for binary floating-point arithmetic also sets out procedures for the implementation of rounding. An IEEE 754 compliant system allows programmers to round to the nearest floating point number; alternatives are rounding towards 0 (truncating), rounding toward positive infinity (i.e. up), or rounding towards negative infinity (i.e. down).

The required ''external rounding'' for interval arithmetic can thus be achieved by changing the rounding settings of the processor in the calculation of the upper limit (up) and lower limit (down). Alternatively, an appropriate small interval <math>[\varepsilon_1, \varepsilon_2]</math> can be added.

=== Dependency problem ===
[[File:Interval-dependence problem-front view.png|right|thumb|Approximate estimate of the value range]]
The so-called ''dependency problem'' is a major obstacle to the application of interval arithmetic.
Although interval methods can determine the range of elementary arithmetic operations and functions very accurately, this is not always true with more complicated functions. If an interval occurs several times in a calculation using parameters, and each occurrence is taken independently then this can lead to an unwanted expansion of the resulting intervals.

[[File:Interval-dependence problem.png|180px|left|thumb|Treating each occurrence of a variable independently]]
As an illustration, take the function <math>f</math> defined by
<math>f(x) = x^2 + x</math>. The values of this function over the interval <math>[-1, 1]</math> are really <math>[-1/4 , 2]</math>. As the natural interval extension, it is calculated as <math>[-1, 1]^2 + [-1, 1] = [0,1] + [-1,1] = [-1,2]</math>, which is slightly larger; we have instead calculated the infimum and supremum of the function <math>h(x, y)= x^2+y</math> over <math>x,y \in [-1,1]</math>.
There is a better expression of <math>f</math> in which the variable <math>x</math> only appears once, namely by rewriting <math>f(x) = x^2 + x</math> as addition and squaring in the quadratic
<math>f(x) = \left(x + \frac{1}{2}\right)^2 -\frac{1}{4}</math>.

So the suitable interval calculation is
:<math> \left([-1,1] + \frac{1}{2}\right)^2 -\frac{1}{4} =
\left[-\frac{1}{2}, \frac{3}{2}\right]^2 -\frac{1}{4} = \left[0, \frac{9}{4}\right] -\frac{1}{4} = \left[-\frac{1}{4},2\right]</math>
and gives the correct values.

In general, it can be shown that the exact range of values can be achieved, if each variable appears only once and if <math>f</math> is continuous inside the box. However, not every function can be rewritten this way.

[[File:Interval-wrapping effect.png|160px|right|thumb|Wrapping effect]]
The dependency of the problem causing over-estimation of the value range can go as far as covering a large range, preventing more meaningful conclusions.

An additional increase in the range stems from the solution of areas that do not take the form of an interval vector. The solution set of the linear system
:<math>
\begin{matrix}
x &=& p\\
y &=& p
\end{matrix}
</math>
for <math> p\in [-1,1]</math>
is precisely the line between the points <math>(-1,-1)</math> and <math>(1,1)</math>.
Interval methods deliver the best case, but in the square <math>[-1,1] \times [-1,1]</math>, The real solution is contained in this square (this is known as the ''wrapping effect'').

=== Linear interval systems ===
A linear interval system consists of a matrix interval extension <math>[\mathbf{A}] \in [\mathbb{R}]^{n\times m}</math> and an interval vector <math>[\mathbf{b}] \in [\mathbb{R}]^{n}</math>. We want the smallest cuboid <math>[\mathbf{x}] \in [\mathbb{R}]^{m}</math>, for all vectors
<math>\mathbf{x} \in \mathbb{R}^{m}</math> which there is a pair <math>(\mathbf{A}, \mathbf{b})</math> with <math>\mathbf{A} \in [\mathbf{A}]</math> and <math>\mathbf{b} \in [\mathbf{b}]</math> satisfying
:<math>\mathbf{A} \cdot \mathbf{x} = \mathbf{b}</math>.

For quadratic systems – in other words, for <math>n = m</math> – there can be such an interval vector <math>[\mathbf{x}]</math>, which covers all possible solutions, found simply with the interval Gauss method. This replaces the numerical operations, in that the linear algebra method known as Gaussian elimination becomes its interval version. However, since this method uses the interval entities<math>[\mathbf{A}]</math> and <math>[\mathbf{b}]</math> repeatedly in the calculation, it can produce poor results for some problems. Hence using the result of the interval-valued Gauss only provides first rough estimates, since although it contains the entire solution set, it also has a large area outside it.

A rough solution <math>[\mathbf{x}]</math> can often be improved by an interval version of the [[Gauss–Seidel method]].
The motivation for this is that the <math>i</math>-th row of the interval extension of the linear equation
:<math>
\begin{pmatrix}
{[a_{11}]} & \cdots & {[a_{1n}]} \\
\vdots & \ddots & \vdots \\
{[a_{n1}]} & \cdots & {[a_{nn}]}
\end{pmatrix}
\cdot
\begin{pmatrix}
{x_1} \\
\vdots \\
{x_n}
\end{pmatrix}
=
\begin{pmatrix}
{[b_1]} \\
\vdots \\
{[b_n]}
\end{pmatrix}
</math>
can be determined by the variable <math>x_i</math> if the division <math>1/[a_{ii}]</math> is allowed. It is therefore simultaneously
:<math>x_j \in [x_j]</math> and <math>x_j \in \frac{[b_i]- \sum\limits_{k \not= j} [a_{ik}] \cdot [x_k]}{[a_{ii}]}</math>.
So we can now replace <math>[x_j]</math> by
:<math>[x_j] \cap \frac{[b_i]- \sum\limits_{k \not= j} [a_{ik}] \cdot [x_k]}{[a_{ii}]}</math>,
and so the vector <math>[\mathbf{x}]</math> by each element.
Since the procedure is more efficient for a [[diagonally dominant matrix]], instead of the system <math> [\mathbf{A}]\cdot \mathbf{x} = [\mathbf{b}]\mbox{,}</math> one can often try multiplying it by an appropriate rational matrix <math>\mathbf{M}</math> with the resulting matrix equation
:<math>(\mathbf{M}\cdot[\mathbf{A}])\cdot \mathbf{x} = \mathbf{M}\cdot[\mathbf{b}]</math>
left to solve. If one chooses, for example, <math>\mathbf{M} = \mathbf{A}^{-1}</math> for the central matrix <math>\mathbf{A} \in [\mathbf{A}]</math>, then <math>\mathbf{M} \cdot[\mathbf{A}]</math> is outer extension of the identity matrix.

These methods only work well if the widths of the intervals occurring are sufficiently small. For wider intervals it can be useful to use an interval-linear system on finite (albeit large) real number equivalent linear systems. If all the matrices <math>\mathbf{A} \in [\mathbf{A}]</math> are invertible, it is sufficient to consider all possible combinations (upper and lower) of the endpoints occurring in the intervals. The resulting problems can be resolved using conventional numerical methods. Interval arithmetic is still used to determine rounding errors.

This is only suitable for systems of smaller dimension, since with a fully occupied <math>n \times n</math> matrix, <math>2^{n^2}</math> real matrices need to be inverted, with <math>2^n</math> vectors for the right hand side. This approach was developed by Jiri Rohn and is still being developed.<ref>[http://www.cs.cas.cz/rohn/publist/000home.htm Jiri Rohn, List of publications]</ref>

===Interval Newton method===
[[File:Interval Newton step.png|250px|right|thumb|Reduction of the search area in the interval Newton step in "thick" functions]]
An interval variant of [[Newton's method]] for finding the zeros in an interval vector <math>[\mathbf{x}]</math> can be derived from the average value extension.<ref name="Hansen">{{cite book|last1=Walster|first1=G. William|last2=Hansen|first2=Eldon Robert|title=Global Optimization using Interval Analysis|edition=2nd|year=2004|publisher=Marcel Dekker|location=New York|isbn=0-8247-4059-9}}</ref> For an unknown vector <math>\mathbf{z}\in [\mathbf{x}]</math> applied to <math>\mathbf{y}\in [\mathbf{x}]</math>, gives
:<math>f(\mathbf{z}) \in f(\mathbf{y}) + [J_f](\mathbf{[x]}) \cdot (\mathbf{z} - \mathbf{y})</math>.
For a zero <math>\mathbf{z}</math>, that is <math>f(z)=0</math>, and thus must satisfy
:<math> f(\mathbf{y}) + [J_f](\mathbf{[x]}) \cdot (\mathbf{z} - \mathbf{y})=0 </math>.
This is equivalent to
<math> \mathbf{z} \in \mathbf{y} - [J_f](\mathbf{[x]})^{-1}\cdot f(\mathbf{y})</math>.
An outer estimate of <math>[J_f](\mathbf{[x]})^{-1}\cdot f(\mathbf{y}))</math> can be determined using linear methods.

In each step of the interval Newton method, an approximate starting value <math>[\mathbf{x}]\in [\mathbb{R}]^n</math> is replaced by <math>[\mathbf{x}]\cap \left(\mathbf{y} - [J_f](\mathbf{[x]})^{-1}\cdot f(\mathbf{y})\right)</math> and so the result can be improved iteratively. In contrast to traditional methods, the interval method approaches the result by containing the zeros. This guarantees that the result will produce all the zeros in the initial range. Conversely, it will prove that no zeros of <math>f</math> were in the initial range <math>[\mathbf{x}]</math> if a Newton step produces the empty set.

The method converges on all zeros in the starting region. Division by zero can lead to separation of distinct zeros, though the separation may not be complete; it can be complemented by the [[#Bisection and covers|bisection method]].

As an example, consider the function <math>f(x)= x^2-2</math>, the starting range <math>[x] = [-2,2]</math>, and the point <math>y= 0</math>. We then have <math> J_f(x) = 2\, x</math> and the first Newton step gives
:<math>[-2,2]\cap \left(0 - \frac{1}{2\cdot[-2,2]} (0-2)\right) = [-2,2]\cap \Big([{-\infty}, {-0.5}]\cup [{0.5}, {\infty}] \Big) = [{-2}, {-0.5}] \cup [{0.5}, {2}]</math>.
More Newton steps are used separately on <math>x\in [{-2}, {-0.5}]</math> and <math>[{0.5}, {2}]</math>. These converge to arbitrarily small intervals around <math>-\sqrt{2}</math> and <math>+\sqrt{2}</math>.

The Interval Newton method can also be used with ''thick functions'' such as <math>g(x)= x^2-[2,3]</math>, which would in any case have interval results. The result then produces intervals containing <math> \left[-\sqrt{3},-\sqrt{2} \right] \cup \left[\sqrt{2},\sqrt{3} \right]</math>.

=== Bisection and covers ===
[[File:Illustration of interval mincing.png|220px|right|thumb|Rough estimate (turquoise) and improved estimates through "mincing" (red)]]
The various interval methods deliver conservative results as dependencies between the sizes of different intervals extensions are not taken into account. However the dependency problem becomes less significant for narrower intervals.

Covering an interval vector <math>[\mathbf{x}]</math> by smaller boxes <math>[\mathbf{x}_1], \dots , [\mathbf{x}_k]\mbox{,}</math> so that <math>\textstyle [\mathbf{x}] = \bigcup_{i=1}^k [\mathbf{x}_i]\mbox{,}</math> is then valid for the range of values
<math>\textstyle f([\mathbf{x}]) = \bigcup_{i=1}^k f([\mathbf{x}_i])\mbox{.}</math>
So for the interval extensions described above,
<math>\textstyle [f]([\mathbf{x}]) \supseteq \bigcup_{i=1}^k [f]([\mathbf{x}_i])</math> is valid.
Since <math>[f]([\mathbf{x}])</math> is often a genuine [[superset]] of the right-hand side, this usually leads to an improved estimate.

Such a cover can be generated by the [[bisection method]] such as thick elements <math>[x_{i1}, x_{i2}]</math> of the interval vector <math>[\mathbf{x}] = ([x_{11}, x_{12}], \dots, [x_{n1}, x_{n2}])</math> by splitting in the centre into the two intervals <math>[x_{i1}, (x_{i1}+x_{i2})/2]</math> and <math>[(x_{i1}+x_{i2})/2, x_{i2}]</math>. If the result is still not suitable then further gradual subdivision is possible. Note that a cover of <math>2^r</math> intervals results from <math>r</math> divisions of vector elements, substantially increasing the computation costs.

With very wide intervals, it can be helpful to split all intervals into several subintervals with a constant (and smaller) width, a method known as ''mincing''. This then avoids the calculations for intermediate bisection steps. Both methods are only suitable for problems of low dimension.

==Application==
Interval arithmetic can be use in various areas (such as [[set inversion]], [[motion planning]], [[set estimation]] or [[Vaimos|stability analysis]]), in order to be treated estimates for which no exact numerical values can stated.<ref>{{cite book|last1=Jaulin|first1=Luc|last2=Kieffer|first2=Michel|last3=Didrit|first3=Olivier|last4=Walter|first4=Eric|title=Applied Interval Analysis|year=2001|publisher=Springer|location=Berlin|isbn=1-85233-219-0}}</ref>

=== Rounding error analysis ===
Interval arithmetic is used with error analysis, to control rounding errors arising from each calculation.
The advantage of interval arithmetic is that after each operation there is an interval which reliably includes the true result. The distance between the interval boundaries gives the current calculation of rounding errors directly:
: Error = <math>\mathrm{abs}(a-b)</math> for a given interval <math>[a,b]</math>.
Interval analysis adds to rather than substituting for traditional methods for error reduction, such as [[pivot element|pivoting]].

===Tolerance analysis===

Parameters for which no exact figures can be allocated often arise during the simulation of technical and physical processes.
The production process of technical components allows certain tolerances, so some parameters fluctuate within intervals.
In addition, many fundamental constants are not known precisely.<ref name="Dreyer"/>

If the behavior of such a system affected by tolerances satisfies, for example, <math>f(\mathbf{x}, \mathbf{p}) = 0</math>, for <math> \mathbf{p} \in [\mathbf{p}]</math> and unknown <math>\mathbf{x}</math> then the set of possible solutions
:<math>\{\mathbf{x}\,|\, \exists \mathbf{p} \in [\mathbf{p}], f(\mathbf{x}, \mathbf{p})= 0\}</math>,
can be found by interval methods. This provides an alternative to traditional [[propagation of error]] analysis.
Unlike point methods, such as [[Monte Carlo simulation]], interval arithmetic methodology ensures that no part of the solution area can be overlooked.
However, the result is always a worst case analysis for the distribution of error, as other probability-based distributions are not considered.

===Fuzzy interval arithmetic===
[[File:Fuzzy arithmetic.png|275px|right|thumb|Approximation of the [[normal distribution]] by a sequence of intervals]]
Interval arithmetic can also be used with affiliation functions for fuzzy quantities as they are used in [[fuzzy logic]]. Apart from the strict statements <math>x\in [x]</math> and <math>x \not\in [x]</math>, intermediate values are also possible, to which real numbers <math>\mu \in [0,1]</math> are assigned. <math>\mu = 1</math> corresponds to definite membership while <math>\mu = 0</math> is non-membership. A distribution function assigns uncertainty which can be understood as a further interval.

For ''fuzzy arithmetic''<ref>[http://www.iam.uni-stuttgart.de/Mitarbeiter/Hanss/hanss_en.htm Application of Fuzzy Arithmetic to Quantifying the Effects of Uncertain Model Parameters, Michael Hanss], [[University of Stuttgart]]</ref> only a finite number of discrete affiliation stages <math>\mu_i \in [0,1]</math> are considered. The form of such a distribution for an indistinct value can then represented by a sequence of intervals
:<math>\left[x^{(1)}\right] \supset \left[x^{(2)}\right] \supset \cdots \supset \left[x^{(k)} \right]</math>. The interval <math>[x^{(i)}]</math> corresponds exactly to the fluctuation range for the stage <math>\mu_i</math>.

The appropriate distribution for a function <math>f(x_1, \cdots, x_n)</math> concerning indistinct values
<math>x_1, \cdots, x_n</math> and the corresponding sequences
<math>\left[x_1^{(1)} \right] \supset \cdots \supset \left[x_1^{(k)} \right], \cdots ,
\left[x_n^{(1)} \right] \supset \cdots \supset \left[x_n^{(k)} \right]
</math> can be approximated by the sequence
<math>\left[y^{(1)}\right] \supset \cdots \supset \left[y^{(k)}\right]</math>.
The values <math>\left[y^{(i)}\right]</math> are given by <math>\left[y^{(i)}\right] = f \left( \left[x_{1}^{(i)}\right], \cdots \left[x_{n}^{(i)}\right]\right)</math> and can be calculated by interval methods. The value <math>\left[y^{(1)}\right]</math> corresponds to the result of an interval calculation.

== History ==
Interval arithmetic is not a completely new phenomenon in mathematics; it has appeared several times under different names in the course of history. For example [[Archimedes]] calculated lower and upper bounds 223/71 < [[Pi#History|π]] < 22/7 in the [[3rd century BC]].
Actual calculation with intervals has neither been as popular as other numerical techniques, nor been completely forgotten.

Rules for calculating with intervals and other subsets of the real numbers were published in a 1931 work by Rosalind Cicely Young, a doctoral candidate at the [[University of Cambridge]]. Arithmetic work on range numbers to improve reliability of digital systems were then published in a 1951 textbook on linear algebra by Paul Dwyer ([[University of Michigan]]); intervals were used to measure rounding errors associated with floating-point numbers. A comprehensive paper on interval algebra in numerical analysis was published by Sunaga (1958).<ref>{{cite book|author=T.Sunaga|title="Theory of interval algebra and its application to numerical analysis", RAAG Memoirs, 2 (1958), pp. 29-46."}}</ref>

The birth of modern interval arithmetic was marked by the appearance of the book ''Interval Analysis'' by Ramon E. Moore in 1966.<ref>{{cite book|last=Moore|first=R. E.|title=Interval Analysis|year=1966|publisher=Prentice-Hall|location=Englewood Cliff, New Jersey|isbn=0-13-476853-1}}</ref><ref name="siam">{{cite book|last1=Cloud|first1=Michael J.|last2=Moore|first2=Ramon E.|last3=Kearfott|first3=R. Baker|title=Introduction to Interval Analysis|year=2009|publisher=Society for Industrial and Applied Mathematics|location=Philadelphia|isbn=0-89871-669-1}}</ref> He had the idea in Spring 1958, and a year later he published an article about computer interval arithmetic.<ref>[http://interval.louisiana.edu/Moores_early_papers/bibliography.html Publications Related to Early Interval Work of R. E. Moore]</ref> Its merit was that starting with a simple principle, it provided a general method for automated error analysis, not just errors resulting from rounding.

Independently in 1956, Mieczyslaw Warmus suggested formulae for calculations with intervals,<ref>[http://www.ippt.gov.pl/~zkulpa/quaphys/warmus.html Precursory papers on interval analysis by M. Warmus]</ref> though Moore found the first non-trivial applications.

In the following twenty years, German groups of researchers carried out pioneering work around Götz Alefeld<ref>{{cite book|last1=Alefeld|first1=Götz|last2=Herzberger|first2=Jürgen|title=Einführung in die Intervallrechnung|series=Reihe Informatik|volume=12|publisher=B.I.-Wissenschaftsverlag|location=Mannheim - Wien - Zürich|isbn=3-411-01466-0|language=German}}</ref> and Ulrich Kulisch<ref name="Kulisch"/> at the [[University of Karlsruhe]] and later also at the [[University of Wuppertal|Bergische University of Wuppertal]].
For example, Karl Nickel explored more effective implementations, while improved containment procedures for the solution set of systems of equations were due to Arnold Neumaier among others.<ref>[http://www.mat.univie.ac.at/~neum/publist.html Publications by Arnold Neumaier]</ref> In the 1960s Eldon R. Hansen dealt with interval extensions for linear equations and then provided crucial contributions to global optimisation, including what is now known as Hansen's method, perhaps the most widely used interval algorithm.<ref name="Hansen"/> Classical methods in this often are have the problem of determining the largest (or smallest) global value, but could only find a local optimum and could not find better values;
Helmut Ratschek and Jon George Rokne developed [[branch and bound]] methods, which till then had only applied to integer values, by using intervals to provide applications for continuous values.<ref>[http://pages.cpsc.ucalgary.ca/~rokne/#SEC3 Some publications of Jon Rokne]</ref>

In 1988, Rudolf Lohner developed [[Fortran]]-based software for reliable solutions for initial value problems using [[ordinary differential equations]].<ref>[http://fam-pape.de/raw/ralph/studium/dgl/dglsem.html Bounds for ordinary differential equations of Rudolf Lohner] (in German)</ref>

The journal ''Reliable Computing'' (originally ''Interval Computations'') has been published since the 1990s, dedicated to the reliability of computer-aided computations. As lead editor, R. Baker Kearfott, in addition to his work on global optimisation, has contributed significantly to the unification of notation and terminology used in interval arithmetic ([[#External links|Web]]: Kearfott).

In recent years work has concentrated in particular on the estimation of [[preimage]]s of parameterised functions and to robust control theory by the COPRIN working group of [[INRIA]] in [[Sophia Antipolis]] in France ([[#External links|Web]]: INRIA).

==Patents==
One of the main sponsors of the interval arithmetic, G. William Walster of [[Sun Microsystems]], has lodged several patents in the field of interval arithmetic at the [[U.S. Patent and Trademark Office]] in the years 2002–04.<ref>[http://www.mat.univie.ac.at/coconut-environment/#patents Patent Issues in Interval Arithmetic]</ref> The validity of these patent applications have been disputed in the interval arithmetic research community, since they may possibly only show the past state of the art.

==Implementations==
There are many software packages that permit the development of numerical applications using interval arithmetic.<ref>[http://www.cs.utep.edu/interval-comp/main.html Software for Interval Computations collected by [[Vladik Kreinovich]] ], [[University of Texas at El Paso]]</ref>
These are usually provided in the form of program libraries.
There are also [[C++]] and Fortran [[compiler]]s that handle interval data types and suitable operations as a language extension, so interval arithmetic is supported directly.

Since 1967 ''Extensions for Scientific Computation'' (XSC) have been developed in the University of Karlsruhe for various [[programming language]]s, such as C++, Fortran and [[Pascal (programming language)|Pascal]].<ref>[http://www.math.uni-wuppertal.de/org/WRST/xsc/history.html History of XSC-Languages]</ref> The first platform was a [[Zuse]] [[Z23 (computer)|Z 23]], for which a new interval data type with appropriate elementary operators was made available. There followed in 1976 Pascal-SC, a Pascal variant on a [[Zilog Z80]] which it made possible to create fast complicated routines for automated result verification. Then came the Fortran 77-based ACRITH XSC for the [[System/370]] architecture, which was later delivered by IBM. Starting from 1991 one could produce code for [[C (programming language)|C]] compilers with [[Pascal-XSC]]; a year later the C++ class library supported C-XSC on many different computer systems. In 1997 all XSC variants were made available under the [[GNU General Public License]]. At the beginning of 2000 C-XSC 2.0 was released under the leadership of the working group for scientific computation at the Bergische University of Wuppertal, in order to correspond to the improved C++ standard.

Another C++-class library was created in 1993 at the [[Hamburg University of Technology]] called ''Profil/BIAS'' (Programmer's Runtime Optimized Fast Interval Library, Basic Interval Arithmetic), which made the usual interval operations more user friendly. It emphasized the efficient use of hardware, portability and independence of a particular presentation of intervals.

The [[Boost (C++ libraries)|Boost collection]] of C++ libraries contains a template class for intervals. Its authors are aiming to have interval arithmetic in the standard C++ language.<ref>[http://www-sop.inria.fr/members/Sylvain.Pion/cxx/ A Proposal to add Interval Arithmetic to the C++ Standard Library]</ref>

Gaol<ref>[http://sourceforge.net/projects/gaol Gaol is Not Just Another Interval Arithmetic Library]</ref> is another C++ interval arithmetic library that is unique in that it offers the relational interval operators used in interval [[constraint programming]].

The [[Frink]] programming language has an implementation of interval arithmetic which can handle [[arbitrary-precision arithmetic|arbitrary-precision number]]s. Programs written in Frink can use intervals without rewriting or recompilation.

In addition computer algebra systems, such as [[Mathematica]], [[Maple (software)|Maple]] and [[MuPAD]], can handle intervals. There is a [[Matlab]] extension ''Intlab'' which builds on [[BLAS]] routines, as well as the Toolbox b4m which makes a Profil/BIAS interface.<ref>[http://www.ti3.tu-harburg.de/~rump/intlab/ INTerval LABoratory] and [http://www.ti3.tu-harburg.de/zemke/b4m/ b4m]</ref> Moreover, the Software [[Euler (software)|Euler Math Toolbox]] includes an interval arithmetic.

== IEEE Interval Standard – P1788 ==
An IEEE Interval Standard<ref>[http://grouper.ieee.org/groups/1788/ IEEE Interval Standard Working Group - P1788]</ref> is currently under development.

== Conferences and Workshop ==
Several international conferences or workshop take place every year in the world.
The main conference is probably SCAN (International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computation), but there is also SWIM (Small Workshop on Interval Methods), PPAM (International Conference on Parallel Processing and Applied Mathematics), REC (International Workshop on Reliable Engineering Computing).

== See also ==
* [[Affine arithmetic]]
* [[Automatic differentiation]]
* [[Multigrid method]]
* [[Monte-Carlo simulation]]
* [[Interval finite element]]
* [[Fuzzy number]]
* [[Significant figures]]

==References==
{{Reflist|2}}

==Further reading==
* {{cite journal|last=Hayes|first=Brian|date=November–December 2003|title=A Lucid Interval|journal=American Scientist|publisher=Sigma Xi|volume=91|issue=6|pages=484–488|url=http://www.cs.utep.edu/interval-comp/hayes.pdf}}

== External links ==
* [http://www-sop.inria.fr/coprin/logiciels/ALIAS/Movie/movie_undergraduate.mpg Introductory Film (mpeg)] of the [http://www-sop.inria.fr/coprin/index_english.html COPRIN] teams of [[INRIA]], [[Sophia Antipolis]]
* [http://interval.louisiana.edu/kearfott.html Bibliography of R. Baker Kearfott], [[University of Louisiana at Lafayette]]
* [http://www.mat.univie.ac.at/~neum/interval.html Interval Methods from Arnold Neumaier], [[University of Vienna]]
* [http://www.ti3.tu-harburg.de/rump/intlab/ INTLAB, Institute for Reliable Computing], [[Hamburg University of Technology]]

{{Data types}}

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Transforming polynomials

2013-03-25T01:59:11Z

129.97.125.145: /* Reciprocals of the roots */

{{about|the Tutte polynomial of a graph|the Tutte polynomial of a matroid|Matroid}}

[[Image:Tutte polynomial and chromatic polynomial of the Bull graph.jpg|thumb|300px|right|The polynomial <math>x^4+x^3+x^2y</math> is the Tutte polynomial of the [[Bull graph]]. The red line shows the intersection with the plane <math>y=0</math>, equivalent to the chromatic polynomial.]]

The '''Tutte polynomial''', also called the '''dichromate''' or the '''Tutte–Whitney polynomial''', is a [[polynomial]] in two variables which plays an important role in [[graph theory]], a branch of [[mathematics]] and [[theoretical computer science]]. It is defined for every [[undirected graph]] and contains information about how the graph is connected.

The importance of the Tutte polynomial <math>T_G</math> comes from the information it contains about ''G''. Though originally studied in [[algebraic graph theory]] as a generalisation of counting problems related to [[graph coloring]] and [[nowhere-zero flow]], it contains several famous other specialisations from other sciences such as the [[Jones polynomial]] from [[knot theory]] and the partition functions of the [[Potts model]] from [[statistical physics]]. It is also the source of several central [[computational problem]]s in [[theoretical computer science]].

The Tutte polynomial has several equivalent definitions. It is equivalent to Whitney’s '''rank polynomial''', Tutte’s own '''dichromatic polynomial''' and Fortuin–Kasteleyn’s '''random cluster model''' under simple transformations. It is essentially a [[generating function]] for the number of edge sets of a given size and connected components, with immediate generalisations to [[matroid]]s. It is also the most general [[graph invariant]] that can be defined by a deletion–contraction recurrence. Several textbooks about graph theory and matroid theory devote entire chapters to it.<ref>Chap. 10 in {{harvtxt|Bollobás|1998}}, chap. 13 in {{harvtxt|Biggs|1993}}, chap. 15 in {{harvtxt|Godsil|Royle|2004}}.</ref>

==Definitions==
<blockquote>'''Definition.''' For an undirected graph <math>G=(V,E)</math> one may define the '''Tutte polynomial''' as

:<math>T_G(x,y)=\sum\nolimits_{A\subseteq E} (x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|},</math>

where <math>k(A)</math> denotes the number of [[connected component (graph theory)|connected component]]s of the graph <math>(V,A)</math>. In this definition it is clear that <math>T_G</math> is well-defined and a polynomial in ''x'' and ''y''.</blockquote>

The same definition can be given using slightly different notation by letting <math>r(A)=|V|-k(A)</math> denote the [[rank (graph theory)|rank]] of the graph <math>(V,A)</math>. Then the '''Whitney rank generating function''' is defined as

:<math>R_G(x,y)=\sum\nolimits_{A\subseteq E} x^{r(E)-r(A)} y^{|A|-r(A)}</math>

the two functions are equivalent under a simple change of variables:

:<math>T_G(x,y)=R_G(x-1,y-1).</math>

Tutte’s '''dichromatic polynomial''' <math>Q_G</math> is the result of another simple transformation:

:<math>T_G(x,y)=(x-1)^{-k(G)} Q_G(x-1,y-1).</math>

Tutte’s original definition of <math>T_G</math> is equivalent but less easily stated. For connected ''G'' we set

:<math>T_G(x,y)=\sum\nolimits_{i,j} t_{ij} x^iy^j,</math>

where <math>t_{ij}</math> denotes the number of [[Spanning tree (mathematics)|spanning tree]]s of “internal activity ''i'' and external activity ''j''.”

A third definition uses a '''deletion–contraction recurrence'''. The [[edge contraction]] ''G''/''uv'' of graph ''G'' is the graph obtained by merging the vertices ''u'' and ''v'' and removing the edge ''uv''. We write ''G'' − ''uv'' for the graph where the edge ''uv'' is merely removed. Then the Tutte polynomial is defined by the recurrence relation

:<math>T_G= T_{G-e}+T_{G/e},</math>

if ''e'' is neither a [[Loop (graph theory)|loop]] nor a [[Bridge (graph theory)|bridge]], with base case

:<math>T_G(x,y)= x^i y^j, </math>

if ''G'' contains ''i'' bridges and ''j'' loops and no other edges. Especially, <math>T_G=1</math> if ''G'' contains no edges.

The '''random cluster model''' from statistical mechanics due to {{harvtxt|Fortuin|Kasteleyn|1972}} provides yet another equivalent definition.<ref>cf. {{harvtxt|Sokal|2005}}</ref> The polynomial

:<math>Z_G(q,w)=\sum\nolimits_{F\subseteq E}q^{k(F)}w^{|F|}</math>

is equivalent to <math>T_G</math> under the transformation<ref>eq. (2.26) in {{harvtxt|Sokal|2005}}</ref>

:<math>T_G(x, y)=(x-1)^{-k(E)}(y-1)^{-|V|} \cdot Z_G\Big((x-1)(y-1),\; y-1\Big).</math>

===Properties===
The Tutte polynomial factors into connected components: If ''G'' is the union of disjoint graphs ''H'' and <math>H'</math> then
: <math>T_G= T_H \cdot T_{H'}</math>

If ''G'' is planar and <math>G^*</math> denotes its [[dual graph]] then

: <math>T_G(x,y)= T_{G^*} (y,x)</math>

Especially, the chromatic polynomial of a planar graph is the flow polynomial of its dual.

===Examples===
Isomorphic graphs have the same Tutte polynomial, but the opposite is not true. For example, the Tutte polynomial of every tree on ''m'' edges is <math>x^m</math>.

Tutte polynomials are often given in tabular form by listing the coefficients <math>t_{ij}</math> of <math>x^iy^j</math> in row ''i'' and column ''j''. For example, the Tutte polynomial of the [[Petersen graph]],

:<math>\begin{align}
36 x &+ 120 x^2 + 180 x^3 + 170x^4+114x^5 + 56x^6 +21 x^7 + 6x^8 + x^9 \\
&+ 36y +84 y^2 + 75 y^3 +35 y^4 + 9y^5+y^6 \\
&+ 168xy + 240x^2y +170x^3y +70 x^4y + 12x^5 y \\
&+ 171xy^2+105 x^2y^2 + 30x^3y^2 \\
&+ 65xy^3 +15x^2y^3 \\
&+10xy^4,
\end{align}</math>

is given by the following table.

{|class="wikitable"
| 0 ||36|| 84|| 75|| 35|| 9|| 1
|-
| 36 ||168 || 171 || 65 || 10
|-
|120 || 240 || 105 || 15
|-
|180 || 170 || 30
|-
|170 || 70
|-
|114 || 12
|-
| 56
|-
| 21
|-
| 6
|-
| 1
|}

==History==
[[W. T. Tutte]]’s interest in the [[deletion–contraction formula]] started in his undergraduate days at [[Trinity College, Cambridge]], originally motivated by [[perfect rectangle]]s and [[Spanning tree (mathematics)|spanning tree]]s. He often applied the formula in his research and “wondered if there were other interesting [[graph invariant|functions of graphs, invariant under isomorphism]], with similar recursion formulae.”<ref name="harvtxt|Tutte|2004">{{harvtxt|Tutte|2004}}</ref> [[R. M. Foster]] had already observed that the [[chromatic polynomial]] is one such function, and Tutte began to discover more. His original terminology for graph invariants that satisfy the delection–contraction recursion was ''W-function'' (and ''V-function'' if multiplicative over component). Tutte writes, “Playing with my ''W-functions'' I obtained a two-variable polynomial from which either the chromatic polynomial or the ﬂow-polynomial could be obtained by setting one of the variables equal to zero, and adjusting signs.”<ref name="harvtxt|Tutte|2004"/> Tutte called this function the ''dichromate'', as he saw it as a generalization of the chromatic polynomial to two variables, but it is usually referred to as the Tutte polynomial. In Tutte’s words, “This may be unfair to [[Hassler Whitney]] who knew and used analogous coefﬁcients without bothering to afﬁx them to two variables.” There is “notable confusion” <ref>Welsh</ref> about the terms ''dichromate'' and ''dichromatic polynomial'', introduced by Tutte in different papers and differ slightly. The generalisation of the Tutte polynomial to matroids was first published by Crapo, though it appears already in Tutte’s thesis.<ref>See {{harvtxt|Farr|2007}}</ref>

Independently of the work in [[algebraic graph theory]], Potts began studying the [[partition function (statistical mechanics)|partition function]] of certain models in [[statistical mechanics]] in 1952. The work of {{harvtxt|Fortuin|Kasteleyn|1972}} on the [[random cluster model]], a generalisation of [[Potts model]], provided a unifying expression that showed the relation to the Tutte polynomial.<ref>{{harvtxt|Farr|2007}}</ref>

== Specialisations==
At various points and lines of the <math>(x,y)</math>-plane, the Tutte polynomial evaluates to quantities that have been studied in their own right in diverse fields of mathematics and physics. Part of the appeal of the Tutte polynomial comes from the unifying framework it provides for analysing these quantities.

===Chromatic polynomial===
{{Main|Chromatic polynomial}}

[[Image:Chromatic in the Tutte plane.jpg|thumb|right|The chromatic polynomial drawn in the Tutte plane]]

At <math>y=0</math>, the Tutte polynomial specialises to the chromatic polynomial,

:<math>\chi_G(\lambda) = (-1)^{|V|-k(G)} \lambda^{k(G)} T_G(1-\lambda,0),</math>
where <math>k(G)</math> denotes the number of connected components of ''G''.

For integer λ the value of chromatic polynomial <math>\chi_G(\lambda)</math> equals the number of [[vertex colouring]]s of ''G'' using a set of λ colours. It is clear that <math>\chi_G(\lambda)</math> does not depend on the set of colours. What is less clear is that it is the evaluation at λ of a polynomial with integer coefficients. To see this, we observe:
# If ''G'' has ''n'' vertices and no edges, then <math>\chi_G(\lambda) = \lambda^n</math>.
# If ''G'' contains a loop (a single edge connecting a vertex to itself), then <math>\chi_G(\lambda) = 0</math>.
# If ''e'' is an edge which is not a loop, then
::<math>\chi_G(\lambda) = \chi_{G\setminus e}(\lambda) - \chi_{G/e}(\lambda).</math>

The three conditions above enable us to calculate <math>\chi_G(\lambda)</math>, by applying a sequence of edge deletions and contractions; but they give no guarantee that a different sequence of deletions and contractions will lead to the same value. The guarantee comes from the fact that <math>\chi_G(\lambda)</math> counts something, independently of the recurrence. In particular,

:<math>T_G(2,0) = (-1)^{|V|} \chi_G(-1)</math>

gives the number of acyclic orientations.

===Jones polynomial===
{{Main|Jones polynomial}}

[[Image:Jones in the Tutte plane.jpg|thumb|right|The Jones polynomial drawn in the Tutte plane]]

Along the hyperbola <math>xy=1</math>, the Tutte polynomial specialises to the [[Jones polynomial]] of an [[alternating knot]] if ''G'' is planar.

===Individual points===
====(2,1)====
<math>T_G(2,1)</math> counts the number of [[tree (graph theory)|forest]]s, i.e., the number of acyclic edge subsets.
====(1,1)====
<math>T_G(1,1)</math> counts the number of spanning forests (edge subsets without cycles and the same number of connected components as ''G'').
====(1,2)====
<math>T_G(1,2)</math> counts the number of spanning subgraphs (edge subsets with the same number of connected components as ''G'').
====(2,0)====
<math>T_G(2,0)</math> counts the number of [[acyclic orientation]]s of ''G''.<ref name="welsh99">{{harvtxt|Welsh|1999}}</ref>
====(0,2)====
<math>T_G(0,2)</math> counts the number of [[Robbins theorem|strongly connected orientations]] of ''G''.<ref>{{harvtxt|Las Vergnas|1980}}.</ref>
====(0,−2)====
If ''G'' is a 4-regular graph, then
:<math>(-1)^{|V|+k(G)}T_G(0,-2)</math>
counts the number of [[Eulerian orientation]]s of ''G''. Here <math>k(G)</math> is the number of connected components of ''G''.<ref name="welsh99"/>
====(3,3)====
If ''G'' is the ''m'' × ''n'' [[grid graph]], then <math>2 T_G(3,3)</math> counts the number of ways to tile a rectangle of width 4''m'' and height 4''n'' with [[tetromino|T-tetrominoes]].<ref>{{harvtxt|Korn|Pak|2004}}, see also {{harvtxt|Korn|Pak|2003}} for combinatorial interpretations of many other points</ref>

If ''G'' is a [[planar graph]], then <math>2 T_G(3,3)</math> equals the sum over weighted Eulerian orientations in a [[medial graph]] of ''G'', where the weight of an orientation is 2 to the number of saddle vertices of the orientation (that is, the number of vertices with incident edges cyclicly ordered "in, out, in out").<ref>{{harvtxt|Las Vergnas|1988}}</ref>

===Potts and Ising models===
{{Main|Ising model|Potts model}}

[[Image:Potts and Ising in the Tutte plane.jpg|thumb|right|The partition functions for the Ising model and the 3- and 4-state Potts models drawn in the Tutte plane.]]

Define the hyperbola in the ''xy''−plane:

:<math> H_2: \quad (x-1)(y-1)=2,</math>

the Tutte polynomial specialises to the partition function, <math>Z(\cdot),</math> of the [[Ising model]] studied in [[statistical physics]]. Specifically, along the hyperbola <math>H_2</math> the two are related by the equation:<ref>{{cite book
| last = Welsh
| first = Dominic
| authorlink = Dominic Welsh
| title = Complexity: Knots, Colourings and Counting
| series = London Mathematical Society Lecture Note Series
| year = 1993
| publisher = Cambridge University Press
| isbn = 978-0521457408
| page = 62
}}</ref>

:<math>Z(G) = 2\left(e^{-\alpha}\right)^{|E| - r(E)} \left(4 \sinh \alpha \right )^{r(E)} T_G \left (\coth \alpha, e^{2 \alpha} \right).</math>

In particular,

:<math>(\coth \alpha - 1) \left(e^{2 \alpha} - 1 \right ) = 2</math>

for all complex α.

More generally, if for any positive integer ''q'', we define the hyperbola:

:<math>H_q: \quad (x-1)(y-1)=q,</math>

then the Tutte polynomial specialises to the partition function of the ''q''-state [[Potts model]]. Various physical quantities analysed in the framework of the Potts model translate to specific parts of the <math>H_q</math>.

{|class="wikitable" style="margin: 1em auto 1em auto"
|+ Correspondences between the Potts model and the Tutte plane <ref>{{harvtxt|Welsh|Merino|2000}}</ref>
! Potts model || Tutte polynomial
|-
| [[Ferromagnetism|Ferromagnetic]]
|| Positive branch of <math>H_q</math>
|-
| [[Antiferromagnetism|Antiferromagnetic]]
|| Negative branch of <math>H_q</math> with <math>y>0</math>
|-
| High temperature
|| Asymptote of <math>H_q</math> to <math>y=1</math>
|-
| Low temperature ferromagnetic
|| Positive branch of <math>H_q</math> asymptotic to <math>x=1</math>
|-
| Zero temperature antiferromagnetic
|| [[Graph coloring|Graph ''q''-colouring]], i.e., <math>x=1-q, y=0</math>
|}

===Flow polynomial===
{{Main|Nowhere-zero flow}}

[[Image:Flow in the Tutte plane.jpg|thumb|right|The flow polynomial drawn in the Tutte plane]]

At <math>x=0</math>, the Tutte polynomial specialises to the flow polynomial studied in combinatorics. For a connected and undirected graph ''G'' and integer ''k'', a nowhere-zero ''k''-flow is an assignment of “flow” values <math>1,2,\dots,k-1</math> to the edges of an arbitrary orientation of ''G'' such that the total flow entering and leaving each vertex is congruent modulo ''k''. The flow polynomial <math>C_G(k)</math> denotes the number of nowhere-zero ''k''-flows. This value is intimately connected with the chromatic polynomial, in fact, if ''G'' is a [[planar graph]], the chromatic polynomial of ''G'' is equivalent to the flow polynomial of its [[dual graph]] <math>G^*</math> in the sense that

<blockquote>'''Theorem (Tutte).'''

:<math>C_G(k)=k^{-1} \chi_{G^*}(k).</math>

The connection to the Tutte polynomial is given by:

: <math>C_G(k)= (-1)^{|E|+|V|+k(G)} T_G(0,1-k).</math></blockquote>

===Reliability polynomial===
{{Main|Connectivity (graph theory)}}

[[Image:Reliability in the Tutte plane.jpg|thumb|right|The reliability polynomial drawn in the Tutte plane]]

At <math>x=1</math>, the Tutte polynomial specialises to the all-terminal reliability polynomial studied in network theory. For a connected graph ''G'' remove every edge with probability ''p''; this models a network subject to random edge failures. Then the reliability polynomial is a function <math>R_G(p)</math>, a polynomial in ''p'', that gives the probability that every pair of vertices in ''G'' remains connected after the edges fail. The connection to the Tutte polynomial is given by

:<math>R_G(p) = (1-p)^{|V|-k(G)} p^{|E|-|V|+k(G)} T_G \left (1, \tfrac{1}{p} \right).</math>

===Dichromatic polynomial===
Tutte also defined a closer 2-variable generalization of the chromatic polynomial, the '''dichromatic polynomial''' of a graph. This is

:<math>Q_G(u,v) = \sum\nolimits_{A \subseteq E} u^{k(A)} v^{|A|-|V|+k(A)},</math>

where <math>k(A)</math> is the number of [[connected component (graph theory)|connected components]] of the spanning subgraph (''V'',''A''). This is related to the '''corank-nullity polynomial''' by

:<math>Q_G(u,v) = u^{k(G)} \, R_G(u,v).</math>

The dichromatic polynomial does not generalize to matroids because ''c''(''A'') is not a matroid property: different graphs with the same matroid can have different numbers of connected components.

==Related polynomials==
===Martin polynomial===
{{main|Martin polynomial}}
The Martin polynomial <math>m_{\vec{G}}(x)</math> of an oriented 4-regular graph <math>\vec{G}</math> was defined by Pierre Martin in his 1977 thesis.<ref> {{Cite thesis |last=Martin |first=Pierre |title=Enumérations Eulériennes dans les multigraphes et invariants de Tutte-Grothendieck
|trans_title=Eulerian Enumerations in multigraphs and Tutte-Grothendieck invariants |language=French |publisher=[[Joseph Fourier University]] |url=http://tel.archives-ouvertes.fr/tel-00287330/en |year=1977}} </ref> In this work, Martin showed that if ''G'' is a plane graph and <math>\vec{G}_m</math> is its [[Medial_graph#Directed_medial_graph|directed medial graph]], then

:<math>T_G(x,x) = m_{\vec{G}_m}(x).</math>

==Algorithms==
===Deletion–contraction===
[[Image:deletion-contraction.svg|thumb|right|300px|The deletion–contraction algorithm applied to the [[diamond graph]]. Red edges are deleted in the left child and contracted in the right child. The resulting polynomial is the sum of the monomials at the leaves, <math>x^3+2x^2 +y^2+2xy+x+y</math>. Based on {{harvtxt|Welsh|Merino|2000}}.]]

The deletion–contraction recurrence for the Tutte polynomial,
: <math>T_G(x,y)= T_{G \setminus e}(x,y) + T_{G/e}(x,y), \qquad e \text{ not a loop nor a bridge.} </math>
immediately yields a recursive algorithm for computing it: choose any such edge ''e'' and repeatedly apply the formula until all edges are either loops or bridges; the resulting base cases at the bottom of the evaluation are easy to compute.

Within a polynomial factor, the running time ''t'' of this algorithm can be expressed in terms of the number of vertices ''n'' and the number of edges ''m'' of the graph,
:<math>t(n+m)= t(n+m-1) + t(n+m-2),</math>
a recurrence relation that scales as the [[Fibonacci numbers]] with solution<ref>{{harvtxt|Wilf|1986}}</ref>

:<math> t(n+m)= \left (\frac{1+\sqrt{5}}{2} \right )^{n+m} = O \left (1.6180^{n+m} \right ).</math>

The analysis can be improved to within a polynomial factor of the number <math>\tau(G)</math> of [[spanning tree (mathematics)|spanning trees]] of the input graph.<ref>{{harvtxt|Sekine|Imai|Tani|1995}}</ref> For sparse graphs with <math>m=O(n)</math> this running time is <math>O(\exp(n))</math>. For regular graphs of degree ''k'', the number of spanning trees can be bounded by

:<math>\tau(G) = O \left (\nu_k^n n^{-1} \log n \right ),</math>

where

:<math>\nu_k = \frac{(k-1)^{k-1}}{(k^2-2k)^{\frac{k}{2}-1}}.</math>

so the deletion–contraction algorithm runs within a polynomial factor of this bound. For example:<ref>{{harvtxt|Chung|Yau|1999}}, following {{harvtxt|Björklund|Husfeldt|Kaski|Koivisto|2008}}</ref>

:<math>\nu_5 \approx 4.4066.</math>

In practice, [[graph isomorphism]] testing is used to avoid some recursive calls. This approach works well for graphs that are quite sparse and exhibit many symmetries; the performance of the algorithm depends on the heuristic used to pick the edge ''e''.<ref>{{harvtxt|Sekine|Imai|Tani|1995}}, {{harvtxt|Imai|2000}}, {{harvtxt|Haggard|Pierce|Royle|2008}}</ref>

===Gaussian elimination===
In some restricted instances, the Tutte polynomial can be computed in polynomial time, ultimately because [[Gaussian elimination]] efficiently computes the matrix operations [[determinant]] and [[Pfaffian]]. These algorithms are themselves important results from [[algebraic graph theory]] and [[statistical mechanics]].

<math>T_G(1,1)</math> equals the number <math>\tau(G)</math> of [[Spanning tree (mathematics)|spanning tree]]s of a connected graph. This is
computable in polynomial time as the [[determinant]] of a maximal principal submatrix of the [[Laplacian matrix]] of ''G'', an early result in algebraic graph theory known as [[Kirchhoff’s Matrix–Tree theorem]]. Likewise, the dimension of the bicycle space at <math>T_G(-1,-1)</math> can be computed in polynomial time by Gaussian elimination.

For planar graphs, the partition function of the Ising model, i.e., the Tutte polynomial at the hyperbola <math>H_2</math>, can be expressed as a Pfaffian and computed efficiently via the [[FKT algorithm]]. This idea was developed by [[Michael Fisher|Fisher]], [[Pieter Kasteleyn|Kasteleyn]], and [[Harold Neville Vazeille Temperley|Temperley]] to compute for the number of [[domino tiling|dimer]] covers of a planar [[Lattice model (physics)|lattice model]].

===Markov chain Monte Carlo===
Using a [[Markov chain Monte Carlo]] method, the Tutte polynomial can be arbitrarily well approximated along the positive branch of <math>H_2</math>, equivalently, the partition function of the ferromagnetic Ising model. This exploits the close connection between the Ising model and the problem of counting [[Matching (graph theory)|matchings]] in a graph. The idea behind this celebrated result of Jerrum and Sinclair<ref>{{harvtxt|Jerrum|Sinclair|1993}}</ref> is to set up a [[Markov chain]] whose states are the matchings of the input graph. The transitions are defined by choosing edges at random and modifying the matching accordingly. The resulting Markov chain is rapidly mixing and leads to “sufficiently random” matchings, which can be used to recover the partition function using random sampling. The resulting algorithm is a [[fully polynomial-time randomized approximation scheme]] (fpras).

==Computational complexity==
Several computational problems are associated with the Tutte polynomial. The most straightforward one is
;Input
:A graph ''G''
;Output
:The coefficients of <math>T_G</math>
In particular, the output allows evaluating <math>T_G(-2,0)</math> which is equivalent to counting the number of 3-colourings of ''G''. This latter question is [[Sharp-P-complete|#P-complete]], even when restricted to the family of [[planar graph]]s, so the problem of computing the coefficients of the Tutte polynomial for a given graph is [[Sharp-P-hard|#P-hard]] even for planar graphs.

Much more attention has been given to the family of problems called Tutte<math>(x,y)</math> defined for every complex pair <math>(x,y)</math>:
;Input
:A graph ''G''
;Output
:The value of <math>T_G(x,y)</math>
The hardness of these problems varies with the coordinates <math>(x,y)</math>.

===Exact computation===
[[Image:Tractable points of the Tutte polynomial in the real plane.svg|thumb|300px|right|
The Tutte plane.
Every point <math>(x,y)</math> in the real plane corresponds to a computational problem <math>T_G(x,y)</math>.
At any red point, the problem is polynomial-time computable;
at any blue point, the problem is #P-hard in general, but polynomial-time computable for planar graphs; and
at any point in the white regions, the problem is #P-hard even for bipartite planar graphs.
]]
If both ''x'' and ''y'' are non-negative integers, the problem <math>T_G(x,y)</math> belongs to [[Sharp-P|#P]]. For general integer pairs, the Tutte polynomial contains negative terms, which places the problem in the complexity class [[GapP]], the closure of [[Sharp-P|#P]] under subtraction. To accommodate rational coordinates <math>(x,y)</math>, one can define a rational analogue of [[Sharp-P|#P]].<ref name="harvtxt|Goldberg |Jerrum |2008"> {{harvtxt|Goldberg|Jerrum|2008}}</ref>

The computational complexity of exactly computing <math>T_G(x,y)</math> falls into one of two classes for any <math>x, y \in \mathbb{C}</math>. The problem is #P-hard unless <math>(x,y)</math> lies on the hyperbola <math>H_1</math> or is one of the points

:<math>\left \{ (1,1), (-1,-1), (0,-1), (-1,0), (i,-i), (-i,i), \left(j,j^2 \right), \left(j^2,j \right) \right \}, \qquad j = e^{\frac{2 \pi i}{3}}.</math>

In which cases it is computable in polynomial time.<ref>{{harvtxt|Jaeger|Vertigan|Welsh|1990}}</ref> If the problem is restricted to the class of planar graphs, the points on the hyperbola <math>H_2</math> become polynomial-time computable as well. All other points remain #P-hard, even for bipartite planar graphs.<ref>{{harvtxt|Vertigan|Welsh|1992}}</ref> In his paper on the dichotomy for planar graphs, Vertigan claims (in his conclusion) that the same result holds when further restricted to graphs with vertex degree at most three, save for the point <math>T_G(0,-2)</math>, which counts nowhere-zero '''Z'''3-flows and is computable in polynomial time.<ref>{{cite journal | last = Vertigan | first = Dirk | year = 2005 | title = The Computational Complexity of Tutte Invariants for Planar Graphs | journal = SIAM J. Comput. | volume = 35 | issue = 3 | pages = 690–712 | doi = 10.1137/S0097539704446797 | url = http://dx.doi.org/10.1137/S0097539704446797 }}</ref>

These results contain several notable special cases. For example, the problem of computing the partition function of the Ising model is #P-hard in general, even though celebrated algorithms of Onsager and Fisher solve it for planar lattices. Also, the Jones polynomial is #P-hard to compute. Finally, computing the number of four-colourings of a planar graph is #P-complete, even though the decision problem is trivial by the [[four color theorem|four colour theorem]]. In contrast, it is easy to see that counting the number of three-colourings for planar graphs is #P-complete because the decision problem is known to be NP-complete via a [[parsimonious reduction]].

===Approximation===
The question which points admit a good approximation algorithm has been very well studied. Apart from the points that can be computed exactly in polynomial time, the only approximation algorithm known for <math>T_G(x,y)</math> is Jerrum and Sinclair’s FPRAS, which works for points on the “Ising” hyperbola <math>H_2</math> for ''y'' > 0. If the input graphs are restricted to dense instances, with degree <math>\Omega(n)</math>, there is an FPRAS if ''x'' ≥ 1, ''y'' ≥ 1.<ref>''x'' ≥ 1, ''y'' = 1 is given by {{harvtxt|Annan|1994}}. ''x'' ≥ 1, ''y'' > 1 is given by {{harvtxt|Alon|Frieze|Welsh|1995}}</ref>

Even though the situation is not as well understood as for exact computation, large areas of the plane are known to be hard to approximate.<ref name="harvtxt|Goldberg |Jerrum |2008" />

==See also==
[[Bollobás–Riordan polynomial]]

==Notes==
{{Reflist|colwidth=25em}}

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{{refend}}

== External links ==
* {{springer|title=Tutte polynomial|id=p/t120210}}
* {{MathWorld | urlname=TuttePolynomial| title=Tutte polynomial}}
* [[PlanetMath]] [http://planetmath.org/encyclopedia/ChromaticPolynomial.html Chromatic polynomial]
* Steven R. Pagano: [http://www.ms.uky.edu/~pagano/Matridx.htm Matroids and Signed Graphs]
* Sandra Kingan: [http://members.aol.com/matroids/ Matroid theory]. Lots of links.
* Code for computing Tutte, Chromatic and Flow Polynomials by Gary Haggard, David J. Pearce and Gordon Royle: [http://www.ecs.vuw.ac.nz/~djp/tutte/]

[[Category:Computational problems]]
[[Category:Duality theories]]
[[Category:Matroid theory]]
[[Category:Polynomials]]
[[Category:Graph invariants]]

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User talk:MCiura

2011-12-16T23:20:31Z

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129.97.144.34: /* Time dependent equation */ That sentence unnecessarily stated the obvious and did not fit the voice of the article.

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129.97.4.50: Made some inline text non-png, for consistancy and page-loading speed.

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