formulasearchengine - User contributions [en]

Multistatic radar

2014-12-30T01:01:13Z

80.65.246.108:

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Rabin cryptosystem

2014-01-10T00:57:26Z

80.65.240.69: /* Computing square roots */

{{redirects|Link number|the logic puzzle|Numberlink}}
[[Image:3D-Link.PNG|thumb|right|The two curves of this (2,8)-[[torus knot|torus link]] have linking number four.]]
In [[mathematics]], the '''linking number''' is a numerical [[invariant (mathematics)|invariant]] that describes the linking of two [[closed curve]]s in [[three-dimensional space]]. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an [[integer]], but may be positive or negative depending on the [[curve orientation|orientation]] of the two curves.

The linking number was introduced by [[Carl Friedrich Gauss|Gauss]] in the form of the '''linking integral'''. It is an important object of study in [[knot theory]], [[algebraic topology]], and [[differential geometry]], and has numerous applications in [[mathematics]] and [[science]], including [[quantum mechanics]], [[electromagnetism]], and the study of [[DNA supercoil]]ing.

==Definition==
Any two closed curves in space, if allowed to pass through themselves but not each other, can be [[homotopy|moved]] into exactly one of the following standard positions. This determines the linking number:
{| border=0 cellpadding=5 align="center"
|-valign="center"
|<math>\cdots</math>
|align="center"|[[Image:Linking Number -2.svg|140px]]
|align="center"|[[Image:Linking Number -1.svg|140px]]
|align="center"|[[Image:Linking Number 0.svg|140px]]
|
|
|-valign="center"
|
|align="center"|linking number -2
|align="center"|linking number -1
|align="center"|linking number 0
|
|
|-valign="center"
|
|
|align="center"|[[Image:Linking Number 1.svg|140px]]
|align="center"|[[Image:Linking Number 2.svg|140px]]
|align="center"|[[Image:Linking Number 3.svg|140px]]
|<math>\cdots</math>
|-valign="center"
|
|
|align="center"|linking number 1
|align="center"|linking number 2
|align="center"|linking number 3
|
|}
Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as [[regular homotopy]], which further requires that each curve be an ''immersion'', not just any map. However, this added condition does not change the definition of linking number (it does not matter if the curves are required to always be immersions or not), which is an example of an [[h-principle|''h''-principle]] (homotopy-principle), meaning that geometry reduces to topology.

=== Proof ===
This fact (that the linking number is the only invariant) is most easily proven by placing one circle in standard position, and then showing that linking number is the only invariant of the other circle. In detail:
* A single curve is regular homotopic to a standard circle (any knot can be unknotted if the curve is allowed to pass through itself). The fact that it is ''homotopic'' is clear, since 3-space is contractible and thus all maps into it are homotopic, though the fact that this can be done through immersions requires some geometric argument.
* The complement of a standard circle is homeomorphic to a solid torus with a point removed (this can be seen by interpreting 3-space as the 3-sphere with the point at infinity removed, and the 3-sphere as two solid tori glued along the boundary), or the complement can be analyzed directly.
* The [[fundamental group]] of 3-space minus a circle is the integers, corresponding to linking number. This can be seen via the [[Seifert–Van Kampen theorem]] (either adding the point at infinity to get a solid torus, or adding the circle to get 3-space, allows one to compute the fundamental group of the desired space).
* Thus homotopy classes of a curve in 3-space minus a circle are determined by linking number.
* It is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument.

==Computing the linking number==
[[Image:Linking Number Example.svg|thumb|With six positive crossings and two negative crossings, these curves have linking number two.]]
There is an [[algorithm]] to compute the linking number of two curves from a link [[knot theory#Knot diagrams|diagram]]. Label each crossing as ''positive'' or ''negative'', according to the following rule:<ref>This is the same labeling used to compute the [[writhe]] of a [[knot (mathematics)|knot]], though in this case we only label crossings that involve both curves of the link.</ref>
<center>[[Image:Link Crossings.svg|350px]]</center>
The total number of positive crossings minus the total number of negative crossings is equal to ''twice'' the linking number. That is:
:<math>\mbox{linking number}=\frac{n_1 + n_2 - n_3 - n_4}{2}</math>
where ''n''1, ''n''2, ''n''3, ''n''4 represent the number of crossings of each of the four types. The two sums <math>n_1 + n_3\,\!</math> and <math>n_2 + n_4\,\!</math> are always equal,<ref>This follows from the [[Jordan curve theorem]] if either curve is simple. For example, if the blue curve is simple, then ''n''1 + ''n''3 and ''n''2 + ''n''4 represent the number of times that the red curve crosses in and out of the region bounded by the blue curve.</ref> which leads to the following alternative formula
:<math>\mbox{linking number}\,=\,n_1-n_4\,=\,n_2-n_3.</math>
Note that <math>n_1-n_4</math> involves only the undercrossings of the blue curve by the red, while <math>n_2-n_3</math> involves only the overcrossings.

==Properties and examples==
[[Image:Labeled Whitehead Link.svg|thumb|The two curves of the [[Whitehead link]] have linking number zero.]]
* Any two unlinked curves have linking number zero. However, two curves with linking number zero may still be linked (e.g. the [[Whitehead link]]).
* Reversing the orientation of either of the curves negates the linking number, while reversing the orientation of both curves leaves it unchanged.
* The linking number is [[chirality (mathematics)|chiral]]: taking the [[mirror image]] of link negates the linking number. The convention for positive linking number is based on a [[right-hand rule]].
* The [[winding number]] of an oriented curve in the ''x''-''y'' plane is equal to its linking number with the ''z''-axis (thinking of the ''z''-axis as a closed curve in the [[3-sphere]]).
* More generally, if either of the curves is [[Curve#Topology|simple]], then the first [[homology (mathematics)|homology group]] of its complement is [[group isomorphism|isomorphic]] to '''[[integer|Z]]'''. In this case, the linking number is determined by the homology class of the other curve.
* In [[physics]], the linking number is an example of a [[topological quantum number]]. It is related to [[quantum entanglement]].

==Gauss's integral definition==
Given two non-intersecting differentiable curves <math>\gamma_1, \gamma_2 \colon S^1 \rightarrow \mathbb{R}^3</math>, define the '''[[Carl Friedrich Gauss|Gauss]] map''' <math>\Gamma</math> from the [[torus]] to the [[unit sphere|sphere]] by
:<math>\Gamma(s,t) = \frac{\gamma_1(s) - \gamma_2(t)}{|\gamma_1(s) - \gamma_2(t)|}.</math>

Pick a point in the unit sphere, ''v'', so that orthogonal projection of the link to the plane perpendicular to ''v'' gives a link diagram. Observe that a point ''(s,t)'' that goes to ''v'' under the Gauss map corresponds to a crossing in the link diagram where <math>\gamma_1</math> is over <math>\gamma_2</math>. Also, a neighborhood of ''(s,t)'' is mapped under the Gauss map to a neighborhood of ''v'' preserving or reversing orientation depending on the sign of the crossing. Thus in order to compute the linking number of the diagram corresponding to ''v'' it suffices to count the ''signed'' number of times the Gauss map covers ''v''. Since ''v'' is a [[regular value]], this is precisely the [[degree of a continuous mapping|degree]] of the Gauss map (i.e. the signed number of times that the [[image (mathematics)|image]] of Γ covers the sphere). Isotopy invariance of the linking number is automatically obtained as the degree is invariant under homotopic maps. Any other regular value would give the same number, so the linking number doesn't depend on any particular link diagram.

This formulation of the linking number of ''γ''1 and ''γ''2 enables an explicit formula as a double [[line integral]], the '''Gauss linking integral''':

:<math>\mbox{linking number}\,=\,\frac{1}{4\pi}
\oint_{\gamma_1}\oint_{\gamma_2}
\frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3}
\cdot (d\mathbf{r}_1 \times d\mathbf{r}_2).</math>

This integral computes the total signed area of the image of the Gauss map (the integrand being the [[Jacobian]] of Γ) and then divides by the area of the sphere (which is 4''π'').

==Generalizations==
[[File:BorromeanRings.svg|thumb|The [[Milnor invariants]] generalize linking number to links with three or more components, allowing one prove that the [[Borromean rings]] are linked, though any two components have linking number 0.]]
* Just as closed curves can be [[link (knot theory)|linked]] in three dimensions, any two [[closed manifold]]s of dimensions ''m'' and ''n'' may be linked in a [[Euclidean space]] of dimension <math>m + n + 1</math>. Any such link has an associated Gauss map, whose [[degree of a continuous mapping|degree]] is a generalization of the linking number.
* Any [[framed knot]] has a [[self-linking number]] obtained by computing the linking number of the knot ''C'' with a new curve obtained by slightly moving the points of ''C'' along the framing vectors. The self-linking number obtained by moving vertically (along the blackboard framing) is known as '''Kauffman's self-linking number'''.
* The linking number is defined for two linked circles; given three or more circles, one can define the [[Milnor invariants]], which are a numerical invariant generalizing linking number.
* In [[algebraic topology]], the [[cup product]] is a far-reaching algebraic generalization of the linking number, with the [[Massey product]]s being the algebraic analogs for the [[Milnor invariants]].
* A [[linkless embedding]] of an [[undirected graph]] is an embedding into three-dimensional space such that every two cycles have zero linking number. The graphs that have a linkless embedding have a [[forbidden graph characterization|forbidden minor characterization]] as the graphs with no [[Petersen family]] [[minor (graph theory)|minor]].

==See also==
* [[Differential geometry of curves]]
* [[Hopf invariant]]
* [[Kissing number problem]]

==Notes==
{{reflist}}

==References==
* {{springer|author=A.V. Chernavskii|title=Linking coefficient|id=L/l059590}}
* {{springer|author=-|title=Writhing number|id=W/w098170}}

{{Knot theory|state=collapsed}}

[[Category:Curves]]
[[Category:Links by linking number| ]]

Photodissociation

2013-08-21T08:28:47Z

80.65.176.20: /* Photolysis in photosynthesis */ Changed "generation of diatomic oxygen from carbon dioxide" into "generation of diatomic oxygen". Oxygen is not coming from carbon dioxide, but from water.

{{refimprove|date=February 2009}}
A '''banked turn''' (aka. '''bank turn'''{{fact|date=July 2013}} or '''banking turn''') is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the inside of the curve. The bank angle is the angle at which the vehicle is [[Grade (slope)|incline]]d about its longitudinal axis with respect to the plane of its curved path.

== Turn on flat surfaces ==
If the bank angle is zero, the surface is flat and the [[normal force]] is vertically upwards. The only force keeping the vehicle turning on its path is [[friction]], or [[traction (engineering)|traction]]. This must be large enough to provide the [[centripetal force]], a relationship which can be expressed as an inequality, assuming the car is driving in a circle of radius ''r'':

:<math>\mu mg > {mv^2\over r}.</math>

The expression on the right hand side is the centripetal acceleration multiplied by mass, the force required to turn the vehicle. The left hand side is the maximum frictional force, which equals the [[coefficient of friction]] ''μ'' multiplied by the normal force. Rearranging the maximum cornering speed is

:<math>v < {\sqrt{r\mu g}}.</math>

Note that ''μ'' can be the coefficient for static or dynamic friction. In the latter case, where the vehicle is skidding around a bend, the friction is at its limit and the inequalities becomes equations. This also ignores effects such as [[downforce]] which can increase the normal force and cornering speed.

== Frictionless banked turn ==
[[File:Banked turn.PNG|thumb|250px|Upper panel: Ball on a banked circular track moving with constant speed ''v''; Lower panel: Forces on the ball. The resultant or [[net force]] on the ball found by [[vector addition]] of the [[normal force]] exerted by the road and vertical force due to [[gravity]] must equal the required force for centripetal acceleration dictated by the need to travel a circular path.]]
As opposed to a vehicle riding along a flat circle, inclined edges add an additional force that keeps the vehicle in its path and prevents a car from being "dragged into" or "pushed out of" the circle (or a railroad wheel from moving sideways so as to nearly rub on the wheel [[flange]]). This force is the horizontal component of the vehicle's normal force. In the absence of friction, the normal force is the only one acting on the vehicle in the direction of the center of the circle. Therefore, as per Newton's second law, we can set the horizontal component of the normal force equal to mass multiplied by centripetal acceleration:

:<math>N\sin \theta ={mv^2\over r}</math>

Because there is no motion in the vertical direction, the sum of all vertical forces acting on the system must be zero. Therefore we can set the vertical component of the vehicles's normal force equal to its weight:

:<math>N\cos \theta =mg</math>

Solving the above equation for the normal force and substituting this value into our previous equation, we get:

:<math>{mv^2\over r}= {mg\tan \theta}</math>

Which is equivalent to:

:<math>{v^2\over r}= {g\tan \theta}</math>

Solving for velocity we have:

:<math>v= {\sqrt{rg\tan \theta}}</math>

This provides the velocity that in the absence of friction and with a given angle of incline and radius of curvature, will ensure that the vehicle will remain in its designated path. The magnitude of this velocity is also known as the "rated speed" (or "balancing speed" for railroads") of a turn or curve.<ref>{{cite book
| last1 = Beer
| first1 = Ferdinand P.
| last2 = Johnston
| first2 = E. Russell
| authorlink = Ferdinand Beer
| title = Vector Mechanics for Engineers: Dynamics
| edition = 7
| publisher = McGraw-Hill
| series = Science/Engineering/Math
| date = July 11, 2003
| isbn = 978-0-07-293079-5
}}</ref> Notice that the rated speed of the curve is the same for all massive objects, and a curve that is not inclined will have a rated speed of 0.

== Banked turn with friction ==

When considering the effects of friction on the system, once again we need to note which way the friction force is pointing. When calculating a maximum velocity for our automobile, friction will point down the incline and towards the center of the circle. Therefore we must add the horizontal component of friction to that of the normal force. The sum of these two forces is our new net force in the direction of the center of the turn (the centripetal force):

:<math>{mv^2\over r}= \mu_s N\cos \theta +N\sin \theta </math>

Once again, there is no motion in the vertical direction, allowing us to set all opposing vertical forces equal to one another. These forces include the vertical component of the normal force pointing upwards and both the car's weight and vertical component of friction pointing downwards:

:<math>N\cos \theta =\mu_s N\sin \theta +mg</math>

By solving the above equation for mass and substituting this value into our previous equation we get:

:<math>{v^2\left(N\cos \theta -\mu_s N\sin \theta \right)\over rg}= \mu_s N\cos \theta +N\sin \theta </math>

Solving for v we get:

:<math>v= {\sqrt{rg\left(\sin \theta +\mu_s \cos \theta \right)\over \cos \theta -\mu_s \sin \theta }}
={\sqrt{rg\left(\tan\theta +\mu_s\right)\over 1 -\mu_s \tan\theta}}</math>

This equation provides the maximum velocity for the automobile with the given angle of incline, [[coefficient of static friction]] and radius of curvature. By a similar analysis of minimum velocity, the following equation is rendered:

:<math>v= {\sqrt{rg\left(\sin \theta -\mu_s \cos \theta \right)\over \cos \theta +\mu_s \sin \theta }}
={\sqrt{rg\left(\tan\theta -\mu_s\right)\over 1 +\mu_s \tan\theta}}</math>

The difference in the latter analysis comes when considering the direction of friction for the minimum velocity of the automobile (towards the outside of the circle). Consequently opposite operations are performed when inserting friction into equations for forces in the centripetal and vertical directions.

Improperly banked road curves increase the risk of run-off-road and head-on crashes. A 2% deficiency in superelevation (say, 4% superelevation on a curve that should have 6%) can be expected to increase crash frequency by 6%, and a 5% deficiency will increase it by 15%.<ref>D.W. Harwood, et al., PREDICTION OF THE EXPECTED SAFETY PERFORMANCE OF RURAL
TWO-LANE HIGHWAYS, Turner-Fairbank Highway Research Center, McLean, VA, December 2000, page 39, http://www.tfhrc.gov/safety/pubs/99207.pdf</ref> Up until now, highway engineers have been without efficient tools to identify improperly banked curves and to design relevant mitigating road actions. A modern [[profilograph]] can provide data of both road [[curvature]] and [[cross slope]] (angle of incline). A practical demonstration of how to evaluate improperly banked turns was developed in the EU Roadex III project, see the linked referenced document below.

== Banked turn in aeronautics == 
[[File:Douglas DC-3, SE-CFP.jpg|thumb|right|[[Douglas DC-3]] banking to make a left turn.]]

When a [[fixed-wing aircraft]] is making a turn (changing its direction) the aircraft must roll to a banked position so that its [[wing]]s are angled towards the desired direction of the turn. When the turn has been completed the aircraft must roll back to the wings-level position in order to resume straight flight.<ref name=FAA>{{cite book |title=Pilot's Encyclopedia of Aeronautical Knowledge |author=Federal Aviation Administration |url=http://books.google.com/books?id=m5V04SXE4zQC&pg=PT33&lpg=PT33&dq=+%22angle+of+bank%22&source=web&ots=iYTi_mZAra&sig=ytjcmr9RStdIdgZzaiBJJ-wxjts&hl=en
|isbn=1-60239-034-7 |year=2007 |publisher=Skyhorse Publishing Inc. |location=Oklahoma City OK |nopp=true |pages=Figure 3–21 }}</ref>

When any moving vehicle is making a turn, it is necessary for the forces acting on the vehicle to add up to a net inward force, to cause [[centripetal acceleration]]. In the case of an aircraft making a turn, the force causing centripetal acceleration is the horizontal component of the [[lift (force)|lift]] acting on the aircraft.

In straight, level flight, the lift acting on the aircraft acts vertically upwards to counteract the weight of the aircraft which acts downwards. During a balanced turn where the angle of bank is ''θ'' the lift acts at an angle ''θ'' away from the vertical. It is useful to resolve the lift into a vertical component and a horizontal component. If the aircraft is to continue in level flight (i.e. at constant [[Altitude#Altitude in aviation|altitude]]), the vertical component must continue to equal the weight of the aircraft and so the pilot must pull back on the stick a little more. The total (now angled) lift is greater than the weight of the aircraft so the vertical component can equal the weight. The horizontal component is unbalanced, and is thus the [[net force]] causing the aircraft to accelerate inward and execute the turn.

[[File:Banked turn.png|thumb|300px|Vector diagram showing lift, weight and centripetal force acting on a fixed-wing aircraft during a banked turn.]]

During a banked turn in ''level flight'' the lift on the aircraft must support the weight of the aircraft, as well as provide the necessary component of horizontal force to cause centripetal acceleration. Consequently, the lift required in a banked turn is greater than that one required in straight, level flight is by increasing the [[angle of attack]] of the wing typically by pulling on the [[Elevator (aircraft)|elevator]] control. The maneuver is usually complemented by an increase in power, in order to maintain airspeed.

Because centripetal acceleration is:
:<math>a = {v^2\over r}</math>

Newton's second law in the horizontal direction can be expressed mathematically as:

:<math>L\sin \theta = {mv^2\over r}</math>

where:
:''L'' is the lift acting on the aircraft
:''θ'' is the angle of bank of the aircraft
:''m'' is the [[mass]] of the aircraft
:''v'' is the [[true airspeed]] of the aircraft
:''r'' is the radius of the turn

In straight level flight, lift is equal to the aircraft weight. In turning flight the lift exceeds the aircraft weight, and is equal to the weight of the aircraft (''mg'') divided by the [[Trigonometric functions#Right-angled triangle definitions|cosine]] of the angle of bank:

:<math>L = {mg\over{\cos \theta}}</math>

where [[Earth's gravity|''g'']] is the gravitational field strength.

The radius of the turn can now be calculated:<ref>Clancy, L.J, Equation 14.9</ref>

:<math>r = {v^2\over{g \tan \theta}}</math>

This formula shows that the radius of turn is proportional to the square of the aircraft’s [[true airspeed]]. With a higher airspeed the radius of turn is larger, and with a lower airspeed the radius is smaller.

This formula also shows that the radius of turn is inversely proportional to the angle of bank. With a higher angle of bank the radius of turn is smaller, and with a lower angle of bank the radius is greater.

== See also ==
* [[Camber angle]]
* [[Coriolis force (perception)]]
* [[Centripetal force#Example: The banked turn|Centripetal force]]
* [[Cant (road/rail)]]
* [[g-force]]
* [[Superelevation]]

==Notes==

{{reflist}}

==References==

;Surface vehicles
* Serway, Raymond. ''Physics for Scientists and Engineers.'' Florida: Saunders College Publishing, 1996.
* [http://www.eurorap.org/library/pdfs/20080412_Health%20Issues.pdf Health and Safety Issues], the EU Roadex III project on health and safety issues raised by poorly maintained road networks.

;Aeronautics
* Kermode, A.C. (1972) ''Mechanics of Flight'', Chapter 8, 10th Edition, Longman Group Limited, London ISBN 0-582-23740-8
* Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London ISBN 0-273-01120-0
* Hurt, H.H. Jr, (1960), ''Aerodynamics for Naval Aviators'', A National Flightshop Reprint, Florida

==External links==

;Surface vehicles
* http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/imgmech/carbank.gif
* http://whitts.alioth.net
* http://www.batesville.k12.in.us/physics/PHYNET/Mechanics/Circular%20Motion/banked_no_friction.htm

;Aeronautics
* [http://www.grc.nasa.gov/WWW/K-12/airplane/turns.html NASA: Guidance on banking turns]
* [http://www.aerospaceweb.org/question/performance/q0146.shtml aerospaceweb.org: Bank Angle and G's (math)]
[[Category:Aerodynamics]]
[[Category:Aerial maneuvers]]
[[Category:Mechanics]]
[[Category:Transport engineering]]

Singular value

2013-04-30T00:48:59Z

80.65.246.4:

In [[mathematics]], a '''Clifford module''' is a [[representation of an algebra|representation]] of a [[Clifford algebra]]. In general a Clifford algebra ''C'' is a [[central simple algebra]] over some [[field extension]] ''L'' of the field ''K'' over which the [[quadratic form]] ''Q'' defining ''C'' is defined.

The [[abstract algebra|abstract theory]] of Clifford modules was founded by a paper of [[Michael Atiyah|M. F. Atiyah]], [[R. Bott]] and [[Arnold S. Shapiro]]. A fundamental result on Clifford modules is that the [[Morita equivalence]] class of a Clifford algebra (the equivalence class of the category of Clifford modules over it) depends only on the signature {{nowrap|''p'' − ''q'' (mod 8)}}. This is an algebraic form of [[Bott periodicity]].

==Matrix representations of real Clifford algebras==
We will need to study ''anticommuting'' [[matrix (mathematics)|matrices]] (''AB'' = −''BA'') because in Clifford algebras orthogonal vectors anticommute
:<math> A \cdot B = \frac{1}{2}( AB + BA ) = 0.</math>

For the real Clifford algebra <math>\mathbb{R}_{p,q}\,</math>, we need ''p'' + ''q'' mutually anticommuting matrices, of which ''p'' have +1 as square and ''q'' have −1 as square.
:<math> \begin{matrix}
\gamma_a^2 &=& +1 &\mbox{if} &1 \le a \le p \\
\gamma_a^2 &=& -1 &\mbox{if} &p+1 \le a \le p+q\\
\gamma_a \gamma_b &=& -\gamma_b \gamma_a &\mbox{if} &a \ne b. \ \\
\end{matrix}</math>

Such a basis of gamma matrices is not unique. One can always obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation.

:<math> \begin{matrix}
\gamma_{a'} &=& S &\gamma_{a } &S^{-1}
\end{matrix}
</math>

where S is a non-singular matrix. The sets γ a' and γ a belong to the same equivalence class.

==Real Clifford algebra R3,1==

Developed by [[Ettore Majorana]], this Clifford module enables the construction of a [[Dirac equation| Dirac-like equation]] without complex numbers, and its elements are called Majorana [[spinors]].

The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix. The [[sign convention| signature]] is (+++−). For the signatures (+−−−) and (−−−+) often used in physics, 4×4 complex matrices or 8×8 real matrices are needed.

== See also ==
* [[Weyl–Brauer matrices]]
* [[Higher-dimensional gamma matrices]]

==References==
*{{citation|first1=Michael|last1=Atiyah|first2=Raoul|last2=Bott|first3=Arnold|last3=Shapiro|title=Clifford Modules|url=http://www.ma.utexas.edu/users/dafr/Index/ABS.pdf|journal=Topology|volume= 3|issue=(Suppl. 1)|year=1964|pages=3–38|doi=10.1016/0040-9383(64)90003-5}}
* {{citation|first=Pierre|last=Deligne|authorlink=Pierre Deligne|chapter=Notes on spinors|title= Quantum Fields and Strings: A Course for Mathematicians|editor= P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, E. Witten|publisher=American Mathematical Society|place= Providence|year=1999|pages=99–135}}. See also [http://www.math.ias.edu/QFT the programme website] for a preliminary version.
* {{citation|title=Spinors and Calibrations|last=Harvey|first= F. Reese|publisher=Academic Press|year=1990|isbn=978-0-12-329650-4}}.
* {{citation|last1=Lawson|first1= H. Blaine|last2=Michelsohn|first2=Marie-Louise|author2-link=Marie-Louise Michelsohn|title=Spin Geometry|publisher= Princeton University Press|year=1989|isbn= 0-691-08542-0}}.

[[Category:Representation theory]]
[[Category:Clifford algebras]]

[[nl:Clifford-algebra]]

Linear dynamical system

2012-07-24T07:08:57Z

80.65.52.67: Small typo in first paragraph, linear system -> dynamical system

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