formulasearchengine - User contributions [en]

Tobit model

2014-11-24T19:37:02Z

128.100.42.106: /* The likelihood function */

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S-matrix theory

2014-09-09T19:39:50Z

128.100.122.177: /* Basic Principles */

The writer's name is Christy. For many years she's been operating as a travel agent. Mississippi is exactly where her house is but her spouse wants them to transfer. To climb is something I truly enjoy performing. My blog post love psychics ([http://www.publicpledge.com/blogs/post/7034 http://www.publicpledge.com/blogs/post/7034])

Langmuir adsorption model

2014-02-07T23:31:24Z

128.100.199.81: /* The Temkin Adsorption Isotherm */

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Prolate spheroidal coordinates

2014-01-22T14:18:21Z

128.100.201.178: /* Definition */ Fixing the range for \nu, based on my vague understanding and http://mathworld.wolfram.com/ProlateSpheroidalCoordinates.html

{{Other uses}}
{{Function
|name = Sine function
|image = Sine_one_period.svg
|heading1 = 1
|parity = odd
|domain = (−∞,∞)<ref group=note name=real>For real (not complex) numbers.</ref>
|codomain = [−1,1]<ref group=note name=real />
|period = 2''π''
|heading2 = 1
|zero = 0
|plusinf =
|minusinf =
|max = ((2''k'' + ½)''π'', 1) <ref group=note name=intk>Variable ''k'' is an [[integer]]. </ref>
|min = ((2''k'' − ½)''π'', −1)
|vr1 =
|f1 =
|vr2 =
|f2 =
|vr3 =
|f3 =
|vr4 =
|f4 =
|vr5 =
|f5 =
|heading3 = 1
|asymptote =
|root = ''kπ''
|critical = ''kπ'' − π/2
|inflection = ''kπ''
|fixed = 0
|notes = {{reflist|group=note}}
}}
[[File:Trigono sine en2.svg|right|thumb|<math>\sin \alpha = \frac {\textrm{opposite}} {\textrm{hypotenuse}} </math> For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.]]
[[Image:Sine.svg|right|thumb|The sine function graphed on the Cartesian plane. In this graph, the angle ''x'' is given in [[radian]]s (π = 180°).]]
[[File:Sine cosine one period.svg|right|thumb|The sine and cosine functions are related in multiple ways. The derivative of <math>\sin(x)</math> is <math>\cos(x)</math>. Also they are out of phase by 90°: <math>\sin(\pi/2 - x)</math> = <math>\cos(x)</math>. And for a given angle, cos and sin give the respective x, y coordinates on a unit circle.]]
In [[mathematics]], the '''sine function''' is a [[trigonometric function]] of an [[angle]]. The sine of an angle is defined in the context of a [[right triangle]]: for the specified angle, it is the ratio of the length of the side that is opposite that angle to (divided by) the length of the [[hypotenuse|longest side of the triangle]] (i.e. the hypotenuse).

Trigonometric functions are commonly defined as [[ratio]]s of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a [[unit circle]]. More modern definitions express them as [[Series (mathematics)|infinite series]] or as solutions of certain [[differential equation]]s, allowing their extension to arbitrary positive and negative values and even to [[complex number]]s.

The sine function is commonly used to model [[periodic function|periodic]] phenomena such as [[sound]] and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.

The function sine can be traced to the [[Jyā, koti-jyā and utkrama-jyā|''jyā'' and ''koṭi-jyā '']] functions used in [[Gupta period]] [[Indian astronomy]] (''[[Aryabhatiya]]'', ''[[Surya Siddhanta]]''), via translation from Sanskrit to Arabic and then from Arabic to Latin.<ref name="Boyer, Carl B. 1991 p. 210">Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7, p. 210.</ref> The word "sine" comes from a [[Latin]] mistranslation of the Arabic ''jiba'', which is a transliteration of the Sanskrit word for half the chord, ''jya-ardha''.<ref>Victor J Katx, A history of mathematics, p210, sidebar 6.1.</ref>

== Right-angled triangle definition ==
For "opposite" and the "adjacent" side ('''[[Trigonometric functions#Sine.2C_cosine_and_tangent|tangent]]'''), etc.

To define the trigonometric functions for an acute angle ''A'', start with any [[right triangle]] that contains the angle ''A''. The three sides of the triangle are named as follows:
* The ''adjacent side'' is the side that is in contact with (adjacent to) both the angle we are interested in (angle ''A'') and the right angle, in this case side '''b'''.
* The ''hypotenuse'' is the side opposite the right angle, in this case side '''h'''. The hypotenuse is always the longest side of a right-angled triangle.
* The ''opposite side'' is the side opposite to the angle we are interested in (angle ''A''), in this case side '''a'''.

In ordinary [[Euclidean geometry]], according to the [[triangle postulate]] the inside angles of every triangle total 180[[degree (angle)|°]] (π [[radian]]s). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be greater than 0° and less than 90°. The following definition applies to such angles.

The angle ''A'' (having measure α) is the angle between the hypotenuse and the adjacent side.

The '''sine''' of an angle is the ratio of the length of the opposite [[Cathetus|side]] to the length of the hypotenuse. In our case

does not depend on the size of the particular right triangle chosen, as long as it contains the angle ''A'', since all such triangles are [[similarity (geometry)|similar]].

== Relation to slope ==
{{Main|Slope}}
The trigonometric functions can be defined in terms of the ''rise'', ''run'', and ''[[slope]]'' of a line segment relative to some horizontal line.

*When the length of the line segment is 1, sine takes an angle and tells the ''rise''
*Sine takes an angle and tells the ''rise'' per unit length of the line segment.
*''Rise'' is equal to sin ''θ'' multiplied by the length of the line segment

In contrast, cosine is used for the telling the ''run'' from the angle; and tangent is used for telling the ''slope'' from the angle. Arctan is used for telling the angle from the ''slope''.

The line segment is the equivalent of the hypotenuse in the right-triangle, and when it has a length of 1 it is also equivalent to the radius of the [[unit circle]].

== Relation to the unit circle ==
In [[trigonometry]], a unit circle is the circle of radius one centered at the origin (0, 0) in the [[Cartesian coordinate system]].

Let a line through the origin, making an angle of ''θ'' with the positive half of the ''x''-axis, intersect the unit circle. The ''x''- and ''y''-coordinates of this point of intersection are equal to cos ''θ'' and sin ''θ'', respectively. The point's distance from the origin is always 1.

Unlike the definitions with the right or left triangle or slope, the angle can be extended to the full set of real arguments by using the [[Unit_circle#Trigonometric_functions_on_the_unit_circle|unit circle]]. This can also be achieved by requiring certain symmetries and that sine be a [[periodic function]].

{|
|valign="top"| [[File:Unit circle3.png|right|thumb|The unit circle.]]
|valign="top"| [[File:Unit circle.svg|Unit circle|right|thumb|Illustration of a unit circle. The radius has a length of 1. The variable ''t'' is an [[angle]] measure.]]
|valign="top"| [[File:Trig functions on unit circle.PNG|thumb|Point ''P''(''x'',''y'') on the circle of unit radius at an [[obtuse angle]] θ > π/2]]
|}

[[File:Sine curve drawing animation.gif|frame|left|Animation showing the graphing process of ''y'' = sin ''x'' (where ''x'' is the angle in radians) using a unit circle. The blue arc around the unit circle (in green) and the blue line at right have the same length, equal to the angle in radians.]]
{{clr}}

== Identities ==
{{See also|List of trigonometric identities}}

Exact identities (using [[radian]]s):

These apply for all values of <math>\theta</math>.

:<math>
\begin{align}
\sin \theta & = \cos \left(\frac{\pi}{2} - \theta \right) \\
& = \frac{1}{\csc \theta}
\end{align}
</math>

=== Reciprocal ===
The [[multiplicative inverse|reciprocal]] of sine is cosecant, i.e. the reciprocal of sin(''A'') is csc(''A''), or cosec(''A''). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:

:<math>\csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}. </math>

=== Inverse ===
[[File:Arcsine.svg|thumb|180px|The usual principal values of the arcsin(x) function graphed on the cartesian plane. Arcsin is the inverse of sin.]]
The [[inverse function]] of sine is arcsine (arcsin or asin) or inverse sine (sin{{sup|−1}}). As sine is non-[[injective]], it is not an exact inverse function but a partial inverse function. For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its [[principal branch]]. With this restriction, for each ''x'' in the domain the expression arcsin(''x'') will evaluate only to a single value, called its [[principal value]].

:<math>\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \sin^{-1} \left( \frac {a} {h} \right).</math>

''k'' is some integer:

:<math>\begin{align}
\sin y = x \ \Leftrightarrow\ & y = \arcsin x + 2k\pi , \text{ or }\\
& y = \pi - \arcsin x + 2k\pi
\end{align}</math>

Or in one equation:

:<math>\sin y = x \ \Leftrightarrow\ y = (-1)^k \arcsin x + k\pi </math>

Arcsin satisfies:

:<math>\sin(\arcsin x) = x\!</math>
and
:<math>\arcsin(\sin \theta) = \theta\quad\text{for }-\pi/2 \leq \theta \leq \pi/2.</math>

=== Calculus ===
{{See also|List of integrals of trigonometric functions|Differentiation of trigonometric functions}}

For the sine function:

:<math> f(x) = \sin x \,</math>

The derivative is:

:<math> f'(x) = \cos x \,</math>

The antiderivative is:

:<math>\int f(x)\,dx = -\cos x + C </math>

''C'' denotes the [[constant of integration]].

=== Other trigonometric functions ===
[[File:Quadrants 01 Pengo.svg|thumb|350px|The four quadrants of a Cartesian coordinate system.]]
It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the [[sign function]]).

Sine in terms of the other common [[trigonometric functions]]:

{| class="wikitable"
|-
!colspan="1" rowspan="3"|
!colspan="1" rowspan="3"|f ''θ''
!colspan="5"|Using plus/minus (±)
!colspan="1"|Using sign function (sgn)
|-
!colspan="1" rowspan=2|f ''θ'' =
!colspan="4"|± per Quadrant
!rowspan="2"|f ''θ'' =
|-
! I
! II
! III
! IV
|-
!rowspan="2"|cos
|<math>\sin \theta </math>
|<math>= \pm\sqrt{1 - \cos^2 \theta}</math>
| +
| +
| -
| -
|<math>= \sgn( \cos (\theta - \frac{\pi}{2})) \sqrt{1 - \cos^2\theta}</math>
|-
|<math>\cos \theta </math>
|<math>= \pm\sqrt{1 - \sin^2\theta}</math>
| +
| -
| -
| +
|<math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \sqrt{1 - \sin^2\theta}</math>
|-
!rowspan="2"|cot
| <math>\sin \theta </math>
| <math>= \pm\frac{1}{\sqrt{1 + \cot^2 \theta}}</math>
| +
| +
| -
| -
| <math>= \sgn( \cot( \frac{\theta}{2})) \frac{1}{\sqrt{1 + \cot^2 \theta}}</math>
|-
|<math>\cot \theta</math>
|<math> = \pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}</math>
| +
| -
| -
| +
| <math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}</math>
|-
!rowspan="2"|tan
| <math>\sin \theta </math>
| <math>= \pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} </math>
| +
| -
| -
| +
| <math>= \sgn( \tan(\frac{2\theta + \pi}{4})) \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} </math>
|-
| <math>\tan \theta </math>
| <math>= \pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}</math>
| +
| -
| -
| +
| <math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}</math>
|-
!rowspan="2"|sec
| <math>\sin \theta </math>
| <math>= \pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta} </math>
| +
| -
| +
| -
| <math>= \sgn( \sec ( \frac{4 \theta - \pi}{2})) \frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta} </math>
|-
| <math>\sec \theta </math>
| <math>= \pm\frac{1}{\sqrt{1 - \sin^2 \theta}}</math>
| +
| -
| -
| +
| <math>= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{1}{\sqrt{1 - \sin^2 \theta}}</math>
|}

Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant.

The basic relationship between the sine and the cosine can also be expressed as the [[Pythagorean trigonometric identity]]:

:<math>\cos^2\theta + \sin^2\theta = 1\!</math>

where sin2''x'' means (sin(''x''))2.

== Properties relating to the quadrants ==
Over the four quadrants of the sine function is as follows.
{| class="wikitable"
|-
![[Cartesian_coordinate_system#Quadrants_and_octants|Quadrant]]
![[Degree (angle)|Degrees]]
![[Radian]]s
!Value
![[Sign (mathematics)|Sign]]
![[Monotonic function|Monotony]]
![[Convex function|Convexity]]
|-
|1st Quadrant
|<math>0^\circ<x<90^\circ</math>
|<math>0<x< \pi/2 </math>
|<math>0<\sin x<1</math>
|<math>+</math>
|increasing
|concave
|-
|2nd Quadrant
|<math>90^\circ<x<180^\circ</math>
|<math>\pi/2<x<\pi</math>
|<math>0<\sin x<1</math>
|<math>+</math>
|decreasing
|concave
|-
|3rd Quadrant
|<math>180^\circ<x<270^\circ</math>
|<math>\pi<x<3\pi/2</math>
|<math>-1<\sin x<0</math>
|<math>-</math>
|decreasing
|convex
|-
|4th Quadrant
|<math>270^\circ<x<360^\circ</math>
|<math>3\pi/2<x<2\pi</math>
|<math>-1<\sin x<0</math>
|<math>-</math>
|increasing
|convex
|}

Points between the quadrants. k is an [[integer]].
[[File:Sine quads 01 Pengo.svg|thumb|390px|The quadrants of the unit circle and of ''sin x'', using the [[Cartesian coordinate system]].]]

{| class="wikitable"
|-
![[Degree (angle)|Degrees]]
![[Radian]]s
0 ≤ ''x'' < 2π
!Radians
!sin ''x''
!Point type
|-
|0°
|0
|<math>2 \pi k</math>
|0
|[[Root of a function|Root]], [[Inflection point|Inflection]]
|-
|90°
|<math>\pi/2</math>
|<math>2 \pi k + \pi/2 </math>
|1
|[[Maxima and minima|Maxima]]
|-
|180°
|<math>\pi</math>
|<math>2 \pi k - \pi</math>
|0
|[[Root of a function|Root]], [[Inflection point|Inflection]]
|-
|270°
|<math>3\pi/2</math>
|<math>2 \pi k - \pi/2 </math>
| -1
|[[Minima]]
|}

For arguments outside those in the table, get the value using the fact the sine function has a period of 360° (or 2π rad): <math>\sin(\alpha + 360^\circ) = \sin(\alpha)</math>, or use <math>\sin(\alpha + 180^\circ) = -\sin(\alpha)</math>.
Or use <math>\cos(x)= \frac{e^{ix}+e^{-ix}}{2} </math> and <math>\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}</math>.
For complement of sine, we have <math>\sin(180^\circ-\alpha) = \sin(\alpha)</math>.

== Series definition ==
[[Image:Taylorsine.svg|thumb|right|The sine function (blue) is closely approximated by its [[Taylor's theorem|Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.]]

[[File:Sine GIF.gif|thumb|right|This animation shows how including more and more terms in the partial sum of its Taylor series gradually builds up a sine curve.]]

Using only geometry and properties of [[limit of a function|limits]], it can be shown that the [[derivative]] of sine is cosine, and that the derivative of cosine is the negative of sine.

Using the reflection from the calculated geometric derivation of the sine is with the 4n + k-th derivative at the point 0:

:<math>\sin^{(4n+k)}(0)=\begin{cases}
0 & \text{when } k=0 \\
1 & \text{when } k=1 \\
0 & \text{when } k=2 \\
-1 & \text{when } k=3 \end{cases}</math>

This gives the following Taylor series expansion at x = 0. One can then use the theory of [[Taylor series]] to show that the following identities hold for all [[real number]]s ''x'' (where x is the angle in radians) :<ref>See Ahlfors, pages 43–44.</ref>

:<math>
\begin{align}
\sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[8pt]
& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt]
\end{align}
</math>

If ''x'' were expressed in degrees then the series would contain messy factors involving powers of π/180: if ''x'' is the number of degrees, the number of radians is ''y'' = π''x'' /180, so

:<math>\begin{align}
\sin x_\mathrm{deg} & = \sin y_\mathrm{rad} \\
& = \frac{\pi}{180} x - \left (\frac{\pi}{180} \right )^3\ \frac{x^3}{3!} + \left (\frac{\pi}{180} \right )^5\ \frac{x^5}{5!} - \left (\frac{\pi}{180} \right )^7\ \frac{x^7}{7!} + \cdots .
\end{align}</math>

The series formulas for the sine and [[cosine]] are uniquely determined, up to the choice of unit for angles, by the requirements that
:<math>
\begin{align}
\sin 0 = 0 & \text{ and } \sin{2x} = 2 \sin x \cos x \\
\cos^2 x + \sin^2 x = 1 & \text{ and } \cos{2x} = \cos^2 x - \sin^2 x \\
\end{align}
</math>
The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that
:<math>
\sin x \approx x \text{ when } x \approx 0.
</math>
The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients.

In general, mathematically important relationships between the sine and cosine functions and the [[exponential function]] (see, for example, [[Euler's formula]]) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.

A similar series is [[Gregory's series]] for [[arctan]], which is obtained by omitting the factorials in the denominator.

==Continued fraction==
The sine function can also be represented as a [[generalized continued fraction]]:

:<math> \sin x =
\cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 +
\cfrac{2\cdot3 x^2}{4\cdot5-x^2 +
\cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}.
</math>

The continued fraction representation expresses the [[real number]] values, both [[rational number|rational]] and [[irrational number|irrational]], of the sine function.

==Fixed point==
[[Image:Sine fixed point.svg|thumb|The fixed point iteration ''x''''n''+1 = sin ''x''''n'' with initial value ''x''0 = 2 converges to 0.]]
Zero is the only real [[Fixed point (mathematics)|fixed point]] of the sine function; in other words the only intersection of the sine function and the [[identity function]] is sin(0) = 0.
{{clear}}

==Arc length==
Strict formulae: <math> \int_a^b{\sqrt{1+\cos(x)^2}} dx </math>

Imprecise formulae for sine curve length from 0 to ''x'': <math> \frac{45}{37} x + \frac{3}{29} \sin(2 x) - \frac{x}{4796} </math>
<ref>[http://sadler.su/gp/sine.html Genetic programming: imprecise length of sin(x) curve]</ref>

==Law of sines==
{{Main|Law of sines}}
The [[law of sines]] states that for an arbitrary [[triangle]] with sides ''a'', ''b'', and ''c'' and angles opposite those sides ''A'', ''B'' and ''C'':

:<math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.</math>

This is equivalent to the equality of the first three expressions below:

:<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,</math>

where ''R'' is the triangle's [[circumscribed circle|circumradius]].

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in ''[[triangulation]]'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

== Values ==
{{See also|Exact trigonometric constants}}
[[File:Sin.svg|thumb|350px|sin(x)]]
[[Image:Unit circle angles.svg|right|thumb|350px|Some common angles (''θ'') shown on the [[unit circle]]. The angles are given in degrees and radians, together with the corresponding intersection point on the unit circle, (cos ''θ'', sin ''θ'').]]
{| border="1" cellpadding="5" cellspacing="0"
|-----
! style="background:#ffdead;" colspan="3" | ''x'' (angle)
! colspan="2" style="background:#ffdead;" | sin ''x''
|-----
| style="background:#efefef;" align="Center" | Degrees
| style="background:#efefef;" | Radians
| style="background:#efefef;" align="Center" | Grads
| style="background:#efefef;" align="Center" | Exact
| style="background:#efefef;" | Decimal
|-----
| align="Center" | 0° || align="Center" | 0
| align="Center" | 0g
| align="Center" rowspan="2" | 0 || rowspan="2" | 0
|-----
| align="Center" | 180° || align="Center" | <math>\pi</math>
| align="Center" | 200g
|-----
| align="Center" | 15°
| align="Center" | <math>\frac{\pi}{12}</math>
| align="Center" | 16{{frac|2|3}}g
| align="Center" rowspan="2" | <math>\frac{\sqrt{6}-\sqrt{2}}{4}</math>
| rowspan="2" | 0.258819045102521
|-----
| align="Center" | 165°
| align="Center" | <math>\frac{11 \cdot \pi}{12}</math>
| align="Center" | 183{{frac|1|3}}g
|-----
| align="Center" | 30°
| align="Center" | <math>\frac{\pi}{6}</math>
| align="Center" | 33{{frac|1|3}}g
| align="Center" rowspan="2" | <math>\frac{1}{2}</math>
| rowspan="2" | 0.5
|-----
| align="Center" | 150°
| align="Center" | <math>\frac{5 \cdot \pi}{6}</math>
| align="Center" | 166{{frac|2|3}}g
|-----
| align="Center" | 45°
| align="Center" | <math>\frac{\pi}{4}</math>
| align="Center" | 50g
| align="Center" rowspan="2" | <math>\sqrt{\frac{1}{2}}</math>
| rowspan="2" | 0.707106781186548
|-----
| align="Center" | 135°
| align="Center" | <math>\frac{3 \cdot \pi}{4}</math>
| align="Center" | 150g
|-----
| align="Center" | 60°
| align="Center" | <math>\frac{\pi}{3}</math>
| align="Center" | 66{{frac|2|3}}g
| align="Center" rowspan="2" | <math>\frac{\sqrt{3}}{2}</math>
| rowspan="2" | 0.866025403784439
|-----
| align="Center" | 120°
| align="Center" | <math>\frac{2 \cdot \pi}{3}</math>
| align="Center" | 133{{frac|1|3}}g
|-----
| align="Center" | 75°
| align="Center" | <math>\frac{5 \cdot \pi}{12}</math>
| align="Center" | 83{{frac|1|3}}g
| align="Center" rowspan="2" | <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| rowspan="2" | 0.965925826289068
|-----
| align="Center" | 105°
| align="Center" | <math>\frac{7 \cdot \pi}{12}</math>
| align="Center" | 116{{frac|2|3}}g
|-----
| align="Center" | 90°
| align="Center" | <math>\frac{\pi}{2}</math>
| align="Center" | 100g || align="Center" | 1
| 1
|}

A memory aid (note it does not include 15° and 75°):

{| border="1" cellpadding="5" cellspacing="0"
|- bgcolor="white" align="center"
|''x'' in degrees||0°||30°||45°||60°||90°
|- bgcolor="white" align="center"
|''x'' in radians||0||π/6||π/4||π/3||π/2
|- bgcolor="white" align="center"
|<math>\mathrm{sin} \, x</math>||<math>\frac {\sqrt 0} 2</math>||<math>\frac {\sqrt 1} 2</math>||<math>\frac {\sqrt 2} 2</math>||<math>\frac {\sqrt 3} 2</math>||<math>\frac {\sqrt 4} 2</math>
|}

90 degree increments:

{| border="1" cellpadding="5" cellspacing="0"
|- bgcolor="white" align="center"
|''x'' in degrees||0°||90°||180°||270°||360°
|- bgcolor="white" align="center"
|''x'' in radians||0||π/2||π||3π/2||2π
|- bgcolor="white" align="center"
|<math>\mathrm{sin} \, x</math>||0||1||0||-1||0
|}

Other values not listed above:
: <math>\sin\frac{\pi}{60}=\sin 3^\circ=\tfrac{1}{16} \left[2(1-\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3+1)\right]\,</math> {{OEIS2C|A019812}}
: <math>\sin\frac{\pi}{30}=\sin 6^\circ=\tfrac{1}{8} \left[\sqrt{6(5-\sqrt5)}-\sqrt5-1\right]\,</math> {{OEIS2C|A019815}}
: <math>\sin\frac{\pi}{20}=\sin 9^\circ=\tfrac{1}{8} \left[\sqrt2(\sqrt5+1)-2\sqrt{5-\sqrt5}\right]\,</math> {{OEIS2C|A019818}}
: <math>\sin\frac{\pi}{15}=\sin 12^\circ=\tfrac{1}{8} \left[\sqrt{2(5+\sqrt5)}-\sqrt3(\sqrt5-1)\right]\,</math> {{OEIS2C|A019821}}
: <math>\sin\frac{\pi}{10}=\sin 18^\circ=\tfrac{1}{4}\left(\sqrt5-1\right)=\tfrac{1}{2}\varphi^{-1}\,</math> {{OEIS2C|A019827}}
: <math>\sin\frac{7\pi}{60}=\sin 21^\circ=\tfrac{1}{16}\left[2(\sqrt3+1)\sqrt{5-\sqrt5}-\sqrt2(\sqrt3-1)(1+\sqrt5)\right]\,</math> {{OEIS2C|A019830}}
: <math>\sin\frac{\pi}{8}=\sin 22.5^\circ=\tfrac{1}{2}(\sqrt{2-\sqrt{2}}),</math>
: <math>\sin\frac{2\pi}{15}=\sin 24^\circ=\tfrac{1}{8}\left[\sqrt3(\sqrt5+1)-\sqrt2\sqrt{5-\sqrt5}\right]\,</math> {{OEIS2C|A019833}}
: <math>\sin\frac{3\pi}{20}=\sin 27^\circ=\tfrac{1}{8}\left[2\sqrt{5+\sqrt5}-\sqrt2\;(\sqrt5-1)\right]\,</math> {{OEIS2C|A019836}}
: <math>\sin\frac{11\pi}{60}=\sin 33^\circ=\tfrac{1}{16}\left[2(\sqrt3-1)\sqrt{5+\sqrt5}+\sqrt2(1+\sqrt3)(\sqrt5-1)\right]\,</math> {{OEIS2C|A019842}}
: <math>\sin\frac{\pi}{5}=\sin 36^\circ=\tfrac14[\sqrt{2(5-\sqrt5)}]\,</math> {{OEIS2C|A019845}}
: <math>\sin\frac{13\pi}{60}=\sin 39^\circ=\tfrac1{16}[2(1-\sqrt3)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3+1)(\sqrt5+1)]\,</math> {{OEIS2C|A019848}}
: <math>\sin\frac{7\pi}{30}=\sin 42^\circ=\frac{\sqrt6\sqrt{5+\sqrt5}-\sqrt5+1}{8}\,</math> {{OEIS2C|A019851}}

For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine [[periodic function]] with period 2π:

: <math>\sin\theta = \sin\left(\theta + 2\pi k \right),\,</math>

for any angle θ and any [[integer]] ''k''.

The primitive period (the ''smallest'' positive period) of sine is a full circle, i.e. 2π radians or 360 degrees.

==Relationship to complex numbers==
{{Main|Trigonometric functions#Relationship to exponential function and complex numbers}}
[[File:Complex picture.svg|thumb|An illustration of the [[complex plane]]. The [[imaginary number]]s are on the vertical coordinate axis.]]
Sine is used to determine the [[imaginary part]] of a [[complex number]] given in [[polar coordinates]] (r,φ):

:<math> z = r(\cos \varphi + i\sin \varphi )\,</math>

the imaginary part is:

:<math>\operatorname{Im}(z) = r \sin \varphi</math>

r and φ represent the magnitude and angle of the complex number respectively. ''i'' is the [[imaginary unit]]. ''z'' is a [[complex number]].

Although dealing with complex numbers, sine's parameter in this usage is still a [[real number]]. Sine can also take a complex number as an argument.
{{clr}}

===Sine with a complex argument===
[[File:ComplexPlot-Sin-z-,1024-.jpg|thumb|<math>\sin z\,</math> [[Domain coloring]] of sin(z) over (-π,π) on x and y axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude.]]
[[File:Sin z vector field 02 Pengo.svg|thumb|sin(z) as a vector field]]

The definition of the sine function for complex arguments ''z'':

: <math>\begin{align}
\sin z & = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} \\
& = \frac{e^{i z} - e^{-i z}}{2i}\, \\
& = \frac{\sinh \left( i z\right) }{i}
\end{align}</math>

where ''i'' 2 = −1, and sinh is [[hyperbolic function|hyperbolic sine]]. This is an [[entire function]]. Also, for purely real ''x'',

: <math>\sin x = \operatorname{Im}(e^{i x}). \,</math>

For purely imaginary numbers:
:<math> \sin iy = i \sinh y. \,</math>

It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument:

: <math>\begin{align}
\sin (x + iy) &= \sin x \cos iy + \cos x \sin iy \\
&= \sin x \cosh y + i \cos x \sinh y.
\end{align}</math>

==== Partial fraction and product expansions of complex sine ====
Using the partial fraction expansion technique in [[Complex Analysis]], one can find that the infinite series

: <math>\begin{align}
\sum_{n = -\infty}^{\infty}\frac{(-1)^n}{z-n} = \frac{1}{z} + \sum_{n = 1}^{\infty}\frac{2z}{n^2-z^2}
\end{align}</math>

both converge and are equal to <math>\frac{\pi}{\sin \pi z}</math>.

Similarly we can find

: <math>\begin{align}
\frac{\pi^2}{\sin^2 \pi z} = \sum_{-\infty}^\infty \frac{1}{(z-n)^2}.
\end{align}</math>

Using product expansion technique, one can derive

: <math>\begin{align}
\sin \pi z = \pi z \prod_{n = 1}^\infty \Bigl( 1- \frac{z^2}{n^2} \Bigr).
\end{align}</math>

==== Usage of complex sine ====
''sin z'' is found in the [[functional equation]] for the [[Gamma function]],
:<math>\Gamma(s)\Gamma(1-s)={\pi\over\sin\pi s},</math>

which in turn is found in the [[functional equation]] for the [[Riemann zeta-function]],
:<math>\zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\pi s/2)\zeta(1-s).</math>

As a [[holomorphic function]], ''sin z'' is a 2D solution of [[Laplace's equation]]:
:<math>\Delta u(x_1, x_2) = 0.</math>

===Complex graphs===

{| style="text-align:center"
|+ '''Sine function in the complex plane'''
|[[File:Complex sin real 01 Pengo.svg|1000x130px|none]]
|[[File:Complex sin imag 01 Pengo.svg|1000x130px|none]]
|[[File:Complex sin abs 01 Pengo.svg|1000x130px|none]]
|-
|real component
|imaginary component
|magnitude
|}
 
{| style="text-align:center"
|+ '''Arcsine function in the complex plane'''
|[[File:Complex arcsin real 01 Pengo.svg|1000x130px|none]]
|[[File:Complex arcsin imag 01 Pengo.svg|1000x130px|none]]
|[[File:Complex arcsin abs 01 Pengo.svg|1000x130px|none]]
|-
|real component
|imaginary component
|magnitude
|}

== History ==
{{Main|History of trigonometric functions}}
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period.
The [[Chord (geometry)|chord]] function was discovered by [[Hipparchus]] of [[İznik|Nicaea]] (180–125 BC) and [[Ptolemy]] of [[Egypt (Roman province)|Roman Egypt]] (90–165 AD).

The function sine (and cosine) can be traced to the [[Jyā, koti-jyā and utkrama-jyā|''jyā'' and ''koṭi-jyā '']] functions used in [[Gupta period]] [[Indian astronomy]] (''[[Aryabhatiya]]'', ''[[Surya Siddhanta]]''), via translation from Sanskrit to Arabic and then from Arabic to Latin.<ref name="Boyer, Carl B. 1991 p. 210"/>

The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician [[Albert Girard]]; these were further promulgated by Euler (see below). The ''Opus palatinum de triangulis'' of [[Georg Joachim Rheticus]], a student of [[Copernicus]], was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In a paper published in 1682, [[Gottfried Leibniz|Leibniz]] proved that sin ''x'' is not an [[algebraic function]] of ''x''.<ref>{{cite book|title=Elements of the History of Mathematics|author=Nicolás Bourbaki|publisher=Springer|year=1994}}</ref> [[Roger Cotes]] computed the derivative of sine in his ''Harmonia Mensurarum'' (1722).<ref>"[http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf Why the sine has a simple derivative]", in ''[http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm Historical Notes for Calculus Teachers]'' by [http://www.math.usma.edu/people/rickey/ V. Frederick Rickey]</ref> [[Leonhard Euler]]'s ''Introductio in analysin infinitorum'' (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "[[Euler's formula]]", as well as the near-modern abbreviations ''sin., cos., tang., cot., sec.,'' and ''cosec.''<ref name=boyer>See Boyer (1991).</ref>

=== Etymology ===
{{Wiktionary|sine}}
[[Etymology|Etymologically]], the word ''sine'' derives from the [[Sanskrit]] word for chord, ''jiva''*(''jya'' being its more popular synonym). This was [[transliteration|transliterated]] in [[Arabic language|Arabic]] as ''jiba'' جــيــب, abbreviated ''jb'' جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word ''jaib'' جــيــب, which means "bosom", when the Arabic text was translated in the 12th century into [[Medieval Latin|Latin]] by [[Gerard of Cremona]]. The translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]'' (which means "bosom" or "bay" or "fold") <ref>See Maor (1998), chapter 3, regarding the etymology.</ref><ref>Victor J Katx, ''A history of mathematics'', p210, sidebar 6.1.</ref> The English form ''sine'' was introduced in the 1590s.

== Software implementations ==
{{See also|Lookup table#Computing sines}}
The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, it is typically abbreviated to <code>sin</code>.

Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.

In programming languages, <code>sin</code> is typically either a built-in function or found within the language's standard math library.

For example, the [[C standard library]] defines sine functions within [[C mathematical functions|math.h]]: <code>sin([[Double-precision floating-point format|double]])</code>, <code>sinf([[Single-precision floating-point format|float]])</code>, and <code>sinl([[long double]])</code>. The parameter of each is a [[floating point]] value, specifying the angle in radians. Each function returns the same [[data type]] as it accepts. Many other trigonometric functions are also defined in [[C mathematical functions|math.h]], such as for cosine, arc sine, and hyperbolic sine (sinh).

Similarly, [[Python (programming language)|Python]], defines <code>math.sin(x)</code> within the built-in <code>math</code> module. Complex sine functions are also available within the <code>cmath</code> module, e.g. <code>cmath.sin(z)</code>. [[CPython]]'s math functions call the [[C (programming language)|C]] <code>math</code> library, and use a [[double-precision floating-point format]].

There is no standard algorithm for calculating sine. [[IEEE 754-2008]], the most widely-used standard for floating-point computation, does not address calculating trigonometric functions such as sine.<ref>Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31 [http://www.jaist.ac.jp/~bjorner/ae-is-budapest/talks/Sept20pm2_Zimmermann.pdf]</ref> Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. <code>sin(10{{sup|22}})</code>.

A once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sine values, for example one value per degree. This allowed results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage. {{citation needed|date=October 2012}} 

== See also ==
{{Commons category|Sine function}}
{{Trigonometry}}
* [[Aryabhata's sine table]]
* [[Bhaskara I's sine approximation formula]]
* [[Discrete sine transform]]
* [[Euler's formula]]
* [[Generalized trigonometry]]
* [[Hyperbolic function]]
* [[Law of sines]]
* [[List of periodic functions]]
* [[List of trigonometric identities]]
* [[Madhava series]]
* [[Madhava's sine table]]
* [[Optical sine theorem]]
* [[Polar sine]] — a generalization to vertex angles
* [[Proofs of trigonometric identities]]
* [[Sine and cosine transforms]]
* [[Sine quadrant]]
* [[Sine wave]]
* [[Sine–Gordon equation]]
* [[Sinusoidal model]]
* [[Trigonometric functions]]

== References ==
{{Reflist}}

[[Category:Trigonometry]]
[[Category:Elementary special functions]]

[[no:Trigonometriske funksjoner#Sinus, cosinus og tangens]]

Adhesion

2013-12-01T20:18:21Z

128.100.37.137:

The '''Malliavin calculus''', named after [[Paul Malliavin]], extends the [[calculus of variations]] from functions to [[stochastic processes]]. The Malliavin calculus is also called the '''[[stochastic calculus]] of variations'''. In particular, it allows the computation of [[derivative]]s of [[random variable]]s.

Malliavin ideas led to a proof that [[Hörmander's condition]] implies the existence and smoothness of a [[probability density function|density]] for the solution of a [[stochastic differential equation]]; [[Lars Hörmander|Hörmander]]'s original proof was based on the theory of [[partial differential equation]]s. The calculus has been applied to [[stochastic partial differential equation]]s as well.

The calculus allows [[integration by parts]] with [[random variable]]s; this operation is used in [[mathematical finance]] to compute the sensitivities of [[derivative (finance)|financial derivative]]s. The calculus has applications for example in [[stochastic filtering]].

==Overview and history==
Paul Malliavin's [[stochastic calculus]] of variations extends the [[calculus of variations]] from functions to [[stochastic processes]]. In particular, it allows the computation of [[derivative]]s of [[random variable]]s.

Malliavin invented his calculus to provide a stochastic proof that [[Hörmander's condition]] implies the existence of a [[probability density function|density]] for the solution of a [[stochastic differential equation]]; [[Lars Hörmander|Hörmander]]'s original proof was based on the theory of [[partial differential equation]]s. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to [[stochastic partial differential equation]]s.

== Invariance principle ==
The usual invariance principle for [[Lebesgue integration]] over the whole real line is that, for any real number ε and integrable function ''f'', the
following holds

:<math> \int_{-\infty}^\infty f(x)\, d \lambda(x) = \int_{-\infty}^\infty f(x+\varepsilon)\, d \lambda(x) .</math>

This can be used to derive the [[integration by parts]] formula since, setting ''f'' = ''gh'' and differentiating with respect to ε on both sides, it implies

:<math> \int_{-\infty}^\infty f' \,d \lambda = \int_{-\infty}^\infty (gh)' \,d \lambda = \int_{-\infty}^\infty g h'\, d \lambda +
\int_{-\infty}^\infty g' h\, d \lambda.</math>

A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let <math>h_s</math> be a square-integrable [[predictable process]] and set

:<math> \varphi(t) = \int_0^t h_s\, d s .</math>

If <math>X</math> is a [[Wiener process]], the [[Girsanov theorem]] then yields the following analogue of the invariance principle:

:<math> E(F(X + \varepsilon\varphi))= E \left [F(X) \exp \left ( \varepsilon\int_0^1 h_s\, d X_s -
\frac{1}{2}\varepsilon^2 \int_0^1 h_s^2\, ds \right ) \right ].</math>

Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:

:<math>E(\langle DF(X), \varphi\rangle) = E\Bigl[ F(X) \int_0^1 h_s\, dX_s\Bigr].
</math>

Here, the left-hand side is the [[Malliavin derivative]] of the random variable <math>F</math> in the direction <math>\varphi</math> and the integral appearing on the right hand side should be interpreted as an [[Itô integral]]. This expression also remains true (by definition) if <math>h</math> is not adapted, provided that the right hand side is interpreted as a [[Skorokhod integral]].{{Citation needed|date=August 2011}}

== Clark-Ocone formula ==
{{Main|Clark–Ocone theorem}}

One of the most useful results from Malliavin calculus is the [[Clark-Ocone theorem]], which allows the process in the [[martingale representation theorem]] to be identified explicitly. A simplified version of this theorem is as follows:

For <math>F: C[0,1] \to \R</math> satisfying <math> E(F(X)^2) < \infty</math> which is Lipschitz and such that ''F'' has a strong derivative kernel, in the sense that
for <math>\varphi</math> in ''C''[0,1]

:<math> \lim_{\varepsilon \to 0} (1/\varepsilon)(F(X+\varepsilon \varphi) - F(X) ) = \int_0^1 F'(X,dt) \varphi(t)\ \mathrm{a.e.}\ X</math>

then

:<math>F(X) = E(F(X)) + \int_0^1 H_t \,d X_t ,</math>

where ''H'' is the previsible projection of ''F''<nowiki>'</nowiki>(''x'', (''t'',1]) which may be viewed as the derivative of the function ''F'' with respect to a suitable parallel shift of the process ''X'' over the portion (''t'',1] of its domain.

This may be more concisely expressed by

:<math>F(X) = E(F(X))+\int_0^1 E (D_t F | \mathcal{F}_t ) \, d X_t .</math>

Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals ''F'' by replacing the derivative kernel used above by the "[[Malliavin derivative]]" denoted <math>D_t</math> in the above statement of the result. {{Citation needed|date=August 2011}}

== Skorokhod integral ==
{{Main|Skorokhod integral}}
The [[Skorokhod integral]] operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative thus for u in the domain of the operator which is a subset of <math>L^2([0,\infty) \times \Omega)</math>,
for '''F''' in the domain of the Malliavin derivative, we require

: <math> E (\langle DF, u \rangle ) = E (F \delta (u) ),</math>

where the inner product is that on <math>L^2[0,\infty)</math> viz

: <math> \langle f, g \rangle = \int_0^\infty f(s) g(s) \, ds.</math>

The existence of this adjoint follows from the [[Riesz representation theorem]] for linear operators on [[Hilbert spaces]].

It can be shown that if ''u'' is adapted then

: <math> \delta(u) = \int_0^\infty u_t\, d W_t ,</math>

where the integral is to be understood in the Itô sense. Thus this provides a method of extending the Itô integral to non adapted integrands.

==Applications==
The calculus allows [[integration by parts]] with [[random variable]]s; this operation is used in [[mathematical finance]] to compute the sensitivities of [[derivative (finance)|financial derivative]]s. The calculus has applications for example in [[stochastic control|stochastic filtering]].

{{No footnotes|date=June 2011}}

== References ==
* Kusuoka, S. and Stroock, D. (1981) "Applications of Malliavin Calculus I", ''Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto'' 1982, pp 271–306
* Kusuoka, S. and Stroock, D. (1985) "Applications of Malliavin Calculus II", ''J. Faculty Sci. Uni. Tokyo Sect. 1A Math.'', 32 pp 1–76
* Kusuoka, S. and Stroock, D. (1987) "Applications of Malliavin Calculus III", ''J. Faculty Sci. Univ. Tokyo Sect. 1A Math.'', 34 pp 391–442
* Malliavin, Paul and Thalmaier, Anton. ''Stochastic Calculus of Variations in Mathematical Finance'', Springer 2005, ISBN 3-540-43431-3
* {{cite book
| last = Nualart
| first = David
| title = The Malliavin calculus and related topics
| edition = Second edition
|publisher = Springer-Verlag
| year = 2006
| isbn = 978-3-540-28328-7
}}
* Bell, Denis. (2007) ''The Malliavin Calculus'', Dover. ISBN 0-486-44994-7
* Schiller, Alex (2009) [http://www.alexschiller.com/media/Thesis.pdf ''Malliavin Calculus for Monte Carlo Simulation with Financial Applications'']. Thesis, Department of Mathematics, Princeton University
* [[Bernt Øksendal|Øksendal, Bernt K.]].(1997) [http://www.quantcode.com/modules/wflinks/visit.php?cid=11&lid=4 ''An Introduction To Malliavin Calculus With Applications To Economics'']. Lecture Notes, Dept. of Mathematics, University of Oslo (Zip file containing Thesis and addendum)
*Di Nunno, Giulia, Øksendal, Bernt, Proske, Frank (2009) "Malliavin Calculus for Lévy Processes with Applications to Finance", Universitext, Springer. ISBN 978-3-540-78571-2

== External links ==

* {{cite web
|url = http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf
|title = An Introduction to Malliavin Calculus
|accessdate = 2007-07-23
|last = Friz
|first = Peter K.
|date = 2005-04-10
|format = PDF
|archiveurl = http://web.archive.org/web/20070417205303/http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf |archivedate = 2007-04-17}} Lecture Notes, 43 pages

[[Category:Stochastic calculus]]
[[Category:Integral calculus]]
[[Category:Mathematical finance]]
[[Category:Calculus of variations]]

Poloidal–toroidal decomposition

2013-11-20T18:07:47Z

128.100.109.30: /* Poloidal and toroidal vector fields */ symbol error

'''DNA sequencing theory''' is the broad body of work that attempts to lay analytical foundations for determining the order of specific [[nucleotide]]s in a sequence of [[DNA]], otherwise known as [[DNA sequencing]]. The practical aspects revolve around designing and optimizing sequencing projects (known as "strategic genomics"), predicting project performance, troubleshooting experimental results, characterizing factors such as sequence bias and the effects of software processing algorithms, and comparing various sequencing methods to one another. In this sense, it could be considered a branch of [[systems engineering]] or [[operations research]]. The permanent archive of work is primarily mathematical, although numerical calculations are often conducted for particular problems too. DNA sequencing theory addresses ''physical processes'' related to sequencing DNA and should not be confused with theories of analyzing resultant DNA sequences, e.g. [[sequence alignment]]. Publications<ref name="waterman95">{{cite book |last=Waterman |first=Michael S. |authorlink=Michael Waterman |year=1995 |title=Introduction to Computational Biology |publisher=Chapman and Hall/CRC |location=Boca Raton |isbn=0-412-99391-0}}</ref> sometimes do not make a careful distinction, but the latter are primarily concerned with [[algorithm]]ic issues. Sequencing theory is based on elements of [[mathematics]], [[biology]], and [[systems engineering]], so it is highly interdisciplinary. The subject may be studied within the context of [[computational biology]].

==Theory and sequencing strategies==

===Sequencing as a covering problem===

All mainstream methods of [[DNA sequencing]] rely on reading small fragments of DNA and subsequently reconstructing these data to infer the original DNA target, either via [[sequence assembly|assembly]] or [[Sequence alignment|alignment]] to a reference. The [[abstraction]] common to these methods is that of a mathematical [[Cover (topology)|covering problem]].<ref name="hall">{{cite book |last=Hall |first=P. |year=1988 |title=Introduction to the Theory of Coverage Processes |publisher=Wiley |location=New York |isbn=0-471-85702-5}}</ref> For example, one can imagine a line segment representing the target and a subsequent process where smaller segments are "dropped" onto random locations of the target. The target is considered "sequenced" when adequate coverage accumulates (e.g., when no gaps remain).

The abstract properties of covering have been studied by mathematicians for over a century.<ref name="solomon">{{cite book |last=Solomon |first=H. |year=1978 |title=Geometric Probability |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |isbn=0-898-71025-1}}</ref> However, direct application of these results has not generally been possible. Closed-form mathematical solutions, especially for probability distributions, often cannot be readily evaluated. That is, they involve inordinately large amounts of computer time for parameters characteristic of [[DNA sequencing]]. Stevens' configuration is one such example.<ref name="stevens">{{cite journal |author=Stevens WL |year=1939 |title=Solution to a Geometrical Problem in Probability |journal=Annals of Eugenics |volume=9 |pages=315–320}}</ref> Results obtained from the perspective of [[pure mathematics]] also do not account for factors that are actually important in sequencing, for instance detectable overlap in sequencing fragments, double-stranding, edge-effects, and target multiplicity. Consequently, development of sequencing theory has proceeded more according to the philosophy of [[applied mathematics]]. In particular, it has been problem-focused and makes expedient use of approximations, simulations, etc.

===Early uses derived from elementary probability theory===

The earliest result was actually borrowed directly from elementary probability theory. If we model the above process and take <math>L</math> and <math>G</math> as the fragment length and target length, respectively, then the probability of "covering" any given location on the target ''with one particular fragment'' is <math>L / G</math>. (Note that this presumes <math>L \ll G</math>, which is valid for many, though not all sequencing scenarios). The probability of '''not''' covering a given location on the target is therefore <math>1 - L / G</math> for a single fragment and <math>\left[1 - L / G\right]^N</math> for <math>N</math> fragments. The probability of covering a given location on the target with ''at least one'' fragment is therefore

:<math>P = 1 - \left[1 - \frac{L}{G}\right]^N.</math>

This equation was first used to characterize plasmid libraries,<ref name="clarkecarbon">{{cite journal |author=Clarke L, Carbon J |year=1976 |title=A colony bank containing synthetic Col-El hybrid plasmids representative of the entire E. coli genome |journal=Cell |volume=9 |issue=1 |pages=91–99 |doi=10.1016/0092-8674(76)90055-6 |pmid=788919 }}</ref> but is often more useful in a modified form. For most projects <math>N \gg 1</math>, so that, to a good degree of approximation

:<math>\left[1 - \frac{L}{G}\right]^N \sim \exp(-NL/G),</math>

where <math>R = NL/G</math> is called the ''redundancy''. Note the significance of redundancy as representing the average number of times a position is covered with fragments. Note also that in considering the covering process over all positions in the target, this probability is identical to the [[expected value]] of the random variable <math>C</math>, which represents the fraction of the target coverage. The final result,

:<math>E\langle C \rangle = 1 - e^{-R},</math>

remains in widespread use as a "[[Back-of-the-envelope calculation|back of the envelope]]" estimator and predicts that coverage for all projects evolves along a universal curve that is a function only of the redundancy.

===Lander-Waterman theory===

In 1988, [[Eric Lander]] and [[Michael Waterman]] published an important paper<ref name="landerwaterman">{{cite journal |author=[[Eric Lander|Lander ES]], [[Michael Waterman|Waterman MS]] |year=1988 |title=Genomic mapping by fingerprinting random clones: a mathematical analysis |journal=Genomics |volume=2 |issue=3 |pages=231–239 |doi=10.1016/0888-7543(88)90007-9 |pmid=3294162 }}</ref> examining the covering problem from the standpoint of gaps. Although they focused on the so-called [[Gene mapping|mapping problem]], the abstraction to sequencing is much the same. They furnished a number of useful results that were adopted as the standard theory from the earliest days of "large-scale" genome sequencing.<ref name="fleischmann">{{cite journal |author=Fleischmann RD, ''et al.'' |year=1995 |title=Whole-genome random sequencing and assembly of haemophilus influenzae Rd |journal=Science |volume=269 |issue=5223 |pages=496–512 |doi=10.1126/science.7542800 |pmid=7542800 |bibcode = 1995Sci...269..496F }}</ref> Their model was also used in designing the [[Human Genome Project]] and continues to play an important role in DNA sequencing.

Ultimately, the main goal of a sequencing project is to close all gaps, so the "gap perspective" was a logical basis of developing a sequencing model. One of the more frequently used results from this model is the expected number of [[contig]]s, given the number of fragments sequenced. If one neglects the amount of sequence that is essentially "wasted" by having to detect overlaps, their theory yields

:<math>E\langle contigs \rangle = N e^{-R}.</math>

In 1995, Roach<ref name="roach95">{{cite journal |author=Roach JC |year=1995 |title=Random subcloning |journal=Genome Research |volume=5 |issue=5|pages=464–473 |doi=10.1101/gr.5.5.464 |pmid=8808467 }}</ref> published improvements to this theory, enabling it to be applied to sequencing projects in which the goal was to completely sequence a target genome. [[Michael Christopher Wendl|Michael Wendl]] and [[Bob Waterston]]<ref name="wendl02">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]], [[Bob Waterston|Waterston RH]] |year=2002 |title=Generalized gap model for bacterial artificial chromosome clone fingerprint mapping and shotgun sequencing |journal=Genome Research |volume=12 |issue=12 |pages=1943–1949 |doi=10.1101/gr.655102 |pmid=12466299 |pmc=187573}}</ref> confirmed, based on Stevens' method,<ref name="stevens"/> that both models produced similar results when the number of contigs was substantial, such as in low coverage mapping or sequencing projects. As sequencing projects ramped up in the 1990s, and projects approached completion, low coverage approximations became inadequate, and the exact model of Roach was necessary. However, as the cost of sequencing dropped, parameters of sequencing projects became easier to directly test empirically, and interest and funding for strategic genomics diminished

The basic ideas of Lander–Waterman theory led to a number of additional results for particular variations in mapping techniques.<ref name="arratia">{{cite journal |author=[[Richard Arratia|Arratia R]], ''et al.'' |year=1991 |title=Genomic mapping by anchoring random clones: a mathematical analysis |journal=Genomics |volume=11 |pages=806–827 |issue=4 |doi=10.1016/0888-7543(91)90004-X |pmid=1783390 }}</ref><ref name="port">{{cite journal |author=Port E, ''et al.'' |year=1995 |title=Genomic mapping by end-characterized random clones: a mathematical analysis |journal=Genomics |volume=26 |pages=84–100 |issue=1 |doi=10.1016/0888-7543(95)80086-2 |pmid=7782090 }}</ref><ref name="zhang">{{cite journal |author=Zhang MQ, Marr TG |year=1993 |title=Genome mapping by nonrandom anchoring: a discrete theoretical analysis |journal=[[Proceedings of the National Academy of Sciences]] |volume=90 |issue=2 |pages=600–604 |doi=10.1073/pnas.90.2.600 |bibcode = 1993PNAS...90..600Z }}</ref> However, technological advancements have rendered mapping theories largely obsolete except in organisms other than highly studied model organisms (e.g., yeast, flies, mice, and humans).

===Parking strategy===

The parking strategy for sequencing resembles the process of parking cars along a curb. Each car is a sequenced clone, and the curb is the genomic target.<ref name="roach2000">{{cite journal |author=Roach JC, ''et al.'' |year=2000 |title=Parking strategies for genome sequencing |doi=10.1101/gr.10.7.1020 |journal=Genome Research |volume=10 |issue=7 |pages=1020–1030 |pmid=10899151 |pmc=310895}}</ref> Each clone sequenced is screened to ensure that subsequently sequenced clones do not overlap any previously sequenced clone. No sequencing effort is redundant in this strategy. However, much like the gaps between parked cars, unsequenced gaps less than the length of a clone accumulate between sequenced clones. There can be considerable cost to close such gaps.

===Pairwise end-sequencing===

In 1995, Roach ''et al.''<ref name="roach1995b">{{cite journal |author=Roach JC, Boysen C, Wang K, [[Leroy Hood|Hood L]] |year=1995 |title=Pairwise end sequencing: a unified approach to genomic mapping and sequencing |journal=Genomics |volume=26 |issue=2 |pages=345–353 |doi=10.1016/0888-7543(95)80219-C |pmid=7601461}}</ref> proposed and demonstrated through simulations a generalization of a set of strategies explored earlier by Edwards and Caskey.<ref name="edwards91">{{cite book |author=Edwards, A.; Caskey, T. |year=1991 |title=Closure strategies for random DNA sequencing |publisher=A Companion to Methods in Enzymology |volume=3 |pages=41–47}}</ref> This [[Shotgun_sequencing#Whole_genome_shotgun_sequencing|whole-genome sequencing]] method became immensely popular as it was championed by Celera and used to sequenced several model organisms before Celera applied it to the human genome. Today, most sequencing projects employ this strategy, often called paired end sequencing.

==Post Human Genome Project advancements==

The physical processes and protocols of DNA sequencing have continued to evolve, largely driven by advancements in bio-chemical methods, instrumentation, and automation techniques. There is now a wide range of problems that [[DNA sequencing]] has made in-roads into, including [[metagenomics]] and [[Cancer Genome Project|medical (cancer) sequencing]]. There are important factors in these scenarios that classical theory does not account for. Recent work has begun to focus on resolving the effects of some of these issues. The level of mathematics becomes commensurately more sophisticated.

===Various artifacts of large-insert sequencing===

Biologists have developed methods to filter highly-repetitive, essentially un-sequenceable regions of genomes. These procedures are important for organisms whose genomes consist mostly of such DNA, for example corn. They yield multitudes of small islands of sequenceable DNA products. Wendl and Barbazuk<ref name="wendl05">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]], Barbazuk WB |year=2005 |title=Extension of Lander–Waterman Theory for sequencing filtered DNA libraries |journal=BMC Bioinformatics |volume=6 |pages=article 245 |doi=10.1186/1471-2105-6-245 |pmid=16216129 |pmc=1280921}}</ref> proposed an extension to Lander–Waterman Theory to account for "gaps" in the target due to filtering and the so-called "edge-effect". The latter is a position-specific sampling bias, for example the terminal base position has only a <math>1 / G</math> chance of being covered, as opposed to <math>L / G</math> for interior positions. For <math>R < 1</math>, classical Lander–Waterman Theory still gives good predictions, but dynamics change for higher redundancies.

Modern sequencing methods usually sequence both ends of a larger fragment, which provides linking information for ''de novo'' assembly and improved probabilities for alignment to reference sequence. Researchers generally believe that longer lengths of data (read lengths) enhance performance for very large DNA targets, an idea consistent with predictions from distribution models.<ref name="wendl06distrib">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]] |year=2006 |title=Occupancy modeling of coverage distribution for whole genome shotgun DNA sequencing |journal=Bulletin of Mathematical Biology |volume=68 |issue=1|pages=179–196 |doi=10.1007/s11538-005-9021-4 |pmid=16794926 }}</ref> However, Wendl<ref name="wendl06gen">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]] |year=2006 |title=A general coverage theory for shotgun DNA sequencing |journal=Journal of Computational Biology |volume=13 |issue=6 |pages=1177–1196 |doi=10.1089/cmb.2006.13.1177 |pmid=16901236 }}</ref> showed that smaller fragments provide better coverage on small, linear targets because they reduce the edge effect in linear molecules. These findings have implications for sequencing the products of DNA filtering procedures. Read-pairing and fragment size evidently have negligible influence for large, whole-genome class targets.

===Individual and population sequencing===

Sequencing is emerging as an important tool in medicine, for example in cancer research. Here, the ability to detect [[Loss of heterozygosity|heterozygous mutations]] is important and this can only be done if the sequence of the [[Diploid#Diploid|diploid genome]] is obtained. In the pioneering efforts to sequence individuals, Levy ''et al.''<ref name="levy07">{{cite journal |author=Levy S, ''et al.'' |year=2007 |title=The diploid genome sequence of an individual human |journal=PLoS Biology |volume=5 |issue=10 |pages=article e254 |doi=10.1371/journal.pbio.0050254 |pmid=17803354 |pmc=1964779}}</ref> and Wheeler ''et al.'',<ref name="wheeler08">{{cite journal |author=Wheeler DA, ''et al.'' |year=2008 |title=The complete genome of an individual by massively parallel DNA sequencing |journal=Nature |volume=452 |issue=7189 |pages=872–876 |doi=10.1038/nature06884 |pmid=18421352 |bibcode=2008Natur.452..872W}}</ref> who sequenced [[Craig Venter]] and [[James D. Watson|Jim Watson]], respectively, outlined models for covering both alleles in a genome. Wendl and Wilson<ref name="wendl08">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]], [[Richard K. Wilson|Wilson RK]] |year=2008 |title=Aspects of coverage in medical DNA sequencing |journal=BMC Bioinformatics |volume=9 |pages=article 239 |doi=10.1186/1471-2105-9-239 |pmid=18485222 |pmc=2430974}}</ref> followed with a more general theory that allowed for an arbitrary number of coverings of each allele and arbitrary [[ploidy]]. These results point to the general conclusion that the amount of data needed for such projects is significantly higher than for traditional haploid projects. Generally, at least 30-fold redundancy, i.e. each nucleotide spanned by an average of 30 sequence reads, is now standard.<ref name="ley2008">{{cite journal |author=[[Timothy Ley|Ley TJ]], ''et al.'' |year=2008 |title=DNA sequencing of a cytogenetically normal acute myeloid leukaemia genome |journal=Nature |volume=456 |issue=7218|pages=66–72 |doi=10.1038/nature07485 |pmid=18987736|bibcode = 2008Natur.456...66L }}</ref>
However, requirements can be even greater, depending upon what kinds of genomic events are to be found. For example, in the so-called "discordant read pairs method", DNA insertions can be inferred if the distance between read pairs is larger than expected. Calculations show that around 50-fold redundancy is needed to avoid [[False positive rate|false-positive errors]] at 1% threshold.<ref name="wendl09a">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]], [[Richard K. Wilson|Wilson RK]] |year=2009 |title=Statistical aspects of discerning indel-type structural variation via DNA sequence alignment |journal=BMC Genomics |volume=10 |pages=article 359 |pmid=19656394 |doi=10.1186/1471-2164-10-359 |pmc=2748092}}</ref>

The advent of [[DNA_sequencing#Next-generation_methods|next-generation sequencing]] has also made large-scale population sequencing feasible, for example the [[1000 Genomes Project]] to characterize variation in human population groups. While common variation is easily captured, rare variation poses a design challenge: too few samples with significant sequence redundancy risks not having a variant in the sample group, but large samples with light redundancy risk not capturing a variant in the read set that is actually in the sample group. Wendl and Wilson<ref name="wendl09b">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]], [[Richard K. Wilson|Wilson RK]] |year=2009 |title=The theory of discovering rare variants via DNA sequencing |journal=BMC Genomics |volume=10 |pages=article 485 |pmid=19843339 |doi=10.1186/1471-2164-10-485 |pmc=2778663}}</ref> report a simple set of optimization rules that maximize the probability of discovery for a given set of parameters. For example, for observing a rare allele at least twice (to eliminate the possibility is unique to an individual) a little less than 4-fold redundancy should be used, regardless of the sample size.

===Metagenomic sequencing===

Next-generation instruments are now also enabling the sequencing of whole uncultured metagenomic communities. The sequencing scenario is more complicated here and there are various ways of framing design theories for a given project. For example, Stanhope<ref name="stanhope2010">{{cite journal |author=Stanhope SA |year=2010 |title=Occupancy modeling maximum contig size probabilities and designing metagenomics experiments |journal=PLoS ONE|volume=5 |pages=article e11652 |pmid= 20686599 |doi=10.1371/journal.pone.0011652|bibcode = 2010PLoSO...511652S }}</ref> developed a probabilistic model for the amount of sequence needed to obtain at least one contig of a given size from each novel organism of the community, while Wendl et al. reported analysis for the average contig size or the probability of completely recovering a novel organism for a given rareness within the community.<ref name="wendl2012">{{cite journal |author=[[Michael Christopher Wendl|Wendl MC]] ''et al.''|year=2012 |title=Coverage theories for metagenomic DNA sequencing based on a generalization of Stevens' theorem |journal=Journal of Mathematical Biology |pmid= 22965653 |doi=10.1007/s00285-012-0586-x}}</ref> Conversely, Hooper et al. propose a semi-empirical model based on the [[Gamma distribution]].<ref name="hooper2010">{{cite journal |author=Hooper SD ''et al.''|year=2010 |title=Estimating DNA coverage and abundance in metagenomes using a gamma approximation |journal=Bioinformatics|volume=26 |pages=295–301 |pmid= 20008478 |doi=10.1093/bioinformatics/btp687}}</ref>

==Limitations==

DNA sequencing theories often invoke the assumption that certain random variables in a model are [[Independent and identically-distributed random variables|independent and identically distributed]]. For example, in Lander–Waterman Theory, a sequenced fragment is presumed to have the same probability of covering each region of a genome and all fragments are assumed to be independent of one another. In actuality, sequencing projects are subject to various types of bias, including differences of how well regions can be cloned, sequencing anomalies, biases in the target sequence (which is ''not'' random), and software-dependent errors and biases. In general, theory will agree well with observation up to the point that enough data have been generated to expose latent biases.<ref name="wendl08"/> The kinds of biases related to the underlying target sequence are particularly difficult to model, since the sequence itself may not be known ''a priori''. This presents a type of [[Chicken or the egg|"chicken and egg"]] closure problem.

==See also==

*[[Computational biology]]
*[[Bioinformatics]]
*[[Mathematical biology]]
*[[Sulston score]]

==References==
{{reflist|2}}

[[Category:Bioinformatics]]
[[Category:Mathematical and theoretical biology]]
[[Category:DNA sequencing]]

Uniformly hyperfinite algebra

2012-08-03T19:20:02Z

128.100.216.171: /* C*-algebras */

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2006-09-15T15:53:35Z

128.100.68.62: /* See also */

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