formulasearchengine - User contributions [en]

Triangular matrix

2014-05-24T16:25:19Z

85.110.56.157:

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Gummel–Poon model

2014-05-19T09:48:01Z

85.110.64.235:

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Antenna (radio)

2014-02-27T14:03:50Z

85.110.21.21:

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Antenna gain

2014-02-25T23:06:07Z

85.110.8.45: /* Power gain */

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(ε, δ)-definition of limit

2014-01-21T07:18:52Z

85.110.52.106: /* Informal statement */

In [[fluid dynamics]], the '''Hadamard–Rybczynski equation''' gives the [[terminal velocity]] of slowly moving [[spherical]] [[liquid bubble|bubble]] through an ambient [[fluid]]. It is named after [[Jacques Hadamard]] and [[Witold Erazm Rybczynski|Witold Rybczynski]]:

:<math> W_b = \frac{2}{3} \frac {R^2 g (\rho_b - \rho_o)}{\mu_o}
\frac {\mu_o + \mu_b}{2\mu_o + 3\mu_b}
</math>
where
* <math>R</math> is the radius of the bubble.
* <math>g</math> the gravitational acceleration.
* <math>\rho_b</math> the density of the bubble.
* <math>\rho_o</math> the density of the ambient fluid.
* <math>\mu_b</math> the viscosity of the bubble.
* <math>\mu_o</math> the viscosity of the ambient fluid.

The Hadamard–Rybczynski equation can be derived from the [[Navier–Stokes equations]] by considering only the [[buoyancy force]] and [[drag force]] acting on the moving bubble. The surface tension force and inertia force of the bubble are neglected.<ref>{{cite book |last=Clift, R. C., Grace, B. J., and Weber, M. E.,|first= |title=Bubbles, Drops, and Particles |publisher=[[Dover Publications]] |year=2005 |isbn=978-0-486-44580-9}}
</ref>

==References==
{{reflist}}

==Further reading==
*{{Cite journal |first=J. S. |last=Hadamard |title={{lang|fr|Mouvement permanent lent d'une sphere liquide et visqueuse dans un liquide visqueux}} |journal=CR Acad. Sci. |volume=152 |year=1911 |pages=1735 }}
*{{Cite journal |first=W. |last=Rybczynski |title={{lang|de|Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium}} |journal=Bull. Acad. Sci. Cracovi, A. |year=1911 |pages=40–46 }}

{{DEFAULTSORT:Hadamard-Rybczynski equation}}
[[Category:Fluid dynamics]]
[[Category:Bubbles]]
[[Category:Equations of fluid dynamics]]
[[Category:Buoyancy]]

Homogeneous differential equation

2014-01-15T18:55:37Z

85.110.110.8: /* Homogeneous linear differential equations */

{{Other uses of|Jot|Jot (disambiguation)}}
{{Notability|date=February 2011}}

'''Iota''' and its successor '''Jot''' (from [[Greek language|Greek]] [[iota]], [[Hebrew]] [[yodh]], the smallest letters in those two alphabets) are extremely minimalist [[formal system]]s, designed to be even simpler than other more popular alternatives, such as the [[lambda calculus]] and [[SKI calculus|SKI combinator calculus]]. They can therefore also be considered minimalist computer programming languages, or [[Turing tarpit]]s, [[esoteric programming language]]s that are designed to be as small as possible but still [[Turing-complete]]. Both systems use only two symbols and involve only two operations. '''Zot''' is a continuized version of Iota that includes input and output.

Iota's universal combinator has a very simple structure (defined here using [[denotational semantics]] in terms of the [[lambda calculus]]):

{{Center|<math>U = \lambda f.((fS)K)</math>}}

From this, one can recover the usual [[SKI_combinator_calculus#SKI_expressions|SKI basis combinators]] as follows:

{{Center|<math>I\, =\, (UU),\; K\, =\, (U(U(UU))), \;S\, =\, (U(U(U(UU))))</math>}}

Because of its minimalism, it has influenced research concerning [[Chaitin's constant]].<ref>Michael Stay, [http://arxiv.org/abs/cs.IT/0508056 "Very Simple Chaitin Machines for Concrete AIT"], ''Fundamenta Informaticae 68 (3). pp. 231--247 CDMTCS Report 265'', 2005</ref>

==See also==
*[[Lambda calculus]]
*[[Combinatory logic]]
*[[Binary combinatory logic]]
*[[SKI calculus|SKI combinator calculus]]

==References==
{{reflist}}

==External links==
* Barker, Chris "[http://semarch.linguistics.fas.nyu.edu/barker/Iota/ Iota and Jot: the simplest languages?]"
* [http://esolangs.org/wiki/Iota Esolang Wiki: Iota]
* [http://esolangs.org/wiki/Jot Esolang Wiki: Jot]

{{DEFAULTSORT:Iota And Jot}}
[[Category:Esoteric programming languages]]

{{Compu-lang-stub}}

Large numbers

2014-01-15T01:20:12Z

85.110.57.118: /* Computers and computational complexity */

{{no footnotes|date=April 2013}}

[[Image:Triangle.Centroid.svg|thumb|right|Centroid of a triangle]]
In [[mathematics]] and [[physics]], the '''centroid''' or '''geometric center''' of a two-dimensional region is, informally, the point at which a cardboard cut-out of the region could be perfectly balanced on the tip of a pencil (assuming uniform density and a uniform gravitational field). Formally, the centroid of a [[plane figure]] or two-dimensional shape is the [[arithmetic mean]] ("average") position of all the points in the shape. The definition extends to any object in ''n''-[[dimension]]al [[space (mathematics)|space]]: its centroid is the mean position of all the points in all of the coordinate directions.

While in geometry the term '''barycenter''' is a synonym for "centroid", in physics "barycenter" may also mean the physical [[center of mass]] or the center of gravity, depending on the context. The center of mass (and center of gravity in a uniform gravitational field) is the arithmetic mean of all points weighted by the local [[density]] or [[specific weight]]. If a physical object has uniform density, then its center of mass is the same as the centroid of its shape.

In [[geography]], the centroid of a radial projection of a region of the Earth's surface to sea level is known as the region's [[geographical center]].

== Properties ==
The geometric centroid of a [[convex set|convex]] object always lies in the object. A non-convex object might have a centroid that is outside the figure itself. The centroid of a [[Annulus (mathematics)|ring]] or a [[bowl (vessel)|bowl]], for example, lies in the object's central void.

If the centroid is defined, it is a [[fixed points of isometry groups in Euclidean space|fixed point of all isometries]] in its [[symmetry group]]. In particular, the geometric centroid of an object lies in the intersection of all its [[hyperplane]]s of [[symmetry]]. The centroid of many figures ([[regular polygon]], [[regular polyhedron]], [[cylinder (geometry)|cylinder]], [[rectangle]], [[rhombus]], [[circle (geometry)|circle]], [[sphere (geometry)|sphere]], [[ellipse]], [[ellipsoid]], [[Lamé curve|superellipse]], [[superellipsoid]], etc.) can be determined by this principle alone.

In particular, the centroid of a [[parallelogram]] is the meeting point of its two [[diagonal]]s. This is not true for other [[quadrilateral]]s.

For the same reason, the centroid of an object with [[translational symmetry]] is undefined (or lies outside the enclosing space), because a translation has no fixed point.

==Triangle centroid==

The centroid of a triangle is the intersection of the three [[median (triangle)|medians]] of the triangle (each median connecting a vertex with the midpoint of the opposite side). It lies on the triangle's [[Euler line]], which also goes through various other key points including the [[orthocenter]] and the [[circumcenter]].<ref>{{harvtxt|Altshiller-Court|1925|p=101}}</ref><ref>{{harvtxt|Kay|1969|pp=18,189,225-226}}</ref>

Let ''M'' be any point in the plane of a triangle with vertices ''A, B,'' and ''C'' and centroid ''G''. Then the sum of the squared distances of ''M'' from the three vertices exceeds the sum of the squared distances of the centroid ''G'' from the vertices by three times the squared distance between ''M'' and ''G'':

:<math>MA^2+MB^2+MC^2=GA^2+GB^2+GC^2+3MG^2.</math><ref>{{harvtxt|Altshiller-Court|1925|pp=70-71}}</ref>

The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:

:<math>AB^2+BC^2+CA^2=3(GA^2+GB^2+GC^2).</math><ref>{{harvtxt|Altshiller-Court|1925|pp=70-71}}</ref>

For other properties of a triangle's centroid, see [[#Of triangle and tetrahedron|below]].

== Locating the centroid ==
=== Plumb line method ===
The centroid of a uniform [[planar lamina]], such as (a) below, may be determined, experimentally, by using a [[plumbline]] and a pin to find the center of mass of a thin body of uniform density having the same shape. The body is held by the pin inserted at a point near the body's perimeter, in such a way that it can freely rotate around the pin; and the plumb line is dropped from the pin (b). The position of the plumbline is traced on the body. The experiment is repeated with the pin inserted at a different point of the object. The intersection of the two lines is the centroid of the figure (c).

{| cellpadding=3
|- valign=center
| [[File:Center gravity 0.svg|147px|center]]
| [[File:Center gravity 1.svg|183px|center]]
| [[File:Center gravity 2.svg|157px|center]]
|-
| align=center | (a)
| align=center | (b)
| align=center | (c)
|}

This method can be extended (in theory) to concave shapes where the centroid lies outside the shape, and to solids (of uniform density), but the positions of the plumb lines need to be recorded by means other than drawing.

=== Balancing method ===

For convex two-dimensional shapes, the centroid can be found by balancing the shape on a smaller shape, such as the top of a narrow cylinder. The centroid occurs somewhere within the range of contact between the two shapes. In principle, progressively narrower cylinders can be used to find the centroid to arbitrary accuracy. In practice air currents make this unfeasible. However, by marking the overlap range from multiple balances, one can achieve a considerable level of accuracy.

=== Of a finite set of points ===
The centroid of a finite set of <math>{k}</math> points <math>\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_k</math> in <math>\mathbb{R}^n</math> is
:<math>\mathbf{C} = \frac{\mathbf{x}_1+\mathbf{x}_2+\cdots+\mathbf{x}_k}{k} </math>.<ref>{{harvtxt|Protter|Morrey, Jr.|1970|p=520}}</ref>
This point minimizes the sum of squared Euclidean distances between itself and each point in the set.

=== By geometric decomposition ===
The centroid of a plane figure <math>X</math> can be computed by dividing it into a finite number of simpler figures <math>X_1,X_2,\dots,X_n</math>, computing the centroid <math>C_i</math> and area <math>A_i</math> of each part, and then computing

: <math> C_x = \frac{\sum C_{i_x} A_i}{\sum A_i} , C_y = \frac{\sum C_{i_y} A_i}{\sum A_i}</math>

Holes in the figure <math>X</math>, overlaps between the parts, or parts that extend outside the figure can all be handled using negative areas <math>A_i</math>. Namely, the measures <math>A_i</math> should be taken with positive and negative signs in such a way that the sum of the signs of <math>A_i</math> for all parts that enclose a given point <math>p</math> is 1 if <math>p</math> belongs to <math>X</math>, and 0 otherwise.

For example, the figure below (a) is easily divided into a square and a triangle, both with positive area; and a circular hole, with negative area (b).
{{multiple images
|align=center
|height=200
|direction=horizontal
|image1=COG 1.svg
|caption1=(a) 2D Object
|image2=COG 2.svg
|caption2=(b) Object described using simpler elements
|image3=COG 3.svg
|caption3=(c) Centroids of elements of the object
}}
The centroid of each part can be found in any [[list of centroids|list of centroids of simple shapes]] (c). Then the centroid of the figure is the weighted average of the three points. The horizontal position of the centroid, from the left edge of the figure is
:<math>x = \frac{5 \times 10^2 + 13.33 \times \frac{1}{2}10^2 - 3 \times \pi2.5^2}{10^2 + \frac{1}{2}10^2 -\pi2.5^2} \approx 8.5 \mbox{ units}.</math>
The vertical position of the centroid is found in the same way.

The same formula holds for any three-dimensional objects, except that each <math>A_i</math> should be the volume of <math>X_i</math>, rather than its area. It also holds for any subset of <math>\R^d</math>, for any dimension <math>d</math>, with the areas replaced by the <math>d</math>-dimensional [[measure (mathematics)|measure]]s of the parts.

=== By integral formula ===
The centroid of a subset ''X'' of <math>\R^n</math> can also be computed by the [[integral]]

:<math>C = \frac{\int x g(x) \; dx}{\int g(x) \; dx}</math>

where the integrals are taken over the whole space <math>\R^n</math>, and ''g'' is the [[Indicator function|characteristic function]] of the subset, which is 1 inside ''X'' and 0 outside it.<ref>{{harvtxt|Protter|Morrey, Jr.|1970|p=526}}</ref> Note that the denominator is simply the [[measure (mathematics)|measure]] of the set ''X''. This formula cannot be applied if the set ''X'' has zero measure, or if either integral diverges.

Another formula for the centroid is

:<math>C_k = \frac{\int z S_k(z) \; dz}{\int S_k(z) \; dz}</math>

where ''C''''k'' is the ''k''th coordinate of ''C'', and ''S''''k''(''z'') is the measure of the intersection of ''X'' with the hyperplane defined by the equation ''x''''k'' = ''z''. Again, the denominator is simply the measure of ''X''.

For a plane figure, in particular, the barycenter coordinates are

:<math>C_{\mathrm x} = \frac{\int x S_{\mathrm y}(x) \; dx}{A}</math>

:<math>C_{\mathrm y} = \frac{\int y S_{\mathrm x}(y) \; dy}{A}</math>

where ''A'' is the area of the figure ''X''; ''S''y(''x'') is the length of the intersection of ''X'' with the vertical line at [[abscissa]] ''x''; and ''S''x(''y'') is the analogous quantity for the swapped axes.

==== Bounded region ====

The centroid <math>(\bar{x},\;\bar{y})</math> of a region bounded by the graphs of the [[continuous function]]s <math>f</math> and <math>g</math> such that <math>f(x) \geq g(x)</math> on the interval <math>[a, b]</math>, <math>a \leq x \leq b</math>, is given by

:<math>\bar{x}=\frac{1}{A}\int_a^b x[f(x) - g(x)]\;dx</math>
:<math>\bar{y}=\frac{1}{A}\int_a^b \left[\frac{f(x) + g(x)}{2}\right][f(x) - g(x)]\;dx,</math>

where <math>A</math> is the area of the region (given by <math>\int_a^b [f(x) - g(x)]\;dx</math>).<ref>{{harvtxt|Larson|1998|pp=458–460}}</ref>

===== Example =====
[[File:Semicircle radius 1 centroid.svg|300px|thumb|Semicircle with a red dot showing the centroid]]

Consider the semicircle bounded by <math>f(x)=\sqrt{1-x^2}</math> and <math>g(x)=0</math>. Its area is <math>A=\frac{\pi r^2}{2}=\frac{\pi}{2}</math>.

:<math>\bar{x}=\frac{1}{A}\int_a^b x[f(x) - g(x)]\;dx=\frac{2}{\pi}\int_{-1}^1 x\sqrt{1-x^2}\;dx=0</math>
:<math>\bar{y}=\frac{1}{A}\int_0^1 2y\sqrt{1-y^2}\;dy=\frac{-2}{3A} (1-y^2)^{\frac{3}{2}}|^1_0=\frac{4}{3\pi}</math>

The centroid is located at <math>(0,\;\frac{4}{3\pi})</math>.

=== Of an L-shaped object ===
This is a method of determining the centroid of an L-shaped object.

[[Image:CoG of L shape.svg|600px|CG of L-shaped object]]

#Divide the shape into two rectangles, as shown in fig 2. Find the centroids of these two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of the shape must lie on this line AB.
#Divide the shape into two other rectangles, as shown in fig 3. Find the centroids of these two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of the L-shape must lie on this line CD.
#As the centroid of the shape must lie along AB and also along CD, it is obvious that it is at the intersection of these two lines, at O. The point O might ''not'' lie inside the L-shaped object.

=== Of triangle and tetrahedron ===
{| align="right" border="0" cellspacing="0" cellpadding="0"
|-
| [[Image:Triangle centroid 1.svg]] [[Image:Triangle centroid 2.svg]]
|}

The centroid of a [[triangle]] is the point of intersection of its [[median (geometry)|medians]] (the lines joining each [[vertex (geometry)|vertex]] with the midpoint of the opposite side). The centroid divides each of the medians in the [[ratio]] 2:1, which is to say it is located ⅓ of the perpendicular distance between each side and the opposing point (see figures at right). Its [[Cartesian coordinates]] are the [[arithmetic mean|means]] of the coordinates of the three vertices. That is, if the three vertices are <math>a = (x_a, y_a)</math>, <math>b = (x_b, y_b)</math>, and <math>c = (x_c, y_c)</math>, then the centroid is

:<math>
C = \frac13(a+b+c) = \left(\frac13 (x_a+x_b+x_c),\;\;
\frac13(y_a+y_b+y_c)\right).</math>

The centroid is therefore at <math>\left(\frac13,\frac13,\frac13\right)</math> in [[Barycentric coordinates (mathematics)|barycentric coordinates]].

The centroid is also the physical center of mass if the triangle is made from a uniform sheet of material; or if all the mass is concentrated at the three vertices, and evenly divided among them. On the other hand, if the mass is distributed along the triangle's perimeter, with uniform [[linear density]], then the center of mass lies at the [[Spieker center]] (the [[incenter]] of the [[medial triangle]]), which does not (in general) coincide with the geometric centroid of the full triangle.

The area of the triangle is 1.5 times the length of any side times the perpendicular distance from the side to the centroid.<ref>{{harvtxt|Johnson|2007|p=173}}</ref>

A triangle's centroid lies on its [[Euler line]] between its [[orthocenter]] and its [[circumcenter]], exactly twice as close to the latter as to the former.

Similar results hold for a [[tetrahedron]]: its centroid is the intersection of all line segments that connect each vertex to the centroid of the opposite face. These line segments are divided by the centroid in the ratio 3:1. The result generalizes to any ''n''-dimensional [[simplex]] in the obvious way. If the set of vertices of a simplex is <math>{v_0,\ldots,v_n}</math>, then considering the vertices as [[vector (geometry)|vectors]], the centroid is

:<math>C = \frac{1}{n+1}\sum_{i=0}^n v_i.</math>

The geometric centroid coincides with the center of mass if the mass is uniformly distributed over the whole simplex, or concentrated at the vertices as ''n'' equal masses.

The [[isogonal conjugate]] of a triangle's centroid is its [[symmedian|symmedian point]].

=== Centroid of polygon ===
The centroid of a non-self-intersecting closed polygon defined by ''n'' vertices (''x''0,''y''0), (''x''1,''y''1), ..., (''x''''n''−1,''y''''n''−1) is the point (''C''x, ''C''y), where

: <math>C_{\mathrm x} = \frac{1}{6A}\sum_{i=0}^{n-1}(x_i+x_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i)</math>

: <math>C_{\mathrm y} = \frac{1}{6A}\sum_{i=0}^{n-1}(y_i+y_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i)</math>

and where ''A'' is the polygon's signed area,

: <math>A = \frac{1}{2}\sum_{i=0}^{n-1} (x_i\ y_{i+1} - x_{i+1}\ y_i)\;</math>.<ref>{{harvtxt|Bourke|July 1997}}</ref>

In these formulas, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter, and the vertex ( ''x''''n'', ''y''''n'' ) is assumed to be the same as ( ''x''0, ''y''0 ). Note that if the points are numbered in clockwise order the area ''A'', computed as above, will have a negative sign; but the centroid coordinates will be correct even in this case.

=== Centroid of cone or pyramid ===
The centroid of a cone or pyramid is located on the line segment that connects the [[Apex (geometry)|apex]] to the centroid of the base. For a solid cone or pyramid, the centroid is 1/4 the distance from the base to the apex. For a cone or pyramid that is just a shell (hollow) with no base, the centroid is 1/3 the distance from the base plane to the apex.

== See also ==
* [[Chebyshev center]]
* [[Fréchet mean]]
* [[K-means algorithm]]
* [[List of centroids]]
* [[Pappus's centroid theorem]]
* [[Triangle center]]

==Notes==
<references/>

== References ==
{{Refimprove|date=August 2010}}
* {{ citation | first1 = Nathan | last1 = Altshiller-Court | year = 1925 | lccn = 52-13504 | title = College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle | edition = 2nd | publisher = [[Barnes and Noble]] | location = New York }}
* {{ cite web | url = http://paulbourke.net/geometry/polygonmesh | last = Bourke | first = Paul | title = Calculating the area and centroid of a polygon | date = July 1997 }}
* {{ citation | first = Roger A. | last1 = Johnson | title = Advanced Euclidean Geometry | publisher = [[Dover]] | year = 2007 }}
* {{ citation | first1 = David C. | last1 = Kay | year = 1969 | lccn = 69-12075 | title = College Geometry | publisher = [[Holt, Rinehart and Winston]] | location = New York }}
* {{ citation | last1 = Larson | first1 = Roland E. | last2 = Hostetler | first2 = Robert P. | last3 = Edwards |first3 = Bruce H. | title = Calculus of a Single Variable | year = 1998 | publisher = [[Houghton Mifflin Company]] | edition = 6th }}
* {{ citation | first1 = Murray H. | last1 = Protter | first2 = Charles B. | last2 = Morrey, Jr. | year = 1970 | lccn = 76-87042 | title = College Calculus with Analytic Geometry | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}

== External links ==
*[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html ''Encyclopedia of Triangle Centers''] by Clark Kimberling. The centroid is indexed as X(2).
* [http://www.cut-the-knot.org/triangle/CharacteristicPropertyOfCentroid.shtml Characteristic Property of Centroid] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/triangle/barycenter.shtml Barycentric Coordinates] at [[cut-the-knot]]
* Interactive animations showing [http://www.mathopenref.com/trianglecentroid.html Centroid of a triangle] and [http://www.mathopenref.com/constcentroid.html Centroid construction with compass and straightedge]
* [http://dynamicmathematicslearning.com/findingtrianglemedian.html Experimentally finding the medians and centroid of a triangle] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], an interactive dynamic geometry sketch using the gravity simulator of Cinderella.

[[Category:Affine geometry]]
[[Category:Geometric centers]]
[[Category:Means]]
[[Category:Triangle centers]]

Almost periodic function

2014-01-14T20:10:17Z

85.110.5.24: /* Quasiperiodic signals in audio and music synthesis */

{{about| the concept from elementary [[differential calculus]]| the generalized advanced mathematical concept from [[differential topology]] and [[differential geometry]] | closed and exact differential forms}}

In [[multivariate calculus]], a [[differential (infinitesimal)|differential]] is said to be '''exact''' (or perfect), as contrasted with an [[inexact differential]], if it is of the form ''dQ'', for some differentiable [[function (mathematics)|function]] ''Q''.

==Overview==

===Definition===
We work in three dimensions, with similar definitions holding in any other number of dimensions. In three dimensions, a form of the type

:<math>A(x,y,z) dx + B(x,y,z) dy + C(x,y,z) dz</math>

is called a [[differential form]]. This form is called ''exact'' on a domain <math>D \subset \mathbb{R}^3</math> in space if there exists some scalar function <math>Q = Q(x,y,z)</math> defined on <math>D</math> such that

:<math>dQ \equiv \left ( \frac{\partial Q}{\partial x} \right )_{y,z} dx + \left ( \frac{\partial Q}{\partial y} \right )_{z,x} dy + \left ( \frac{\partial Q}{\partial z} \right )_{x,y} dz,</math> {{pad|3em}} <math>dQ = A dx + B dy + C dz</math>

throughout D. This is equivalent to saying that the vector field <math>(A, B, C)</math> is a [[conservative vector field]], with corresponding potential <math>Q</math>.

===One dimension===
In one dimension, a differential form

:<math>A(x) \, dx</math>

is exact as long as <math>A</math> has an [[antiderivative]]; in this case let <math>Q</math> be the antiderivative of <math>A</math>. Otherwise, if <math>A</math> does ''not'' have an antiderivative, we cannot write <math>dQ = A(x) \, dx</math> and so the differential form is inexact.

===Two and three dimensions===
By [[symmetry of second derivatives]], for any "nice" (non-[[Pathological (mathematics)|pathological]]) function <math>Q</math> we have

:<math>\frac{\partial ^2 Q}{\partial x \partial y} = \frac{\partial ^2 Q}{\partial y \partial x}</math>

Hence, it follows that in a [[simply-connected]] region ''R'' of the ''xy''-plane, a differential

:<math>A(x, y)\,dx + B(x, y)\,dy</math>

is an exact differential [[if and only if]] the following holds:

:<math>\left( \frac{\partial A}{\partial y} \right)_x = \left( \frac{\partial B}{\partial x} \right)_y</math>

For three dimensions, a differential

:<math>dQ = A(x, y, z) \, dx + B(x, y, z) \, dy + C(x, y, z) \, dz</math>

is an exact differential in a simply-connected region ''R'' of the ''xyz''-coordinate system if between the functions ''A'', ''B'' and ''C'' there exist the relations:

:<math>\left( \frac{\partial A}{\partial y} \right)_{x,z} \!\!\!= \left( \frac{\partial B}{\partial x} \right)_{y,z}</math>   ''';'''   <math>\left( \frac{\partial A}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial x} \right)_{y,z}</math>   ''';'''   <math>\left( \frac{\partial B}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial y} \right)_{x,z}</math>

::Note: The subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the [[partial derivative]], these subscripts are not required, but they are included as a reminder.

These conditions are equivalent to the following one: If ''G'' is the graph of this vector valued function then for all tangent vectors ''X'',Y of the ''surface'' ''G'' then ''s''(''X'', ''Y'') = 0 with ''s'' the [[symplectic form]].

These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives. So, in order for a differential ''dQ'', that is a function of four variables to be an exact differential, there are six conditions to satisfy.

In summary, when a differential ''dQ'' is exact:

*the function ''Q'' exists;
*<math>\int_i^f dQ=Q(f)-Q(i),</math> independent of the path followed.

In [[thermodynamics]], when ''dQ'' is exact, the function ''Q'' is a state function of the system. The thermodynamic functions ''[[Internal energy|U]]'', ''[[Entropy|S]]'', ''[[Enthalpy|H]]'', ''[[Helmholtz free energy|A]]'' and ''[[Gibbs free energy|G]]'' are [[state function]]s. Generally, neither [[Work (thermodynamics)|work]] nor [[heat]] is a state function. An ''exact differential'' is sometimes also called a 'total differential', or a 'full differential', or, in the study of [[differential geometry]], it is termed an [[exact form]].

==Partial differential relations==
If three variables, <math>x</math>, <math>y</math> and <math>z</math> are bound by the condition <math>F(x,y,z) = \text{constant}</math> for some differentiable function <math>F(x,y,z)</math>, then the following [[total differential]]s exist<ref name="Cengel1998">{{cite book |last=Çengel |first=Yunus A. |authorlink= |coauthors=Boles, Michael A. |title=Thermodynamics - An Engineering Approach |origyear=1989 |edition=3rd |series=McGraw-Hill Series in [[Mechanical Engineering]] |year=1998 |publisher=McGraw-Hill |location=Boston, MA. |isbn=0-07-011927-9 |chapter=Thermodynamics Property Relations}}</ref>{{rp|667&669}}

:<math>d x = {\left ( \frac{\partial x}{\partial y} \right )}_z \, d y + {\left ( \frac{\partial x}{\partial z} \right )}_y \,dz</math>
:<math>d z = {\left ( \frac{\partial z}{\partial x} \right )}_y \, d x + {\left ( \frac{\partial z}{\partial y} \right )}_x \,dy.</math>

Substituting the first equation into the second and rearranging, we obtain<ref name="Cengel1998"/>{{rp|669}}

:<math>d z = {\left ( \frac{\partial z}{\partial x} \right )}_y \left [ {\left ( \frac{\partial x}{\partial y} \right )}_z d y + {\left ( \frac{\partial x}{\partial z} \right )}_y dz \right ] + {\left ( \frac{\partial z}{\partial y} \right )}_x dy,</math>
:<math>d z = \left [ {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial y} \right )}_z + {\left ( \frac{\partial z}{\partial y} \right )}_x \right ] d y + {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial z} \right )}_y dz,</math>
:<math>\left [ 1 - {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial z} \right )}_y \right ] dz = \left [ {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial y} \right )}_z + {\left ( \frac{\partial z}{\partial y} \right )}_x \right ] d y.</math>

Since <math>y</math> and <math>z</math> are independent variables, <math>d y</math> and <math>d z</math> may be chosen without restriction. For this last equation to hold in general, the bracketed terms must be equal to zero.<ref name="Cengel1998"/>{{rp|669}}

===Reciprocity relation===
Setting the first term in brackets equal to zero yields<ref name="Cengel1998"/>{{rp|60฿฿฿70}}

:<math>{\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial z} \right )}_y = 1.</math>

A slight rearrangement gives a reciprocity relation,<ref name="Cengel1998"/>{{rp|670}}

:<math>{\left ( \frac{\partial z}{\partial x} \right )}_y = \frac{1}{{\left ( \frac{\partial x}{\partial z} \right )}_y}.</math>

There are two more [[permutations]] of the foregoing derivation that give a total of three reciprocity relations between <math>x</math>, <math>y</math> and <math>z</math>. [[Inverse functions and differentiation|Reciprocity relations]] show that the inverse of a partial derivative is equal to its reciprocal.

===Cyclic relation===
The cyclic relation is also known as the cyclic rule or the [[Triple product rule]]. Setting the second term in brackets equal to zero yields<ref name="Cengel1998"/>{{rp|670}}

:<math>{\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial y} \right )}_z = - {\left ( \frac{\partial z}{\partial y} \right )}_x.</math>

Using a reciprocity relation for <math>\tfrac{\partial z}{\partial y}</math> on this equation and reordering gives a cyclic relation (the [[triple product rule]]),<ref name="Cengel1998"/>{{rp|670}}

:<math>{\left ( \frac{\partial x}{\partial y} \right )}_z {\left ( \frac{\partial y}{\partial z} \right )}_x {\left ( \frac{\partial z}{\partial x} \right )}_y = -1.</math>

If, ''instead'', a reciprocity relation for <math>\tfrac{\partial x}{\partial y}</math> is used with subsequent rearrangement, a [[Implicit function#Formula for two variables|standard form for implicit differentiation]] is obtained:

:<math>{\left ( \frac{\partial y}{\partial x} \right )}_z = - \frac { {\left ( \frac{\partial z}{\partial x} \right )}_y }{ {\left ( \frac{\partial z}{\partial y} \right )}_x }.</math>

== Some useful equations derived from exact differentials in two dimensions ==

(See also [[Bridgman's thermodynamic equations]] for the use of exact differentials in the theory of [[thermodynamic equations]])

Suppose we have five state functions <math>z,x,y,u</math>, and <math>v</math>. Suppose that the state space is two dimensional and any of the five quantities are exact differentials. Then by the [[chain rule]]

<math>(1)~~~~~
dz =
\left(\frac{\partial z}{\partial x}\right)_y dx+
\left(\frac{\partial z}{\partial y}\right)_x dy
=
\left(\frac{\partial z}{\partial u}\right)_v du
+\left(\frac{\partial z}{\partial v}\right)_u dv
</math>

but also by the chain rule:

<math>(2)~~~~~
dx =
\left(\frac{\partial x}{\partial u}\right)_v du
+\left(\frac{\partial x}{\partial v}\right)_u dv
</math>

and

<math>(3)~~~~~
dy=
\left(\frac{\partial y}{\partial u}\right)_v du
+\left(\frac{\partial y}{\partial v}\right)_u dv
</math>

so that:

<math>(4)~~~~~
dz =
\left[
\left(\frac{\partial z}{\partial x}\right)_y
\left(\frac{\partial x}{\partial u}\right)_v
+
\left(\frac{\partial z}{\partial y}\right)_x
\left(\frac{\partial y}{\partial u}\right)_v
\right]du
</math>

:::<math>+
\left[
\left(\frac{\partial z}{\partial x}\right)_y
\left(\frac{\partial x}{\partial v}\right)_u
+
\left(\frac{\partial z}{\partial y}\right)_x
\left(\frac{\partial y}{\partial v}\right)_u
\right]dv
</math>

which implies that:

<math>(5)~~~~~
\left(\frac{\partial z}{\partial u}\right)_v
=
\left(\frac{\partial z}{\partial x}\right)_y
\left(\frac{\partial x}{\partial u}\right)_v
+
\left(\frac{\partial z}{\partial y}\right)_x
\left(\frac{\partial y}{\partial u}\right)_v
</math>

Letting <math>v=y</math> gives:

<math>(6)~~~~~
\left(\frac{\partial z}{\partial u}\right)_y
=
\left(\frac{\partial z}{\partial x}\right)_y
\left(\frac{\partial x}{\partial u}\right)_y
</math>

Letting <math>u=y</math> gives:

<math>(7)~~~~~
\left(\frac{\partial z}{\partial y}\right)_v
=
\left(\frac{\partial z}{\partial y}\right)_x
+
\left(\frac{\partial z}{\partial x}\right)_y
\left(\frac{\partial x}{\partial y}\right)_v
</math>

Letting <math>u=y</math>, <math>v=z</math> gives:

<math>(8)~~~~~
\left(\frac{\partial z}{\partial y}\right)_x
= -
\left(\frac{\partial z}{\partial x}\right)_y
\left(\frac{\partial x}{\partial y}\right)_z
</math>

using (<math>\partial a/\partial b)_c = 1/(\partial
b/\partial a)_c</math> gives the [[triple product rule]]:

<math>(9)~~~~~
\left(\frac{\partial z}{\partial x}\right)_y
\left(\frac{\partial x}{\partial y}\right)_z
\left(\frac{\partial y}{\partial z}\right)_x
=-1
</math>

== See also ==
*[[Closed and exact differential forms]] for a higher-level treatment
*[[Differential (mathematics)]]
*[[Inexact differential]]
*[[Integrating factor]] for solving non-exact differential equations by making them exact
*[[Exact differential equation]]

== References ==
<references/>
*Perrot, P. (1998). ''A to Z of Thermodynamics.'' New York: Oxford University Press.
*Zill, D. (1993). ''A First Course in Differential Equations, 5th Ed.'' Boston: PWS-Kent Publishing Company.

==External links==
*[http://mathworld.wolfram.com/InexactDifferential.html Inexact Differential] – from Wolfram MathWorld
*[http://www.chem.arizona.edu/~salzmanr/480a/480ants/e&idiff/e&idiff.html Exact and Inexact Differentials] – University of Arizona
*[http://farside.ph.utexas.edu/teaching/sm1/lectures/node36.html Exact and Inexact Differentials] – University of Texas
*[http://mathworld.wolfram.com/ExactDifferential.html Exact Differential] – from Wolfram MathWorld

{{DEFAULTSORT:Exact Differential}}
[[Category:Thermodynamics]]
[[Category:Multivariable calculus]]

Parameter

2014-01-10T22:41:09Z

85.110.26.25: /* Mathematical functions */

[[Image:Airspeed p1230157.jpg|thumb|250px|Aircraft use pitot tubes to measure airspeed. The example, from an [[Airbus A380]], combines a pitot tube (pencil point shape) with a static port and an angle-of-attack vane (black). Air-flow, relative to this device, is right to left.]]
[[Image:Pitot tube types.svg|thumb|250px|right|Types of pitot tubes]]
[[Image:Pitot-cső helikopteren 4.jpg|thumb|250px|Pitot tube on [[Kamov Ka-26]] helicopter]]

A '''pitot''' {{IPAc-en|ˈ|p|iː|t|oʊ}} '''tube''' is a [[pressure measurement]] instrument used to measure [[fluid]] flow [[velocity]]. The pitot tube was invented by the [[France|French]] engineer [[Henri Pitot]] in the early 18th century<ref>{{cite journal|first= Henri |last=Pitot |year= 1732| title = Description d'une machine pour mesurer la vitesse des eaux courantes et le sillage des vaisseaux | journal = Histoire de l'Académie royale des sciences avec les mémoires de mathématique et de physique tirés des registres de cette Académie | pages = 363–376 | url = http://gallica.bnf.fr/ark:/12148/bpt6k35294.image.f543.langFR | format = PDF| accessdate = 2009-06-19}}</ref> and was modified to its modern form in the mid-19th century by French scientist [[Henry Darcy]].<ref>{{cite journal|first= Henry |last=Darcy |year= 1858| title = Note relative à quelques modifications à introduire dans le tube de Pitot | journal = Annales des Ponts et Chaussées | pages = 351–359 | url = http://gallica.bnf.fr/ark:/12148/bpt6k408489d.image.f354 | format = PDF| accessdate = 2009-07-31}}</ref> It is widely used to determine the [[airspeed]] of an [[aircraft]], water speed of a boat, and to measure liquid, air and gas velocities in industrial applications. The pitot tube is used to measure the local velocity at a given point in the flow stream and not the average velocity in the pipe or conduit.<ref>{{cite book|first= C.J.|last=Geankoplis |year=2003|title=Transport processes and separation process principles (includes unit operations)|edition=4th |publisher=Prentice Hall |location=New Jersey}}</ref>

==Theory of operation==
The basic pitot tube consists of a tube pointing directly into the fluid flow. As this tube contains fluid, a pressure can be measured; the moving fluid is brought to rest (stagnates) as there is no outlet to allow flow to continue. This pressure is the [[stagnation pressure]] of the fluid, also known as the total pressure or (particularly in aviation) the pitot pressure.

The measured stagnation pressure cannot itself be used to determine the fluid velocity (airspeed in aviation). However, [[Bernoulli's equation]] states:

<dl><dd>Stagnation pressure = static pressure + dynamic pressure</dd></dl>

Which can also be written

:<math>p_t = p_s + \left(\frac{\rho V^2}{2}\right)</math>

Solving that for velocity we get:

:<math>V = \sqrt{\frac{2 (p_t - p_s)}{\rho}}</math>

'''NOTE:''' The above equation applies ONLY to fluids that can be treated as incompressible. Liquids are treated as incompressible under almost all conditions. Gases under certain conditions can be approximated as incompressible. See [[Compressibility]].

where:
* <math>V</math> is fluid velocity;
* <math>p_t</math> is stagnation or total pressure;
* <math>p_s</math> is static pressure;
* and <math>\rho</math> is fluid density.

The value for the pressure drop <math>p_2</math> – <math>p_1</math> or <math>\Delta p</math> due to <math>\Delta h</math>, the reading on the manometer:

:<math>\Delta p = \rho g \Delta h</math>

Where:
* <math>\rho</math> is the density of the fluid in the manometer
* <math>\Delta h</math> is the manometer reading

The [[dynamic pressure]], then, is the difference between the stagnation pressure and the [[static pressure]]. The static pressure is generally measured using the [[static port]]s on the side of the fuselage. The dynamic pressure is then determined using a diaphragm inside an enclosed container. If the air on one side of the diaphragm is at the static pressure, and the other at the stagnation pressure, then the deflection of the diaphragm is proportional to the dynamic pressure, which can then be used to determine the [[indicated airspeed]] of the aircraft. The diaphragm arrangement is typically contained within the [[airspeed indicator]], which converts the dynamic pressure to an airspeed reading by means of mechanical levers.

Instead of separate pitot and static ports, a pitot-static tube (also called a [[Prandtl]] tube) may be employed, which has a second tube coaxial with the pitot tube with holes on the sides, outside the direct airflow, to measure the static pressure.<ref>[http://books.google.com/books?id=0SkDAAAAMBAJ&pg=PA116&dq=Popular+Science+The+lads+who+fly+Britain%27s+1944&hl=en&sa=X&ei=UHyfT6aEIYOs9ATwh8C1AQ&ved=0CDkQ6AEwAA#v=onepage&q&f=true "How Aircraft Instruments Work."] ''Popular Science'', March 1944, pp. 116.</ref>

==Operation==
Pitot tubes on aircraft commonly have heating elements called pitot heat to prevent the tube from becoming clogged with ice. The failure of these systems can have catastrophic consequences, as in the case of [[Austral Líneas Aéreas Flight 2553]], [[Birgenair Flight 301]] (investigators suspected that some kind of insect could have created a nest inside the pitot tube: the prime suspect is a species called the [[black and yellow mud dauber]] wasp), [[Northwest Airlines Flight 6231]], [[Aeroperú Flight 603]] (blocked static port), and of one [[X-31]].<ref>[http://www.nasa.gov/centers/dryden/news/NewsReleases/1995/95-33_pf.html NASA Dryden news releases. (1995)]</ref> The French air safety authority [[Bureau d'Enquêtes et d'Analyses pour la Sécurité de l'Aviation Civile|BEA]] said that pitot tube icing was a contributing factor in the crash of [[Air France Flight 447]] into the [[Atlantic Ocean]].<ref>{{cite web |url=http://www.reuters.com/article/2012/07/05/uk-airfrance-crash-idUSLNE86402T20120705 |title=Training flaws exposed in Rio-Paris crash report |date=5 July 2012 |publisher=Reuters |accessdate=5 October 2012}}</ref>
In 2008 [[Air Caraïbes]] reported two incidents of [[pitot tube]] icing malfunctions on its A330s.<ref>{{cite web |url= http://www.flightglobal.com/news/articles/air-caraibes-atlantique-memo-details-pitot-icing-incidents-327738/ |title=Air Caraibes Atlantique memo details pitot icing incidents |first=Kieran |last=Daly |work=Flight International |date=11 June 2009 |accessdate=19 February 2012}}</ref>

== Industry applications ==
[[Image:FA18pitot.png|thumb|250px|Pitot tube from an [[F/A-18]]]]

In industry, the velocities being measured are often those flowing in ducts and tubing where measurements by an [[anemometer]] would be difficult to obtain. In these kinds of measurements, the most practical instrument to use is the pitot tube. The pitot tube can be inserted through a small hole in the duct with the pitot connected to a U-tube [[water gauge]] or some other differential [[pressure gauge]] for determining the velocity inside the ducted wind tunnel. One use of this technique is to determine the volume of air that is being delivered to a conditioned space.

The fluid flow rate in a duct can then be estimated from:

:Volume flow rate (cubic feet per minute) = duct area (square feet) × velocity (feet per minute)
:Volume flow rate (cubic meters per second) = duct area (square meters) × velocity (meters per second)

In aviation, airspeed is typically measured in [[knot (unit)|knot]]s.

==See also==
{{col-begin}}{{col-break}}
*[[Air data boom]]
*[[Altimeter]]
*[[Annubar]]
*[[Anti-icing]]
*[[Atmospheric icing]]
*[[Calibrated airspeed]] {{nb10}}
{{col-break}}
*[[Deicing]]
*[[Flow measurement]] {{nb10}}
*[[Gyrocompass]]
*[[Icing conditions]]
*[[Kiel probe]]
*[[Mach number]]
{{col-break}}
*[[Piezometer]]
*[[Pitot-static system]]
*[[Position error]]
*[[Stagnation pressure]]
*[[True airspeed]]
{{col-end}}

==References==
'''Notes'''
{{reflist}}

'''Bibliography'''
{{refbegin}}
*{{cite book|last=Kermode|first=A.C.|others=Barnard, R.H. (Ed.) and Philpott, D.R. (Ed.)|title=Mechanics of Flight|edition=10th|origyear=1972|year=1996|publisher=Prentice Hall|isbn=0-582-23740-8|pages=63–67}}
*{{cite book|last=Pratt|first=Jeremy M.|edition=3rd|title=The Private Pilot's Licence Course: Principles of Flight, Aircraft General Knowledge, Flight Performance and Planning|origyear=1997|year=2005|isbn=1-874783-23-3|pages=gen108-gen111|nopp=true}}
*{{cite book|author=Tietjens, O.G.|title=Applied Hudro- and Aeromechanics, based on lectures of L. Prandtl, Ph.D| publisher=Dove Publications, Inc. | year=1934 | isbn=0-486-60375-X | pages=226–239}}
*{{cite book|last=Saleh |first=J.M. |year=2002 |title=Fluid Flow Handbook |publisher=McGraw-Hill Professional}}
{{refend}}

==External links==
{{Commons|Pitot tube}}
*[http://www.youtube.com/watch?v=D6sbzkYq3_c 3D animation of the Pitot Tube Differential Pressure Flow Measuring Principle]
*[http://www.wired.com/autopia/2009/06/airspeed-sensors/ How 18th Century Technology Could Down an Airliner] (wired.com)



{{Flight instruments}}

{{DEFAULTSORT:Pitot Tube}}
[[Category:Flow meters]]
[[Category:Aircraft instruments]]
[[Category:Speed sensors]]
[[Category:Measuring instruments]]
[[Category:French inventions]]

{{Link FA|de}}

Symbolic computation

2014-01-09T19:13:22Z

85.110.80.94:

{{infobox graph
| name = Harries graph
| image = [[Image:Harries_graph.svg|220px]]
| image_caption = The Harries graph
| namesake =
| vertices = 70
| edges = 105
| automorphisms = 120 ([[Symmetric group|''S''5]])
| girth = 10
| diameter = 6
| radius = 6
| chromatic_number = 2
| chromatic_index = 3
| properties = [[Cubic graph|Cubic]] [[Cage (graph theory)|Cage]] [[Triangle-free graph|Triangle-free]] [[Hamiltonian graph|Hamiltonian]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Harries graph''' or '''Harries (3-10)-cage''' is a 3-[[regular graph|regular]] [[undirected graph]] with 70 vertices and 105 edges.<ref> {{MathWorld|urlname=HarriesGraph|title=Harries Graph}}</ref>

The Harries graph has [[chromatic number]] 2, [[chromatic index]] 3, radius 6, diameter 6, girth 10 and is [[Hamiltonian graph|Hamiltonian]]. It is also a 3-[[k-vertex-connected graph|vertex-connected]] and 3-[[k-edge-connected graph|edge-connected]] [[planar graph|non-planar]] [[cubic graph]].

The [[characteristic polynomial]] of the Harries graph is

: <math>(x-3) (x-1)^4 (x+1)^4 (x+3) (x^2-6) (x^2-2) (x^4-6x^2+2)^5 (x^4-6x^2+3)^4 (x^4-6x^2+6)^5. \, </math>

==History==
In 1972, A. T. Balaban published a (3-10)-[[cage graph]], a cubic graph that has as few vertices as possible for girth 10.<ref>A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1-5. 1972.</ref> It was the first (3-10)-cage discovered but it was not unique.<ref>Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [http://citeseer.ist.psu.edu/448980.html].</ref>

The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong in 1980.<ref>M. O'Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory Ser. B 29 (1980) 91-105.</ref> There exist three distinct (3-10)-cage graphs—the [[Balaban 10-cage]], the Harries graph and the [[Harries–Wong graph]].<ref>Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.</ref> Moreover, the Harries–Wong graph and Harries graph are [[Spectral graph theory|cospectral graphs]].

==Gallery==
<gallery>
Image:Harries graph 2COL.svg|The chromatic number of the Harries graph is 2.
Image:Harries graph 3color edge.svg|The chromatic index of the Harries graph is 3.
Image:harries_graph_alternative_drawing.svg|Alternative drawing of the Harries graph.
Image:Harries graph petersen drawing.jpg|Alternative drawing emphasizing the graph's 4 orbits.

</gallery>

== References ==
{{reflist}}

[[Category:Individual graphs]]
[[Category:Regular graphs]]

Vis-viva equation

2014-01-06T17:10:04Z

85.110.65.58: /* Derivation */

In the mathematical discipline of [[model theory]], the '''Ehrenfeucht–Fraïssé game''' (also called back-and-forth games)
is a technique for determining whether two [[structure (mathematical logic)|structure]]s
are [[elementarily equivalent]]. The main application of Ehrenfeucht–Fraïssé games is in proving the inexpressibility of certain properties in first-order logic. Indeed, Ehrenfeucht–Fraïssé games provide a complete methodology for proving inexpressibility results for first-order logic. In this role, these games are of particular importance in [[finite model theory]] and its applications in computer science (specifically [[Computer Aided Verification]] and [[database theory]]), since Ehrenfeucht–Fraïssé games are one of the such few techniques from model theory that remain valid in the context of finite models. Other widely used techniques for proving inexpressibility results, such as the [[compactness theorem]], do not work in finite models.

Ehrenfeucht–Fraïssé like games can also be defined for other logics, e.g. [[pebble game]]s for finite variable logics and [[fixpoint logic]]s.

== Main idea ==
The main idea behind the game is that we have two structures, and two players (defined below). One of the players wants to show that the two structures are different, whereas the other player wants to show that they are somewhat similar (according to first-order logic). The game is played in turns and rounds; A round proceeds as follows: First the first player (Spoiler) chooses any element from one of the structures, and the other player chooses an element from the other structure. The other player's task is to always pick an element that is "similar" to the one that Spoiler chose. The second player (Duplicator) wins if there exists an isomorphism between the elements chosen in the two different structures.

The game lasts for a fixed amount of steps (<math>\gamma</math>) (an ordinal, but usually a finite number or <math>\omega</math>).

== Definition ==
Suppose that we are given two structures <math>\mathfrak{A}</math>
and <math>\mathfrak{B}</math>, each with no [[function (mathematics)|function]] symbols and the same set of [[Relation (mathematics)|relation]] symbols,
and a fixed [[natural number]] ''n''. We can then define the Ehrenfeucht–Fraïssé
game <math>G_n(\mathfrak{A},\mathfrak{B})</math> to be a game between two players, Spoiler and Duplicator,
played as follows:
# The first player, Spoiler, picks either a member <math>a_1</math> of <math>\mathfrak{A}</math> or a member <math>b_1</math> of <math>\mathfrak{B}</math>.
# If Spoiler picked a member of <math>\mathfrak{A}</math>, Duplicator picks a member <math>b_1</math> of <math>\mathfrak{B}</math>; otherwise, Duplicator picks a member <math>a_1</math> of <math>\mathfrak{A}</math>.
# Spoiler picks either a member <math>a_2</math> of <math>\mathfrak{A}</math> or a member <math>b_2</math> of <math>\mathfrak{B}</math>.
# Duplicator picks an element <math>a_2</math> or <math>b_2</math> in the model from which Spoiler did not pick.
# Spoiler and Duplicator continue to pick members of <math>\mathfrak{A}</math> and <math>\mathfrak{B}</math> for <math>n-2</math> more steps.
# At the conclusion of the game, we have chosen distinct elements <math>a_1, \dots, a_n</math> of <math>\mathfrak{A}</math> and <math>b_1, \dots, b_n</math> of <math>\mathfrak{B}</math>. We therefore have two structures on the set <math>\{1, \dots,n\}</math>, one induced from <math>\mathfrak{A}</math> via the map sending <math>i</math> to <math>a_i</math>, and the other induced from <math>\mathfrak{B}</math> via the map sending <math>i</math> to <math>b_i</math>. Duplicator wins if these structures are the same; Spoiler wins if they are not.

It is easy to prove that if Duplicator wins this game for all ''n'', then <math>\mathfrak{A}</math> and <math>\mathfrak{B}</math> are elementarily equivalent. If the set of relation symbols being considered is finite, the converse is also true.

== History ==

The [[back-and-forth method]] used in the Ehrenfeucht–Fraïssé game to verify elementary equivalence was given by [[Roland Fraïssé]]
in his thesis;<ref>''Sur une nouvelle classification des systèmes de relations'', Roland Fraïssé, ''Comptes Rendus'' '''230''' (1950), 1022–1024.</ref><ref>''Sur quelques classifications des systèmes de relations'', Roland Fraïssé, thesis, Paris, 1953;
published in ''Publications Scientifiques de l'Université d'Alger'', series A '''1''' (1954), 35–182.</ref>
it was formulated as a game by [[Andrzej Ehrenfeucht]].<ref>An application of games to the completeness problem for formalized theories, A. Ehrenfeucht, ''Fundamenta Mathematicae'' '''49''' (1961), 129–141.</ref> The names Spoiler and Duplicator are due to [[Joel Spencer]].<ref>[http://plato.stanford.edu/entries/logic-games/ Stanford Encyclopedia of Philosophy, entry on Logic and Games.]</ref> Other usual names are Eloise [sic] and Abelard (and often denoted by <math>\exists</math> and <math>\forall</math>) after [[Heloise and Abelard]], a naming scheme introduced by [[Wilfrid Hodges]] in his book ''Model Theory''.

== Further reading ==

Chapter 1 of [[Bruno Poizat|Poizat]]'s model theory text<ref>''A Course in Model Theory'', Bruno Poizat, tr. Moses Klein, New York:
Springer-Verlag, 2000.</ref> contains an introduction to the Ehrenfeucht–Fraïssé game, and so do Chapters 6, 7, and 13 of Rosenstein's book on [[Linear ordering|linear orders]].<ref>''Linear Orderings'', Joseph G. Rosenstein, New York: Academic Press, 1982.</ref> A simple example of the Ehrenfeucht–Fraïssé game is given in .<ref>[http://www.maa.org/mathland/mathtrek_3_19_01.html Example of the Ehrenfeucht-Fraïssé game.]</ref>

Phokion Kolaitis' slides<ref>[http://www.cs.ucsc.edu/~kolaitis/talks/essllif.ps Course on combinatorial games in finite model theory by Phokion Kolaitis (.ps file)]</ref> and [[Neil Immerman]]'s book chapter<ref name="Immerman1999">{{cite book|author=Neil Immerman|title=Descriptive complexity|url=http://books.google.com/books?id=kWSZ0OWnupkC&pg=PA91|year=1999|publisher=Springer|isbn=978-0-387-98600-5}}, chapter 6</ref> on Ehrenfeucht–Fraïssé games discuss applications in computer science, the methodology theorem for proving inexpressibility results, and several simple inexpressibility proofs using this methodology.

== References ==
<references />

==External links==
* [http://www.math.cornell.edu/~mec/Summer2009/Raluca/index.html Six Lectures Ehrenfeucht-Fraïssé games] at MATH EXPLORERS' CLUB, Cornell Department of Mathematics.

{{DEFAULTSORT:Ehrenfeucht-Fraisse Game}}
[[Category:Model theory]]

Talk:Three-phase electric power

2013-12-03T06:04:22Z

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