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Sylow theorems

2014-02-24T17:54:12Z

128.101.152.31: /* Theorems */ Fixed run-on sentence

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Jackson network

2014-02-19T16:22:28Z

128.101.152.60: supplying a conspicuously missing comma

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Conformal map

2014-02-10T23:39:04Z

128.101.152.75:

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Major index

2014-02-07T23:06:29Z

128.101.152.69:

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Kronecker limit formula

2014-01-28T20:09:21Z

128.101.152.60:

In [[mathematics]], in the theory of [[integrable systems]], a '''Lax pair''' is a pair of time-dependent matrices or [[operator (mathematics)|operator]]s that satisfy a corresponding [[differential equation]], called the ''Lax equation''. Lax pairs were introduced by [[Peter Lax]] to discuss [[soliton]]s in [[continuous media]]. The [[inverse scattering transform]] makes use of the Lax equations to solve such systems.

==Definition==
A Lax pair is a pair of matrices or operators <math>L(t), P(t)</math> dependent on time and acting on a fixed [[Hilbert space]], and satisfying '''Lax's equation''':

:<math>\frac{dL}{dt}=[P,L]</math>

where <math>[P,L]=PL-LP</math> is the [[commutator]].
Often, as in the example below, <math>P</math> depends on <math>L</math> in a prescribed way, so this is a nonlinear equation for <math>L</math> as a function of <math>t</math>.

==Isospectral property==
It can then be shown that the [[eigenvalue]]s and more generally the [[Operator spectrum|spectrum]] of ''L'' are independent of ''t''. The matrices/operators ''L'' are said to be ''[[isospectral]]'' as <math>t</math> varies.

The core observation is that the matrices <math>L(t)</math> are all similar by virtue of

:<math>L(t)=U(t,s) L(s) U(t,s)^{-1}</math>

where <math>U(t,s)</math> is the solution of the [[Cauchy problem]]

:<math> \frac{d}{dt} U(t,s) = P(t) U(t,s), \qquad U(s,s) = I,</math>

where ''I'' denotes the identity matrix. Note that if ''L(t)'' is [[self-adjoint]] and ''P(t)'' is [[skew-adjoint]], then ''U(t,s)'' will be [[unitary operator|unitary]].

In other words, to solve the eigenvalue problem ''Lψ = λψ'' at time ''t'', it is possible to solve the same problem at time 0 where L is generally known better, and to propagate the solution with the following formulas:
:<math>\lambda(t)=\lambda(0)</math> (no change in spectrum)
:<math>\frac{\partial \psi}{\partial t}=P \psi.</math>

=== Link with the inverse scattering method ===
The above property is the basis for the inverse scattering method. In this method, ''L'' and ''P'' act on a [[functional space]] (thus ''ψ = ψ(t,x)''), and depend on an unknown function ''u(t,x)'' which is to be determined. It is generally assumed that ''u(0,x)'' is known, and that ''P'' does not depend on ''u'' in the scattering region where <math>\Vert x \Vert\to \infty</math>.
The method then takes the following form:
# Compute the spectrum of <math>L(0)</math>, giving <math>\lambda</math> and <math>\psi(0,x)</math>,
# In the scattering region where <math>P</math> is known, propagate <math>\psi</math> in time by using <math>\frac{\partial \psi}{\partial t}(t,x)=P \psi(t,x)</math> with initial condition <math>\psi(0,x)</math>,
# Knowing <math>\psi</math> in the scattering region, compute <math>L(t)</math> and/or <math>u(t,x)</math>.

==Example==
The [[Korteweg–de Vries equation]] is
:<math>u_t=6uu_x-u_{xxx}.\,</math>
It can be reformulated as the Lax equation
:<math>L_t=[P,L]\,</math>
with
:<math>L=-\partial_{x}^2+u\,</math> (a [[Sturm–Liouville operator]])
:<math>P= -4\partial_{x}^3+3(u\partial_{x}+\partial_{x} u)\,</math>
where all derivatives act on all objects to the right. This accounts for the infinite number of first integrals of the KdV equation.

==Equations with a Lax pair==
Further examples of systems of equations that can be formulated as a Lax pair include:

* [[Benjamin–Ono equation]]
* One dimensional cubic [[non-linear Schrödinger equation]]
* [[Davey-Stewartson system]]
* [[Kadomtsev–Petviashvili equation]]
* [[Korteweg–de Vries equation]]
* [[KdV hierarchy]]
* [[Modified Korteweg-de Vries equation]]
* [[Sine-Gordon equation]]
* [[Toda lattice]]

==References==
* {{citation|first=P.|last= Lax|title=Integrals of nonlinear equations of evolution and solitary waves|journal=Comm. Pure Applied Math.|volume=21|year=1968|pages= 467–490|doi=10.1002/cpa.3160210503|issue=5 }} [http://archive.org/details/integralsofnonli00laxp archive]
* P. Lax and R.S. Phillips, ''Scattering Theory for Automorphic Functions'', (1976) Princeton University Press.

[[Category:Differential equations]]
[[Category:Automorphic forms]]
[[Category:Spectral theory]]
[[Category:Exactly solvable models]]

Incomplete gamma function

2014-01-28T20:07:45Z

128.101.152.60: fixing some incorrect capitals and some punctuation errors

[[File:Carbon lattice diamond.png|thumb|The [[Diamond cubic|diamond crystal structure]] belongs to the face-centered [[Cubic crystal system|cubic lattice]], with a repeated 2-atom pattern.]]
In [[crystallography]], the terms '''crystal system''', '''crystal family''', and '''lattice system''' each refer to one of several classes of [[space group]]s, [[Bravais lattice|lattice]]s, [[point group]]s, or [[crystal]]s. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.

Crystal systems, crystal families, and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the [[trigonal crystal system]] is often confused with the [[rhombohedral lattice system]], and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".

[[Space group]]s and crystals are divided into 7 crystal systems according to their [[point group]]s, and into 7 lattice systems according to their [[Bravais lattice]]s. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems.
The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.

==Overview==
[[Image:Hanksite.JPG|thumb|Hexagonal [[hanksite]] crystal, with three-fold c-axis symmetry]]
A '''lattice system''' is a class of lattices with the same point group. In three dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. The lattice system of a crystal or space group is determined by its lattice but not always by its point group.

A '''crystal system''' is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system,
and in these cases the crystal system corresponds to a lattice system and is given the same name. However, for the five point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.

A ''' crystal family''' also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.

In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called oblique, rectangular, square, and hexagonal.

The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table:
{|class="wikitable" cellpadding=0 style="margin: 1em auto; text-align: center;"
|-
![[Crystal family]]
!Crystal system
!Required symmetries of point group
![[Crystallographic point group|Point groups]]
![[Space group]]s
![[Bravais lattice]]s
![[Lattice system]]
|-
|colspan=2|[[Triclinic]]
|None
|2
|2
|1
|[[Triclinic]]
|-
|colspan=2|[[Monoclinic]]
|1 twofold [[Rotational symmetry|axis of rotation]] or 1 [[Reflection symmetry|mirror plane]]
|3
|13
|2
|[[Monoclinic]]
|-
|colspan=2|[[Orthorhombic]]
| 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.
|3
|59
|4
|[[Orthorhombic]]
|-
|colspan=2|[[Tetragonal]]
| 1 fourfold axis of rotation
|7
|68
|2
|[[Tetragonal]]
|-
|rowspan=3|[[Hexagonal crystal family|Hexagonal]]
|rowspan=2|[[Trigonal]]
|rowspan=2|1 threefold axis of rotation
|rowspan=2|5
|7
|1
|[[Rhombohedral lattice system|Rhombohedral]]
|-
|18
|rowspan=2|1
|rowspan=2|[[Hexagonal lattice system|Hexagonal]]
|-
|[[hexagonal crystal system|Hexagonal]]
|1 sixfold axis of rotation
|7
|27
|-
|colspan=2|[[cubic crystal system|Cubic]]
|4 threefold axes of rotation
|5
|36
|3
|[[cubic crystal system|Cubic]]
|- bgcolor=#e0e0e0
|'''Total:''' 6
|7
|
|32
|230
|14
|7
|}

==Crystal classes==
The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 [[crystallographic point group]]s) as shown in the following table:

{| class=wikitable
|-
! crystal family
! crystal system
! [[point group]] / crystal class
! [[Schönflies notation|Schönflies]]
! [[Hermann–Mauguin notation|Hermann-Mauguin]]
! [[Orbifold notation|Orbifold]]
! [[Coxeter notation|Coxeter]]
! Point symmetry
! [[Symmetry number|Order]]
! [[Group_theory#Abstract_groups|Abstract group]]
|-
| rowspan=2 colspan=2| [[triclinic crystal system|triclinic]]
| triclinic-pedial
| C1
| 1
| 11
| [ ]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 1
| trivial <math>\mathbb{Z}_1</math>
|-
| triclinic-pinacoidal
| Ci
| {{overline|1}}
| 1x
| [2,1+]
| [[centrosymmetric]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]]
| monoclinic-sphenoidal
| C2
| 2
| 22
| [2,2]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| monoclinic-domatic
| Cs
| m
| *11
| [ ]
| [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| monoclinic-[[prism (geometry)|prismatic]]
| C2h
| 2/m
| 2*
| [2,2+]
| [[centrosymmetric]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]]
| orthorhombic-sphenoidal
| D2
| 222
| 222
| [2,2]+
| [[Chirality (chemistry)|enantiomorphic]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| orthorhombic-[[Pyramid (geometry)|pyramidal]]
| C2v
| mm2
| *22
| [2]
| [[Polar point group|polar]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| orthorhombic-[[bipyramid]]al
| D2h
| mmm
| *222
| [2,2]
| [[centrosymmetric]]
| 8
| <math>\mathbb{V}\times\mathbb{Z}_2</math>
|-
| rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]]
| tetragonal-pyramidal
| C4
| 4
| 44
| [4]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-disphenoidal
| S4
| {{overline|4}}
| 2x
| [2+,2]
| [[non-centrosymmetric]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-dipyramidal
| C4h
| 4/m
| 4*
| [2,4+]
| [[centrosymmetric]]
| 8
| <math>\mathbb{Z}_4\times\mathbb{Z}_2</math>
|-
| tetragonal-trapezoidal
| D4
| 422
| 422
| [2,4]+
| [[Chirality (chemistry)|enantiomorphic]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-pyramidal
| C4v
| 4mm
| *44
| [4]
| [[Polar point group|polar]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| tetragonal-scalenoidal
| D2d
| {{overline|4}}2m or {{overline|4}}m2
| 2*2
| [2+,4]
| [[non-centrosymmetric]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-dipyramidal
| D4h
| 4/mmm
| *422
| [2,4]
| [[centrosymmetric]]
| 16
| <math>\mathbb{D}_8\times\mathbb{Z}_2</math>
|-
| rowspan=12|[[hexagonal crystal family|hexagonal]] || rowspan=5 | [[trigonal crystal system|trigonal]]
| trigonal-pyramidal
| C3
| 3
| 33
| [3]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 3
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math>
|-
| rhombohedral
| S6 (C3i)
| {{overline|3}}
| 3x
| [2+,3+]
| [[centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-trapezoidal
| D3
| 32 or 321 or 312
| 322
| [3,2]+
| [[Chirality (chemistry)|enantiomorphic]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-pyramidal
| C3v
| 3m or 3m1 or 31m
| *33
| [3]
| [[Polar point group|polar]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-scalahedral
| D3d
| {{overline|3}}m or {{overline|3}}m1 or {{overline|3}}1m
| 2*3
| [2+,6]
| [[centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| rowspan=7 | [[Hexagonal crystal system|hexagonal]]
| hexagonal-pyramidal
| C6
| 6
| 66
| [6]+
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-dipyramidal
| C3h
| {{overline|6}}
| 3*
| [2,3+]
| [[non-centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| hexagonal-dipyramidal
| C6h
| 6/m
| 6*
| [2,6+]
| [[centrosymmetric]]
| 12
| <math>\mathbb{Z}_6\times\mathbb{Z}_2</math>
|-
| hexagonal-trapezoidal
| D6
| 622
| 622
| [2,6]+
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-pyramidal
| C6v
| 6mm
| *66
| [6]
| [[Polar point group|polar]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-dipyramidal
| D3h
| {{overline|6}}m2 or {{overline|6}}2m
| *322
| [2,3]
| [[non-centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-dipyramidal
| D6h
| 6/mmm
| *622
| [2,6]
| [[centrosymmetric]]
| 24
| <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math>
|-
| rowspan=5 colspan=2 | [[cubic crystal system|cubic]]
| tetrahedral
| T || 23
| 332
| [3,3]+
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[alternating group|alternating]] <math>\mathbb{A}_4</math>
|-
| hextetrahedral
| Td
| {{overline|4}}3m
| *332
| [3,3]
| [[non-centrosymmetric]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| diploidal
| Th
| m{{overline|3}}
| 3*2
| [3+,4]
| [[centrosymmetric]]
| 24
| <math>\mathbb{A}_4\times\mathbb{Z}_2</math>
|-
| gyroidal
| O
| 432
| 432
| [4,3]+
| [[Chirality (chemistry)|enantiomorphic]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hexoctahedral
| Oh
| m{{overline|3}}m
| *432
| [4,3]
| [[centrosymmetric]]
| 48
| <math>\mathbb{S}_4\times\mathbb{Z}_2</math>
|}

Point symmetry can be thought of in the following fashion: consider the coordinates which make up the structure, and project them all through a single point, so that (x,y,z) becomes (-x,-y,-z). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is '''''centrosymmetric'''''. Otherwise it is '''''non-centrosymmetric'''''. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure. This is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral (enantiomorphic) and its symmetry group is '''''enantiomorphic'''''.<ref>{{cite journal|author=Howard D. Flack|year=2003|title=Chiral and Achiral Crystal Structures|journal=Helvetica Chimica Acta |volume=86|pages= 905–921|doi=10.1002/hlca.200390109}}</ref>

A direction is called polar if its two directional senses are geometrically or physically different. A polar symmetry direction of a crystal is called a polar axis.<ref>E. Koch , W. Fischer , U. Müller , in ‘International Tables for Crystallography, Vol. A, Space-Group Symmetry’, 5th edn., Ed. T. Hahn, Kluwer Academic Publishers, Dordrecht, 2002, Chapt. 10, p. 804.</ref> Groups containing a polar axis are called '''''[[polar point group|polar]]'''''. A polar crystal possess a "unique" axis (found in no other directions) such that some geometrical or physical property is different at the two ends of this axis. It may develop a [[Polarization density|dielectric polarization]], e.g. in [[Pyroelectricity|pyroelectric crystals]]. A polar axis can occur only in non-centrosymmetric structures. There should also not be a mirror plane or 2-fold axis perpendicular to the polar axis, because they will make both directions of the axis equivalent.

The [[crystal structure]]s of chiral biological molecules (such as [[protein]] structures) can only occur in the 11 [[Chirality (chemistry)|enantiomorphic]] point groups (biological molecules are usually [[Chirality (chemistry)|chiral]]).


==Lattice systems ==
The distribution of the 14 Bravais lattice types into 7 lattice systems is given in the following table.

{| class=wikitable
|align=center|'''The 7 lattice systems'''
|colspan=4 align=center| '''The 14 Bravais Lattices'''
|-
|colspan=1 align=center| [[triclinic crystal system|triclinic]] ([[parallelepiped]])
|| [[Image:Triclinic.svg|80px|Triclinic]]
|-
|rowspan=2 align=center| [[monoclinic crystal system|monoclinic]] (right [[prism (geometry)|prism]] with [[parallelogram]] base; here seen from above)
|align=center| simple
|align=center| base-centered
|-
|| [[Image:Monoclinic.svg|80px|Monoclinic, simple]]
|| [[Image:Monoclinic-base-centered.svg|80px|Monoclinic, centered]]
|-
|rowspan=2 align=center| [[orthorhombic crystal system|orthorhombic]] ([[cuboid]])
|align=center| simple
|align=center| base-centered
|align=center| body-centered
|align=center| face-centered
|-
|| [[Image:Orthorhombic.svg|80px|Orthohombic, simple]]
|| [[Image:Orthorhombic-base-centered.svg|80px|Orthohombic, base-centered]]
|| [[Image:Orthorhombic-body-centered.svg|80px|Orthohombic, body-centered]]
|| [[Image:Orthorhombic-face-centered.svg|80px|Orthohombic, face-centered]]
|-
|rowspan=2 align=center| [[tetragonal crystal system|tetragonal]] (square [[cuboid]])
|align=center|simple
|align=center| body-centered
|-
|| [[Image:Tetragonal.svg|80px|Tetragonal, simple]]
|| [[Image:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]]
|-
|align=center| [[rhombohedral lattice system|rhombohedral]] ([[trigonal trapezohedron]])
| [[Image:Rhombohedral.svg|80px|Rhombohedral]]
|-
|align=center| [[Hexagonal crystal system|hexagonal]] (centered regular [[hexagon]])
| [[Image:Hexagonal lattice.svg|80px|Hexagonal]]
|-
|rowspan=2 align=center| [[Cubic crystal system|cubic]] (isometric; [[cube]])
|align=center| simple
|align=center| body-centered
|align=center| face-centered
|-
|| [[Image:Cubic.svg|80px|Cubic, simple]]
| [[Image:Cubic-body-centered.svg|80px|Cubic, body-centered]]
| [[Image:Cubic-face-centered.svg|80px|Cubic, face-centered]]
|}
{{-}}
In [[geometry]] and [[crystallography]], a '''Bravais lattice''' is a category of [[symmetry group]]s for [[translational symmetry]] in three directions, or correspondingly, a category of translation [[Lattice (group)|lattice]]s.

Such symmetry groups consist of translations by vectors of the form

:<math>\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math>

where ''n''1, ''n''2, and ''n''3 are [[integer]]s and ''a''1, ''a''2, and ''a''3 are three non-coplanar vectors, called ''primitive vectors''.

These lattices are classified by [[space group]] of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.

All crystalline materials must, by definition fit in one of these arrangements (not including [[quasicrystal]]s).

For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3 or 4 larger than the [[primitive cell]]. Depending on the symmetry of a crystal or other pattern, the [[fundamental domain]] is again smaller, up to a factor 48.

The Bravais lattices were studied by [[Moritz Ludwig Frankenheim]] (1801–1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by [[Auguste Bravais|A. Bravais]] in 1848.

==Crystal systems in four-dimensional space==

The four-dimensional unit cell is defined by four edge lengths (<math>a, b, c, d</math>) and six interaxial angles (<math>\alpha, \beta, \gamma, \delta, \epsilon, \zeta</math>). The following conditions for the lattice parameters define 23 crystal families:

1 Hexaclinic: <math>a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne \delta \ne \epsilon \ne \zeta \ne 90 ^\circ</math>

2 Triclinic: <math>a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne 90 ^\circ, \delta = \epsilon = \zeta = 90 ^\circ</math>

3 Diclinic: <math>a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta \ne 90 ^\circ</math>

4 Monoclinic: <math>a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

5 Orthogonal: <math>a\ne b \ne c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

6 Tetragonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

7 Hexagonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>

8 Ditetragonal Diclinic: <math>a = d \ne b = c, \alpha = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne 90 ^\circ, \delta = 180 ^\circ - \gamma </math>

9 Ditrigonal (Dihexagonal) Diclinic: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne \delta \ne 90 ^\circ, cos \delta = cos \beta - cos \gamma</math>

10 Tetragonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

11 Hexagonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>

12 Ditetragonal Monoclinic: <math>a = d \ne b = c, \alpha = \gamma = \delta = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ</math>

13 Ditrigonal (Dihexagonal) Monoclinic: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma = \delta \ne 90 ^\circ, cos \gamma = -\color{Black}\tfrac{1}{2} cos \beta</math>

14 Ditetragonal Orthogonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

15 Hexagonal Tetragonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>

16 Dihexagonal Orthogonal: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, </math>

17 Cubic Orthogonal: <math>a = b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

18 Octagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne 90 ^\circ, \beta = \epsilon = 90 ^\circ, \delta = 180 ^\circ - \alpha</math>

19 Decagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne \beta = \delta = \epsilon, cos \beta = -0.5 - cos \alpha</math>

20 Dodecagonal: <math>a = b = c = d, \alpha = \zeta = 90 ^\circ, \beta = \epsilon = 120 ^\circ, \gamma = \delta \ne 90 ^\circ</math>

21 Di-isohexagonal Orthogonal: <math>a = b = c = d, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ</math>

22 Icosagonal (Icosahedral): <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta, cos \alpha = -\color{Black}\tfrac{1}{4}</math>

23 Hypercubic: <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>

The names here are given according to Whittaker.<ref name="Whittaker">E. J. W. Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.</ref> They are almost the same as in Brown ''et al'',<ref name="Brown">H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978.</ref> with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown ''et al'' are given in parenthesis.

The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table.<ref name="Whittaker"/><ref name="Brown"/> Enantiomorphic systems are marked with asterisk. The number of enantiomorphic pairs are given in parentheses. Here the term "enantiomorphic" has different meaning than in table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means, that group itself (considered as geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P31 and P32, P4122 and P4322. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
{|class="wikitable" cellpadding=4 cellspacing=0
|-align=center
!No. of Crystal family
!Crystal family
!Crystal system
!No. of Crystal system
!Point groups
!width=120|Space groups
!Bravais lattices
!Lattice system
|-
| I ||colspan=2| Hexaclinic|| 1
|2
|2
|1
|Hexaclinic P
|-
| II || colspan=2| Triclinic|| 2
|3
|13
|2
|Triclinic P, S
|-
| III ||colspan=2| Diclinic|| 3
|2
|12
|3
|Diclinic P, S, D
|-
|IV || colspan=2| Monoclinic|| 4
|4
|207
|6
|Monoclinic P, S, S, I, D, F
|-
|rowspan=3| V ||rowspan=3| Orthogonal
|rowspan=2|Non-axial Orthogonal|| rowspan=2| 5
|rowspan=2|2
|2
|1
|Orthogonal KU
|-
|112
|rowspan=2|8
|rowspan=2|Orthogonal P, S, I, Z, D, F, G, U
|-
|Axial Orthogonal|| 6
|3
|887
|-
| VI || colspan=2| Tetragonal Monoclinic || 7
|7
|88
|2
|Tetragonal Monoclinic P, I
|-
|rowspan=3| VII ||rowspan=3| Hexagonal Monoclinic
|rowspan=2|Trigonal Monoclinic ||rowspan=2| 8
|rowspan=2|5
|9
|1
|Hexagonal Monoclinic R
|-
|15
|rowspan=2|1
|rowspan=2|Hexagonal Monoclinic P
|-
|Hexagonal Monoclinic || 9
|7
|25
|-
| VIII || colspan=2| Ditetragonal Diclinic* ||10
|1 (+1)
|1 (+1)
|1 (+1)
|Ditetragonal Diclinic P*
|-
|IX || colspan=2| Ditrigonal Diclinic* ||11
|2 (+2)
|2 (+2)
|1 (+1)
|Ditrigonal Diclinic P*
|-
|rowspan=3| X ||rowspan=3| Tetragonal Orthogonal
|rowspan=2|Inverse Tetragonal Orthogonal ||rowspan=2| 12
|rowspan=2|5
|7
|1
|Tetragonal Orthogonal KG
|-
|351
|rowspan=2|5
|rowspan=2|Tetragonal Orthogonal P, S, I, Z, G
|-
|Proper Tetragonal Orthogonal || 13
|10
|1312
|-
|rowspan=3|XI ||rowspan=3| Hexagonal Orthogonal
|rowspan=2|Trigonal Orthogonal ||rowspan=2| 14
|rowspan=2|10
|81
|2
|Hexagonal Orthogonal R, RS
|-
|150
|rowspan=2|2
|rowspan=2|Hexagonal Orthogonal P, S
|-
|Hexagonal Orthogonal || 15
|12
|240
|-
| XII || colspan=2| Ditetragonal Monoclinic* || 16
|1 (+1)
|6 (+6)
|3 (+3)
|Ditetragonal Monoclinic P*, S*, D*
|-
| XIII || colspan=2| Ditrigonal Monoclinic* || 17
|2 (+2)
|5 (+5)
|2 (+2)
|Ditrigonal Monoclinic P*, RR*
|-
|rowspan=3| XIV ||rowspan=3| Ditetragonal Orthogonal
|rowspan=2|Crypto-Ditetragonal Orthogonal ||rowspan=2| 18
|rowspan=2|5
|10
|1
|Ditetragonal Orthogonal D
|-
|165 (+2)
|rowspan=2|2
|rowspan=2|Ditetragonal Orthogonal P, Z
|-
|Ditetragonal Orthogonal ||19
|6
|127
|-
|XV ||colspan=2| Hexagonal Tetragonal || 20
|22
|108
|1
|Hexagonal Tetragonal P
|-
|rowspan=5| XVI || rowspan=5| Dihexagonal Orthogonal
|rowspan=2| Crypto-Ditrigonal Orthogonal* || rowspan=2|21
|rowspan=2|4 (+4)
|5 (+5)
|1 (+1)
|Dihexagonal Orthogonal G*
|-
|5 (+5)
|rowspan=3|1
|rowspan=3|Dihexagonal Orthogonal P
|-
|Dihexagonal Orthogonal || 23
|11
|20
|-
|rowspan=2| Ditrigonal Orthogonal || rowspan=2| 22
|rowspan=2| 11
|41
|-
|16
|1
|Dihexagonal Orthogonal RR
|-
|rowspan=3| XVII ||rowspan=3| Cubic Orthogonal
|rowspan=2|Simple Cubic Orthogonal ||rowspan=2| 24
|rowspan=2|5
|9
|1
|Cubic Orthogonal KU
|-
|96
|rowspan=2|5
|rowspan=2|Cubic Orthogonal P, I, Z, F, U
|-
|Complex Cubic Orthogonal || 25
|11
|366
|-
| XVIII ||colspan=2| Octagonal* || 26
|2 (+2)
|3 (+3)
|1 (+1)
| Octagonal P*
|-
| XIX ||colspan=2| Decagonal || 27
|4
|5
|1
| Decagonal P
|-
| XX ||colspan=2| Dodecagonal* ||28
|2 (+2)
|2 (+2)
|1 (+1)
| Dodecagonal P*
|-
|rowspan=3| XXI ||rowspan=3| Di-isohexagonal Orthogonal
|rowspan=2| Simple Di-isohexagonal Orthogonal || rowspan=2| 29
|rowspan=2|9 (+2)
|19 (+5)
|1
|Di-isohexagonal Orthogonal RR
|-
|19 (+3)
|rowspan=2|1
|rowspan=2|Di-isohexagonal Orthogonal P
|-
|Complex Di-isohexagonal Orthogonal ||30
|13 (+8)
|15 (+9)
|-
|XXII ||colspan=2| Icosagonal|| 31
|7
|20
|2
| Icosagonal P, SN
|-
|rowspan=3| XXIII ||rowspan=3| Hypercubic
|rowspan=2| Octagonal Hypercubic||rowspan=2|32
|rowspan=2|21 (+8)
|73 (+15)
|1
|Hypercubic P
|-
|107 (+28)
|rowspan=2|1
|rowspan=2|Hypercubic Z
|-
|Dodecagonal Hypercubic|| 33
|16 (+12)
|25 (+20)
|- bgcolor=#e0e0e0
|'''Total:'''
|23 (+6)
|33 (+7)
|
|227 (+44)
|4783 (+111)
|64 (+10)
|33 (+7)
|}

==See also==
*[[Crystal cluster]]
*[[Crystal structure]]
*[[Space group#Classification systems for space groups|List of the 230 crystallographic 3D space groups]]
*[[Polar point group]]

==Notes==

<div class="references-small" >
<references/>
</div>

==References==
*{{Cite book | editor1-last=Hahn | editor1-first=Theo | title=International Tables for Crystallography, Volume A: Space Group Symmetry | url=http://it.iucr.org/A/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=5th | isbn=978-0-7923-6590-7 | doi=10.1107/97809553602060000100 | year=2002 | volume=A}}

==External links==
*[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Overview of the 32 groups]
*[http://mineral.galleries.com/minerals/symmetry/symmetry.htm Mineral galleries – Symmetry]
*[http://www.ifg.uni-kiel.de/kubische_Formen all cubic crystal classes, forms and stereographic projections (interactive java applet)]
*[http://reference.iucr.org/dictionary/Crystal_system Crystal system] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
*[http://reference.iucr.org/dictionary/Crystal_family Crystal family] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
*[http://reference.iucr.org/dictionary/Lattice_system Lattice system] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
*[http://materials.duke.edu/awrapper.html Conversion Primitive to Standard Conventional for VASP input files]
*[http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Learning Crystallography]

{{Crystal systems}}
{{Mineral identification}}

[[Category:Symmetry]]
[[Category:Euclidean geometry]]
[[Category:Crystallography]]
[[Category:Morphology]]
[[Category:Mineralogy]]

[[ru:Сингония]]

Complex plane

2014-01-28T19:51:33Z

128.101.152.60: corrections per WP:MOS and WP:MOSMATH

{{Redirect|Carbon burning|combustion of carbon containing compounds|combustion}}

The '''carbon-burning process''' or '''carbon fusion''' is a set of [[nuclear fusion]] reactions that take place in massive [[star]]s (at least 8 [[Solar mass|<math>\begin{smallmatrix}M_\odot\end{smallmatrix}</math>]] at birth) that have used up the lighter elements in their cores. It requires high temperatures (> 5×108 [[kelvin|K]] or 50 [[keV]]) and [[density|densities]] (> 3×109 kg/m3).<ref name=Ryan>{{cite book| author=Ryan, Sean G.; Norton, Andrew J. | title=Stellar Evolution and Nucleosynthesis | year=2010 | page=135| isbn=978-0-521-13320-3|publisher=Cambridge University Press|url=http://books.google.com/?id=PE4yGiU-JyEC&q=carbon+burning#v=onepage&q=carbong%20burning&f=false}}</ref>

These figures for temperature and density are only a guide. More massive stars burn their nuclear fuel more quickly, since they have to offset greater gravitational forces to stay in (approximate) [[hydrostatic equilibrium]]. That generally means higher temperatures, although lower densities, than for less massive stars.<ref name="Clayton">{{cite book|last=Clayton|first=Donald| url=http://books.google.com/books?id=8HSGFThnbvkC|title=Principles of Stellar Evolution and Nucleosynthesis|year=1983|publisher=University of Chicago Press|isbn=978-0-226-10953-4}}</ref> To get the right figures for a particular mass, and a particular stage of evolution, it is necessary to use a numerical [[stellar structure|stellar model]] computed with computer algorithms.<ref name="Siess">{{cite journal
| author=Siess L.
| title = Evolution of massive AGB stars. I. Carbon burning phase
| journal = Astronomy and Astrophysics
| year= 2007
| volume = 476
| issue=2
| pages = 893–909
| doi = 10.1051/0004-6361:20053043
| bibcode = 2006A&A...448..717S}}</ref> Such models are continually being refined based on [[particle physics experiments]] (which measure nuclear reaction rates) and astronomical observations (which include direct observation of mass loss, detection of nuclear products from spectrum observations after convection zones develop from the surface to fusion-burning regions – known as 'dredge-up' events – and so bring nuclear products to the surface, and many other observations relevant to models).<ref>{{cite journal
| title = Rubidium-Rich Asymptotic Giant Branch Stars
| journal = Science
| eprint = arXiv:astro-ph/0611319
| year = 2006
| month = dec,
| volume = 314
| issue = 5806
| pages = 1751–1754
| doi = 10.1126/science.1133706
| bibcode = 2006Sci...314.1751G
| author = Hernandez, G. et al
| pmid = 17095658
|arxiv = astro-ph/0611319 }}</ref>

==Fusion reactions==
The principal reactions are:<ref name="Camiel1992">{{cite book|last=Camiel|first=W. H.|coauthors=de Loore, C. Doom|chapter=Structure and evolution of single and binary stars|editor=Camiel W. H. de Loore|title=Volume 179 of Astrophysics and space science library|year=1992|publisher=Springer|isbn=978-0-7923-1768-5|url=http://books.google.com/?id=LJgNIi0vkeYC&dq=%22carbon+burning%22&q=carbon+burning#v=snippet&q=carbon%20burning&f=false|pages=95–97}}</ref>
:{| border="0"
|- style="height:3em;"
||[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Neon-20|{{nuclide|neon|20}}]] ||+ ||[[Helium|{{nuclide|helium|4}}]] ||+ ||4.617 [[electron volt|MeV]]
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Sodium-23|{{nuclide|sodium|23}}]] ||+ ||[[Hydrogen|{{nuclide|hydrogen|1}}]] ||+ ||2.241 [[electron volt|MeV]]
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Magnesium-23|{{nuclide|magnesium|23}}]] ||+ ||[[Neutron|n]] ||− ||2.599 [[electron volt|MeV]]
|- style="height:3em;"
|colspan=99|Alternatively:
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Magnesium-24|{{nuclide|magnesium|24}}]] ||+ ||{{SubatomicParticle|link=yes|Gamma}} ||+ ||13.933 [[electron volt|MeV]]
|- style="height:3em;"
|[[Carbon-12|{{nuclide|carbon|12}}]] ||+ ||[[Carbon-12|{{nuclide|carbon|12}}]] ||→ ||[[Oxygen-16|{{nuclide|oxygen|16}}]] ||+ ||2 [[Helium|{{nuclide|helium|4}}]] ||colspan=2|−   0.113 [[electron volt|MeV]]
|}

==Reaction products==
This sequence of reactions can be understood by thinking of the two interacting carbon nuclei as coming together to form an [[excited state]] of the Mg-24 nucleus, which then decays in one of the five ways listed above.<ref name=Rose1998>{{cite book|url=http://books.google.com/books?id=yaX0etDmbXMC|last=Rose,|first=William K.|title=Advanced Stellar Astrophysics|publisher=Cambridge University Press|year=1998|isbn=978-0-521-58833-1|pages=227–229}}</ref> The first two reactions are strongly exothermic, as indicated by the large positive energies released, and are the most frequent results of the interaction. The third reaction is strongly endothermic, as indicated by the large negative energy indicating that energy is absorbed rather than emitted. This makes it much less likely, yet still possible in the high-energy environment of carbon burning.<ref name=Camiel1992/> But the production of a few neutrons by this reaction is important, since these neutrons can combine with heavy nuclei, present in tiny amounts in most stars, to form even heavier isotopes in the [[s-process]].<ref name=Rose1998-229>Rose (1998), pp. 229–234</ref>

The fourth reaction might be expected to be the most common from its large energy release, but in fact it is extremely improbable because it proceeds via the electromagnetic interaction,<ref name=Camiel1992/> as it produces a gamma ray photon, rather than utilising the strong force between nucleons as do the first two reactions. Nucleons look a lot bigger to each other than they do to photons of this energy. However, the Mg-24 produced in this reaction is the only magnesium left in the core when the carbon-burning process ends, as Mg-23 is radioactive.

The last reaction is also very unlikely since it involves three reaction products,<ref name=Camiel1992/> as well as being endothermic—think of the reaction proceeding in reverse, it would require the three products all to converge at the same time, which is less likely than two-body interactions.

The protons produced by the second reaction can take part in the [[proton-proton chain reaction]], or the [[CNO cycle]], but they can also be captured by Na-23 to form Ne-20 plus a He-4 nucleus.<ref name=Camiel1992/> In fact, a significant fraction of the Na-23 produced by the second reaction gets used up this way.<ref name=Rose1998/> The oxygen (O-16) already produced by [[helium fusion]] in the previous stage of stellar evolution manages to survive the carbon-burning process pretty well, despite some of it being used up by capturing He-4 nuclei, in stars between 9 and 11 [[solar mass]]es.<ref name=Ryan/><ref name=Camiel1992-97>Camiel (1992), pp.97–98</ref> So the end result of carbon burning is a mixture mainly of oxygen, neon, sodium and magnesium.<ref name="Siess"/><ref name=Camiel1992/>

The fact that the mass-energy sum of the two carbon nuclei is similar to that of an excited state of the magnesium nucleus is known as 'resonance'. Without this resonance, carbon burning would only occur at temperatures one hundred times higher.
The experimental and theoretical investigation of such resonances is still a subject of research.<ref>{{cite journal
| author = Strandberg, E. et al
| title = Mg24(α,γ)Si28 resonance parameters at low α-particle energies
| journal = Physical Review C
| date = May 2008
| volume = 77
| issue = 5
| pages = 055801-+
| doi = 10.1103/PhysRevC.77.055801
| bibcode = 2008PhRvC..77e5801S
}}</ref> A similar resonance increases the probability of the [[triple-alpha process]], which is responsible for the original production of carbon.

==Neutrino losses==
[[Neutrino]] losses start to become a major factor in the fusion processes in stars at the temperatures and densities of carbon burning. Though the main reactions don't involve neutrinos, the side reactions such as the [[proton-proton chain reaction]] do. But the main source of neutrinos at these high temperatures involves a process in quantum theory known as [[pair production]]. A high energy [[gamma ray]] which has a greater energy than the [[rest mass]] of two [[electron]]s ([[mass-energy equivalence]]) can interact with electromagnetic fields of the atomic nuclei in the star, and become a particle and [[anti-particle]] pair of an electron and positron.

Normally, the positron quickly annihilates with another electron, producing two photons, and this process can be safely ignored at lower temperatures. But around 1 in 1019 pair productions<ref name="Clayton"/> end with a weak interaction of the electron and positron, which replaces them with a [[neutrino]] and anti-neutrino pair. Since they move at virtually the speed of light and interact very weakly with matter, these neutrino particles usually escape the star without interacting, carrying away their mass-energy. This energy loss is comparable to the energy output from the carbon fusion.

Neutrino losses, by this and similar processes, play an increasingly important part in the evolution of the most massive stars. They force the star to burn its fuel at a higher temperature to offset them.<ref name="Clayton"/> Fusion processes are very sensitive to temperature so the star can produce more energy to retain [[hydrostatic equilibrium]], at the cost of burning through successive nuclear fuels ever more rapidly. Fusion produces less energy per unit mass as the fuel nuclei get heavier, and the core of the star contracts and heats up when switching from one fuel to the next, so both these processes also significantly reduce the lifetime of each successive fusion-burning fuel.

Up to helium burning, the neutrino losses are negligible, but from carbon burning the reduction in lifetime due to them roughly matches that due to fuel change and core contraction. In successive fuel changes in the most massive stars, the reduction in lifetime is dominated by the neutrino losses. For example, a star of 25 solar masses burns hydrogen in the core for 107 years, helium for 106 years and carbon for only 103 years.<ref name="WoosleyJanka">{{cite journal
| last=Woosley | first=S. | coauthors=Janka, H.-T.
| date=2006-01-12
| bibcode=2005NatPh...1..147W
| title=The Physics of Core-Collapse Supernovae
| journal=Nature Physics
| year=2005 | month=December
| volume=1 | issue=3 | pages=147–154
| doi=10.1038/nphys172|arxiv = astro-ph/0601261 }}</ref>

==Stellar evolution==
{{main|Stellar evolution}}
During [[helium fusion]], stars build up an inert core rich in carbon and oxygen. The inert core eventually reaches sufficient mass to collapse due to gravitation, whilst the helium burning moves gradually outward. This decrease in the inert core volume raises the temperature to the carbon ignition temperature. This will raise the temperature around the core and allow helium to burn in a shell around the core.<ref name="Ostlie">Ostlie, Dale A. and Carrol, Bradley W., [http://www.amazon.com/dp/0805304029 ''An introduction to Modern Stellar Astrophysics''], Addison-Wesley (2007)</ref> Outside this is another shell burning hydrogen. The resulting carbon burning provides energy from the core to restore the star's [[mechanical equilibrium]]. However, the balance is only short-lived; in a star of 25 solar masses, the process will use up most of the carbon in the core in only 600 years. The duration of this process varies significantly depending on the mass of the star.<ref name="Lecture19">Anderson, Scott R.,
[http://www.opencourse.info/astronomy/introduction/19.stars_death_high-mass/ ''Open Course: Astronomy: Lecture 19: Death of High-Mass Stars''], GEM (2001)</ref>

Stars with below 8–9 [[Solar mass]]es never reach high enough core temperature to burn carbon, instead ending their lives as carbon-oxygen [[white dwarf]]s after shell [[helium flash]]es gently expel the outer envelope in a [[planetary nebula]].<ref name="Siess"/><ref name=Ryan-147/>

In the late stages of carbon burning, stars with masses between 8 and 11 solar masses develop a massive stellar wind, which quickly ejects the outer envelope in a [[planetary nebula]] leaving behind an O-Ne-Na-Mg [[white dwarf]] core of about 1.1 solar masses.<ref name="Siess"/> The core never reaches high enough temperature for further fusion burning of heavier elements than carbon.<ref name=Ryan-147/>

Stars with more than 11 solar masses proceed with the [[neon-burning process]] after contraction of the inert (O, Ne, Na, Mg) core raises the temperature sufficiently.<ref name=Ryan-147>Ryan (2010), pp.147–148</ref>

==See also==
*[[Proton–proton chain reaction]]
*[[CNO process]]
*[[Triple alpha process]]
*[[Alpha process]]
*[[Carbon detonation]]
*[[Neon burning]]

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

{{Nuclear processes}}

[[Category:Nucleosynthesis]]

Hoop Conjecture

2014-01-20T20:29:18Z

128.101.152.70:

'''Network calculus''' is "a set of mathematical results which give insights into man-made systems such as [[concurrent program]]s, [[digital circuit]]s and [[communication network]]s."<ref name="leboudec">{{cite doi|10.1007/3-540-45318-0}}</ref> Network calculus gives a theoretical framework for analysing performance guarantees in [[computer network]]s. As traffic flows through a network it is subject to [[Constraint (mathematics)|constraints]] imposed by the system components, for example:
* [[data link|link]] capacity
* traffic shapers ([[leaky bucket]]s)
* [[congestion control]]
* background traffic

These constraints can be expressed and analysed with network calculus methods. Constraint curves can be ''combined'' using [[convolution]] under [[min-plus algebra]]. Network calculus can also be used to express traffic arrival and departure functions as well as service curves.

The calculus uses "alternate algebras ... to transform complex non-linear network systems into analytically tractable linear systems."<ref>{{cite doi|10.1007/978-1-84800-127-5}}</ref>

==Min-plus algebra==
In filter theory respectively linear systems theory the [[convolution]] of two functions <math>f</math> and <math>g</math> is defined as

<math>
(f \ast g) (t) := \sum_{\tau} f(\tau) \cdot g(t-\tau).
</math>

In '''min-plus algebra''' the ''sum'' is replaced by the minimum respectively ''[[infimum]]'' operator and the ''product'' is replaced by the ''sum''. So the min-plus convolution of two functions <math>f</math> and <math>g</math> becomes

<math>
(f \otimes g) (t) := \inf_{0 \leq \tau \leq t}\left\{f(\tau) + g(t-\tau)\right\}
</math>

e.g. see the definition of service curves. Convolution and min-plus convolution share many algebraic properties. In particular both are commutative and associative.

A so-called min-plus de-convolution operation is defined as

<math>
(f \oslash g) (t) := \sup_{\tau \ge 0}\left\{f(t+\tau) - g(\tau)\right\}
</math>

e.g. as used in the definition of traffic envelopes.

==Traffic envelopes==
Traffic flows in networks are described as cumulative functions. For example, <math>A(t)</math>
is the number of bits in the interval <math>[0,t)</math>. <math>A(t)</math> is said to conform to an envelope arrival curve <math>E(t)</math>, if for all <math>t</math> it holds that

<math>
E(t) \ge \sup_{\tau \ge 0} \{A(t+\tau) - A(\tau) \} = (A \oslash A)(t).
</math>

Thus, <math>E(t)</math> places an upper constraint on flow <math>A(t)</math>, i.e. an envelope <math>E(t)</math> specifies an upper bound on the number of bits of flow <math>A(t)</math> seen in any interval of length <math>t</math> starting at an arbitrary <math>\tau</math>. The above equation can be rephrased for all <math>t</math> as

<math>
A(t) \le \inf_{0 \leq \tau \leq t} \{ A(\tau) + E(t-\tau) \} = (A \otimes E)(t).
</math>

==Service curves==
In order to provide performance guarantees to traffic flows it may be necessary to implement reservations in the network. Service curves provide a means of expressing resource allocations.

Let <math>A(t)</math> be a flow arriving at the ingress of a system, e.g. a link, a scheduler, a traffic shaper, or a whole network, and <math>D(t)</math> be the flow departing at the egress. The system is said to provide a service curve <math>S(t)</math>, if for all <math>t</math> it holds that

<math>
D(t) \ge \inf_{0 \le \tau \le t} \{A(\tau) + S(t-\tau) \} = (A \otimes S)(t).
</math>

==Concatenation==
Consider two systems with service curve <math>S_1(t)</math> and <math>S_2(t)</math> in series. Let <math>A(t)</math> be the arrival function at system 1 and <math>D(t)</math> the departure function of system 2. By iterative application of the definition of service curves we have

<math>D(t) \geq ((A \otimes S_1) \otimes S_2)(t)</math>.

By associativity of the min-plus convolution it follows that

<math>D(t) \geq (A \otimes (S_1 \otimes S_2))(t)</math>

i.e. the tandem of systems <math>S_1(t)</math> and <math>S_2(t)</math> is equivalent to a single, lumped system <math>S_{e2e}(t)</math> where

<math>S_{e2e}(t) = (S_1 \otimes S_2)(t)</math>.

==Performance bounds==
The virtual backlog is defined as the vertical deviation of <math>A(t)</math> and <math>D(t)</math>

<math>
B(t) = A(t) - D(t), \forall t \ge 0.
</math>

Using the concepts of traffic envelopes and service curves the maximum backlog is bounded by

<math>
B_{\max} \le \sup_{\tau \ge 0} \{E(\tau) - S(\tau) \} = (E \oslash S)(0).
</math>

The delay <math>W(t)</math> is defined as the horizontal deviation of <math>A(t)</math> and <math>D(t)</math>

<math>
W(t) = \inf \{w : A(t) - D(t+w) \le 0\} , \forall t \ge 0.
</math>

Using the concepts of traffic envelopes and service curves the maximum delay is bounded by

<math>
W_{\max} \le \inf \{w : \sup_{\tau \ge w} \{ E(\tau-w) - S(\tau) \} \le 0 \} = \inf \{w : (E \oslash S)(-w) \le 0 \}.
</math>

==References==

{{Reflist}}

Books that cover Network Calculus
* C.-S. Chang: ''Performance Guarantees in Communications Networks'', Springer, 2000.
* J.-Y. Le Boudec and P. Thiran: ''[http://ica1www.epfl.ch/PS_files/NetCal.htm Network Calculus: A Theory of Deterministic Queuing Systems for the Internet]'', Springer, LNCS, 2001.
* A. Kumar, D. Manjunath, and J. Kuri: ''Communication Networking: An Analytical Approach'', Elsevier, 2004.
* Y. Jiang and Y. Liu: ''Stochastic Network Calculus'', Springer, 2008.

Related books on the max-plus algebra or on [[convex minimization]]
* [[R. T. Rockafellar]]: ''[[Convex analysis]]'', Princeton University Press, 1972.
* F. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat: ''Synchronization and Linearity: An Algebra for Discrete Event Systems'', Wiley, 1992.
* V. N. Kolokol'tsov, Victor P. Maslov: ''Idempotent Analysis and Its Applications'', Springer, 1997. ISBN 0792345096.

Some related research papers
* R. L. Cruz: {{doi-inline|10.1109/18.61109|A Calculus for Network Delay. Part I: Network Elements in Isolation}} and {{doi-inline|10.1109/18.61110|Part II: Network Analysis}}, IEEE Transactions on Information Theory, 37(1):114-141, Jan. 1991.
* O. Yaron and M. Sidi: ''Performance and Stability of Communication Networks via Robust Exponential Bounds'', IEEE/ACM Transactions on Networking, 1(3):372-385, Jun. 1993.
* C.-S. Chang: ''Stability, Queue Length and Delay of Deterministic and Stochastic Queueing Networks'', IEEE Transactions on Automatic Control, 39(5):913-931, May 1994.
* D. E. Wrege, E. W. Knightly, H. Zhang, and J. Liebeherr: ''Deterministic delay bounds for VBR video in packet-switching networks: Fundamental limits and practical tradeoffs'', IEEE/ACM Transactions on Networking, 4(3):352-362, Jun. 1996.
* R. L. Cruz: ''SCED+: Efficient Management of Quality of Service Guarantees'', IEEE INFOCOM, pp. 625-634, Mar. 1998.
* J.-Y. Le Boudec: ''Application of Network Calculus to Guaranteed Service Networks'', IEEE Transactions on Information Theory, 44(3):1087-1096, May 1998.
* C.-S. Chang: ''On Deterministic Traffic Regulation and Service Guarantees: A Systematic Approach by Filtering'', IEEE Transactions on Information Theory, 44(3):1097-1110, May 1998.
* R. Agrawal, R. L. Cruz, C. Okino, and R. Rajan: ''Performance Bounds for Flow Control Protocols'', IEEE/ACM Transactions on Networking, 7(3):310-323, Jun. 1999.
* D. Starobinski and M. Sidi: ''Stochastically Bounded Burstiness for Communication Networks'', IEEE Transactions on Information Theory, 46(1):206-212, Jan. 2000.
* A. Charny and J.-Y. Le Boudec: ''Delay Bounds in a Network with Aggregate Scheduling'', QoFIS, pp. 1-13, Sep. 2000.
* R.-R. Boorstyn, A. Burchard, J. Liebeherr, and C. Oottamakorn: ''Statistical Service Assurances for Traffic Scheduling Algorithms'', IEEE Journal on Selected Areas in Communications, 18(12):2651-2664, Dec. 2000.
* J.-Y. Le Boudec: ''Some properties of variable length packet shapers'', IEEE/ACM Transactions on Networking, 10(3):329-337, Jun. 2002.
* Q. Yin, Y. Jiang, S. Jiang, and P. Y. Kong: ''Analysis of Generalized Stochastically Bounded Bursty Traffic for Communication Networks'', IEEE LCN, pp. 141-149, Nov. 2002.
* C.-S. Chang, R. L. Cruz, J.-Y. Le Boudec, and P. Thiran: ''A Min, + System Theory for Constrained Traffic Regulation and Dynamic Service Guarantees'', IEEE/ACM Transactions on Networking, 10(6):805-817, Dec. 2002.
* D. Starobinski, M. Karpovsky, and L. Zakrevski: ''Application of Network Calculus to General Topologies using Turn-Prohibition'', IEEE/ACM Transactions on Networking, 11(3):411-421, Jun. 2003.
* C. Li, A. Burchard, and J. Liebeherr: ''A Network Calculus with Effective Bandwidth'', University of Virginia, Technical Report CS-2003-20, Nov. 2003.
* F. Ciucu, A. Burchard, and J. Liebeherr: ''A Network Service Curve Approach for the Stochastic Analysis of Networks'', IEEE/ACM Transactions on Networking, 52(6):2300–2312, Jun. 2006.
* M. Fidler and S. Recker: ''Conjugate network calculus: A dual approach applying the Legendre transform'', Computer Networks, 50(8):1026-1039, Jun. 2006.
* M. Fidler: ''An End-to-End Probabilistic Network Calculus with Moment Generating Functions'', IEEE IWQoS, Jun. 2006.
* A. Burchard, J. Liebeherr, and S. D. Patek: ''A Min-Plus Calculus for End-to-end Statistical Service Guarantees'', IEEE Transactions on Information Theory, 52(9):4105–4114, Sep. 2006.
* Eitan Altman, Kostya Avrachenkov, and Chadi Barakat: ''TCP network calculus: The case of large bandwidth-delay product'', In proceedings of IEEE INFOCOM, NY, June 2002.
* Kym Watson, Juergen Jasperneite: ''Determining End-to-End Delays using Network Calculus'', in 5th IFAC International Conference on Fieldbus Systems and their Applications (FeT´2003) S.: 255-260, Aveiro, Portugal, Jul 2003

[[Category:Network performance]]
[[Category:Computer network analysis]]

Capillary number

2013-12-31T17:47:16Z

128.101.142.152: Changed imprecise term "shear viscosity" to technical term "dynamic viscosity".

In [[mathematics]], an '''invariant subspace''' of a [[linear mapping]]

:''T'' : ''V'' → ''V''

from some [[vector space]] ''V'' to itself is a [[linear subspace|subspace]] ''W'' of ''V'' such that ''T''(''W'') is contained in ''W''. An invariant subspace of ''T'' is also said to be ''' ''T'' invariant'''.

If ''W'' is ''T''-invariant, we can restrict ''T'' to ''W'' to arrive at a new linear mapping

:''T''|''W'' : ''W'' → ''W''.

Next we give a few immediate examples of invariant subspaces.

Certainly ''V'' itself, and the subspace {0}, are trivially invariant subspaces for every linear operator ''T'' : ''V'' → ''V''. For certain linear operators there is no ''non-trivial'' invariant subspace; consider for instance a [[rotation (mathematics)|rotation]] of a two-dimensional real vector space.

Let '''''v''''' be an [[eigenvector]] of ''T'', i.e. ''T'' '''''v''''' = λ'''''v'''''. Then ''W'' = [[linear span|span]] {'''''v'''''} is ''T'' invariant. As a consequence of the [[fundamental theorem of algebra]], every linear operator on a [[complex number|complex]] finite-[[dimension (vector space)|dimensional]] vector space with dimension at least 2 has an eigenvector. Therefore every such linear operator has a non-trivial invariant subspace. The fact that the complex numbers are [[algebraically closed]] is required here. Comparing with the previous example, one can see that the invariant subspaces of a linear transformation are dependent upon the underlying scalar field of ''V''.

An '''invariant vector''' ([[Fixed point (mathematics)|fixed point]] of ''T''), other than 0, spans an invariant subspace of dimension 1. An invariant subspace of dimension 1 will be acted on by ''T'' by a scalar, and consists of invariant vectors if and only if that scalar is 1.

As the above examples indicate, the invariant subspaces of a given linear transformation ''T'' shed light on the structure of ''T''. When ''V'' is a finite dimensional vector space over an algebraically closed field, linear transformations acting on ''V'' are characterized (up to similarity) by the [[Jordan canonical form]], which decomposes ''V'' into invariant subspaces of ''T''. Many fundamental questions regarding ''T'' can be translated to questions about invariant subspaces of ''T''.

More generally, invariant subspaces are defined for sets of operators as subspaces invariant for each operator in the set. Let ''L''(''V'') denote the algebra of linear transformations on ''V'', and Lat(''T'') be the family of subspaces invariant under ''T'' ∈ ''L''(''V''). (The "Lat" notation refers to the fact that Lat(''T'') forms a [[lattice (order)|lattice]]; see discussion below.) Given a nonempty set Σ ⊂ ''L''(''V''), one considers the invariant subspaces invariant under each ''T'' ∈ Σ. In symbols,

:<math>\mbox{Lat}(\Sigma) = \bigcap_{T \in \Sigma} \mbox{Lat}( T ) \;.</math>

For instance, it is clear that if Σ = ''L''(''V''), then Lat(Σ) = { {0}, ''V''}.

Given a [[Group representation|representation]] of a group ''G'' on a vector space ''V'', we have a linear transformation ''T''(''g'') : ''V'' → ''V'' for every element ''g'' of ''G''. If a subspace ''W'' of ''V'' is invariant with respect to all these transformations, then it is a [[group representation|subrepresentation]] and the group ''G'' acts on ''W'' in a natural way.

As another example, let ''T'' ∈ ''L''(''V'') and Σ be the algebra generated by {1, ''T''}, where 1 is the identity operator. Then Lat(''T'') = Lat(Σ). Because ''T'' lies in Σ trivially, Lat(Σ) ⊂ Lat(''T''). On the other hand, Σ consists of polynomials in 1 and ''T'', therefore the reverse inclusion holds as well.

== Matrix representation ==

Over a finite dimensional vector space every linear transformation ''T'' : ''V'' → ''V'' can be represented by a [[matrix (math)|matrix]] once a [[basis (linear algebra)|basis]] of ''V'' has been chosen.

Suppose now ''W'' is a ''T'' invariant subspace. Pick a basis ''C'' = {'''''v'''''1, ..., '''''v'''''''k''} of ''W'' and complete it to a basis ''B'' of ''V''. Then, with respect to this basis, the matrix representation of ''T'' takes the form:

:<math> T = \begin{bmatrix} T_{11} & T_{12} \\ 0 & T_{22} \end{bmatrix} </math>

where the upper-left block ''T''11 is the restriction of ''T'' to ''W''.

In other words, given an invariant subspace ''W'' of ''T'', ''V'' can be decomposed into the [[direct sum of vector spaces|direct sum]]

:<math>V = W \oplus W'.</math>

Viewing ''T'' as an operator matrix

:<math>
T = \begin{bmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{bmatrix} : \begin{matrix}W \\ \oplus \\ W' \end{matrix} \rightarrow \begin{matrix}W \\ \oplus \\ W' \end{matrix},
</math>

it is clear that ''T''21: ''W'' → ''W' '' must be zero.

Determining whether a given subspace ''W'' is invariant under ''T'' is ostensibly a problem of geometric nature. Matrix representation allows one to phrase this problem algebraically. The [[projection operator]] ''P'' onto ''W'', is defined by
''P''(''w + w' '') = ''w'', where ''w'' ∈ ''W'' and ''w' '' ∈ ''W' ''. The projection ''P'' has matrix representation

:<math>
P = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} : \begin{matrix}W \\ \oplus \\ W' \end{matrix} \rightarrow \begin{matrix}W \\ \oplus \\ W' \end{matrix}.
</math>

A straightforward calculation shows that ''W'' = Ran ''P'', the range of ''P'', is invariant under ''T'' if and only of ''PTP'' = ''TP''. In other words, a subspace ''W'' being an element of Lat(''T'') is equivalent to the corresponding projection satisfying the relation ''PTP'' = ''TP''.

If ''P'' is a projection (i.e. ''P''2 = ''P''), so is 1 - ''P'', where 1 is the identity operator. It follows from the above that ''TP = PT'' if and only if both Ran ''P'' and Ran (1 - ''P'') are invariant under ''T''. In that case, ''T'' has matrix representation

:<math>
T = \begin{bmatrix} T_{11} & 0 \\ 0 & T_{22} \end{bmatrix} : \begin{matrix} \mbox{Ran}P \\ \oplus \\ \mbox{Ran}(1-P) \end{matrix} \rightarrow \begin{matrix} \mbox{Ran}P \\ \oplus \\ \mbox{Ran}(1-P) \end{matrix} \;.
</math>

Colloquially, a projection that commutes with ''T'' "diagonalizes" ''T''.

== Invariant subspace problem ==
:{{main|Invariant subspace problem}}

The invariant subspace problem concerns the case where ''V'' is a separable [[Hilbert space]] over the [[complex number]]s, of dimension > 1, and ''T'' is a [[bounded operator]]. The problem is to decide whether every such ''T'' has a non-trivial, closed, invariant subspace. This problem is unsolved {{As of|2013|lc=on}}.

In the more general case where ''V'' is hypothesized to be a [[Banach space]], there is an example of an [[Invariant subspace problem|operator without an invariant subspace]] due to [[Per Enflo]] (1976). A [[Invariant subspace problem|concrete example]] of an operator without an invariant subspace was produced in 1985 by [[Charles Read (mathematician)|Charles Read]].

== Invariant-subspace lattice ==

Given a nonempty Σ ⊂ ''L''(''V''), the invariant subspaces invariant under each element of Σ form a [[lattice (order)|lattice]], sometimes called the '''invariant-subspace lattice''' of Σ and denoted by Lat(Σ).

The lattice operations are defined in a natural way: for Σ' ⊂ Σ, the ''meet'' operation is defined by

:<math>\bigwedge_{W \in \Sigma'} W = \bigcap_{W \in \Sigma'} W</math>

while the ''join'' operation is

:<math>\bigvee_{W \in \Sigma'} W = \mbox{span} \bigcup_{W \in \Sigma'} W \;.</math>

A minimal element in Lat(Σ) in said to be a '''minimal invariant subspace'''.

== Fundamental theorem of noncommutative algebra ==

Just as the fundamental theorem of algebra ensures that every linear transformation acting on a finite dimensional complex vector space has a nontrivial invariant subspace, the ''fundamental theorem of noncommutative algebra'' asserts that Lat(Σ) contains nontrivial elements for certain Σ.

'''Theorem (Burnside)''' Assume ''V'' is a complex vector space of finite dimension. For every proper subalgebra Σ of ''L''(''V''), Lat(Σ) contain a nontrivial element.

Burnside's theorem is of fundamental importance in linear algebra. One consequence is that every commuting family in ''L''(''V'') can be simultaneously upper-triangularized.

A nonempty Σ ⊂ ''L''(''V'') is said to be '''triangularizable''' if there exists a basis {''e''1...''en''} of ''V'' such that

:<math>\mbox{span} \{ e_1, \cdots, e_k \} \in \mbox{Lat}(\Sigma) \quad \forall k \geq 1 \;.</math>

In other words, Σ is triangularizable if there exists a basis such that every element of Σ has an upper-triangular matrix representation in that basis. It follows from Burnside's theorem that every commutative algebra Σ in ''L''(''V'') is triangularizable. Hence every commuting family in ''L''(''V'') can be simultaneously upper-triangularized.

== Left ideals ==

If ''A'' is an algebra, one can define a [[regular representation|''left regular representation'']] Φ on ''A'': Φ(''a'')''b'' = ''ab'' is a homomorphism from ''A'' to ''L''(''A''), the algebra of linear transformations on ''A''

The invariant subspaces of Φ are precisely the left ideals of ''A''. A left ideal ''M'' of ''A'' gives a subrepresentation of ''A'' on ''M''.

If ''M'' is a left ideal of ''A''. Consider the quotient vector space ''A''/''M''. The left regular representation Φ on ''M'' now descends to a representation Φ' on ''A''/''M''. If [''b''] denotes an equivalence class in ''A''/''M'', Φ'(''a'')[''b''] = [''ab'']. The kernel of the representation Φ' is the set {''a'' ∈ ''A''| ''ab'' ∈ ''M'' for all ''b''}.

The representation Φ' is [[irreducible representation|irreducible]] if and only if ''M'' is a maximal left ideal, since a subspace ''V'' ⊂ ''A''/''M'' is an invariant under {Φ'(''a'')| ''a'' ∈ ''A''} if and only if its preimage under the quotient map, ''V'' + ''M'', is a left ideal in ''A''.

==See also==
* [[Invariant manifold]]

==Bibliography==
* {{cite book
|author=Yuri A. Abramovich and [[Charalambos D. Aliprantis]],
|title=An Invitation to Operator Theory
|publisher=[http://www.ams.org/bookstore-getitem/item=GSM-50 American Mathematical Society]
|year=2002
|isbn=978-0-8218-2146-6}}
*{{cite book
|author=Beauzamy, Bernard
|title=Introduction to Operator Theory and Invariant Subspaces
|year=1988
|publisher=North Holland
}}
* {{cite book
|author=[[Per Enflo|Enflo, Per]] and Lomonosov, Victor
|chapter=Some aspects of the invariant subspace problem
|title=Handbook of the geometry of Banach spaces
|volume=I
|pages=533–559
|publisher=North-Holland
|location=Amsterdam|year=2001
}}
*{{cite book
|title=Invariant Subspaces of Matrices with Applications
|author=Israel Gohberg, Peter Lancaster, and Leiba Rodman
|edition=Reprint, with list of [[errata]] and new preface, of the 1986 Wiley
|series=Classics in Applied Mathematics
|volume=51
|publisher=[http://www.ec-securehost.com/SIAM/CL51.html Society for Industrial and Applied Mathematics (SIAM)]
|year=2006
|pages=xxii+692
|isbn=978-0-89871-608-5
}}
* Yurii I. Lyubich. ''Introduction to the Theory of Banach Representations of Groups''. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.
* {{cite book
|author=Heydar Radjavi and Peter Rosenthal
|title=Invariant Subspaces
|year=2003
|edition=Update of 1973 Springer-Verlag
|isbn=0-486-42822-2
|publisher=[http://store.doverpublications.com/0486428222.html Dover]
}}

[[Category:Linear algebra]]
[[Category:Operator theory]]
[[Category:Representation theory]]

Bapat–Beg theorem

2013-12-13T02:10:11Z

128.101.152.127: /* Statement of theorem */

{{sources|date=January 2012}}
'''All-electric range''' (AER) is the driving range of a vehicle using only power from its electric [[battery pack]] to traverse a given [[United_States_Environmental_Protection_Agency#Fuel_economy_testing_and_results|driving cycle]]. In the case of a [[battery electric vehicle]], it means the total range per charge. For a [[plug-in hybrid]] (PHEV), it means the range of the vehicle in [[charge-depleting]] mode. PHEVs can travel considerably further in [[charge-sustaining]] mode which uses both on-board fuel and the battery pack.

Calculating AER is made more complicated because of variations in PHEV design. A vehicle like the [[Fisker Karma]] that utilizes a serial hybrid design has a clear AER. Similarly a vehicle like the Chevy Volt which disengages the internal combustion engine (ICE) from the drive train while in electric mode has a clear AER, however blended mode PHEVs which utilize the ICE and electric motor in conjunction do not have a clear AER because they utilize gasoline and grid provided electricity at the same time. Equivalent AER is a term used to denote the AER of vehicles following this architecture. One example of this calculation can be found in Argonne National Labs report titled "TEST PROCEDURES AND BENCHMARKING Blended-Type and EV-Capable Plug-In Hybrid Electric Vehicles."[http://www.transportation.anl.gov/pdfs/HV/457.pdf]

This procedure uses the formula below to calculate an equivalent AER for vehicles that operate in blended mode:

:<math>\text{AER}_\text{Equivalent} = \left (1-\frac{GPM_{CD} }{GPM_{CS} }\right ) d^{CD}</math>

Where GPMCD designates efficiency in [[charge-depleting]] mode, and GPMCS [[charge-sustaining]] mode as designated and ''d''CD is distance in charge depleting mode.

A [[plug-in hybrid]]'s all-electric range is designated by PHEV-''(miles)'' or PHEV''(kilometers)''km representing the distance the vehicle can travel on battery power alone. For example, a PHEV-20 can travel 20 miles without using its internal combustion engine, or about 32 kilometers, so it may also be designated as PHEV32km.

[[Category:Electric vehicles]]

{{Automotive-tech-stub}}

Singular homology

2013-10-25T20:45:48Z

128.101.152.69:

{{good article}}
{{Infobox scientist
| name = Arthur Holly Compton
| image = Arthur Compton.jpg
| image_size =
| caption =
| birth_date = {{birth date|mf=yes|1892|9|10}}
| birth_place = [[Wooster, Ohio]], [[United States|USA]]
| death_date = {{death date and age|mf=yes|1962|3|15|1892|9|10}}
| death_place = [[Berkeley, California]], [[United States|USA]]
| nationality = [[United States]]
| field = [[Physics]]
| work_institution = [[Washington University in St. Louis]] [[University of Chicago]] [[University of Minnesota]] [[Forman Christian College]]
| alma_mater = [[College of Wooster]] [[Princeton University]]
| doctoral_advisor = Hereward L. Cooke
| doctoral_students = [[Luis Walter Alvarez]] [[Winston H. Bostick]] [[Robert S. Shankland]] Joyce A. Bearden
| known_for = [[Compton effect]] [[Compton length]] [[Compton scattering]] [[Compton wavelength]] [[Compton shift]]
| prizes = {{nowrap|[[File:Nobel prize medal.svg|20px]] [[Nobel Prize for Physics]] (1927) [[Franklin Medal]] (1940) }}
| religion = [[Presbyterian]]
| footnotes = Brother of [[Wilson Compton]] and [[Karl Compton]]
| signature = Arthur Compton signature.png
}}

'''Arthur Holly Compton''' (September 10, 1892 – March 15, 1962) was an American physicist who won the [[Nobel Prize in Physics]] in 1927 for his discovery of the [[Compton effect]]. He is also known for his leadership of the [[Manhattan Project]]'s [[Metallurgical Laboratory]]. He served as Chancellor of [[Washington University in St. Louis]] from 1945 to 1953.

In 1919, Compton was awarded one of the first two [[United States National Research Council|National Research Council]] Fellowships that allowed students to study abroad. He chose to go to [[Cambridge University]]'s [[Cavendish Laboratory]] in England, where he studied the [[scattering]] and [[Absorption (electromagnetic radiation)|absorption]] of [[gamma rays]]. Further research along these lines led to the discovery of the Compton effect, which demonstrated the [[particle physics|particle]] nature of [[electromagnetic radiation]]. It was a sensational discovery at the time, for the wave nature of light had been well-demonstrated, but the idea that light could have a dual nature was not easily accepted.

During [[World War II]], Compton was a key figure in the [[Manhattan Project]] that developed the first [[nuclear weapon]]s. His reports were important in launching the project. In 1942 he became head of the [[Metallurgical Laboratory]], with responsibility for producing [[nuclear reactor]]s to convert [[uranium]] into plutonium, finding ways to separate the plutonium from the uranium and to design an atomic bomb. Compton oversaw [[Enrico Fermi]]'s creation of [[Chicago Pile-1]], the first nuclear reactor, which went critical on December 2, 1942. The Metallurgical Laboratory was also responsible for the design and operation of the [[X-10 Graphite Reactor]] at [[Oak Ridge, Tennessee]]. Plutonium began being produced in the [[Hanford Site]] reactors in 1945.

After the war, Compton became Chancellor of [[Washington University in St. Louis]]. During his time as Chancellor, the university formally desegregated its undergraduate divisions, named its first female full professor, and enrolled a record number of students as wartime veterans returned to the United States.

==Early life==
[[File:Compton Heisenberg 1929 Chicago.jpg|thumb|left|Arthur Compton and [[Werner Heisenberg]] in 1929 in Chicago]]
Arthur Compton was born in [[Wooster, Ohio]] in 1892 to Elias and Otelia Catherine (née Augspurger) Compton.{{sfn|Hockey|2009}} She was named American Mother of the Year in 1939.<ref>{{cite web |url=http://www.americanmothers.org/past-national-mothers-of-the-year |title=Past National Mothers of the Year |accessdate=July 23, 2013 |publisher=American Mothers, Inc.}}</ref> They were an academic family. Elias was dean of the University of Wooster (later [[The College of Wooster]]), which Arthur also attended. Arthur's eldest brother, [[Karl Compton|Karl]], who also attended Wooster, earned a [[PhD]] in physics from [[Princeton University]] in 1912, and was president of [[MIT]] from 1930 to 1948. His second brother [[Wilson Martindale Compton|Wilson]] likewise attended Wooster, earned his PhD in economics from Princeton in 1916 and was president of the State College of Washington, later [[Washington State University]] from 1944 to 1951.{{sfn|Compton|1967|p=425}} All three brothers were members of the [[Alpha Tau Omega]] fraternity.<ref>{{cite web |url=https://depaul.digication.com/atohistory/The_1920_s |accessdate=August 10, 2013 |publisher=Alpha Tau Fraternity |title=The Official History of the Beta Beta Chapter of the Alpha Tau Omega Fraternity }}</ref>

Compton was initially interested in astronomy, and took a photograph of [[Halley's Comet]] in 1910.{{sfn|Compton|1967|pp=11–12}} Around 1913, Compton described an experiment where an examination of the motion of water in a circular tube demonstrated the rotation of the earth.<ref>{{cite journal|journal=Science|volume=37|issue=960|doi=10.1126/science.37.960.803 |pages=803–806|title=A Laboratory Method of Demonstrating the Earth's Rotation|date=May 23, 1913|last1=Compton|first1=A. H.|bibcode = 1913Sci....37..803C }}</ref> He graduated from Wooster that year with a [[Bachelor of Science]] degree. He then also entered Princeton, where he received his [[Master of Arts]] degree in 1914.<ref name="Nobel"/> He then studied for his PhD in physics under the supervision of Hereward L. Cooke, writing his dissertation on "The intensity of X-ray reflection, and the distribution of the electrons in atoms".<ref>{{cite web |url=http://library.nd.edu/chemistry/resources/genealogy/physics/documents/ComptonAH.pdf |accessdate=July 24, 2013 |title=Arthur Holly Compton (1892–1962) |publisher=[[University of Notre Dame]]}}</ref>

When he earned his PhD in 1916, Arthur, Karl and Wilson Compton became the first group of three brothers to earn PhDs from Princeton. Later, they would become the first three to simultaneously head American colleges.{{sfn|Compton|1967|p=425}} They had a sister, Mary, who married a missionary, C. Herbert Rice, who became the principal of [[Forman Christian College]] in [[Lahore]].{{sfn|Allison|1965|p=82}} In June 1916 Compton married Betty Charity McCloskey, a Wooster classmate and fellow graduate.{{sfn|Allison|1965|p=82}} They had two sons, Arthur Alan and John Joseph Compton.{{sfn|Allison|1965|p=94}}

He became a physics instructor at the [[University of Minnesota]], then spent two years a research engineer with the [[Westinghouse Electric (1886)|Westinghouse Lamp Company]] in [[Pittsburgh]], where he worked on the development of the [[sodium-vapor lamp]]. During [[World War I]] he developed aircraft instrumentation for the [[Signal Corps (United States Army)|Signal Corps]].{{sfn|Allison|1965|p=82}}

In 1919, Compton was awarded one of the first two [[United States National Research Council|National Research Council]] Fellowships that allowed students to study abroad. He chose to go to [[Cambridge University]]'s [[Cavendish Laboratory]] in England. Working with [[George Paget Thomson]], the son of [[J.J. Thomson]], he studied the [[scattering]] and [[Absorption (electromagnetic radiation)|absorption]] of [[gamma rays]].{{sfn|Allison|1965|p=83}}{{sfn|Compton|1967|p=27}} He observed that the scattered rays were more easily absorbed than the original source.<ref name="Nobel"/> Compton was greatly impressed by the Cavendish scientists, especially [[Ernest Rutherford]], [[Charles Galton Darwin]] and [[Arthur Eddington]], and he ultimately named his second son after J. J. Thomson.{{sfn|Compton|1967|p=27}}

For a time Compton was a deacon at a Baptist church. "Science can have no quarrel," he said "with a religion which postulates a God to whom men are as His children."<ref name="Time">{{cite news |url=http://www.time.com/time/magazine/article/0,9171,755635-5,00.html |newspaper=[[Time Magazine]] |title=Science: Cosmic Clearance|date=January 13, 1936}}</ref>

==Physics professor==
[[File:Time Cover Arthur H Compton.jpg|thumb|Arthur Holly Compton on the cover of ''Time Magazine'' on January 13, 1936 holding his cosmic ray detector]]
Compton returned to the United States, where he was appointed Wayman Crow Professor of Physics, and Head of the Department of Physics at [[Washington University in St. Louis]] in 1920.<ref name="Nobel"/> In 1923 he moved to the [[University of Chicago]] as Professor of Physics,<ref name="Nobel">{{cite web |url=http://www.nobelprize.org/nobel_prizes/physics/laureates/1927/compton-bio.html |title=Arthur H. Compton – Biography |publisher=Nobel Foundation |accessdate=March 19, 2013}}</ref> where he would stay for the next 22 years.

===Compton effect===
In 1922, he found that X-rays [[Quantum|quanta]] scattered by free electrons had longer [[wavelength]]s and in accordance with [[Planck constant|Planck's relation]] less [[energy]] than the incoming X-rays, the surplus energy having been transferred to the electrons.This discovery, known as the "[[Compton effect]]" or "Compton scattering" demonstrated the [[particle physics|particle]] concept of [[electromagnetic radiation]].{{sfn|Allison|1965|pp=84–86}}<ref name="Compton effect">{{cite journal|author=Compton, Arthur H.|title=A Quantum Theory of the Scattering of X-Rays by Light Elements |journal=[[Physical Review]] |volume=21|issue=5|pages=483–502|date=May 1923|doi=10.1103/PhysRev.21.483|bibcode = 1923PhRv...21..483C | url = http://prola.aps.org/abstract/PR/v21/i5/p483_1 |accessdate=September 26, 2013}}</ref>

In 1923, Compton published a paper in the ''[[Physical Review]]'' which explained the X-ray shift by attributing particle-like momentum to [[photon]]s which Einstein had invoked the use of for his 1905 Nobel prize winning explanation of the photo-electric effect. First postulated by [[Max Planck]], these conceptualized as elements of light "quantized" by containing a specific amount of energy depending only on the frequency of the light. In his paper, Compton derived the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays by assuming that each scattered X-ray photon interacted with only one electron. His paper concludes by reporting on experiments which verified his derived relation:
:<math>\lambda' - \lambda = \frac{h}{m_e c}(1-\cos{\theta}),</math>
where
:<math>\lambda</math> is the initial wavelength,
:<math>\lambda'</math> is the wavelength after scattering,
:<math>h</math> is the [[Planck constant]],
:<math>m_e</math> is the [[Electron rest mass]],
:<math>c</math> is the [[speed of light]], and
:<math>\theta</math> is the scattering angle.<ref name="Compton effect"/>

The quantity {{frac|''h''|''mec''}} is known as the [[Compton wavelength]] of the electron; it is equal to {{val|2.43|e=-12|u=m}}. The wavelength shift {{nowrap|''λ′'' − ''λ''}} lies between zero (for {{nowrap|''θ'' {{=}} 0°}}) and twice the Compton wavelength of the electron (for {{nowrap|''θ'' {{=}} 180°}}).<ref>{{cite web |url=http://math.ucr.edu/home/baez/lengths.html#compton_wavelength |accessdate=August 18, 2013 |title=The Compton wavelength of the electron |publisher=[[University of California Riverside]] }}</ref>

Compton found that some X-rays experienced no wavelength shift despite being scattered through large angles; in each of these cases the photon failed to eject an electron. Thus the magnitude of the shift is related not to the Compton wavelength of the electron, but to the Compton wavelength of the entire atom, which can be upwards of 10,000 times smaller.<ref name="Compton effect"/>

This was a sensational discovery at the time, for the wave nature of light had been well-demonstrated, and the idea that it could have a dual nature was not easily accepted. It earned Compton the [[Nobel Prize in Physics]] in 1927. Compton and Alfred W. Simon developed the method for observing at the same instant individual scattered [[X-ray]] [[photon]]s and the [[recoil]] [[electron]]s. In Germany, [[Walther Bothe]] and [[Hans Geiger]] independently developed a similar method.{{sfn|Allison|1965|pp=84–86}}

===X-rays===
[[File:Alvarez and Compton.jpeg|thumb|Compton at the University of Chicago in 1933 with graduate student [[Luis Walter Alvarez|Luis Alvarez]] next to his cosmic ray telescope.]]
In 1925 Compton demonstrated that the scattering of 130,000-volt X-rays from the first sixteen elements on the [[periodic table]] (hydrogen to sulfur) were [[polarized]], a result predicted by [[J. J. Thomson]]. William Duane from [[Harvard University]] spearheaded an effort to prove Compton's interpretation of the Compton effect was wrong. Duane carried out a series of experiments to disprove Compton, but merely would up accumulating evidence that Compton was correct. In 1924, Duane conceded that this was the case.{{sfn|Allison|1965|pp=84–86}}

Compton investigated the effect of X-rays on the sodium and chlorine nuclei in [[salt]]. He also used X-rays to investigate [[ferromagnetism]], concluding that it was a result of the alignment of [[electron spin]]s.{{sfn|Allison|1965|pp=87–88}} In 1926, he became a consultant for the Lamp Department at [[General Electric]]. In 1934, he returned to England as Eastman visiting professor at [[Oxford University]]. While there General Electric asked him to report on activities at [[General Electric Company plc]]'s research laboratory at [[Wembley]]. Compton was intrigued by the possibilities of the research there into [[fluorescent lamp]]s. His report prompted a research program in America, which developed the fluorescent lamp.{{sfn|Allison|1965|pp=88–89}}<ref>{{cite web |url=http://www.americanrhodes.org/info.html |title=Eastman Professorship |publisher=The Association of American Rhodes Scholars |accessdate=July 26, 2013 }}</ref>

Compton's first book, ''X-Rays and Electrons'', was published in 1926. In it he showed how to calculate the densities of diffracting materials from their X-ray diffraction patterns.{{sfn|Allison|1965|pp=87–88}} He revised his book with the help of [[Samuel K. Allison]] to produce ''X-Rays in Theory and Experiment'' (1935). This work remained a standard reference for the next three decades.{{sfn|Allison|1965|p=90}}

===Cosmic rays===
By the early 1930s, Compton had become interested in [[cosmic ray]]s. At the time, their existence was known but their origin and nature remained speculative. Their presence could be detected using a spherical "bomb" containing compressed air or argon gas and measuring its electrical conductivity. Trips to Europe, India, Mexico, Peru and Australia gave Compton the opportunity to measure cosmic rays at different altitudes and latitudes. Along with other groups who made observations around the globe, they found that cosmic rays were 15 per cent more intense at the poles than at the equator. Compton attributed this to the effect of cosmic rays being made up principally of charged particles, rather than photons as [[Robert Millikan]] had suggested, with the latitude effect being due to [[Earth's magnetic field]].{{sfn|Compton|1967|pp=157–163}}

==Manhattan Project==
[[File:Compton ID badge.JPG|thumb|left|300px|Arthur Compton's ID badge from the Hanford Site. For security reasons, he used a fake name.]]
In April 1941, [[Vannevar Bush]], head of the wartime [[National Defense Research Committee]] (NDRC), created a special committee headed by Compton to report on the NDRC uranium program. Compton's report, which was submitted in May 1941, foresaw the prospects of developing [[radiological weapon]]s, [[nuclear propulsion]] for ships, and [[nuclear weapons]] using [[uranium-235]] or the recently discovered [[plutonium]].{{sfn|Hewlett|Anderson|1962|pp=36-38}} In October he wrote another report on the practicality of an atomic bomb. For this report, he worked with [[Enrico Fermi]] on calculations of the [[critical mass]] of uranium-235, conservatively estimating it to be between {{convert|20|kg}} and {{convert|2|t}}. He also discussed the prospects for [[uranium enrichment]] with [[Harold Urey]], spoke with [[Eugene Wigner]] about how plutonium might be produced in a [[nuclear reactor]], and with [[Robert Serber]] about how the plutonium produced in a reactor might be separated from uranium. His report, submitted in November, stated that a bomb was feasible, although he was more conservative about its destructive power than [[Mark Oliphant]] and his British colleagues.{{sfn|Hewlett|Anderson|1962|pp=46-49}}

The final draft of Compton's November report made no mention of using plutonium, but after discussing the latest plutonium research with [[Ernest Lawrence]], Compton became convinced that a plutonium bomb was also feasible. In December, Compton was placed in charge of the plutonium project.{{sfn|Hewlett|Anderson|1962|pp=50-51}} He hoped to achieved a controlled [[chain reaction]] by January 1943, and to have a bomb by January 1945. To tackle the problem, he had all the different research groups working on plutonium and nuclear reactor design at [[Columbia University]], [[Princeton University]] and the [[University of California, Berkeley]], concentrated together as the [[Metallurgical Laboratory]] in Chicago. Its objectives were to produce reactors to convert uranium to plutonium, to find ways to chemically separate the plutonium from the uranium and to design a bomb.{{sfn|Hewlett|Anderson|1962|pp=54-55}} In June 1942, the [[United States Army Corps of Engineers]] assumed control of the nuclear weapons program and Compton's Metallurgical Laboratory became part of the [[Manhattan Project]].{{sfn|Hewlett|Anderson|1962|pp=74-75}} That month, Compton gave [[Robert Oppenheimer]] responsibility for bomb design.{{sfn|Hewlett|Anderson|1962|p=103}} It fell to Compton to decide which of the different types of reactor designs that the Metallurgical Laboratory scientists had devised should be built, even though a successful reactor had not yet been built.{{sfn|Hewlett|Anderson|1962|pp=180-181}}

When labor disputes delayed construction of the Metallurgical Laboratory's new home in the [[Red Gate Woods]], Compton decided to build [[Chicago Pile-1]], the first nuclear reactor, under the stands at [[Stagg Field]].{{sfn|Hewlett|Anderson|1962|pp=108-109}} Under Fermi's direction, it went critical on December 2, 1942.{{sfn|Hewlett|Anderson|1962|p=174}} Compton arranged for [[Mallinckrodt]] to undertake the purification of uranium ore,{{sfn|Allison|1965|p=92}} and with [[DuPont]] to build the plutonium semi-works at [[Oak Ridge, Tennessee]].{{sfn|Hewlett|Anderson|1962|pp=190-191}}

A major crisis for the plutonium program occurred in July 1943, when [[Emilio Segrè]]'s group confirmed that plutonium created in the [[X-10 Graphite Reactor]] at Oak Ridge contained high levels of [[plutonium-240]]. Its [[spontaneous fission]] ruled out the use of plutonium in a [[gun-type nuclear weapon]]. Oppenheimer's [[Los Alamos Laboratory]] met the challenge by designing and building an [[implosion-type nuclear weapon]].{{sfn|Hewlett|Anderson|1962|pp=46-49}}

Compton was at the [[Hanford site]] in September 1944 to watch the first reactor being brought online. The first batch of uranium slugs was fed into Reactor B at Hanford in November 1944, and shipments of plutonium to Los Alamos began in February 1945. {{sfn|Hewlett|Anderson|1962|pp=304-310}} Throughout the war, Compton would remain a prominent scientific adviser and administrator. In 1945, he served, along with Lawrence, Oppenheimer, and Fermi, on the Scientific Panel which recommended military use of the atomic bomb against Japan.<ref>{{cite web |url=http://www.nuclearfiles.org/menu/key-issues/nuclear-weapons/history/pre-cold-war/interim-committee/interim-committee-recommendations_1945-06-16.htm |title=Recommendations on the Immediate Use of Nuclear Weapons |publisher=nuclearfiles.org |accessdate=July 27, 2013}}</ref> He was awarded the [[Medal for Merit]] for his services to the Manhattan Project.{{sfn|Allison|1965|p=93}}

==Return to Washington University==
After the war ended, Compton resigned his chair as Charles H. Swift Distinguished Service Professor of Physics at the University of Chicago and returned to Washington University in St. Louis, where he was inaugurated as the university's ninth Chancellor in 1946.{{sfn|Allison|1965|p=93}} During Compton's time as Chancellor, the university formally desegregated its undergraduate divisions in 1952, named its first female full professor, and enrolled a record number of students as wartime veterans returned to the United States. His reputation and connections in national scientific circles allowed him to recruit many nationally renowned scientific researchers to the university. Despite Compton's accomplishments, he was criticized then, and subsequently by historians, for moving slowly toward full [[racial integration]], making Washington University the last major institution of higher learning in [[St. Louis, Missouri|St. Louis]] to open its doors to [[African American]]s.<ref>{{cite journal |first=Amy M. |last=Pfeiffenberger |title=Democracy at Home: The Struggle to Desegregate Washington University in the Postwar Era |journal=Gateway-Heritage |publisher=Missouri Historical Society|volume=10 |issue=3 |date=Winter 1989 |pages=17–24}}</ref>

Compton retired as Chancellor in 1954, but remained on the faculty as Distinguished Service Professor of Natural Philosophy until his retirement from the full-time faculty in 1961. In retirement he wrote ''Atomic Quest'', a personal account of his role in the Manhattan Project, which was published in 1956.{{sfn|Allison|1965|p=93}}

==Philosophy==
Compton was one of a handful of scientists and philosophers to propose a [[two-stage model of free will]]. Others include [[William James]], [[Henri Poincaré]], [[Karl Popper]], [[Henry Margenau]], and [[Daniel Dennett]].<ref>{{cite web |url=http://www.informationphilosopher.com/freedom/two-stage_models.html |title=Two-Stage Models for Free Will |publisher=The Information Philosopher |accessdate=July 27, 2013}}</ref> In 1931, Compton championed the idea of human freedom based on [[quantum indeterminacy]] and invented the notion of amplification of microscopic quantum events to bring [[Indeterminism|chance]] into the macroscopic world. In his somewhat bizarre mechanism, he imagined sticks of dynamite attached to his amplifier, anticipating the [[Schrödinger's cat]] paradox, which was published in 1935.<ref>{{cite journal|journal=Science|volume= 74|issue=1911|pmid=17808216|doi=10.1126/science.74.1911.172 |page=172|title=The Uncertainty Principle and Free Will |date=August 14, 1931|last1=Compton|first1=A. H.|bibcode = 1931Sci....74..172C }}</ref>

Reacting to criticisms that his ideas made chance the direct cause of our actions, Compton clarified the two-stage nature of his idea in an ''Atlantic Monthly'' article in 1955. First there is a range of random possible events, then one adds a determining factor in the act of choice.{{quote|A set of known physical conditions is not adequate to specify precisely what a forthcoming event will be. These conditions, insofar as they can be known, define instead a range of possible events from among which some particular event will occur. When one exercises freedom, by his act of choice he is himself adding a factor not supplied by the physical conditions and is thus himself determining what will occur. That he does so is known only to the person himself. From the outside one can see in his act only the working of physical law. It is the inner knowledge that he is in fact doing what he intends to do that tells the actor himself that he is free.{{sfn|Compton|1967|p=115}}}}

==Death and legacy==
Compton died in [[Berkeley, California]], from a [[cerebral hemorrhage]] on March 15, 1962. He was survived by his wife and sons.{{sfn|Allison|1965|p=94}} He is buried in the Wooster Cemetery in Wooster, Ohio.<ref>{{findagrave|22551|accessdate=July 27, 2013}}</ref>

Compton received many awards in his lifetime, including the Nobel Prize for Physics in 1927, the Matteucci Gold Medal in 1933, the [[Royal Society]]'s [[Hughes Medal]] and the [[Franklin Institute]]'s [[Benjamin Franklin Medal (Franklin Institute)|Benjamin Franklin Medal]] in 1940.{{sfn|Allison|1965|p=97}} He is commemorated in various ways. The [[Compton (crater)|Compton crater]] on the [[Moon]] is co-named for Compton and his brother Karl.<ref>{{cite web |url=http://the-moon.wikispaces.com/Compton |title=Compton |publisher=Tangient LLC |accessdate=July 27, 2013}}</ref> The physics research building at Washington University in St Louis is named in his honor.<ref>{{cite web |url=http://wustl.edu/community/visitors/tour/danforth/compton-laboratory.html |title=Arthur Holly Compton Laboratory of Physics |publisher=[[Washington University]] |accessdate=July 27, 2013}}</ref> Compton invented a more gentle, elongated, and ramped version of the [[speed bump]] called a "Holly hump," many of which are on the roads of the [[Washington University in St. Louis]] campus.<ref>{{cite web |url=http://library.wustl.edu/units/spec/archives/facts/compton_bumps.html |title=Compton Speed Bumps for Traffic Control, 1953 |publisher=[[Washington University]] |accessdate=July 27, 2013}}</ref> The University of Chicago Residence Halls remembered Compton and his achievements by dedicating [[Arthur H. Compton House]] in his honor.<ref>{{cite web |url=http://compton-house.uchicago.edu |title=Compton House |publisher=[[University of Chicago]] |accessdate=July 27, 2013}}</ref> Compton also has a star on the [[St. Louis Walk of Fame]].<ref>{{cite web|url=http://www.stlouiswalkoffame.org/inductees/?view=achievement|title=St. Louis Walk of Fame Inductees|last=St. Louis Walk of Fame|publisher=stlouiswalkoffame.org|accessdate=25 April 2013}}</ref>[[Arthur H. Compton House]] in [[Chicago]] is listed as a [[National Historic Landmark]].<ref>{{cite web|url=http://tps.cr.nps.gov/nhl/detail.cfm?ResourceId=1613&ResourceType=Building|title=Compton, Arthur H., House |accessdate=July 27, 2013|work=National Historic Landmark summary listing|publisher=National Park Service}}</ref> [[NASA]]'s [[Compton Gamma Ray Observatory]] was named in honor of Compton. The Compton effect is central to the [[gamma ray]] detection instruments aboard the observatory.<ref>{{cite web |url=http://heasarc.gsfc.nasa.gov/docs/cgro/index.html |publisher=[[NASA]] |title=The CGRO Mission (1991 - 2000) |accessdate=July 27, 2013}}</ref>

==Bibliography==
* {{cite book |last=Compton |first=Arthur |year=1926 |title=X-Rays and Electrons: an Outline of Recent X-Ray Theory |location=New York |publisher=D. Van Nostrand Company, Inc |oclc=1871779 }}
* {{cite book |last=Compton |first=Arthur |year=1935 |first2=With S. K. Allison |authorlink2=Samuel King Allison |title=X-Rays in Theory and Experiment |location=New York |publisher=D. Van Nostrand Company, Inc. |oclc=853654 }}
* {{cite book |last=Compton |first=Arthur |year=1935 |title=The Freedom of Man |location=New Haven |publisher=Yale University Press |oclc=5723621 }}
* {{cite book |last=Compton |first=Arthur |year=1940 |title=The Human Meaning of Science |location=Chapel Hill |publisher=University of North Carolina Press |oclc=311688}}
* {{cite book |last=Compton |first=Arthur |year=1949 |title=Man's Destiny in Eternity |location=Boston |publisher=Beacon Press |oclc=4739240 }}
* {{cite book |last=Compton |first=Arthur |year=1956 |title=Atomic Quest |location=New York |publisher=Oxford University Press |oclc=173307 }}
* {{cite book |last=Compton |first=Arthur |year=1967 |title=The Cosmos of Arthur Holly Compton |location=New York |publisher=Alfred A. Knopf |editor-first=Marjorie |editor-last=Johnston |oclc=953130 |ref=harv }}
* {{cite book |last=Compton |first=Arthur |year=1973 |title=Scientific Papers of Arthur Holly Compton |location=Chicago: University of Chicago Press; |editor-first=Robert S. |editor-last=Shankland |editor-link=Robert S. Shankland |isbn=9780226114309 |oclc=962635 }}

==Notes==
{{reflist|30em}}

==References==
* {{cite book |last=Allison |first =Samuel K. |authorlink=Samuel King Allison |year=1965 |title=Arthur Holly Compton 1892—1962 |series=Biographical Memoirs |publisher=National Academy of Sciences |ref=harv }}
*{{cite book |last=Hewlett |first=Richard G. |authorlink=Richard G. Hewlett |last2=Anderson |first2=Oscar E. |title=The New World, 1939–1946 |url=http://www.governmentattic.org/5docs/TheNewWorld1939-1946.pdf|accessdate=26 March 2013|location=University Park|publisher=Pennsylvania State University Press |year=1962 |isbn=0-520-07186-7|oclc=637004643 |ref=harv}}
*{{cite book |last=Hockey |first=Thomas |title=The Biographical Encyclopedia of Astronomers |year=2009 |publisher=[[Springer Publishing]] |isbn=978-0-387-31022-0 |accessdate=August 22, 2012 |url=http://www.springerreference.com/docs/html/chapterdbid/58299.html |ref=harv}}

==External links==
{{Portal|Biography|History of Science|Nuclear technology|Physics|World War II}}
*[http://books.google.com/books?id=QSgDAAAAMBAJ&pg=PA60 "Strange Instrument Built To Solve Mystery Of Cosmic Rays", April 1932, Popular Science] article about Compton on world wide research on cosmic rays
*[http://www.wustl.edu/about/facts/chancellors/compton.html Arthur Compton biographical entry] at [[Washington University in Saint Louis]]
*[http://alsos.wlu.edu/qsearch.aspx?browse=people/Compton,+Arthur Annotated bibliography for Arthur Compton from the Alsos Digital Library for Nuclear Issues]
*[http://www.informationphilosopher.com/solutions/scientists/compton/ Arthur Holly Compton on Information Philosopher]
*[http://nobelprize.org/nobel_prizes/physics/laureates/1927/compton-bio.html Nobelprize.com Biography]
*[http://www.osti.gov/accomplishments/compton.html Biography and Bibliographic Resources], from the [[Office of Scientific and Technical Information]], [[United States Department of Energy]]
*{{Find a Grave|22551}}
*[http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/compton-arthur-h.pdf National Academy of Sciences Biographical Memoir]

{{WUSTL Chancellors}}
{{Nobel Prize in Physics Laureates 1926-1950}}
{{Presidents of the American Physical Society}}
{{Manhattan Project}}
{{WUSTL |state=expanded}}

{{Authority control|VIAF=51769421|LCCN=n/50/030032|GND=118676695}}

{{Persondata 
| NAME = Compton, Arthur Holly
| ALTERNATIVE NAMES = Arthur Compton
| SHORT DESCRIPTION = [[USA|American]] [[Physicist]]
| DATE OF BIRTH = 1892-09-10
| PLACE OF BIRTH = Wooster, Ohio, United States
| DATE OF DEATH = 1962-03-15
| PLACE OF DEATH = Berkeley, California, United States
}}
{{DEFAULTSORT:Compton, Arthur}}
[[Category:1892 births]]
[[Category:1962 deaths]]
[[Category:American Nobel laureates]]
[[Category:American physicists]]
[[Category:American Presbyterians]]
[[Category:Chancellors of Washington University in St. Louis]]
[[Category:College of Wooster alumni]]
[[Category:Forman Christian College faculty]]
[[Category:Manhattan Project people]]
[[Category:Nobel laureates in Physics]]
[[Category:Particle physicists]]
[[Category:People from Wayne County, Ohio]]
[[Category:Princeton University alumni]]
[[Category:Quantum physicists]]
[[Category:Optical physicists]]
[[Category:Fellows of the American Association for the Advancement of Science]]
[[Category:Recipients of the Pour le Mérite (civil class)]]
[[Category:College of Wooster]]
[[Category:Washington University in St. Louis faculty]]
[[Category:Guggenheim Fellows]]
[[Category:Theoretical physicists]]
[[Category:Spectroscopists]]
[[Category:Naval Consulting Board]]

James–Stein estimator

2013-10-09T20:04:07Z

128.101.152.128: /* Extensions */ double indenting since the paragraph is already indented

{{lowercase}}
In [[combinatorics|combinatorial]] [[mathematics]], the '''q-difference polynomials''' or '''q-harmonic polynomials''' are a [[polynomial sequence]] defined in terms of the [[q-derivative]]. They are a type of [[Brenke polynomial]], and generalize the [[Appell polynomial]]s. See also [[Sheffer sequence]].

==Definition==
The q-difference polynomials satisfy the relation

:<math>\left(\frac {d}{dz}\right)_q p_n(z) =
\frac{p_n(qz)-p_n(z)} {qz-z} = p_{n-1}(z)</math>

where the derivative symbol on the left is the q-derivative. In the limit of <math>q\to 1</math>, this becomes the definition of the Appell polynomials:

:<math>\frac{d}{dz}p_n(z) = p_{n-1}(z).</math>

==Generating function==
The [[generating function]] for these polynomials is of the type of generating function for Brenke polynomials, namely

:<math>A(w)e_q(zw) = \sum_{n=0}^\infty p_n(z) w^n</math>

where <math>e_q(t)</math> is the [[q-exponential]]:
:<math>e_q(t)=\sum_{n=0}^\infty \frac{t^n}{[n]_q!}=
\sum_{n=0}^\infty \frac{t^n (1-q)^n}{(q;q)_n}.</math>

Here, <math>[n]_q!</math> is the [[q-factorial]] and

:<math>(q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q)</math>

is the [[q-Pochhammer symbol]]. The function <math>A(w)</math> is arbitrary but assumed to have an expansion

:<math>A(w)=\sum_{n=0}^\infty a_n w^n \mbox{ with } a_0 \ne 0. </math>

Any such <math>A(w)</math> gives a sequence of q-difference polynomials.

==References==
* A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", ''Riv. Mat. Univ. Parma'', '''5''' (1954) 325-337.
* Ralph P. Boas, Jr. and R. Creighton Buck, ''Polynomial Expansions of Analytic Functions (Second Printing Corrected)'', (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. ''(Provides a very brief discussion of convergence.)''

[[Category:Q-analogs]]
[[Category:Polynomials]]

Notation in probability and statistics

2013-08-20T03:23:46Z

128.101.167.121: fixed the sigma character

In [[algebraic geometry]], a branch of [[mathematics]], a [[morphism]] ''f'' : ''X'' → ''Y'' of [[scheme (mathematics)|schemes]] is '''quasi-finite''' if it is [[Glossary of scheme theory|finite type]] and satisfies any of the following equivalent conditions:<ref>EGA II, Définition 6.2.3</ref>

* Every point ''x'' of ''X'' is isolated in its fiber ''f''−1(''f''(''x'')). In other words, every fiber is a discrete (hence finite) set.
* For every point ''x'' of ''X'', the scheme {{nowrap|''f''−1(''f''(''x'')) {{=}} ''X'' ×YSpec κ(''f''(''x''))}} is a finite κ(''f''(''x'')) scheme. (Here κ(''p'') is the residue field at a point ''p''.)
* For every point ''x'' of ''X'', <math>\mathcal{O}_{X,x}\otimes \kappa(f(x))</math> is finitely generated over <math>\kappa(f(x))</math>.

Quasi-finite morphisms were originally defined by [[Alexander Grothendieck]] in [[Séminaire de géométrie algébrique|SGA]] 1 and did not include the finite type hypothesis. This hypothesis was added to the definition in [[Éléments de géométrie algébrique|EGA]] II 6.2 because it makes it possible to give an algebraic characterization of quasi-finiteness in terms of [[Stalk (sheaf)|stalks]].

For a general morphism {{nowrap|''f'' : ''X'' → ''Y''}} and a point ''x'' in ''X'', ''f'' is said to be '''quasi-finite''' at ''x'' if there exist open affine neighborhoods ''U'' of ''x'' and ''V'' of ''f''(''x'') such that ''f''(''U'') is contained in ''V'' and such that the restriction {{nowrap|''f'' : ''U'' → ''V''}} is quasi-finite. ''f'' is '''locally quasi-finite''' if it is quasi-finite at every point in ''X''.<ref>EGA III, ErrIII, 20.</ref> A quasi-compact locally quasi-finite morphism is quasi-finite.

== Properties ==
For a morphism ''f'', the following properties are true.<ref name=autogenerated1>EGA II, Proposition 6.2.4.</ref>
* If ''f'' is quasi-finite, then the induced map ''f''red between [[reduced scheme]]s is quasi-finite.
* If ''f'' is a closed immersion, then ''f'' is quasi-finite.
* If ''X'' is noetherian and ''f'' is an immersion, then ''f'' is quasi-finite.
* If {{nowrap|g : ''Y'' → ''Z''}}, and if {{nowrap|''g'' ∘ ''f''}} is quasi-finite, then ''f'' is quasi-finite if any of the following are true:
*#''g'' is separated,
*#''X'' is noetherian,
*#{{nowrap|''X'' ×''Z'' ''Y''}} is locally noetherian.

Quasi-finiteness is preserved by base change. The composite and fiber product of quasi-finite morphisms is quasi-finite.<ref name=autogenerated1 />

If ''f'' is [[Glossary of scheme theory#Unramified and étale morphisms|unramified]] at a point ''x'', then ''f'' is quasi-finite at ''x''. Conversely, if ''f'' is quasi-finite at ''x'', and if also <math>\mathcal{O}_{f^{-1}(f(x)),x}</math>, the local ring of ''x'' in the fiber ''f''−1(''f''(''x'')), is a field and a finite separable extension of κ(''f''(''x'')), then ''f'' is unramified at ''x''.<ref>EGA IV4, Théorème 17.4.1.</ref>

[[Finite morphism]]s are quasi-finite.<ref>EGA II, Corollaire 6.1.7.</ref> A quasi-finite [[proper morphism]] locally of finite presentation is finite.<ref>EGA IV3, Théorème 8.11.1.</ref>

A generalized form of '''[[Zariski Main Theorem]]''' is the following:<ref>EGA IV3, Théorème 8.12.6.</ref> Suppose ''Y'' is [[compact space#Introduction and definition|quasi-compact]] and quasi-separated. Let ''f'' be quasi-finite, separated and of finite presentation. Then ''f'' factors as <math> X \hookrightarrow X' \to Y</math> where the first morphism is an open immersion and the second is finite. (''X'' is open in a finite scheme over ''Y''.)

== Notes ==
{{reflist}}

== References ==
*{{cite book
| last = Grothendieck
| first = Alexandre
| authorlink = Alexandre Grothendieck
| coauthors = Michèle Raynaud
| title = Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques '''3''')
| origyear = 1971
| edition = Updated
| year = 2003
| publisher = Société Mathématique de France
| language = French
| isbn = 2-85629-141-4
| nopp = true
| page = xviii+327
}}
*{{cite journal
| last = Grothendieck
| first = Alexandre
| authorlink = Alexandre Grothendieck
| coauthors = [[Jean Dieudonné]]
| year = 1961
| title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes | journal = Publications Mathématiques de l'IHÉS
| volume = 8
| pages = 5–222
| url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_
}}
*{{cite journal
| last = Grothendieck
| first = Alexandre
| authorlink = Alexandre Grothendieck
| coauthors = [[Jean Dieudonné]]
| year = 1966
| title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie
| journal = Publications Mathématiques de l'IHÉS
| volume = 28
| pages = 5–255
| url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_
}}

[[Category:Morphisms of schemes]]

Structure constants

2012-08-26T03:20:46Z

128.101.152.65:

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