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Inequality of arithmetic and geometric means

2014-10-17T18:49:10Z

146.186.134.244: Explicit equations using natural log, to show why convexity is important

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Majorization

2014-10-04T16:29:44Z

146.186.130.194: /* Equivalent conditions */

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Debye sheath

2014-02-25T19:29:10Z

146.186.210.22: /* The planar sheath equation */

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David Harbater

2013-09-07T23:35:08Z

146.186.130.209:

{{Expert-subject|Physics|date=November 2008}}

In [[particle physics]] the extra symmetry of the Higgs potential in the [[Standard Model]]
:<math>V_{SM} = -\lambda (H^\dagger H) + \mu(H^\dagger H)^2</math>
responsible for keeping <math>\rho</math> ≈ 1 and insuring small corrections to <math>\rho</math> is called a '''custodial symmetry'''.<ref>P. Sikivie, L. Susskind, M. B. Voloshin and V. I. Zakharov, Nucl. Phys. B 173, 189 (1980).</ref>
(Note <math>\rho</math> is a ratio involving the masses of the weak bosons and the [[Weinberg angle]]).

With one or more [[electroweak]] Higgs doublets in the [[Higgs sector]], the [[effective action]] term <math>\left|H^\dagger D_\mu H\right|^2/\Lambda^2</math> which generically arises whenever we have new physics [[beyond the Standard Model]] at the scale Λ contributes to the [[Peskin-Takeuchi]] T parameter. However, current precision electroweak measurements restrict Λ to more than a few [[TeV]]. This will not be a problem if we have no new physics right up to at least that scale. However, attempts to solve the [[gauge hierarchy problem]] generically require the addition of new particles below that scale. The preferred way of preventing the nasty <math>\left|H^\dagger D_\mu H\right|^2/\Lambda^2</math> term from being generated is to introduce an [[approximate symmetry]] which acts upon the Higgs sector. In addition to the gauged SU(2)W which acts exactly upon the Higgs doublets, we will also introduce another approximate global SU(2)R symmetry which also acts upon the Higgs doublet. The Higgs doublet is now a [[real representation]] (2,2) of SU(2)L × SU(2)R with four real components. Here, we have relabeled W as L following the standard convention. Such a symmetry will not forbid Higgs kinetic terms like <math>D^\mu H^\dagger D_\mu H</math> or tachyonic mass terms like <math>H^\dagger H</math> or self-coupling terms like <math>\left(H^\dagger H\right)^2</math> (fortunately!) but will outlaw <math>\left|H^\dagger D_\mu H\right|^2/\Lambda^2</math>. On the other hand, such an SU(2)R symmetry can never be exact and unbroken because otherwise, the up-type and the down-type Yukawa couplings will be exactly identical. Besides, SU(2)R does not map the [[hypercharge]] symmetry U(1)Y to itself but this is not too much of a problem because the hypercharge gauge coupling strength is small and in the limit as it goes to zero, we won't have a problem. In the parlance of model building, we say that U(1)Y is weakly gauged and this explicitly breaks SU(2)R. After the Higgs doublet acquires a nonzero [[vacuum expectation value]], the (approximate) SU(2)L × SU(2)R symmetry is spontaneously broken to the (approximate) [[diagonal subgroup]] SU(2)V. This approximate symmetry is called the '''custodial symmetry'''.<ref>B. Grzadkowski, M. Maniatis, Jose Wudka, "Note on Custodial Symmetry in the Two-Higgs-Doublet Model", [http://arxiv.org/abs/1011.5228 arXiv:1011.5228].</ref>

==See also==
*[[Peskin-Takeuchi parameter]]
*[[left-right model]]
*[[little Higgs]]

==References==
{{Reflist}}

[[Category:Electroweak theory]]

Thermal ellipsoid

2013-04-02T18:47:34Z

146.186.10.12:

In [[number theory]], a '''Durfee square''' is an attribute of an [[Partition (number theory)|integer partition]]. A partition of ''n'' has a Durfee square of side ''s'' if ''s'' is the largest number such that the partition contains at least ''s'' parts with values ≥ ''s''.<ref>{{Cite book
| last = Andrews
| first = George E.
| coauthors = Eriksson, Kimmo
| title = Integer Partitions
| publisher = Cambridge University Press
| year = 2004
| pages = 76
| isbn = 0-521-60090-1}}</ref> An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's [[Ferrers diagram]].<ref>{{MathWorld |urlname=DurfeeSquare |title=Durfee Square }}</ref>

The '''Durfee symbol''' consists of the two partitions represented by the points to the right or below the Durfee square.

==Examples==
The partition 4 + 3 + 3 + 2 + 1 + 1:

:{|
|- style="vertical-align:top; text-align:left;"
| [[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:GrayDot.svg|16px|*]] [[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]] [[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]] [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]] [[File:GrayDot.svg|16px|*]] [[File:GrayDot.svg|16px|*]]
|}

has a Durfee square of side 3 (in red) because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. Its Durfee symbol consists of the 2 partitions 1 and 3+1.

==History==
Durfee squares are named after [[William Durfee]], a student of English mathematician [[James Joseph Sylvester]]. In a letter to [[Arthur Cayley]] in 1883, Sylvester wrote:<ref>{{Cite book
| last = Parshall
| first = Karen Hunger
| title = James Joseph Sylvester: life and work in letters
| publisher = Oxford University Press
| year = 1998
| pages = 224
| isbn = 0-19-850391-1}}</ref>

{{quote|"''Durfee's square is a great invention of the importance of which its author has no conception.''"}}

==Properties==
It is clear from the visual definition that the Durfee square of a partition and its conjugate partition have the same size. The partitions of an integer ''n'' contain Durfee squares with sides up to and including <math>\lfloor \sqrt{n} \rfloor</math>.

==See also==
*[[H-index]]

==References==
{{reflist}}

[[Category:Number theory]]

Superprocess

2012-10-15T18:52:32Z

146.186.131.40: /* References */

In [[mathematical physics]], the '''Penrose transform''', introduced by {{harvs|txt|authorlink=Roger Penrose|first=Roger |last=Penrose|year1=1967|year2=1968|year3=1969}}, is a complex analogue of the [[Radon transform]] that relates [[massless field]]s on spacetime to [[sheaf cohomology|cohomology]] of [[sheaf (mathematics)|sheaves]] on [[complex projective space]]. The projective space in question is the [[twistor space]], a geometrical space naturally associated to the original spacetime, and the twistor transform is also geometrically natural in the sense of [[integral geometry]]. The Penrose transform is a major component of classical [[twistor theory]].

==Overview==

Abstractly, the Penrose transform operates on a double [[fibration]] of a space ''Y'', over two spaces ''X'' and ''Z''

:<math>Z\xleftarrow{\eta} Y \xrightarrow{\tau} X.</math>

In the classical Penrose transform, ''Y'' is the [[spin bundle]], ''X'' is a compactified and complexified form of [[Minkowski space]] and ''Z'' is the twistor space. More generally examples come from double fibrations of the form

:<math>G/H_1\xleftarrow{\eta} G/(H_1\cap H_2) \xrightarrow{\tau} G/H_2</math>
where ''G'' is a complex semisimple Lie group and ''H''1 and ''H''2 are parabolic subgroups.

The Penrose transform operates in two stages. First, one [[pullback|pulls back]] the sheaf cohomology groups ''H''''r''(''Z'','''F''') to the sheaf cohomology ''H''''r''(''Y'',η−1'''F''') on ''Y''; in many cases where the Penrose transform is of interest, this pullback turns out to be an isomorphism. One then pushes the resulting cohomology classes down to ''X''; that is, one investigates the [[direct image]] of a cohomology class by means of the [[Leray spectral sequence]]. The resulting direct image is then interpreted in terms of differential equations. In the case of the classical
Penrose transform, the resulting differential equations are precisely the massless field equations for a given spin.

==Example==

The classical example is given as follows
*The "twistor space" ''Z'' is complex projective 3-space '''CP'''3, which is also the Grassmannian Gr1('''C'''4) of lines in 4-dimensional complex space.
*''X'' = Gr2('''C'''4), the Grassmannian of 2-planes in 4-dimensional complex space. This is a compactification of complex Minkowski space.
*''Y'' is the flag manifold whose elements correspond to a line in a plane of '''C'''4.
*''G'' is the group SL4('''C''') and ''H''1 and ''H''2 are the parabolic subgroups fixing a line or a plane containing this line.

The maps from ''Y'' to ''X'' and ''Z'' are the natural projections.

==Penrose–Ward transform==

The Penrose–Ward transform is a non-linear modification of the Penrose transform, introduced by {{harvtxt|Ward|1977}}, that (among other things) relates holomorphic vector bundles on 3-dimensional complex projective space '''CP'''3 to solutions of the [[self-dual Yang–Mills equations]] on '''S'''4.
{{harvtxt|Atiyah|Ward|1977}} used this to describe instantons in terms of algebraic vector bundles on complex projective 3-space. and {{harvtxt|Atiyah|1979}} explained how this could be used to classify instantons on a 4-sphere.

==References==

*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Ward | first2=R. S. | title=Instantons and algebraic geometry | doi=10.1007/BF01626514 | publisher=Springer Berlin / Heidelberg | DUPLICATE DATA: doi=10.1007/BF01626514 | mr=0494098 | year=1977 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=55 | pages=117–124|bibcode = 1977CMaPh..55..117A }}
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=Geometry of Yang-Mills fields | publisher=Scuola Normale Superiore Pisa, Pisa | series=Lezioni Fermiane | isbn=978-88-7642-303-1 | mr=554924 | year=1979}}
*{{Citation | last1=Baston | first1=Robert J. | last2=Eastwood | first2=Michael G. | title=The Penrose transform | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-853565-2 | mr=1038279 | year=1989}}.
*{{Citation | last1=Eastwood | first1=Michael | editor1-last=Eastwood | editor1-first=Michael | editor2-last=Wolf | editor2-first=Joseph | editor3-last=Zierau. | editor3-first=Roger | title=The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992) | url=http://www.ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-154 | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Contemp. Math. | isbn=978-0-8218-5176-0 | mr=1246377 | year=1993 | volume=154 | chapter=Introduction to Penrose transform | pages=71–75}}
*{{eom|id=P/p120100|first=M.G.|last= Eastwood}}
*{{Citation | last=David | first=Liana | title=The Penrose transform and its applications|publisher=[[University of Edinburgh]]|year=2001|url=http://www.maths.ed.ac.uk/pg/thesis/david.pdf}}; Doctor of Philosophy thesis.
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Twistor algebra | url=http://link.aip.org/link/JMAPAQ/v8/i2/p345/s1 | doi=10.1063/1.1705200 | mr=0216828 | year=1967 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=8 | pages=345–366|bibcode = 1967JMP.....8..345P }}
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Twistor quantisation and curved space-time | doi=10.1007/BF00668831 | publisher=Springer Netherlands | DUPLICATE DATA: doi=10.1007/BF00668831 | year=1968 | journal=International Journal of Theoretical Physics | issn=0020-7748 | volume=1 | pages=61–99|bibcode = 1968IJTP....1...61P }}
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Solutions of the Zero‐Rest‐Mass Equations | url=http://link.aip.org/link/JMAPAQ/v10/i1/p38/s1 | doi=10.1063/1.1664756 | year=1969 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=10 | issue=1 | pages=38–39|bibcode = 1969JMP....10...38P }}
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | last2=Rindler | first2=Wolfgang | author2-link=Wolfgang Rindler | title=Spinors and space-time. Vol. 2 | publisher=[[Cambridge University Press]] | series=Cambridge Monographs on Mathematical Physics | isbn=978-0-521-25267-6 | mr=838301 | year=1986}}.
*{{Citation | last1=Ward | first1=R. S. | title=On self-dual gauge fields | doi=10.1016/0375-9601(77)90842-8 | mr=0443823 | year=1977 | journal=Physics Letters A | issn=0375-9601 | volume=61 | issue=2 | pages=81–82|bibcode = 1977PhLA...61...81W }}

{{Topics of twistor theory}}

[[Category:Integral geometry]]

Talk:Projective vector field

2008-02-15T01:06:53Z

146.186.134.176:

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