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| {{About|the mathematical process|the industrial OMEGA process|OMEGA process}}
| | I'm Stewart and I live in Esprick. <br>I'm interested in Asian Studies, Singing and Turkish art. I like to travel and watching Game of Thrones.<br><br>Here is my web-site :: [http://quietpyramid.com/?p=347356 белковая диета для похудения] |
| In mathematics, '''Cayley's Ω process''', introduced by {{harvs|txt|authorlink=Arthur Cayley|first=Arthur |last=Cayley|year=1846}}, is a relatively invariant [[differential operator]] on the [[general linear group]], that is used to construct [[invariant (mathematics)|invariants]] of a [[group action]].
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| As a [[partial differential operator]] acting on functions of ''n''<sup>2</sup> variables ''x''<sub>''ij''</sup>, the omega operator is given by the [[determinant]]
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| :<math>
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| \Omega = \begin{vmatrix} \frac{\partial}{\partial x_{11}} & \cdots &\frac{\partial}{\partial x_{1n}} \\ \vdots& \ddots & \vdots\\ \frac{\partial}{\partial x_{n1}} & \cdots &\frac{\partial}{\partial x_{nn}} \end{vmatrix}.
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| </math>
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| For [[Invariant of a binary form|binary forms]] ''f'' in ''x''<sub>1</sub>, ''y''<sub>1</sub> and ''g'' in ''x''<sub>2</sub>, ''y''<sub>2</sub> the Ω operator is <math>\frac{\partial^2 fg}{\partial x_1 \partial y_2} - \frac{\partial^2 fg}{\partial x_2 \partial y_1}</math>. The ''r''-fold Ω process Ω<sup>''r''</sup>(''f'', ''g'') on two forms ''f'' and ''g'' in the variables ''x'' and ''y'' is then
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| # Convert ''f'' to a form in ''x''<sub>1</sub>, ''y''<sub>1</sub> and ''g'' to a form in ''x''<sub>2</sub>, ''y''<sub>2</sub>
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| # Apply the Ω operator ''r'' times to the function ''fg'', that is, ''f'' times ''g'' in these four variables
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| # Substitute ''x'' for ''x''<sub>1</sub> and ''x''<sub>2</sub>, ''y'' for ''y''<sub>1</sub> and ''y''<sub>2</sub> in the result
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| The result of the ''r''-fold Ω process Ω<sup>''r''</sup>(''f'', ''g'') on the two forms ''f'' and ''g'' is also called the ''r''-th [[transvectant]] and is commonly written (''f'', ''g'')<sup>''r''</sup>.
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| ==Applications==
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| Cayley's Ω process appears in [[Capelli's identity]], which
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| {{harvtxt|Weyl|1946}} used to find generators for the invariants of various classical groups acting on natural polynomial algebras.
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| {{harvtxt|Hilbert|1890}} used Cayley's Ω process in his proof of finite generation of rings of invariants of the general linear group. His use of the Ω process gives an explicit formula for the [[Reynolds operator]] of the special linear group.
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| Cayley's Ω process is used to define [[transvectant]]s.
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| ==References==
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| *{{citation|first=Arthur|last=Cayley|title=On linear transformations|journal=Cambridge and Dublin mathematical journal|volume=1|year=1846|pages=104–122|url=http://books.google.com/books?id=PBcAAAAAMAAJ&dq=Cambridge%20and%20Dublin%20mathematical%20journal%201846&pg=PR3#v=onepage&q=Cambridge%20and%20Dublin%20mathematical%20journal%201846&f=false}} Reprinted in {{citation|last=Cayley|title=The collected mathematical papers|volume=1|year=1889|publisher=Cambridge University press|place=Cambridge|pages=95–112}}
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| *{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Theorie der algebraischen Formen | doi=10.1007/BF01208503 | year=1890 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=36 | issue=4 | pages=473–534}}
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| *{{Citation | doi=10.1090/S0002-9947-1989-0986027-X | last1=Howe | first1=Roger | author1-link=Roger Evans Howe | title=Remarks on classical invariant theory. | jstor=2001418 | mr=0986027 | year=1989 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=313 | issue=2 | pages=539–570 | publisher=American Mathematical Society}}
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| *{{Citation | last1=Olver | first1=Peter J. | title=Classical invariant theory | publisher=[[Cambridge University Press]] | isbn=978-0-521-55821-1 | year=1999}}
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| *{{Citation | last1=Sturmfels | first1=Bernd | author1-link=Bernd Sturmfels | title=Algorithms in invariant theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Texts and Monographs in Symbolic Computation | isbn=978-3-211-82445-0 | mr=1255980 | year=1993}}
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| *{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The Classical Groups: Their Invariants and Representations | url=http://books.google.com/?id=zmzKSP2xTtYC | accessdate=03/2007/26 | publisher=[[Princeton University Press]] | isbn=978-0-691-05756-9 | mr=0000255 | year=1946}}
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| [[Category:Invariant theory]]
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I'm Stewart and I live in Esprick.
I'm interested in Asian Studies, Singing and Turkish art. I like to travel and watching Game of Thrones.
Here is my web-site :: белковая диета для похудения