“Transvectant”的意思、由来-开放百科全书

词条

Transvectant

释义

Definition
Partial transvectants
Examples
Footnotes
References

In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's Ω process.

Definition

If Q₁,...,Q_n are functions of n variables x = (x₁,...,x_n) and r ≥ 0 is an integer then the r^th transvectant of these functions is a function of n variables given by

where Ω is Cayley's Ω process, the tensor product means take a product of functions with different variables x¹,..., xⁿ, and tr means set all the vectors x^k equal.

Partial transvectants

Examples

The zeroth transvectant is the product of the n functions.

The first transvectant is the Jacobian determinant of the n functions.

The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.

Footnotes

References

{{Citation | last1=Olver | first1=Peter J. |author1-link=Peter J. Olver | title=Classical invariant theory | publisher=Cambridge University Press | isbn=978-0-521-55821-1 | year=1999}}
{{Citation | last1=Olver | first1=Peter J. |author1-link=Peter J. Olver | last2=Sanders | first2=Jan A. | title=Transvectants, modular forms, and the Heisenberg algebra | doi=10.1006/aama.2000.0700 | mr=1783553 | year=2000 | journal=Advances in Applied Mathematics | issn=0196-8858 | volume=25 | issue=3 | pages=252–283}}

1 : Invariant theory

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