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This seems like a really easy question but I can't seem to find the answer anywhere...
Is there such a thing as a "fractional/fractal/continuum exterior product" of a positive non-integer number of vectors? I.e. not just for a discrete number of vectors as in
but a continuum of vectors, in analogy with the continuous product, Gamma function, fractional calculus, or such like? Many thanks, Maschen ( talk) 09:58, 25 November 2012 (UTC)
This seems like a really easy question but I can't seem to find the answer anywhere... Really easy questions tend to be the hardest of all. :-) So I agree with Quondum that you are pretty ambitious. Some late-night speculations, no guarantee they make the slightest sense: exterior algebra is a quotient of Tensor algebra. It might be easier to think about the same questions for tensor algebras first. Second, following Quondum, there are two things desired here, corresponding to the 2 desired operations of "addition" and "multiplication" - a continuum of grades and a continuum operation respecting the grades. Now there is a well-known construction that might be thought of as corresponding to the adding of the various homogeneous grade spaces - the Direct integral. If you could develop a tensor/exterior analog of that, and take direct integrals of the various spaces you get, then that could be what you are looking for. A category-theoretic treatment of the direct integral might help, and the above musings might give some hint where to look to see if somebody has concocted this stuff already - operator algebra theory, non-commutative geometry after Connes. John Z ( talk) 09:16, 29 November 2012 (UTC)
If G is an open set then curve γ is homologous to zero if for all ω belonging to ₵-G η(γ;ω)=0. How i can prove this? — Preceding unsigned comment added by 14.99.166.149 ( talk) 17:13, 25 November 2012 (UTC)
I want to know what is orientation? If (1,0,∞) is the orientation of R,then the cross ratio (z,1,0,∞)=? — Preceding unsigned comment added by 14.99.166.149 ( talk) 17:25, 25 November 2012 (UTC)
Mathematics desk | ||
---|---|---|
< November 24 | << Oct | November | Dec >> | November 26 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
This seems like a really easy question but I can't seem to find the answer anywhere...
Is there such a thing as a "fractional/fractal/continuum exterior product" of a positive non-integer number of vectors? I.e. not just for a discrete number of vectors as in
but a continuum of vectors, in analogy with the continuous product, Gamma function, fractional calculus, or such like? Many thanks, Maschen ( talk) 09:58, 25 November 2012 (UTC)
This seems like a really easy question but I can't seem to find the answer anywhere... Really easy questions tend to be the hardest of all. :-) So I agree with Quondum that you are pretty ambitious. Some late-night speculations, no guarantee they make the slightest sense: exterior algebra is a quotient of Tensor algebra. It might be easier to think about the same questions for tensor algebras first. Second, following Quondum, there are two things desired here, corresponding to the 2 desired operations of "addition" and "multiplication" - a continuum of grades and a continuum operation respecting the grades. Now there is a well-known construction that might be thought of as corresponding to the adding of the various homogeneous grade spaces - the Direct integral. If you could develop a tensor/exterior analog of that, and take direct integrals of the various spaces you get, then that could be what you are looking for. A category-theoretic treatment of the direct integral might help, and the above musings might give some hint where to look to see if somebody has concocted this stuff already - operator algebra theory, non-commutative geometry after Connes. John Z ( talk) 09:16, 29 November 2012 (UTC)
If G is an open set then curve γ is homologous to zero if for all ω belonging to ₵-G η(γ;ω)=0. How i can prove this? — Preceding unsigned comment added by 14.99.166.149 ( talk) 17:13, 25 November 2012 (UTC)
I want to know what is orientation? If (1,0,∞) is the orientation of R,then the cross ratio (z,1,0,∞)=? — Preceding unsigned comment added by 14.99.166.149 ( talk) 17:25, 25 November 2012 (UTC)