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Maury Barbato
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 Posted: Sun May 27, 2007 10:15 am Post subject: Families of Curves |
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Hello,
let f(x,y,s,t) a C^1 function f:R^4->R. Let us suppose
that
(I) (f_x,f_y)=/=0 for every (x,y,s,t) in R^4
(II) X(s,t)={(x,y) in R^2 | f(x,y,s,t)=0 } is not empty for every (s,t) in R^2
Then I suppose that the sets X(s,t) are not disjoint,
that is, there are two distinct points (s,t), (s',t')
such that X(s,t) /\ X(s',t') is not empty.
What do you think about?
Thank you very very very .. much for your attention.
My Best Regards,
Maury |
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Maury Barbato
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| Posted: Mon May 28, 2007 7:30 am Post subject: Re: Families of Curves |
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I wrote:
| Quote: |
Hello,
let f(x,y,s,t) a C^1 function f:R^4->R. Let us
suppose
that
(I) (f_x,f_y)=/=0 for every (x,y,s,t) in R^4
(II) X(s,t)={(x,y) in R^2 | f(x,y,s,t)=0 } is not
empty for every (s,t) in R^2
Then I suppose that the sets X(s,t) are not
disjoint,
that is, there are two distinct points (s,t),
(s',t')
such that X(s,t) /\ X(s',t') is not empty.
What do you think about?
Thank you very very very .. much for your attention.
My Best Regards,
Maury
|
Maybe, we can use the following topological result.
If n, m are two positive integers, with n>m, and U is
an opne subset of R^n, there's no injective continuous
map f:U->R^m.
Have you some idea?
Thank you very much for your help.
My Best Regards,
Maury |
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