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## Rearrangements of vector valued functions, with application to atmospheric and oceanic ows

Citations: | 5 - 2 self |

### Citations

610 |
Classical descriptive set theory
- Kechris
- 1995
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Citation Context ...e Ai = 'n (bi) for each i =1;:::; m. Write Xn i for XnjAi . For a given i, Xn i = bi +(1=n)x. Let E be a Lebesgue negligible subset of R3 . Then (Xn i ) ,1 \ (E) = Ai (nE , nbi) (nE , nbi) = (nE) =0: =-=(9)-=- By way of explanation, we haveused translation invariance of Lebesgue measure to obtain the rst equality in (9), and properties of Lebesgue measure to obtain the second. This demonstrates that Xn i i... |

582 |
Measure Theory
- Halmos
- 1964
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Citation Context ...s x. Rewriting in terms of X and x, wehave DX Dt DY Dt DZ Dt The geostrophic energy E is de ned as Z 1 E = 2 u2 1 + g 2 v2 2 1 = f 2 Henceforth we ignore the constant f 2 . = f(y , Y ) (6) = f(X , x) =-=(7)-=- = 0: (8) , g g z d (x) 0 Z Z 2 2 2 2 2 X + x + Y + y d (x) , f x:Xd (x) At any time t, X is found on particles by predicting (X; Y; Z) on particles using the equations (6), (7) and (8). The Cullen{No... |

123 |
Functional Analysis, 2nd Edition
- Rudin
- 1991
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Citation Context ...(U)). We recall the de nition of measure preserving transformation in the next section. The restriction to nite measure spaces (U; ) isomorphic to (0; (U)) with Lebesgue measure is not severe: Royden =-=[12]-=- yields that any separable complete metric space U, equipped with a Borel measure such that (U) < 1 and (fxg) = 0 for each x 2 U, is isomorphic to ((0; (U)); ). De nition Let (U; ) be a measure space ... |

78 |
The geostrophic momentum approximation and the semigeostrophic equations.
- Hoskins
- 1975
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Citation Context .... These are a standard model for slowly varying ows constrained by rotation and strati cation, and are used to study front formation in meteorology. We state the equations in the form used by Hoskins =-=[8]-=-. Du g Dt , fv ag =0; Dv g D Dt r:u =0; Dt + fu ag =0; (3) =0; (4) r = fv g;,fu g; g 0 (5) 8where u (u; v; w) ug + uag; ug D Dt (u g;v g;0); @ @t + u:r f is the Coriolis parameter, assumed constant, ... |

47 |
TROMBETTI: On optimization problems with prescribed rearrangements, Nonlinear Anal
- ALVINO, LIONS, et al.
- 1989
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Citation Context ...rea: recent advances have been made by Brenier [2]. In comparison the theory of rearrangements of scalar valued functions is well developed: for example see Burton [3] and Alvino, Trombetti and Lions =-=[1]-=-. Some results for scalar valued rearrangements do not have vector valued equivalents. For example the monotone rearrangement of a vector valued function does not satisfy some of the inequalities whic... |

22 |
Generalised Lagrangian solutions for atmospheric and oceanic ° ows.
- Cullen, Norbury, et al.
- 1991
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Citation Context ... valued functions f and g (belonging to L p ( Rn ; ;Rd)) to be rearrangements if R F (f) = R F(g)for each F in a subclass of continuous functions from Rd to R. In contrast, Cullen, Norbury and Purser =-=[5]-=- made a direct extension of the de nition of scalar valued rearrangement, requiring that fx : f(x) cg = fx : g(x) cg for each c 2 Rd , where the inequalities are calculated component by component. Sec... |

8 |
A duality principle in semi-geostrophic theory.
- Purser, Cullen
- 1987
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Citation Context ...d bounded connected set R3 , with normal velocity u:n given on @ . For x =(x; y; z) 2 , by making the the substitution X (X; Y; Z) (x + v g=f; y , u g=f; (g=f 2 0) ); it is shown in Purser and Cullen =-=[11]-=- that we may replace (3) and (4) by DX Dt = ug: We think of X as a function of the physical space co{ordinates x. Rewriting in terms of X and x, wehave DX Dt DY Dt DZ Dt The geostrophic energy E is de... |

7 |
Existence and uniqueness of monotone measure preserving maps
- McCann
- 1995
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Citation Context ...tood to be the restriction to of a convex function de ned on R3 . We call u the monotone rearrangement of u. The name comes from the fact that u is a cyclically monotone function. We note that McCann =-=[10]-=- has generalised the rst part of this result (concerning the existence of an essentially unique rearrangement equal to the gradient of a convex function) to more general measures than Lebesgue measure... |

5 |
Polar factorisation and monotone rearrangement of vector-valued functions
- Brenier
- 1991
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Citation Context ...vector valued function. For a prescribed f0 we write R(f0) to denote the set of rearrangements of f0. Di erent de nitions have been given for two vector valued functions being rearrangements. Brenier =-=[2]-=- de ned vector valued functions f and g (belonging to L p ( Rn ; ;Rd)) to be rearrangements if R F (f) = R F(g)for each F in a subclass of continuous functions from Rd to R. In contrast, Cullen, Norbu... |

5 |
Turkington,Multiconstrained variational problems of nonlinear eigenvalue type: new formulation and algorithms
- Eydeland, Spruck, et al.
- 1990
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Citation Context ...lds a characterisation for the set of rearrangements of a 1prescribed vector valued function. This is a vector valued extension of the real valued characterisation of Eydeland, Spruck and Turkington =-=[6]-=-. These results are stated in Theorem 1. The proofs require a technical lemma from the theory of analytic sets. Section 3 studies a variational problem arising from a model for atmospheric and oceanic... |

3 |
Rearrangement of functions, maximisation of convex functionals, and vortex rings
- Burton
- 1987
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Citation Context ...ained by rotation and strati cation, and are used to study front formation in meteorology. We state the equations in the form used by Hoskins [8]. Du g Dt , fv ag =0; Dv g D Dt r:u =0; Dt + fu ag =0; =-=(3)-=- =0; (4) r = fv g;,fu g; g 0 (5) 8where u (u; v; w) ug + uag; ug D Dt (u g;v g;0); @ @t + u:r f is the Coriolis parameter, assumed constant, g denotes the acceleration due to gravity, 0 is a referenc... |

1 |
New Mathematical tools for the Lagrangian description of atmospheric dynamics. U.K. Meteorological O ce Forecasting Research Scienti c Paper No
- CULLEN, DOUGLAS
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Citation Context ...knowledge the insight and help of G.R. Burton. The research was carried out under the guidance of M.J.P. Cullen, and the meteorological signi cance of these results is discussed in Cullen and Douglas =-=[4]-=-. The author would also like to thank J. Norbury, Y. Brenier and J-D. Benamou for stimulating discussions. The author is grateful for the hospitality and support of the Isaac Newton Institute for Math... |