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MULTIDIMENSIONAL EXACT CLASSES, SMOOTH APPROXIMATION AND BOUNDED 4-TYPES
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 07 September 2020, pp. 1305-1341
- Print publication:
- December 2020
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In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class (R-mec), a special kind of multidimensional asymptotic class (R-mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatization [13] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language
$\mathcal {L}$ and any positive integer d the class 
$\mathcal {C}(\mathcal {L},d)$ of all finite 
$\mathcal {L}$-structures with at most d 4-types is a polynomial exact class in 
$\mathcal {L}$, where a polynomial exact class is a multidimensional exact class with polynomial measuring functions.
Lagrangian conditions for a nonsmooth vector-valued minimax
- Part of
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- Journal:
- Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 65 / Issue 2 / October 1998
- Published online by Cambridge University Press:
- 09 April 2009, pp. 163-175
- Print publication:
- October 1998
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