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A formula for calculation of metric dimension of converging sequences

Ladislav, Jr. Mišík, Tibor Žáčik (1999)

Commentationes Mathematicae Universitatis Carolinae

Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived....

A new algorithm for approximating the least concave majorant

Martin Franců, Ron Kerman, Gord Sinnamon (2017)

Czechoslovak Mathematical Journal

The least concave majorant, F ^ , of a continuous function F on a closed interval, I , is defined by F ^ ( x ) = inf { G ( x ) : G F , G concave } , x I . We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on I . Given any function F 𝒞 4 ( I ) , it can be well-approximated on I by a clamped cubic spline S . We show that S ^ is then a good approximation to F ^ . We give two examples, one to illustrate, the other to apply our algorithm.

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