Upper and lower bounds

Summary

In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S. Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, 5 is a lower bound for the set S = (as a subset of the integers or of the real numbers, etc.), and so is 4. On the other hand, 6 is not a lower bound for S since it is not smaller than every

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https://en.wikipedia.org/wiki/Upper_and_lower_bounds

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In this paper we consider the class of anti-uniform Huffman codes and derive tight lower and upper hounds on the average length, entropy, and redundancy of such codes in terms of the alphabet size of the source. Also an upper bound on the entropy of AUH codes is also presented in terms of the average cost of the code. The Fibonacci distributions are introduced which play a fundamental role in AUH codes. It is shown that such distributions maximize the average length and the entropy of the code for a given alphabet size. Another previously known bound on the entropy for given average length follows immediately from our results.

Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa2008

Algorithmes d'adaptation de maillages anisotropes et application à l'aérodynamique

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The goal of this thesis is to study an anisotropic adaptive algorithm for transonic compressible viscous flow around an airwing. A convection-diffusion model problem is considered, an anisotropic a posteriori error estimator for the H1 semi-norm of the error is derived. The equivalence between the error and the estimator is proved, which provides the efficiency and the reliability of this estimator. A goal oriented anisotropic a posteriori error estimator is introduced. The equivalence with the error is not proved but lower and upper bounds are obtained. Based on this error estimator, an anisotropic mesh algorithm is proposed and applied to the transonic compressible flow around an airwing. The mesh is structured close to the boundary layer and is kept as is, while the mesh outside the boundary layer is adapted according to the anisotropic error estimator. This anisotropic adaptive algorithm allows shocks to be captured accurately, while keeping the number of vertices as low as possible.

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