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Kraft's inequality limits the lengths of codewords in a prefix code: if one takes an exponential of the length of each valid codeword, the resulting set of values must look like a probability mass function, that is, it must have total measure less than or equal to one. Kraft's inequality can be thought of in terms of a constrained budget to be ...
McMillan was born in Minneapolis, Minnesota, in 1915, the only child of Franklin Richardson McMillan, a civil engineer, and Luvena Lucille Brockway McMillan, a schoolteacher. [3] He received his B.S. in 1936 and a Ph.D. 1939 from Massachusetts Institute of Technology (MIT) on a thesis entitled The calculus of discrete homogenous chaos ...
The Shannon–McMillan–Breiman theorem, due to Claude Shannon, Brockway McMillan, and Leo Breiman, states that we have convergence in the sense of L1. [2] Chung Kai-lai generalized this to the case where X {\displaystyle X} may take value in a set of countable infinity, provided that the entropy rate is still finite.
Call a full subtree of height whose leaves are a subset of the leaves of the full binary tree of depth , an -triangle. Identify a codeword of length l {\displaystyle l} with a node in the tree at depth l {\displaystyle l} , as usual, and also with the ( l m − l ) {\displaystyle (l_{m}-l)} -triangle rooted at that node.
In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [4]Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel).
The Atkinson index is defined as: (, …,) = {(=) / (=) / = (,...,) = +where is individual income (i = 1, 2, ..., N) and is the mean income.. In other words, the Atkinson index is the complement to 1 of the ratio of the Hölder generalized mean of exponent 1−ε to the arithmetic mean of the incomes (where as usual the generalized mean of exponent 0 is interpreted as the geometric mean).
The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for f to be operator monotone. [5] An elementary proof of the theorem is discussed in [1] and a more general version of it in. [6]