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If Kraft's inequality holds with strict inequality, the code has some redundancy. If Kraft's inequality holds with equality, the code in question is a complete code. [2] If Kraft's inequality does not hold, the code is not uniquely decodable. For every uniquely decodable code, there exists a prefix code with the same length distribution.
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.
Lagrange reversion theorem; Laplace principle (large deviations theory) Lax equivalence theorem; Lax–Milgram theorem; Lax–Wendroff theorem; Lebesgue integrability condition; Levi–Lechicki theorem; Lévy–Steinitz theorem; Linearity of differentiation; Littlewood's 4/3 inequality
Choi's theorem on completely positive maps; Closed graph theorem; Closed graph theorem (functional analysis) Closed range theorem; Cohen–Hewitt factorization theorem; Commutant lifting theorem; Commutation theorem for traces; Continuous functional calculus; Convex series; Cotlar–Stein lemma
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In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value out of a set of possibilities X can be seen as representing an implicit probability distribution () = over X, where is the length of the code for in bits.
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 ...