scholarly journals Generalizations of Fano’s Inequality for Conditional Information Measures via Majorization Theory

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 288
Author(s):  
Yuta Sakai
Keyword(s):  
Asymptotic Behavior ◽  
Marginal Distribution ◽  
Broad Class ◽  
Acta Math ◽  
Infinite Dimensional ◽  
Information Measures ◽  
Asymptotic Equipartition Property ◽  
Dimensional Version ◽  
Conditional Information ◽  
Error Probabilities

Fano’s inequality is one of the most elementary, ubiquitous, and important tools in information theory. Using majorization theory, Fano’s inequality is generalized to a broad class of information measures, which contains those of Shannon and Rényi. When specialized to these measures, it recovers and generalizes the classical inequalities. Key to the derivation is the construction of an appropriate conditional distribution inducing a desired marginal distribution on a countably infinite alphabet. The construction is based on the infinite-dimensional version of Birkhoff’s theorem proven by Révész [Acta Math. Hungar. 1962, 3, 188–198], and the constraint of maintaining a desired marginal distribution is similar to coupling in probability theory. Using our Fano-type inequalities for Shannon’s and Rényi’s information measures, we also investigate the asymptotic behavior of the sequence of Shannon’s and Rényi’s equivocations when the error probabilities vanish. This asymptotic behavior provides a novel characterization of the asymptotic equipartition property (AEP) via Fano’s inequality.

scholarly journals Quasi-regularity in Optimization

1986 ◽  
Vol 41 (2) ◽  
pp. 188-192 ◽  
Author(s):  
Kung-Fu Ng ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Lagrange Multiplier ◽  
Optimization Problems ◽  
General Situation ◽  
Lagrange Multiplier Rule ◽  
Infinite Dimensional ◽  
Space Setting ◽  
Multiplier Rule ◽  
Dimensional Version ◽  
Definition Of

AbstractThe notion of quasi-regularity, defined for optimization problems in Rn, is extended to the Banach space setting. Examples are given to show that our definition of quasi-regularity is more natural than several other possibilities in the general situation. An infinite dimensional version of the Lagrange multiplier rule is established.

scholarly journals AN INFINITE DIMENSIONAL VERSION OF THE SCHUR CONVEXITY PROPERTY AND APPLICATIONS

2007 ◽  
Vol 05 (02) ◽  
pp. 123-136 ◽  
Author(s):  
CLAUDE VALLÉE ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Symmetric Matrix ◽  
Liouville Problem ◽  
Convexity Property ◽  
Infinite Dimensional ◽  
Selfadjoint Operators ◽  
Dimensional Version ◽  
Sturm Liouville ◽  
Schur Convexity

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the present paper are illustrated by several examples from mechanics or quantum mechanics, including the Sturm–Liouville problem, the Schrödinger equation, and the harmonic oscillator.

STATIONARY SOLUTIONS OF A SELECTION MUTATION MODEL: THE PURE MUTATION CASE

2005 ◽  
Vol 15 (07) ◽  
pp. 1091-1117 ◽  
Author(s):  
ÀNGEL CALSINA ◽  
SÍLVIA CUADRADO
Keyword(s):  
Stationary Solutions ◽  
Banach Lattices ◽  
Age At Maturity ◽  
Frobenius Theorem ◽  
Positive Semigroups ◽  
Infinite Dimensional ◽  
Mutation Model ◽  
Dimensional Version

A selection mutation equations model for the distribution of individuals with respect to the age at maturity is considered. In this model we assume that a mutation, perhaps very small, occurs in every reproduction where the noncompactness of the domain of the structuring variable and the two-dimensionality of the environment are the main features. Existence of stationary solutions is proved using the theory of positive semigroups and the infinite-dimensional version in Banach lattices of the Perron Frobenius theorem. The behavior of these stationary solutions when the mutation is small is studied.

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