Note on the kolmogorov statistic in the discrete case

We refine a classical logarithmic inequality using a discrete case of Bernoulli inequality, and then we refine furthermore two information inequalities between information measures for graphs, based on information functionals, presented by Dehmer and Mowshowitz in (2010) as Theorems 4.7 and 4.8. The inequalities refer to entropy-based measures of network information content and have a great impact for information processing in complex networks (a subarea of research in modeling of complex systems).

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BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

Reviews in Mathematical Physics ◽

10.1142/s0129055x94000407 ◽

1994 ◽

Vol 06 (05a) ◽

pp. 1147-1161 ◽

Cited By ~ 97

Author(s):

MARY BETH RUSKAI

Keyword(s):

Relative Entropy ◽

Discrete Case ◽

Sobolev Inequalities ◽

Completely Positive ◽

Logarithmic Sobolev Inequalities ◽

Strong Subadditivity

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

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On the regularity of one-sided fractional maximal functions

Mathematica Slovaca ◽

10.1515/ms-2017-0171 ◽

2018 ◽

Vol 68 (5) ◽

pp. 1097-1112 ◽

Cited By ~ 4

Author(s):

Feng Liu

Keyword(s):

Bounded Variation ◽

Maximal Functions ◽

Continuous Case ◽

Discrete Case ◽

Maximal Operators ◽

Functions Of Bounded Variation ◽

Sharp Bounds ◽

Regularity Properties ◽

Natural Extensions ◽

The One

Abstract In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators $ \mathcal{M}_{\beta}^{+} $ and $ \mathcal{M}_{\beta}^{-} $ map $ W^{1,p}(\mathbb{R}) $ into $ W^{1,q}(\mathbb{R}) $ with 1 <p <∞, 0≤β<1/p and q=p/(1-pβ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators $ M_{\beta}^{+} $ and $ M_{\beta}^{-} $ from $ \ell^{1}(\mathbb{Z}) $ to $ {\rm BV}(\mathbb{Z}) $. Here $ {\rm BV}(\mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.

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