A Physical Application of a Rearrangement Inequality

1970 ◽  
Vol 77 (1) ◽  
pp. 68 ◽  
Author(s):  
M. S. Klamkin
Keyword(s):  
Physical Application ◽  
Rearrangement Inequality

The Rearrangement Inequality for the Ergodic Maximal Function

2001 ◽  
Vol 8 (4) ◽  
pp. 727-732
Author(s):  
L. Ephremidze
Keyword(s):  
Maximal Function ◽  
Rearrangement Inequality ◽  
Decreasing Rearrangement ◽  
Exact Constants

Abstract The equivalence of the decreasing rearrangement of the ergodic maximal function and the maximal function of the decreasing rearrangement is proved. Exact constants are obtained in the corresponding inequalities.

scholarly journals Some Order Preserving Inequalities for Cross Entropy and Kullback–Leibler Divergence

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 959 ◽  
Author(s):  
Mateu Sbert ◽  
Min Chen ◽  
Jordi Poch ◽  
Anton Bardera
Keyword(s):  
Machine Learning ◽  
Information Theory ◽  
Geometric Mean ◽  
Cross Entropy ◽  
Rearrangement Inequality ◽  
Numerical Examples ◽  
Leibler Divergence ◽  
Recent Theorem ◽  
The Difference ◽  
New Inequalities

Cross entropy and Kullback–Leibler (K-L) divergence are fundamental quantities of information theory, and they are widely used in many fields. Since cross entropy is the negated logarithm of likelihood, minimizing cross entropy is equivalent to maximizing likelihood, and thus, cross entropy is applied for optimization in machine learning. K-L divergence also stands independently as a commonly used metric for measuring the difference between two distributions. In this paper, we introduce new inequalities regarding cross entropy and K-L divergence by using the fact that cross entropy is the negated logarithm of the weighted geometric mean. We first apply the well-known rearrangement inequality, followed by a recent theorem on weighted Kolmogorov means, and, finally, we introduce a new theorem that directly applies to inequalities between K-L divergences. To illustrate our results, we show numerical examples of distributions.

CHAPTER 3 A PHYSICAL APPLICATION

1982 ◽  
Vol 60 (sup1) ◽  
pp. 44-63
Author(s):  
Leonard Goddard ◽  
Brenda Judge
Keyword(s):  
Physical Application

Rearrangement inequality for periodic functions

1976 ◽  
Vol 61 (1) ◽  
pp. 35-44 ◽  
Author(s):  
R. Friedberg ◽  
J. M. Luttinger
Keyword(s):  
Periodic Functions ◽  
Rearrangement Inequality

A physical application of the function exp(−1/x2)

1975 ◽  
Vol 43 (3) ◽  
pp. 268-269
Author(s):  
Mary L. Boas
Keyword(s):  
Physical Application
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