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On some functional equations from additive and non-additive measures-I

Published online by Cambridge University Press:  20 January 2009

PL Kannappan
PL Kannappan
Affiliation:
University of WaterlooWaterloo, Ontario
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It is known that while the Shannon and the Rényi entropies are additive, the measure entropy of degree β proposed by Havrda and Charvat (7) is non-additive. Ever since Chaundy and McLeod (4) considered the following functional equation

which arose in statistical thermodynamics, (1.1) has been extensively studied (1, 5, 6, 8). From the algebraic properties of symmetry, expansibility and branching of the entropy (viz. Shannon entropy Hn, etc.) one obtains the sum representation

which with the property of additivity yields the functional equation (1.1), (9, 10).

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 23 , Issue 2 , June 1980 , pp. 145 - 150
Copyright
Copyright © Edinburgh Mathematical Society 1980

References

REFERENCES

(1)Aczél, J. und Daróczy, Z., Charakterisierung der Entropien positiver Ordnung und der Shannonschen Entropie, Acta Math. Acad. Sci. Hungar. 14 (1963), 95–121.CrossRefGoogle Scholar
(2)Behara, M., Additive and non-additive measures of entropy with applications (McMaster University report, 1978).Google Scholar
(3)Behara, M. and Nath, P., Additive and non-additive entropies affinite measurable partitions, Probability and Information Theory, Vol. 2, Springer-Verlag, 1973, V296, 102–138.Google Scholar
(4)Chaundy, T. W. and McLeod, J. B., On a functional equation, Edin. Math. Notes 43 (1960), 7–8.CrossRefGoogle Scholar
(5)Daróczy, Über die Charakterisierung der Shannonschen Entropy, Proc. Colloq. on Information Theory, I, Jámos Bo ly ai Math. Soc., Budapest (1968), 135–139.Google Scholar
(6)Daróczy, Z., On the measurable solutions of a functional equation, Acta Math. Acad. Sci. Hungar. 22 (1971), 11–14.CrossRefGoogle Scholar
(7)Havrda, J. and Charvat, F., Quantification method of classification process, Concept of structural a-entropy, Kybernetika (Prague), 3 (1967), 30–35.Google Scholar
(8)Kannappan, Pl., On Shannon's entropy, directed divergence and inaccuracy, Z. Wahr. Verw. Geb. 22 (1972), 95–100.CrossRefGoogle Scholar
(9)Kannappan, Pl., On directed divergence and inaccuracy, Z. Wahr. Verw. Geb. 25 (1972), 49–55.CrossRefGoogle Scholar
(10)Kannappan, Pl., On a functional equation connected with generalized directed divergence, Aeq. Math. (1974), 81–93.Google Scholar
(11)Kannappan, Pl., On generalization of some measures in information theory, Glasnik Mat. 29 (1974), 81–93.Google Scholar
(12)Kannappan, Pl., On a generalization of sum form functional equation – 1 (submitted).Google Scholar
(13)Ng, C. T., Representation of measures of information with the branching property, Information and Control 25 (1974), 45–56.CrossRefGoogle Scholar