Measures of uncertainty mathematical programming and physics

The first section gives the measure of uncertainty given by Shannon (1948) and the generalizations thereof by Schvitzenberger (1954), Kullback (1959), Renyi (1961,1965), Kapur (1967, 1968), and Rathie (1970). It gives some postulates characterizing Shannon's entropy, Renyi's entropy of order a and our entropy of order a and type P. It also gives some properties of this most general type of entropy. In the second section an optimization problem is formulated and solved in the case of Shannon's and Renyi's entropies by the use of the principle of optimality. It is shown that this principle fails to solve the problem in the case of entropy of order a and type (3 and this leads to an interesting problem in non-linear integer fractional functional programming. In the third section, we discuss the connection between the concepts of entropy in information theory and physics and show how Shannon's entropy leads to Boltzman distribution of statistical mechanics but fails to give the Fermi-Dirac and BoseEinstein distributions of quantum mechanics. We find the entropies which lead to these distributions, but these do not satisfy an important property satisfied by Shannons entropy. This may give us some insight into quantum mechanical systems . In the fourth and last section, we obtain some properties of Bose-Einstein and Fermi-Dirac entropies obtained in the third section.