Integral transforms of the κ-Hilfer fractional derivative

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

The Fundamental Theorem of Natural Selection

Suppose we have n different types of self-replicating entity, with the population Pi of the ith type changing at a rate equal to Pi times the fitness fi of that type. Suppose the fitness fi is any continuous function of all the populations P1,⋯,Pn. Let pi be the fraction of replicators that are of the ith type. Then p=(p1,⋯,pn) is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher’s fundamental theorem of natural selection. We compare it to Fisher’s original result as interpreted by Price, Ewens and Edwards.

Study of Fractional Analytic Functions and Local Fractional Calculus

In this present paper, the role of fractional analytic function in local fractional calculus is studied. Some important properties and theorems in local fractional calculus are discussed, such as product rule, quotient rule, chain rule, fundamental theorem of local fractional calculus, change of variable, integration by parts and so on. In addition, we propose several examples and formulas of local fractional calculus.