Principal parts of a vector bundle on projective line and the fractional derivative

Abstract

This work is an exposition on computational aspects of principal parts of a vector bundle on projective line over the field of characteristic zero. Principal parts help determine the possibility of algebraically formalizing infinitesimal-neighborhoods of subschemes inside some ambient scheme. The purpose of this study is to look for the possibility of formalizing the algebraic geometric interpretation of fractional derivative. For the latter, this study follows the approach proposed by Vasily Tarasov. The difference is that Tarasov proposed a geometric interpretation using finiteorder jet bundles from differential geometry. Present study proposes finite-order principal parts of the structure-sheaf of real projective line as its formal algebraic geometric parallel.

Entropy Interpretation of Hadamard Type Fractional Operators: Fractional Cumulative Entropy

“…There are various interpretations of the fractional integrals and derivatives [ 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 ] such as geometric interpretations [ 35 , 36 , 37 , 38 ] and [ 39 , 40 , 41 , 42 ], physical interpretations [ 39 , 40 , 41 , 42 ] and [ 30 , 31 , 32 , 33 , 34 ], economic interpretations [ 43 , 44 ], probabilistic interpretations [ 45 , 46 , 47 , 48 , 49 ], information interpretation [ 50 ] and some others.…”