Rough Derivatives in Rough Function Model and Their Application

Abstract

On the theory and application of rough derivatives in rough function model, the notion of generalized rough functions is defined to improve the definition of higher order rough derivatives Pawlak proposed, and analyze the functional features of roughly derived functions and higher order roughly derived functions. The rough extremum definition of discrete functions is given, Fermat theorem and Rolle theorem of roughly smooth discrete functions are proposed and proved, which complete and perfect the theoretical foundation of rough derivatives application in rough function model. The concepts that are rough monotone and rough convexity of discrete functions are defined. By comparing with the derivative application of real continuous function, a series of theorems are put forward which are the relation theorem of rough derivatives and rough monotone, the two sufficient conditions of rough extrema, the two relation theorems of rough derivatives and rough convexity. Moreover, some new results are achieved such as the sufficient condition of rough smoothing of discrete functions, etc. Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007) 0-7695-2874-0/07 $25.00

Notes for rough derivatives and rough continuity in rough function model

“…The rough derivative f ′ in definition 1 has the following characteristics [18] : The discrete function f: [n]→[m] is a rough function defined on n +1 points. It has n f ′(i), where i∈[n-1], and has n-k+1 f (k) (i), where i∈ [n-k].…”

“…The development and achievement of rough set theory are outstandingly manifested in various deformation and generalization of generalized rough set models [2][3][4][5][6][7][8] from the initial classical rough set model. Rough function model [9][10][11] is one of the important generalized rough set models, which are constructed by generalizing rough set theory to real numbers field and study mathematical analysis problems such as rough numbers [9,12,13] , rough continuousness [10,11,14,15] , rough calculus [10,11,[15][16][17][18][19] , rough differential equations [10,11] , etc.. Rough function model, based on rough set theory, is a more effective mathematical tool and technical means transacting indefinite and incomplete knowledge.…”

“…Are they equivalent? Pawlak did not analyze further, while other quite few relative references [12][13][14][15][16][17][18][19] did not investigate this theory either. The completeness of related theory and applications about rough function model and the rough continuity is badly needed.…”