The Classification of Generalized Riemann Derivatives Stefan Catoiu - PowerPoint PPT Presentation

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Generalized Riemann Derivatives Stefan Catoiu DePaul University, Chicago Groups, Rings and the Yang-Baxter Equation Spa, Belgium, June 18-24, 2017 Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Outline Definition and basic properties 1 Definition and basic properties Generalized vs. Ordinary Differentiation 2 Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives 3 The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann 4 Derivatives The Classification of Complex Generalized Riemann Derivatives Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Definition and basic properties The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Outline Definition and basic properties 1 Definition and basic properties Generalized vs. Ordinary Differentiation 2 Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives 3 The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann 4 Derivatives The Classification of Complex Generalized Riemann Derivatives Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Definition and basic properties The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Definition An n th generalized Riemann derivative or the A -derivative of a real function f at x is defined by the limit � m i = 0 A i f ( x + a i h ) D A f ( x ) = lim , h n h → 0 where the data vector A = { A i , a i : i = 1 , . . . , m } of 2 m real numbers satisfies the n th Vandermonde relations m � 0 if j = 0 , 1 , . . . , n − 1 , A i a j � i = n ! if j = n . i = 1 The numerator ∆ A f ( x , h ) is an n th generalized Riemann difference. Linear algebra forces m > n . Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Definition and basic properties The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Vandermonde conditions assure that the ordinary derivative implies the generalized derivative and these are equal whenever they both exist. Indeed, for n = 1, suppose that f is differentiable at x , and let A be a data vector of a first GRD. We have � m i = 0 A i f ( x + a i h ) D A f ( x ) = lim h h → 0 � m i = 0 A i [ f ( x + a i h ) − f ( x )] + � m i = 0 A i f ( x ) = lim h h → 0 � m m � f ( x + a i h ) − f ( x ) f ( x ) � � = lim A i a i + A i a i h h h → 0 i = 0 i = 0 � m � � f ′ ( x ) + 0 = f ′ ( x ) . = A i a i i = 0 Thus f differentiable at x implies f is A -differentiable at x . Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Definition and basic properties The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Vandermonde conditions assure that the ordinary derivative implies the generalized derivative and these are equal whenever they both exist. Indeed, for n = 1, suppose that f is differentiable at x , and let A be a data vector of a first GRD. We have � m i = 0 A i f ( x + a i h ) D A f ( x ) = lim h h → 0 � m i = 0 A i [ f ( x + a i h ) − f ( x )] + � m i = 0 A i f ( x ) = lim h h → 0 � m m � f ( x + a i h ) − f ( x ) f ( x ) � � = lim A i a i + A i a i h h h → 0 i = 0 i = 0 � m � � f ′ ( x ) + 0 = f ′ ( x ) . = A i a i i = 0 Thus f differentiable at x implies f is A -differentiable at x . Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Definition and basic properties The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Examples of Generalized Riemann Derivatives First order: the ordinary derivative has A = { 1 , − 1 ; 1 , 0 } : f ( x + h ) − f ( x ) f ′ ( x ) = lim ; h h → 0 the symmetric derivative has A = { 1 , − 1 ; 1 2 , − 1 2 } : f ( x + h 2 ) − f ( x − h 2 ) f ′ s ( x ) = lim ; h h → 0 the “crazy” derivative has A = { 2 , − 3 , 1 ; 1 , 0 , − 1 } : 2 f ( x + h ) − 3 f ( x ) + f ( x − h ) f ′ ∗ ( x ) = lim ; h h → 0 Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Definition and basic properties The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Examples of Generalized Riemann Derivatives Higher order: the second symmetric (Schwarz) derivative has A = { 1 , − 2 , 1 ; 1 , 0 , − 1 } : f ( x + h ) − 2 f ( x ) + f ( x − h ) f ′′ s ( x ) = lim ; h 2 h → 0 the n th forward Riemann derivative � n k = 0 ( − 1 ) k � n � f ( x + kh ) k R n f ( x ) = lim h n h → 0 the n th symmetric Riemann derivative � n k = 0 ( − 1 ) k � n f ( x + ( n � 2 − k ) h ) R s k n f ( x ) = lim h n h → 0 Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Definition and basic properties The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Examples of Generalized Riemann Derivatives Quantum Riemann derivatives (after Ash, C, Rios, 2002): the n th forward quantum Riemann derivative q ( k 2 ) f ( q n − k x ) � n k = 0 ( − 1 ) k � n � k R [ n ] f ( x ) = lim ( qx − x ) n q → 1 the n th symmetric quantum Riemann derivative q ( k 2 ) f ( q n � n k = 0 ( − 1 ) k � n 2 − k x ) � R s k [ n ] f ( x ) = lim ( qx − x ) n q → 1 Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Outline Definition and basic properties 1 Definition and basic properties Generalized vs. Ordinary Differentiation 2 Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives 3 The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann 4 Derivatives The Classification of Complex Generalized Riemann Derivatives Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation I Ordinary n th differentiability ⇒ n th A -differentiability for functions f at x comes by Taylor expansion. The converse is false: Function f ( x ) = | x | has f ( 0 + h 2 ) − f ( 0 − h � h � − � − h � � � � 2 ) � f ′ 2 2 s ( 0 ) = lim = lim = 0 , h h h → 0 h → 0 while f ′ ( 0 ) does not exist. Moreover, any even function is symmetric Riemann differentiable of any odd order but not symmetric Riemann defferentiable of any even order. In particular, higher order generalized differentiation does not imply lower order generalized differentiation. Since the pointwise relation between the ordinary and generalized derivative seemed pointless, the research focus moved to the a.e. relation. Stefan Catoiu The Classification of Generalized Riemann Derivatives

Definition and basic properties Generalized vs. Ordinary Differentiation Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives Generalized vs. Ordinary Differentiation I Ordinary n th differentiability ⇒ n th A -differentiability for functions f at x comes by Taylor expansion. The converse is false: Function f ( x ) = | x | has f ( 0 + h 2 ) − f ( 0 − h � h � − � − h � � � � 2 ) � f ′ 2 2 s ( 0 ) = lim = lim = 0 , h h h → 0 h → 0 while f ′ ( 0 ) does not exist. Moreover, any even function is symmetric Riemann differentiable of any odd order but not symmetric Riemann defferentiable of any even order. In particular, higher order generalized differentiation does not imply lower order generalized differentiation. Since the pointwise relation between the ordinary and generalized derivative seemed pointless, the research focus moved to the a.e. relation. Stefan Catoiu The Classification of Generalized Riemann Derivatives