Some approximation results on Bernstein-Schurer operators defined by (p,q)-integers (Revised)
aa r X i v : . [ m a t h . C A ] N ov Some approximation results onBernstein-Schurer operators defined by ( p, q ) -integers (Revised) M. Mursaleen , Md. Nasiruzzaman and
Ashirbayev Nurgali
Department of Mathematics, Aligarh Muslim University, Aligarh–202002, [email protected]; [email protected]
Abstract
In the present article, we have given a corrigendum to our paper Some approximationresults on Bernstein-Schurer operators defined by ( p, q )-integers published in Journal of In-equalities and Applications (2015) 2015:249.
Keywords and phrases : q -integers; ( p, q )-integers; Bernstein operator; ( p, q )-Bernstein op-erator; q -Bernstein-Schurer operator; ( p, q )-Bernstein-Schurer operator; Modulus of continuity. AMS Subject Classification (2010):
Introduction and Preliminaries
In 1912, S.N Bernstein [4] introduced the following sequence of operators B n : C [0 , → C [0 ,
1] defined for any n ∈ N and for any f ∈ C [0 ,
1] such as B n ( f ; x ) = n X k =0 (cid:18) nk (cid:19) x k (1 − x ) n − k f (cid:18) kn (cid:19) , x ∈ [0 , . (1 . q -Bernstein operators by applying the idea of q -integers, and in 1997 another generalization of these operators introduced byPhilip [19]. Later on, many authors introduced q -generalization of various opera-tors and investigated several approximation properties. For instance, q -analogueof Stancu-Beta operators in [3] and [12]; q -analogue of Bernstein-Kantorovichoperators in [20]; q - Baskakov-Kantorovich operators in [7]; q -Sz´ a sz-Mirakjanoperators in [18]; q -Bleimann, Butzer and Hahn operators in [2] and in [6]; q -analogue of Baskakov and Baskakov-Kantorovich operators in [9]; q -analogue ofSz´ a sz-Kantorovich operators in [10]; and q -analogue of generalized Bernstein-Shurer operators in [13].We recall certain notations on ( p, q )-calculus.The ( p, q )-integer was introduced in order to generalize or unify several formsof q -oscillator algebras well known in the earlier physics literature related to therepresentation theory of single parameter quantum algebras [5]. The ( p, q )-integer[ n ] p,q is defined by[ n ] p,q = p n − q n p − q , n = 0 , , , · · · , < q < p ≤ . The ( p, q )-Binomial expansion is( ax + by ) np,q := n X k =0 p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q a n − k b k x n − k y k ( x + y ) np,q := ( x + y )( px + qy )( p x + q y ) · · · ( p n − x + q n − y ) . Also, the ( p, q )-binomial coefficients are defined by (cid:20) nk (cid:21) p,q := [ n ] p,q ![ k ] p,q ![ n − k ] p,q ! . Details on ( p, q )-calculus can be found in [22]. For p = 1, all the notions of( p, q )-calculus are reduced to q -calculus [1].In 1962, Schurer [21] introduced and studied the operators S m,ℓ : C [0 , ℓ + 1] → C [0 ,
1] defined for any m ∈ N and ℓ be fixed in N and any function f ∈ C [0 , ℓ + 1]as follows S m,ℓ ( f ; x ) = m + ℓ X k =0 (cid:20) m + ℓk (cid:21) x k (1 − x ) m + ℓ − k f (cid:18) km (cid:19) , x ∈ [0 , . (1.1)For any m ∈ and f ∈ C [0 , ℓ +1], ℓ is fixed, then q − analogue of Bernstein-Schureroperators in [11] defined as follows˜ B m,ℓ ( f ; q ; x ) = m + ℓ X k =0 (cid:20) m + ℓk (cid:21) q x k m + ℓ − k − Y s =0 (1 − q s x ) f (cid:18) [ k ] q [ m ] q (cid:19) , x ∈ [0 , . (1.2)Our aim is to introduce a ( p, q )-analogue of these operators. We investigate theapproximation properties of this class and we estimate the rate of convergence andsome theorem by using the modulus of continuity. We study the approximationproperties based on Korovkin’s type approximation theorem and also establishthe some direct theorem.2. Construction of ( p, q ) -Bernstein-Schurer operators (Revised) Mursaleen et. al [14] has defined ( p, q )-analogue of Bernstein operators as: B p,qn ( f ; x ) = n X k =0 (cid:20) nk (cid:21) p,q x k n − k − Y s =0 ( p s − q s x ) f (cid:18) [ k ] p,q [ n ] p,q (cid:19) , x ∈ [0 , . (2.1)But B p,qm,ℓ ( f ; x ) = 1, for all x ∈ [0 , p, q ) Bern-stein operators [15] as follows: B p,qm,ℓ ( f ; x ) = 1 p n ( n − n X k =0 (cid:20) nk (cid:21) p,q p k ( k − x k n − k − Y s =0 ( p s − q s x ) f (cid:18) [ k ] p,q p k − n [ n ] p,q (cid:19) , x ∈ [0 , . (2.2) Mursaleen et. al [17] introduced the ( p, q )-analogue of Bernstein Schurer operatorsas: B p,qm,ℓ ( f ; x ) = m + ℓ X k =0 (cid:20) m + ℓk (cid:21) p,q x k m + ℓ − k − Y s =0 ( p s − q s x ) f (cid:18) [ k ] p,q [ m ] p,q (cid:19) , x ∈ [0 , . (2.3)But B p,qm,ℓ ( f ; x ) = 1, for all x ∈ [0 , < q < p ≤ m ∈ N , f ∈ C [0 , ℓ + 1], ℓ is fixed, weconstruct a revised generalized ( p, q )-Bernstein Schurer operators: B p,qm,ℓ ( f ; x ) = 1 p ( m + ℓ )( m + ℓ − m + ℓ X k =0 (cid:20) m + ℓk (cid:21) p,q p k ( k − x k m + ℓ − k − Y s =0 ( p s − q s x ) f (cid:18) [ k ] p,q p k − m − ℓ [ m ] p,q (cid:19) , x ∈ [0 , . (2.4)Clearly, the operator defined by (2.4) is linear and positive. And if we put p = 1 in(2.4), then ( p, q ) Shurer operator given by (2.4) turn out the q - Bernstein Shureroperators [11]. Lemma 2.1.
Let B p,qm,ℓ ( . ; . ) be given by (2.4) , then for any x ∈ [0 , and < q
Let B p,qm,ℓ ( . ; . ) be given by lemma (2.1) , then for any x ∈ [0 , and < q < p ≤ we have the following identities (i) B p,qm,ℓ ( e − x ) = [ m + ℓ ] p,q [ m ] p,q x − B p,qm,ℓ ( e − x ; x ) = (cid:16) [ m + ℓ ] p,q [ m ] p,q − (cid:17) x (iii) B p,qm,ℓ (( e − x ) ; x ) = p m + ℓ − [ m + ℓ ] p,q [ m ] p,q x + (cid:16) − [ m + ℓ ] p,q [ m ] p,q + q [ m + ℓ − p,q [ m + ℓ ] p,q [ m ] p,q (cid:17) x . On the convergence of ( p, q ) -Bernstein-Schurer operators Let f ∈ C [0 , γ ], and the modulus of continuity of f denoted by ω ( f, δ ) gives themaximum oscillation of f in any interval of length not exceeding δ > ω ( f, δ ) = sup | y − x |≤ δ | f ( y ) − f ( x ) | , x, y ∈ [0 , γ ] . It is known that lim δ → ω ( f, δ ) = 0 for f ∈ C [0 , γ ] and for any δ > | f ( y ) − f ( x ) |≤ (cid:18) | y − x | δ + 1 (cid:19) ω ( f, δ ) . (3.1)For q ∈ (0 ,
1) and p ∈ ( q,
1] obviously have lim m →∞ [ m ] p,q = p − q . In order to reachto the convergence results of the operator B p,qm,ℓ , we take a sequence q m ∈ (0 , p m ∈ ( q m ,
1] such that lim m →∞ p m = 1 and lim m →∞ q m = 1, so we getlim m →∞ [ m ] p m ,q m = ∞ . Theorem 3.1.
Let p = p m , q = q m satisfying < q m < p m ≤ such that lim m →∞ p m = 1 , lim m →∞ q m = 1 . Then for each f ∈ C [0 , ℓ + 1] , lim m →∞ B p m ,q m m,ℓ ( f ; x ) = f, (3.2) is uniformly on [0 , .Proof. The proof is based on the well known Korovkin theorem regarding theconvergence of a sequence of linear and positive operators, so it is enough toprove the conditions B p m ,q m m,ℓ (( e j ; x ) = x j , j = 0 , , , { as m → ∞} uniformly on [0 , m →∞ B p m ,q m m,ℓ ( e ; x ) = 1 . By taking the simple calculation we getlim m →∞ [ m + ℓ ] p m ,q m [ m ] p m ,q m = 1 , as 0 < q m < p m ≤ . Since as 0 < q m < p m ≤
1, then we get,lim m →∞ [ m + ℓ ] p m ,q m [ m ] p m ,q m = 0 . Hence we have lim m →∞ B p m ,q m m,ℓ ( e ; x ) = x lim m →∞ B p m ,q m m,ℓ ( e ; x ) = x (cid:3) Theorem 3.2. If f ∈ C [0 , ℓ + 1] , then | B p,qm,ℓ ( f ; x ) − f ( x ) |≤ ω f ( δ m ) , where δ m = x (cid:12)(cid:12)(cid:12)(cid:12) [ m + ℓ ] p,q [ m ] p,q − (cid:12)(cid:12)(cid:12)(cid:12) + s [ m + ℓ ] p,q [ m ] p,q . s ( q [ m + ℓ − p,q − [ m + ℓ ] p,q ) x + p ( m + ℓ − x [ m ] p,q . Proof. | B p,qm,ℓ ( f ; x ) − f ( x ) |≤ p ( m + ℓ )( m + ℓ − m + ℓ X k =0 (cid:20) m + ℓk (cid:21) p,q p k ( k − x k m + ℓ − k − Y s =0 ( p s − q s x ) (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) [ k ] p,q p k − m − ℓ [ m ] p,q (cid:19) − f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ p ( m + ℓ )( m + ℓ − m + ℓ X k =0 (cid:20) m + ℓk (cid:21) p,q p k ( k − x k m + ℓ − k − Y s =0 ( p s − q s x ) (cid:12)(cid:12)(cid:12)(cid:12) [ k ] p,q p k − m − ℓ [ m ] p,q − x (cid:12)(cid:12)(cid:12)(cid:12) δ + 1 ω ( f, δ ) . By using the Cauchy inequality and lemma (2.1) we have | B p,qm,ℓ ( f ; x ) − f ( x ) |≤ δ ( p ( m + ℓ )( m + ℓ − m + ℓ X k =0 (cid:20) m + ℓk (cid:21) p,q p k ( k − x k (cid:18) [ k ] p,q [ m ] p,q − x (cid:19) m + ℓ − k − Y s =0 ( p s − q s x ) ) × (cid:0) B p,qm,ℓ ( e ; x ) (cid:1) ω ( f, δ )= (cid:26) δ (cid:0) B p,qm,ℓ ( e ; x ) − xB p,qm,ℓ ( e ; x ) + x B p,qm,ℓ ( e ; x ) (cid:1) + 1 (cid:27) ω ( f, δ )= ( δ (cid:18) [ m + ℓ ] p,q p m + ℓ − [ m ] p,q x + (cid:18) [ m + ℓ ] p,q [ m + ℓ − p,q [ m ] p,q q − m + ℓ ] p,q [ m ] p,q + 1 (cid:19) x (cid:19) + 1 ) ω ( f, δ )= δ (cid:16) x (cid:16) [ m + ℓ ] p,q [ m ] p,q − (cid:17)(cid:17) + (cid:18)q [ m + ℓ ] p,q [ m ] p,q . q ( q [ m + ℓ − p,q − [ m + ℓ ] p,q ) x + p ( m + ℓ − x [ m ] p,q (cid:19) ! + 1 ω ( f, δ ) ≤ (cid:26) δ (cid:18) x (cid:12)(cid:12)(cid:12)(cid:12) [ m + ℓ ] p,q [ m ] p,q − (cid:12)(cid:12)(cid:12)(cid:12) + q [ m + ℓ ] p,q [ m ] p,q . q ( q [ m + ℓ − p,q − [ m + ℓ ] p,q ) x + p ( m + ℓ − x [ m ] p,q (cid:19) + 1 (cid:27) ω ( f, δ ). { by using ( a + b ) ≤ ( | a | + | b | ) } .Hence we obtain the desired result by choosing δ = δ m . (cid:3) Direct Theorems on ( p, q ) -Bernstein-Schurer operators The Peetre’s K -functional is defined by K ( f, δ ) = inf (cid:8) ( k f − g k + δ k g ′′ k ) : g ∈ W (cid:9) , where W = { g ∈ C [0 , ℓ + 1] : g ′ , g ′′ ∈ C [0 , ℓ + 1] } . Then there exits a positive constant C > K ( f, δ ) ≤ C ω ( f, δ ) , δ > ω ( f, δ ) = sup Let f ∈ C [0 , ℓ + 1] , g ′ ∈ C [0 , ℓ + 1] and satisfying < q < p ≤ .Then for all n ∈ N there exits a constant C > such that (cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( f ; x ) − f ( x ) − xg ′ ( x ) (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ω ( f, δ m ( x )) , where δ m ( x ) = [ m + ℓ ] p,q [ m ] p,q p ( m + ℓ − x + (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) + [ m + ℓ ] p,q [ m ] p,q ( q [ m + ℓ − p,q − [ m + ℓ ] p,q ) ! x Proof. Let g ∈ W , then from the Taylor’s expansion, we get g ( t ) = g ( x ) + g ′ ( x )( t − x ) + Z tx ( t − u ) g ′′ ( u )d u, t ∈ [0 , A ] , A > . Now by lemma (2.2), we have B p,qm,ℓ ( g ; x ) = g ( x ) + xg ′ ( x ) (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) + B p,qm,ℓ (cid:18)Z tx ( e − u ) g ′′ ( u )d u ; p, q ; x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( g ; x ) − g ( x ) − xg ′ ( x ) (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≤ B p,qm,ℓ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z tx | ( e − u ) | | g ′′ ( u ) | d u ; p, q ; x (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ B p,qm,ℓ (cid:0) ( e − x ) ; p, q ; x (cid:1) k g ′′ k Hence we get (cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( g ; x ) − g ( x ) − xg ′ ( x ) (cid:16) [ m + ℓ ] p,q [ m ] p,q − (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤k g ′′ k (cid:18) ( [ m + ℓ ] p,q [ m ] p,q p ( m + ℓ − x + (cid:18)(cid:16) [ m + ℓ ] p,q [ m ] p,q − (cid:17) + [ m + ℓ ] p,q [ m ] p,q ( q [ m + ℓ − p,q − [ m + ℓ ] p,q ) (cid:19) x (cid:19) .On the other hand we have (cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( f ; x ) − f ( x ) − xg ′ ( x ) (cid:16) [ m + ℓ ] p,q [ m ] p,q − (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | B p,qm,ℓ (( f − g ); x ) − ( f − g )( x ) | + (cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( g ; x ) − g ( x ) − xg ′ ( x ) (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) . Since we know the relation | B p,qm,ℓ ( f ; x ) |≤k f k . Therefore (cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( f ; x ) − f ( x ) − xg ′ ( x ) (cid:16) [ m + ℓ ] p,q [ m ] p,q − (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k f − g k + k g ′′ k (cid:18) ( [ m + ℓ ] p,q [ m ] p,q p ( m + ℓ − x + (cid:18)(cid:16) [ m + ℓ ] p,q [ m ] p,q − (cid:17) + [ m + ℓ ] p,q [ m ] p,q ( q [ m + ℓ − p,q − [ m + ℓ ] p,q ) (cid:19) x (cid:19) , Now taking the infimum on the right hand side over all g ∈ W , we get (cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( f ; x ) − f ( x ) − xg ′ ( x ) (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C K (cid:0) f, δ m ( x ) (cid:1) . In the view of the property of K -functional, we get (cid:12)(cid:12)(cid:12)(cid:12) B p,qm,ℓ ( f ; x ) − f ( x ) − xg ′ ( x ) (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ω ( f, δ m ( x )) . This completes the proof. (cid:3) Theorem 4.2. Let f ∈ C [0 , ℓ + 1] be such that f ′ , f ′′ ∈ C [0 , ℓ + 1] , and thesequence { p m } , { q m } satisfying < q m < p m ≤ such that p m → , q m → and p mm → α, q mm → β as m → ∞ , where ≤ α, β < . Then lim m →∞ [ m ] p m ,q m (cid:0) B p m ,q m m,ℓ ( f ; x ) − f ( x ) (cid:1) = x ( λ − αx )2 f ′′ ( x ) , is uniformly on [0 , ℓ + 1] , where < λ ≤ . Proof. From the Taylors formula we have f ( t ) = f ( x ) + f ′ ( x )( t − x ) + 12 f ′′ ( x )( t − x ) + r ( t, x )( e − x ) , where r ( t, x ) is the remainder term and lim t → x r ( t, x ) = 0, therefore we have[ m ] p m ,q m (cid:0) B p m ,q m m,ℓ ( f ; x ) − f ( x ) (cid:1) = [ m ] p m ,q m (cid:16) f ′ ( x ) B p m ,q m m,ℓ (( e − x ); x ) + f ′′ ( x )2 B p m ,q m m,ℓ (( e − x ) ; x ) + B p m ,q m m,ℓ ( r ( t, x )( t − x ) ; x ) (cid:17) . Now by applying the Cauchy-Schwartz inequality, we have B p m ,q m m,ℓ (cid:0) r ( t, x )( t − x ) ; x ) (cid:1) ≤ q B p m ,q m m,ℓ ( r ( t, x ); x )) . q B p m ,q m m,ℓ (( t − x ) ; x )) . Since r ( x, x ) = 0, and r ( t, x ) ∈ C [0 , ℓ + 1], then for from the Theorem 3.1 wehave B p m ,q m m,ℓ (cid:0) r ( t, x ); x ) (cid:1) = r ( x, x ) = 0 , which imply that B p m ,q m m,ℓ (cid:0) r ( t, x )( t − x ) ; x ) (cid:1) = 0lim m →∞ [ m ] p m ,q m (cid:0) B p m ,q m m,ℓ (( e − x ); x )) (cid:1) = x lim m →∞ [ m ] p m ,q m (cid:18) [ m + ℓ ] p m ,q m [ m ] p m ,q m − (cid:19) = 0lim m →∞ [ m ] p m ,q m (cid:0) B p m ,q m m,ℓ (( e − x ) ; x )) (cid:1) = x lim m →∞ [ m ] p m ,q m [ m + ℓ ] p m ,q m [ m ] p m ,q m p m + ℓ − m + x lim m →∞ [ m ] p m ,q m (cid:18) [ m + ℓ ] p m ,q m [ m ] p m ,q m − (cid:19) + [ m + ℓ ] p m ,q m [ m ] p m ,q m ( q m [ m + ℓ − p m ,q m − [ m + ℓ ] p m ,q m ) ! lim m →∞ [ m ] p m ,q m (cid:0) B p m ,q m m,ℓ (cid:0) ( e − x ) ; x ) (cid:1)(cid:1) = λx − αx = x ( λ − αx ) , where λ ∈ (0 , 1] depending on the sequence { p m } .Hence we havelim m →∞ [ m ] p m ,q m (cid:0) B p m ,q m m,ℓ ( f ; x ) − f ( x ) (cid:1) = x ( λ − αx )2 f ′′ ( x ) . This completes the proof. (cid:3) Now we give the rate of convergence of the operators B p,qm,ℓ ( f ; x ) in terms of theelements of the usual Lipschitz class Lip M ( ν ).Let f ∈ C [0 , m + ℓ ], M > < ν ≤ 1. We recall that f belongs to the class Lip M ( ν ) if the inequality | f ( t ) − f ( x ) |≤ M | t − x | ν ( t, x ∈ (0 , Theorem 4.3. Let < q < p ≤ . Then for each f ∈ Lip M ( ν ) we have | B p,qm,ℓ ( f ; x ) − f ( x ) |≤ M δ νm ( x ) where δ m ( x ) = [ m + ℓ ] p,q [ m ] p,q p ( m + ℓ − x + (cid:18) [ m + ℓ ] p,q [ m ] p,q − (cid:19) + [ m + ℓ ] p,q [ m ] p,q ( q [ m + ℓ − p,q − [ m + ℓ ] p,q ) ! x . Proof. By the monotonicity of the operators B p,qm,ℓ ( f ; x ), we can write | B p,qm,ℓ ( f ; x ) − f ( x ) |≤ B p,qm,ℓ ( | f ( t ) − f ( x ) | ; p, q ; x ) ≤ p ( m + ℓ )( m + ℓ − m + ℓ X k =0 (cid:20) m + ℓk (cid:21) p,q p k ( k − x k m + ℓ − k − Y s =0 ( p s − q s x ) (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) [ k ] p,q p k − m − ℓ [ m ] p,q (cid:19) − f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M p ( m + ℓ )( m + ℓ − m + ℓ X k =0 (cid:20) m + ℓk (cid:21) p,q p k ( k − x k m + ℓ − k − Y s =0 ( p s − q s x ) (cid:12)(cid:12)(cid:12)(cid:12) [ k ] p,q p k − m − ℓ [ m ] p,q − x (cid:12)(cid:12)(cid:12)(cid:12) ν = M m + ℓ X k =0 p ( m + ℓ )( m + ℓ − P m,ℓ,k ( x ) (cid:18) [ k ] p,q p k − m − ℓ [ m ] p,q − x (cid:19) ! ν (cid:18) p ( m + ℓ )( m + ℓ − P m,ℓ,k ( x ) (cid:19) − ν , where P m,ℓ,k ( x ) = (cid:20) m + ℓk (cid:21) p,q p k ( k − x k Q m + ℓ − k − s =0 ( p s − q s x ) Now applying the H¨older’s inequality for the sum with p = ν and q = − ν | B p,qm,ℓ ( f ; x ) − f ( x ) |≤ M p ( m + ℓ )( m + ℓ − m + ℓ X k =0 P m,ℓ,k ( x ) (cid:18) [ k ] p,q [ m ] p,q − x (cid:19) ! ν p ( m + ℓ )( m + ℓ − m + ℓ X k =0 P m,ℓ,k ( x ) ! − ν = M (cid:0) B p,qm,ℓ (cid:0) ( e − x ) ; x (cid:1)(cid:1) ν Choosing δ : δ m ( x ) = q B p,qm,ℓ (( e − x ) ; x ),we obtain | B p,qm,ℓ ( f ; x ) − f ( x ) |≤ M δ νm ( x ) . 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