FINSLER SPACE SUBJECTED TO A KROPINA CHANGE WITH AN. 1. Introduction

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1 FACTA UNIVERSITATIS (NIŠ) Se.Math.Infom. Vol. 30 No 4 (2015) FINSER SPACE SUBJECTE TO A KROPINA CHANGE WITH AN h-vector M. K. Gupta and P. N. Pandey Abstact. In this pape we discuss the Finsle spaces (M n ) and (M n ) whee (x y) is obtained fom (x y) by Kopina change (x y) = and b i (x y) isanh-vecto 2 (xy) b i (xy) y i in (M n ). We find the necessay and sufficient condition when the Catan connection coefficients fo both spaces (M n ) and (M n ) ae the same. We also find the necessay and sufficient condition fo Kopina change with an h-vecto to be pojective. Keywods: Finsle space Kopina change h-vecto. 1. Intoduction In 1984 C. Shibata [16] dealt with a change of Finsle metic which is called a - change of metic. A emakable class of -change is Kopina change (x y) = 2 (xy) b i. (x) y i If (x y) is a metic function of a Riemannian space then (x y) educes to the metic function of a Kopina space. Kopina metic was fist intoduced by. Bewald in connection with a two-dimensional Finsle space with ectilinea extemals and was investigated by V.K. Kopina [8 9]. Kopina metic is the simplest non-tivial Finsle metic having many inteesting applications in physics electon optics with a magnetic field plants study fo fungal fusion hypothesis dissipative mechanics and ievesible themodynamics [ ]. In 1978 C. Shibata [15] studied some basic local geometic popeties of Kopina spaces. In 1991 M. Matsumoto obtained a set of necessay and sufficient conditions fo a Kopina space to be of constant cuvatue [12]. H. Izumi [7] while studying the confomal tansfomation of Finsle spaces intoduced the concept of h-vecto b i which is v-covaiant constant with espect to the Catan connection and satisfies C h ij b h = ρ h ij whee ρ is a non-zeo scala function C h ij ae components of Catan tenso and h ij ae components of angula metic tenso. Thus if b i is an h-vecto then (1.1) (i) b i k = 0 (ii) C h ij b h = ρ h ij. Received Januay ; Accepted Apil Mathematics Subject Classification. 53B40 Coesponding autho. 513

2 514 Manish Kuma Gupta and Paas Nath Pandey This gives (1.2) j b i = ρ h ij. Since ρ 0 and h ij 0 the h-vecto b i depends not only on positional coodinates but also on diectional aguments. Izumi [7] showed that ρ is independent of diectional aguments. M. Matsumoto [11] discussed the Catan connection of Randes change of Finsle metic while B. N. Pasad [14] obtained the Catan connection of (M n ) whee (x y) is given by (x y) = (x y) + b i (x y) y i and b i (x y)isanh-vecto. Pesent authos [4 5] discussed the hypesuface of a Finsle space whose metic is given by cetain tansfomations with an h-vecto. In this pape we obtain the elation between the Catan connections of F n = (M n ) and F n = (M n ) whee (x y) is obtained by the tansfomation (1.3) (x y) = 2 (x y) b i (x y) y i and b i (x y)isanh-vecto in (M n ). The pape is oganized as follows: In Section 2 we study how the fundamental metic tenso and the Catan tenso change by Kopina change with an h-vecto. The elation between the Catan connection coefficients of both spaces is obtained in the Section 3 and we find the necessay and sufficient condition when these connection coefficients ae the same. In Section 4 we find the necessay and sufficient condition fo the Kopina change with an h-vecto to be pojective. 2. The Finsle space F n = (M n ) et F n = (M n )beann-dimensional Finsle space equipped with the fundamental function (x y). We conside a change of Finsle metic (x y) which is defined by (1.3) and have anothe Finsle space F n = (M n ). If we denote b i y i by then the indicatoy popety of h ij yields i = b i. Thoughout this pape the geometic objects associated with F n will be maked by the asteisk. We shall use the notation i = i = l i ij = i j ijk = k ij... etc. Fom (1.3) we get (2.1) i = 2 τ i τ 2 b i (2.2) (2.3) ij = (2τ ρτ 2 ) ij + 2 τ2 m i m j ijk =(2τ ρτ 2 ) ijk + 2 τ (ρτ 1) (m i jk + m j ik + m k ij ) 2τ2 (m im j l k + m j m k l i + m k m i l j ) 6τ2 2 m im j m k

3 Finsle Space Subjected to a Kopina Change with an h-vecto 515 whee τ = / m i = b i 1 τ l i. The nomalized suppoting element the metic tenso and Catan tenso of F n ae obtained as (2.4) l i = 2 τl i τ 2 b i (2.5) ij = (2τ 2 ρτ 3 ) ij + 3τ 4 b i b j 4τ 3 (l i b j + b i l j ) + (4τ 2 + ρτ 3 )l i l j (2.6) C ijk = (2τ 2 ρτ 3 ) C ijk τ2 2 (4 3ρτ)(h ijm k + h jk m i + h ki m j ) 6τ2 m im j m k. Fo the computation of the invese metic tenso we use the following lemma [10]: emma 2.1. et (m ij ) be a non-singula matix and l ij = m ij + n i n j. The elements l ij of the invese matix and the deteminant of the matix (l ij ) ae given by l ij = m ij (1 + n k n k ) 1 n i n j det(l ij ) = (1 + n k n k ) det(m ij ) espectively whee m ij ae elements of the invese matix of (m ij ) and n k = m ki n i. (2.7) The invese metic tenso of F n is deived as follows: ij = (2τ 2 ρτ 3 ) 1[ ij whee b is the magnitude of the vecto b i = ij b j. Fom (2.6) and (2.7) we get C h ij = Ch ij 2τ 2b 2 τ ρ bi b j + 4 ρτ 2b 2 τ ρ (li b j + b i l j ) 3ρb2 τ 3 ρ 2 τ 2 4b 2 τ 2 2ρτ + 8 l i l j] τ(2b 2 τ ρ) (4 3ρτ)τ ( hij m h +h h j 2(2 ρτ) m i + h h i m 6τ j) (2 ρτ) m im j m h (2.8) + 2τ bh (4 ρτ)l h [ 1 hij (2 ρτ)(2b 2 τ ρ) 2 m2 τ(4 3ρτ) ρ(2 ρτ) + m i m j 6τ m 2 + τ(4 3ρτ) ]. 3. Catan connection of the space F n et C Γ=( F i jk N i j C i jk ) be the Catan connection fo the space F n = (M n ). Since fo a Catan connection i j = 0 we obtain (3.1) j i = i N j + F ij.

4 516 Manish Kuma Gupta and Paas Nath Pandey iffeentiating (2.1) with espect to x j we get (3.2) j i = 2τ j i + 2 i j τ τ 2 j b i 2τ b i j τ. This equation may be witten in tensoial fom as (3.3) i N j + F ij = 2τ ( in j + F ij ) + 2 i (N j τ j τn j b ) τ 2 (b i j + ρ i N j + b F ij ) 2τ b i(n j τ j τn j b ) whee j = j. If we put (3.4) F i jk = F i jk + i jk then in view of (2.2) equation (3.3) may be witten as (3.5) (2τl τ 2 b ) ij + (2τ ρτ 2 ) i + 2τ2 m im 0j = 2τ2 m i j τ 2 b i j whee the subscipt 0 denote the contaction by y i. In ode to find the diffeence tenso i we constuct supplementay equations jk to (3.5). Fom (2.2) we obtain k ij =(2τ ρτ 2 ) k ij + ij (2 k τ 2ρτ k τ τ 2 k ρ) (3.6) + 2 τ2 m i k m j + 2 τ2 m j k m i + 2m i m j 1 2 ( 2τ k τ τ 2 k ). Fom ij k = 0 equation (3.6) is witten in the fom ij N k + j F ik + i F jk =(2τ ρτ2 ) ij N k + j F ik + i F jk 2 + ij (1 ρτ)(τ k τ N k m ) τ 2 ρ k + 2τ2 m i F jk m + N k ( 1 j ρ τ F + 2τ2 m j ik m + N k i ( ρ 1 τ ) 1 τ N k l jm ) 1 τ N k l im m im j 2τ(τk τ N k m ) τ 2 ( k + N k b )

5 Finsle Space Subjected to a Kopina Change with an h-vecto 517 whee ρ k = ρ k = k ρ. In view of (2.2) (2.3) and (3.4) above equation is witten as (2τ ρτ 2 ) ij 0k + j ik + i jk + 2τ2 m (m j ik + m i jk ) (3.7) 2τ2 (m im 2 j b + m j m b i + m m i b j ) 0k + 2τ k ij 2ρτ2 k ij Now we will pove: + 2τ2 (ρ 1 τ )( jm i + i m j + ij m ) 0k + 6τ2 2 k m im j + τ 2 ρ k ij = 0. Poposition 3.1. The diffeence tenso i jk (3.5) and (3.7). is completely detemined by the equations To pove this fist we will pove a lemma: emma 3.1. The system of algebaic equations (i) i A = B i (ii) A = B has a unique solution A fo given B and B i. Poof. It follows fom (2.2) that (i) is witten in the fom (3.8) (2τ ρτ 2 ) 1 ( i l i l ) + 2τ2 m im A = B i. Contacting by b i we get i.e. (2τ ρτ 2 ) 1 ( b 1 τ l ) + 2τ 2 (3.9) m A = B ( 2τ 2 b 2 m2 m A = B ρτ2 ) 1 whee the subscipt denote the contaction by b i i.e. B = B i b i. Also fom (2.1) equation (ii) is witten in the fom i.e. (2τ l τ 2 b ) A = B (3.10) τ 2 m A τ l A = B.

6 518 Manish Kuma Gupta and Paas Nath Pandey Using (3.9) in (3.10) we get Then (3.8) is witten as i A = This gives (3.11) A i = l A = τ 1 B + τ B ( 2τ 2 b 2 2τ ρτ B ( 2τ 2 i + l i τ 1 2 b 2 B + τ B 2τ ρτ 2 Bi + l ( 2τ i τ 1 2 b 2 B + τ B which is the concete fom of the solution A i. ρτ2 ) 1. ρτ2 ) 1 2τ 2 2 ρτ m ( 2τ 2 b 2 i B ρτ2 ) 1 2τ 2 ( 2τ 2 b 2 2 ρτ mi B We ae now in a position to pove the poposition. Taking the symmetic and anti-symmetic pats of (3.5) we get ρτ2 ) 1. ρτ2 ) 1 (3.12) and 2(2τl τ 2 b ) ij + (2τ ρτ 2 ) i + 2τ2 m im 0j + (2τ ρτ 2 ) j + 2τ2 m jm 0i = 2τ2 (m i j + m j i ) 2τ 2 E ij (3.13) (2τ ρτ 2 ) i + 2τ2 m im 0j (2τ ρτ 2 ) j + 2τ2 m jm 0i = 2τ2 (m i j m j i ) 2τ 2 F ij whee we put 2E ij = b i j + b j i and 2F ij = b i j b j i. On the othe hand applying Chistoffel pocess with espected to indices i j k in equation (3.7) we get (3.14) (2τ ρτ 2 ) ij 0k + jk 0i ki 0j + 2 ik (2τ ρτ 2 ) j + 2τ2 m m j 2τ k ( (ρτ 1)ij 3τ m im j ) + i ( (ρτ 1)jk 3τ m jm k ) j ( (ρτ 1)ki 3τ m km i ) + 2τ 0k S (ij)m i (ρτ 1)j τ m jb + 2τ 0i S (jk)m j (ρτ 1)k τ m kb 2τ 0j S (ki)m k (ρτ 1)i τ m ib + τ 2 (ρ k ij + ρ i jk ρ j ki ) = 0

7 Finsle Space Subjected to a Kopina Change with an h-vecto 519 whee S (ijk) denote cyclic intechange of indices i j k and summation. Contacting (3.12) and (3.13) by y j we get (3.15) (4τ l 2τ 2 b ) 0i + (2τ ρτ 2 ) i + 2τ2 m im 00 = 2τ2 0 m i 2τ 2 E i0 and i.e. (3.16) (2τ ρτ 2 ) i + 2τ2 m im 00 = 2τ2 0 m i 2τ 2 F i0 i 00 = 2τ2 0 m i 2τ 2 F i0 which on contaction by b i gives m 00 = ( 2 )( 2 0 m2 2F 0 b2 ρ ) 1 whee 0 = j y j. Similaly contaction of (3.14) by y k gives (2τ ρτ 2 ) ij 00 j 0i + i 0j + 2 0i (2τ ρτ 2 ) j + 2τ2 m m j (3.17) + 2τ 00 S (ij)m i (ρτ 1)j τ m jb 2τ 2 0i m jm + 2τ2 0j m im 2τ 0( (ρτ 1)ij 3τ m im j ) + τ 2 ρ 0 ij = 0 Contaction of (3.15) by y i gives i.e. (2τ l τ 2 b ) 00 = τ2 E 00 (3.18) 00 = E 00. We can apply emma 3.1 to equations (3.16) and (3.18) to obtain (3.19) i 00 =li τ ( 2 )( 2 0 m2 2F 0 b2 ρ ) 1 E00 whee F i 0 = ij F j0. Also note that 2τ2 2 ρτ( 2 0 m2 2F 0 )( 2 b2 ρ ) 1 m i + 2τ 1 2 ρτ( ) 0 mi F i 0 E 00 = E ij y i y j = b i j y i y j = (b i y i ) j y j = 0 = 0.

8 520 Manish Kuma Gupta and Paas Nath Pandey Now adding (3.13) and (3.17) we obtain i.e. 0j (2τ ρτ 2 ) i + 2τ2 m im = Gij (3.20) i 0j = G ij whee we put (3.21) G ij = τ2 (m i j m j i ) τ 2 F ij 1 2 (2τ ρτ2 ) ij 00 τ2 2 ρ 0 ij τ 00 S (ij)m i (ρτ 1)j τ m τ jb + 0 (ρτ 1)ij 3τ m im j. The equation (3.15) is witten in the fom i.e. (2τ l τ 2 b ) 0j = G j (3.22) 0j = 1 τ 2 G j whee G j = τ2 0 m j τ 2 E j0 1 2 (2τ ρτ2 ) j + τ2 m jm 00. In view of (3.16) G j ae witten as (3.23) G j = τ 2 (F j0 E j0 ). Applying emma 3.1 to equations (3.20) and (3.22) to obtain (3.24) i 0j = li 1 2 τ( b2 ρ ) 1Gj + 1 τ G 2 m i ( 2 j 2 ρτ b2 ρ ) 1Gj + 2τ ρτ G i 2 j whee G i j = ik G kj. Finally we solve (3.12) and (3.14) fo i. These equations may be witten as jk (3.25) j ik = H jik and (3.26) ik = H ik

9 whee Finsle Space Subjected to a Kopina Change with an h-vecto 521 (3.27) H jik = (ρτ2 2τ) ij 2 0k + jk 0i ki 0j τ 0k S (ij)m i (ρτ 1)j τ m τ jb 0i S (jk)m j (ρτ 1)k τ m kb and + τ 0j S (ki)m k (ρτ 1)i τ m τ 2 ib 2 (ρ k ij + ρ i jk ρ j ki ) + τ ( k (ρτ 1)ij 3τ m ) ( im j + i (ρτ 1)jk 3τ m ) jm k j ( (ρτ 1)ki 3τ m km i ) (3.28) H ik = τ2 (m i k + m k i ) τ 2 E ik 1 2 (G ik + G ki ). Again applying emma 3.1 to equations (3.25) and (3.26) to obtain (3.29) j ik = lj 1 2 τ( b2 ρ ) 1Hik + 1 τ H 2 m ik j 2 2 ρτ( b2 ρ ) 1Hik + 2τ ρτ H j 2 ik whee we put H j ik = jm H mik. This completes the Poposition 3.1. We now popose a lemma: emma 3.2. If the h-vecto is gadient then the scala ρ is constant. Poof. Taking h-covaiant deivative of (1.2) and using k = 0 and h ij k = 0 we get ( j b i ) k = ρ k h ij. Utilizing the commutation fomula exhibited by k (T i j h ) ( k T i j ) h = T j k F i h T i k F jh ( T i j )C hk 0 ; we get 2 j F ik = ρ k h ij ρ i h jk. If b i is a gadient vecto i.e. 2F ij = b i j b j i = 0. Then above equation becomes ρ k h ij ρ i h jk = 0 which afte contaction by y k gives ρ k y k = 0. iffeentiating ρ k y k = 0 patially with espect to y j and using the commutation fomula j (ρ k ) ( j ρ) k = ( ρ)c jk 0 and the fact that ρ is a function of position only we get ρ j = 0 and theefoe j ρ = 0. This completes the poof.

10 522 Manish Kuma Gupta and Paas Nath Pandey Now we find the condition fo which the Catan connection coefficients fo both spaces F n and F n ae the same i.e. F i jk = F i jk then i = 0. Theefoe (3.15) and jk (3.16) gives E i0 = F i0. This will give (3.30) b 0 i = 0 i.e. i = 0. iffeentiating i = 0 patially with espect to y j and using the commutation fomula j ( i ) ( j ) i = ( )C we get ij 0 (3.31) b j i = b C ij 0. This gives F ij = 0 and then in view of emma 3.2 F ij = 0 implies ρ i = ρ i = 0. Taking h-covaiant deivative of (1.1)(ii) and using k = 0 ρ k = 0 and h ij k = 0 we get (b C ij ) k = ( ρ h ) ij = 0. This gives k b k C ij + b C ij k = 0. Fom (3.31) we get b k = b k then above equation becomes b k C ij + b C ij k = 0. Contacting by y k we get b 0 C ij + b C = 0. Using (3.30) and (3.31) this gives ij 0 b i j = 0 i.e. the h-vecto b i is paallel with espect to the Catan connection of F n. Convesely if b i j = 0 then we get E ij = 0 = F ij and i = i = b j i y j = 0. In view of emma 3.2 F ij = 0 implies ρ i = ρ i = 0. Theefoe fom (3.19) we get i 00 = 0 and then G ij = 0 and G j = 0. This gives i 0j = 0 and then H jik = 0 and H = 0. Theefoe ik (3.29) implies i jk = 0 and then F i jk = F i. Thus we have: jk Theoem 3.1. Fo the Kopina change with an h-vecto the Catan connection coefficients fo both spaces F n and F n ae the same if and only if the h-vecto b i is paallel with espect to the Catan connection of F n. Tansvecting (3.4) by y j and using F i jk yj = G i we get k (3.32) G i k = G i k + i 0k. Futhe tansvecting (3.32) by y k and using G i k yk = 2 G i we get (3.33) 2 G i = 2 G i + i 00. iffeentiating (3.32) patially with espect to y h and using h G i k = G i we have kh (3.34) G i kh = G i kh + h i 0k

11 Finsle Space Subjected to a Kopina Change with an h-vecto 523 whee G i ae the Bewald connection coefficients. kh Now if the h-vecto b i is paallel with espect to the Catan connection of F n then by Theoem 3.1 the Catan connection coefficients fo both spaces F n and F n ae the same theefoe i jk = 0. Then fom (3.34) we get G i kh = G i kh. Thus we have: Theoem 3.2. Fo the Kopina change with an h-vecto if the h-vecto b i is paallel with espect to the Catan connection of F n. Then the Bewald connection coefficients fo both the spaces F n and F n ae the same. 4. Relation between Pojective change and Kopina change with an h-vecto We conside two Finsle spaces F n = (M n ) and F n = (M n ). If any geodesic on F n is also a geodesic on F n and the invese is tue the change of the metic is called pojective. A geodesic on F n is given by dy i dt + 2 Gi (x y) = τ y i ; τ = d2 s/dt 2 ds/dt. The change is a pojective change if and only if thee exists a scala P(x y) which is positively homogeneous of degee one in y i and satisfies [13] G i (x y) = G i (x y) + P(x y) y i. Now we find condition fo the Kopina change (1.3) with h-vecto to be pojective. Fom (3.33) it follows that the Kopina change with an h-vecto is pojective if and only if i 00 = 2 Pyi. Then fom (3.19) we get (4.1) 2 Py i =l i τ ( 2 )( 2 0 m2 2F 0 b2 ρ ) 1 E00 2τ2 2 ρτ( 2 0 m2 2F 0 )( 2 b2 ρ ) 1 m i + 2τ ( 1 2 ρτ ) 0 mi F i 0. Contacting (4.1) by y i and using m i y i = 0 = F i 0 y we get i 2 P 2 = τ ( 2 )( 2 0 m2 2F 0 b2 ρ ) 1 E00 i.e. (4.2) P = τ ( 2 2 )( 2 0 m2 2F 0 b2 ρ ) 1 E00. Putting the value of P in (4.1) we get 2τ2 2 2 ρτ( )( 2 0 m2 2F 0 b2 ρ ) 1 m i + 2τ 1 2 ρτ( ) 0 mi F i 0 = 0

12 524 Manish Kuma Gupta and Paas Nath Pandey i.e. Tansvecing by ij we get F i 0 = 0 mi 1 m 00 mi. (4.3) F i0 = 0 m i 1 m 00 m i. Using (4.3) in (3.16) and efeing 2τ ρτ 2 0 we get i = 0 which tansvecting by m i and using i m i = 1 m we get m = 0 and then (4.3) becomes (4.4) F i0 = 0 m i. This equation (4.4) is a necessay condition fo the Kopina change with an h-vecto to be a pojective change. Convesely if (4.4) satisfies then (3.16) gives (2τ ρτ 2 ) i + 2τ2 m im 00 = 0. Tansvecting by m i and efeing (2τρτ2 ) + 2τ2 m2 0 we get m = 0 and then 00 (3.19) gives i 00 = E 00 τ li. Theefoe F n is pojective to F n. Thus we have: Theoem 4.1. The Kopina change (1.3) with an h-vecto is pojective if and only if the condition (4.4) is satisfied. Acknowledgement. M. K. Gupta gatefully acknowledges the financial suppot povided by the Univesity Gants Commission (UGC) Govenment of India though UGC-BSR Reseach Stat-up-Gant. REFERENCES [1] P.. Antonelli R. S. Ingaden and M. Matsumoto The theoy of Spays and Finsle spaces with applications in Physics and Biology Kluwe Acad. Publishes odecht / Boston / ondon [2] P.. Antonelli S. F. Rutz and K. T. Fonseca The mathematical theoy of endosymbiosis II: Models of the Fungal Fusion hypothesis Nonlinea Analysis RWA 13 (2012) [3] G. S. Asanov Finsle Geomety Relativity and Gauge Theoies. Reidel Publishing Company odecht Holland [4] M. K. Gupta and P. N. Pandey On hypesuface of a Finsle space with a special metic Acta Math. Hunga. 120(1-2) (2008)

13 Finsle Space Subjected to a Kopina Change with an h-vecto 525 [5] M. K. Gupta and P. N. Pandey Hypesufaces of confomally and h-confomally elated Finsle spaces Acta Math. Hunga. 123(3) (2009) [6] R. S. Ingaden Geomety of themodynamics iff. Geom. Methods in The. Phys. (ed. H.. oebne et al.) XV Inten. Conf. Clausthal 1986 Wold Scientific Singapoe [7] H. Izumi Confomal tansfomations of Finsle spaces II. An h-confomally flat Finsle space Tenso N.S. 34 (1980) [8] V. K. Kopina On pojective Finsle spaces with a cetain special fom Naučn oklady vyss. Skoly fiz.-mat. Nauki 1959(2) (1960) (in Russian). [9] V. K. Kopina On pojective two-dimensional Finsle spaces with special metic Tudy Sem. Vekto. Tenzo. Anal. 11 (1961) (in Russian). [10] M. Matsumoto On C-educible Finsle spaces Tenso N.S. 24 (1972) [11] M. Matsumoto On Finsle spaces with Randes metic and special foms of special tensos J. Math. Kyoto Univ (1974) [12] M. Matsumoto Finsle spaces of constant cuvatue with Kopina metic Tenso N.S. 50 (1991) [13] M. Matsumoto Theoy of Finsle spaces with (α )-metic Rep. Math. Phys (1992) [14] B.N. Pasad On the tosion tensos R hjk and P hjk of Finsle spaces with a metic ds = ( ij (dx) dx i dx j ) 1/2 + b i (x y) dx i Indian J. pue appl. Math (1990) [15] C. Shibata On Finsle spaces with Kopina metic Rep. Math. Phys. 13 (1978) [16] C. Shibata On invaiant tensos of -changes of Finsle metic J. Math. Kyoto Univ (1984) Manish Kuma Gupta epatment of Pue & Applied Mathematics Guu Ghasidas Vishwavidyalaya Bilaspu (C.G.) India mkgiaps@gmail.com Paas Nath Pandey epatment of Mathematics Univesity of Allahabad Allahabad India pnpiaps@gmail.com

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