# Space time with generalized covariant recurrent energy momentum tensor.

1. Introduction

In the general theory of relativity, energy momentum tensor plays an important role and the condition on energy momentum tensor for a perfect fluid space time changes the nature of space time (5). In 1996 M.C.Chaki and S.Roy in (2) have shown that if energy momentum tensor is covariant constant, the space time is Ricci-symmetric. In 2004, A.Konar and B.Biswas studied energy momentum tensor which is covariant recurrent in (6). Also generalized Ricci-recurrent spaces were introduced in (3) by U.C.De, N.Guha and D.Kamilya. In (1) and (3) authors have studied Kenmotsu and Sasakian manifolds which are generalized Ricci-recurrent. So in this paper we have defined generalized covariant recurrent energy momentum tensor and studied in a general relativistic space time. It is shown that the space time becomes generalized Ricci-recurrent. Further we have defined a semi-Einstein space, given an example and proved that in a Ricci-recurrent Riemannian manifold energy momentum tensor T is generalized recurrent if and only if the Einstein tensor G is generalized Ricci-recurrent and G will be generalized Ricci-recurrent if and only if the manifold is semi-Einstein.

We know an energy momentum tensor T will be covariant recurrent (6) if

([[nabla].sub.Z]T)(X,Y) = [psi](Z)T(X,Y), (1)

where [psi] is non-zero 1-form such that, [psi](Z) = g(Z,[rho]).

So we like to define generalized covariant recurrent energy momentum tensor as follows:

Definition. An energy momentum tensor T is said to be generalized covariant recurrent if

([[nabla].sub.Z]T)(X,Y) = [psi](Z)T(X,Y) + [phi](Z)g(X,Y) (2)

where [psi] and [phi] are two non-zero 1-forms such that,

[psi](Z) = g(Z,[rho]) and [phi](Z) = g(Z,[bar.[rho]]) (3)

[rho] and [bar.[rho]] being unit orthogonal vector fields, [nabla] denotes the operator of covariant differentiation with respect to the metric tensor g.

Next we define a semi-Einstein manifold which is the generalization of Einstein manifold.

Definition. A Riemannian manifold M is said to be semi-Einstein if Ricci tensor S, which is non-zero, satisfies

S(X,Y) = P(Z)g(X,Y) [for all]X,Y,Z[member of]TM (4)

and P is a non-zero 1-form, such that the vector fields are not scalar multiple of a single vector.

Example of semi-Einstein space: Let M be an odd dimensional Riemannian manifold with a Riemannian metric g defined as follows--

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where G is Kaehlerian metric for a Kaehlerian manifold F of dimension 2n and f(x) = [ce.sup.x], which is a function on real line for a non-zero constant c.

We define a 1-form P(Z) = g(Z, W), where Z is a vector field, not a scalar multiple of W . Then M is a semi-Einstein manifold, when G satisfies the relation, G = [e.sup.[QX-X-Z]] where Q is a symmetric endomorphism of the tangent space of M.

We also state some other definitions which are given in [3], [6] and will be required in the next section.

A non-flat Riemannian manifold is said to be generalized Ricci-recurrent (3), if the non-zero Ricci-tensor S satisfies

([[nabla].sub.Z]S)(X,Y) = [alpha](Z)S(X,Y) + [beta](Z)g(X,Y), (5)

where [alpha], [beta] are non-zero 1-forms, such that

[alpha](Z) = g(Z,[gamma]) and [beta](Z) = g(Z,[bar.[gamma]]), (6)

[gamma] and [bar.[gamma]] being unit orthogonal vector fields and [nabla] has its usual means.

An Einstein equation without any cosmological constant is given by

S(X,Y)-[1/2]rg(X,Y) = KT(X,Y) = G(X,Y), [for all]X,Y[member of]TM, (7)

where r is the scalar curvature at any point of the space time and K is the gravitational constant [6]. G is called Einstein's tensor.

Again an Einstein equation with cosmological constant is given by

S(X,Y)-[1/2]rg(X,Y) + [[lambda]g](X,Y) = KT(X,Y) = K[([sigma] + [rho])[omega](X)[omega](Y) + pg(X,Y)] (8)

where [lambda] is the cosmological constant, [sigma] is the energy density and p is isotropic pressure of the fluid.

2. Generalized recurrent energy momentum tensor in a general relativistic space time

In this section it is assumed that the Einstein equation in a general relativistic space time is without cosmological constant. Here we prove,

Theorem 1. A necessary and sufficient condition that the energy momentum tensor T without cosmological constant in a general relativistic space time is generalized recurrent is that space time is generalized Ricci-recurrent.

Proof. Differentiating (7) covariantly with respect to Z and using (2) we get

([[nabla].sub.Z]S)(X, Y) - [1/2]dr(Z)g(X, Y) = K{[psi](Z)T(X, Y) + [phi](Z)g(X, Y)}. (9)

Replacing (7) in (9) and simplifying we have

([[nabla].sub.Z]S)(X, Y) = A(Z)S(X, Y) + B(Z)g(X, Y) (10)

where,

A(Z) = [psi](Z) and K [phi](Z) + [1/2]dr(Z) - [1/2]r[psi](Z) = B(Z)

which are 1-forms. Equation (10) shows that the space time is generalized Ricci-recurrent.

Again in (10) if we take X = Y = [e.sub.i], i = 1, 2, 3, 4, where {[e.sub.i]} are orthonormal basis of space time, we obtain

dr(Z) = A(Z)r + 4B(Z). (11)

Conversely, we assume that the space time is generalized Ricci-recurrent. Then from (2),(9) and (10) we get

K([[nabla].sub.Z]T)(X, Y) = A(Z)S(X, Y) + B(Z)g(X, Y) - [1/2]dr(Z)g(X, Y). (12)

Using (11) in (12) and by (7) we have

([[nabla].sub.Z]T)(X, Y) = A(Z)T(X, Y) - [1/K]B(Z)g(X, Y). (13)

which implies T is generalized recurrent.

Theorem 2. In a 4-dimensional generalized Ricci recurrent Riemannian manifold energy momentum tensor is generalized recurrent if and only if the Einstein tensor is generalized Ricci-recurrent.

Proof. We assume that a Riemannian manifold M of dimension 4 is generalized Ricci-recurrent. Using theorem-1. we have energy momentum tensor T is generalized recurrent.

From (7) we have

T(X, Y) = [1/K][S(X, Y) - [1/2]rg(X, Y)]. (14)

Differentiating (14) covariantly, using (2), (5) and after a brief calculation, we obtain

KT(X, Y) = [[A(Z)]/[[psi](Z)]]S(X, Y) + [1/[[psi](Z)]][-[r/2]A(Z) - B(Z) - K[phi](Z)]g(X, Y). (15)

(15) can be written as

G(X, Y) = [alpha](Z)S(X, Y) + [beta](Z)g(X, Y), (16)

where [alpha](Z) = [A(Z)/[psi](Z)] and [beta](Z) = [1/[psi](Z)][-[r/2]A(Z) - B(Z) - K[phi](Z)]

are the 1-forms, as [psi](z) [not equal to] 0.

By virtue of (5) and equation (16) we can assert that the Einstein tensor G is generalized Ricci-recurrent.

Conversely, if G is generalized Ricci-recurrent, then

G(X, Y) = [omega](Z)S(X, Y) + [delta](Z)g(X, Y), (17)

where [omega](Z) and [delta](Z) are non-zero 1-forms.

Taking [omega](Z) = [[[alpha](Z)]/[[beta](Z)]] and [delta](Z) = [1/[beta](Z)]][m(Z) - [1/2]dr(Z) - K N(Z),

where [alpha], [beta], m, N are also non-zero 1-forms, and substituting in (17) we get

[beta](Z)G(X, Y) = [[alpha](Z)S(X, Y) + m(Z)g(X, Y)] + [- [1/2]dr(Z) - KN(Z)]g(X, Y). (18)

As the Riemannian manifold M is generalized Ricci-recurrent, using (5) and (7) in (18) we have

[beta](Z)T(X, Y) + N(Z)g(X, Y) = [1/K][([[nabla].sub.Z]S)(X, Y) - [1/2]dr(Z)g(X, Y)]. (19)

By virtue of covariant differentiation of (7) with respect to Z, (19) reduces to

([[nabla].sub.Z]T)(X, Y) = [beta](Z)T(X, Y) + N(Z)g(X, Y) (20)

which shows that T is generalized recurrent.

Theorem 3. In a 4-dimensional generalized Ricci recurrent Riemannian manifold with generalized recurrent energy momentum tensor which is without cosmological constant, an Einstein tensor is generalized Ricci-recurrent if and only if the manifold is semi-Einstein.

Proof. Further, if G is generalized Ricci-recurrent,then

G(X,Y) = ([[nabla].sub.Z]S)(X,Y). (21)

By virtue of covariant derivative of (7) with respect to Z we get

G(X,Y) = K([[nabla].sub.Z]T)(X,Y) + [1/2]dr(Z)g(X,Y). (22)

Using (2), (7) and (11) in (22) we obtain

KT(X,Y) = E(Z)g(X,Y), (23)

where E(Z) = [1/1-[psi](Z)][K[phi](Z) + [1/2]{rA(Z) + 4B(Z)}] provided [psi](Z)[not equal to]1.

From (7) and (23) we have

S(X,Y) = P(Z)g(X,Y), (24)

where P(Z) = E(Z) +[1/2]r, is a 1-form.

By virtue of definition (3), we assert that the manifold is semi-Einstein.

Theorem 4. A necessary and sufficient condition that the energy momentum tensor T with cosmological constant in a general relativistic space time is generalized recurrent is that space time is generalized Ricci-recurrent.

Proof. Differentiating equation (8) covariantly with respect to Z, we obtain

([[nabla].sub.Z]S)(X,Y)-[1/2]dr(Z)g(X,Y) + [d[lambda]](Z)g(X,Y) = K([[nabla].sub.Z]T)(X,Y) = K{[psi](Z)T(X,Y) + [phi](Z)g(X,Y)}. (25)

Using (2), (8) we get

([[nabla].sub.Z]S)(X,Y) = M(Z)S(X,Y) + N(Z)g(X,Y), (26)

where M and N are 1-forms such that

M(Z) = [psi](Z) and N(Z) = [1/2]dr(Z) + K[phi](Z) + ([lambda] - [1/2]r)[psi](Z) - [d[lambda]](Z).

Equation (26) shows that the space time is generalized Ricci-recurrent. Putting X = Y = [e.sub.i] in (26) we have

dr(Z) = rM(Z) + 4N(Z). (27)

Conversely, we assume that the space time is generalized Ricci-recurrent, then from (2), (25) and (26) we obtain

K([[nabla].sub.Z]T)(X,Y) = M(Z)S(X,Y) + {N(Z)-[1/2]dr(Z) + d[lambda](Z)}g(X,Y). (28)

From equations (8), (27) and (28) we have

([[nabla].sub.Z]T)(X,Y) = A(Z)T(X,Y)-[1/K]B(Z)g(X,Y), (29)

where A(Z) = M(Z) and B(Z) = {[lambda]M(Z) + N(Z)-d[lambda](Z)} and M, N are 1-forms. Equation (29) shows that T is generalized recurrent. Hence the proof.

References

(1) A. Bhattacharyya, On generalized Ricci-recurrent Kenmotsu manifold, Bull. Cal. Math. Soc., 95(6)(2003), 463-466.

(2) M. C. Chaki, S. Ray, Space time with covariant constant energy momentum tensor, Int. I. Theor. Phys, 35(1027), 1996.

(3) U. C. De, N. Guha, and D. Kamilya, On generalized Ricci-recurrent manifolds, Tensor. N. S., 56(1995), 312.

(4) L. P. Eisenhart, Riemannian geometry, Princaton Univ. press,(1947), 1925.

(5) Hanstephani, General relativity, an introduction to the theory of the gravitational field, Cambridge Univ. Press.

(6) A. Konar and B. Biswas, Space-timewith covariant recurrent energy momentum tensor, Jr.of Cal. Math. Soc., 1(2004), 75-90.

A. Bhattacharyya [dagger], T. De [double dagger], and D. Debnath [section]

Dept. of Mathematics, Jadavpur University, Kolkata-32, India

Received November 23, 2007, Accepted February 20, 2008.

* 2000 Mathematics Subject Classification. 83C47.

[dagger] E-mail:bhattachar1968@yahoo.co.in

[double dagger] E-mail:tapan97@indiatimes.com

[section] E-mail: dipankardebnath123@rediffmail.com

In the general theory of relativity, energy momentum tensor plays an important role and the condition on energy momentum tensor for a perfect fluid space time changes the nature of space time (5). In 1996 M.C.Chaki and S.Roy in (2) have shown that if energy momentum tensor is covariant constant, the space time is Ricci-symmetric. In 2004, A.Konar and B.Biswas studied energy momentum tensor which is covariant recurrent in (6). Also generalized Ricci-recurrent spaces were introduced in (3) by U.C.De, N.Guha and D.Kamilya. In (1) and (3) authors have studied Kenmotsu and Sasakian manifolds which are generalized Ricci-recurrent. So in this paper we have defined generalized covariant recurrent energy momentum tensor and studied in a general relativistic space time. It is shown that the space time becomes generalized Ricci-recurrent. Further we have defined a semi-Einstein space, given an example and proved that in a Ricci-recurrent Riemannian manifold energy momentum tensor T is generalized recurrent if and only if the Einstein tensor G is generalized Ricci-recurrent and G will be generalized Ricci-recurrent if and only if the manifold is semi-Einstein.

We know an energy momentum tensor T will be covariant recurrent (6) if

([[nabla].sub.Z]T)(X,Y) = [psi](Z)T(X,Y), (1)

where [psi] is non-zero 1-form such that, [psi](Z) = g(Z,[rho]).

So we like to define generalized covariant recurrent energy momentum tensor as follows:

Definition. An energy momentum tensor T is said to be generalized covariant recurrent if

([[nabla].sub.Z]T)(X,Y) = [psi](Z)T(X,Y) + [phi](Z)g(X,Y) (2)

where [psi] and [phi] are two non-zero 1-forms such that,

[psi](Z) = g(Z,[rho]) and [phi](Z) = g(Z,[bar.[rho]]) (3)

[rho] and [bar.[rho]] being unit orthogonal vector fields, [nabla] denotes the operator of covariant differentiation with respect to the metric tensor g.

Next we define a semi-Einstein manifold which is the generalization of Einstein manifold.

Definition. A Riemannian manifold M is said to be semi-Einstein if Ricci tensor S, which is non-zero, satisfies

S(X,Y) = P(Z)g(X,Y) [for all]X,Y,Z[member of]TM (4)

and P is a non-zero 1-form, such that the vector fields are not scalar multiple of a single vector.

Example of semi-Einstein space: Let M be an odd dimensional Riemannian manifold with a Riemannian metric g defined as follows--

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where G is Kaehlerian metric for a Kaehlerian manifold F of dimension 2n and f(x) = [ce.sup.x], which is a function on real line for a non-zero constant c.

We define a 1-form P(Z) = g(Z, W), where Z is a vector field, not a scalar multiple of W . Then M is a semi-Einstein manifold, when G satisfies the relation, G = [e.sup.[QX-X-Z]] where Q is a symmetric endomorphism of the tangent space of M.

We also state some other definitions which are given in [3], [6] and will be required in the next section.

A non-flat Riemannian manifold is said to be generalized Ricci-recurrent (3), if the non-zero Ricci-tensor S satisfies

([[nabla].sub.Z]S)(X,Y) = [alpha](Z)S(X,Y) + [beta](Z)g(X,Y), (5)

where [alpha], [beta] are non-zero 1-forms, such that

[alpha](Z) = g(Z,[gamma]) and [beta](Z) = g(Z,[bar.[gamma]]), (6)

[gamma] and [bar.[gamma]] being unit orthogonal vector fields and [nabla] has its usual means.

An Einstein equation without any cosmological constant is given by

S(X,Y)-[1/2]rg(X,Y) = KT(X,Y) = G(X,Y), [for all]X,Y[member of]TM, (7)

where r is the scalar curvature at any point of the space time and K is the gravitational constant [6]. G is called Einstein's tensor.

Again an Einstein equation with cosmological constant is given by

S(X,Y)-[1/2]rg(X,Y) + [[lambda]g](X,Y) = KT(X,Y) = K[([sigma] + [rho])[omega](X)[omega](Y) + pg(X,Y)] (8)

where [lambda] is the cosmological constant, [sigma] is the energy density and p is isotropic pressure of the fluid.

2. Generalized recurrent energy momentum tensor in a general relativistic space time

In this section it is assumed that the Einstein equation in a general relativistic space time is without cosmological constant. Here we prove,

Theorem 1. A necessary and sufficient condition that the energy momentum tensor T without cosmological constant in a general relativistic space time is generalized recurrent is that space time is generalized Ricci-recurrent.

Proof. Differentiating (7) covariantly with respect to Z and using (2) we get

([[nabla].sub.Z]S)(X, Y) - [1/2]dr(Z)g(X, Y) = K{[psi](Z)T(X, Y) + [phi](Z)g(X, Y)}. (9)

Replacing (7) in (9) and simplifying we have

([[nabla].sub.Z]S)(X, Y) = A(Z)S(X, Y) + B(Z)g(X, Y) (10)

where,

A(Z) = [psi](Z) and K [phi](Z) + [1/2]dr(Z) - [1/2]r[psi](Z) = B(Z)

which are 1-forms. Equation (10) shows that the space time is generalized Ricci-recurrent.

Again in (10) if we take X = Y = [e.sub.i], i = 1, 2, 3, 4, where {[e.sub.i]} are orthonormal basis of space time, we obtain

dr(Z) = A(Z)r + 4B(Z). (11)

Conversely, we assume that the space time is generalized Ricci-recurrent. Then from (2),(9) and (10) we get

K([[nabla].sub.Z]T)(X, Y) = A(Z)S(X, Y) + B(Z)g(X, Y) - [1/2]dr(Z)g(X, Y). (12)

Using (11) in (12) and by (7) we have

([[nabla].sub.Z]T)(X, Y) = A(Z)T(X, Y) - [1/K]B(Z)g(X, Y). (13)

which implies T is generalized recurrent.

Theorem 2. In a 4-dimensional generalized Ricci recurrent Riemannian manifold energy momentum tensor is generalized recurrent if and only if the Einstein tensor is generalized Ricci-recurrent.

Proof. We assume that a Riemannian manifold M of dimension 4 is generalized Ricci-recurrent. Using theorem-1. we have energy momentum tensor T is generalized recurrent.

From (7) we have

T(X, Y) = [1/K][S(X, Y) - [1/2]rg(X, Y)]. (14)

Differentiating (14) covariantly, using (2), (5) and after a brief calculation, we obtain

KT(X, Y) = [[A(Z)]/[[psi](Z)]]S(X, Y) + [1/[[psi](Z)]][-[r/2]A(Z) - B(Z) - K[phi](Z)]g(X, Y). (15)

(15) can be written as

G(X, Y) = [alpha](Z)S(X, Y) + [beta](Z)g(X, Y), (16)

where [alpha](Z) = [A(Z)/[psi](Z)] and [beta](Z) = [1/[psi](Z)][-[r/2]A(Z) - B(Z) - K[phi](Z)]

are the 1-forms, as [psi](z) [not equal to] 0.

By virtue of (5) and equation (16) we can assert that the Einstein tensor G is generalized Ricci-recurrent.

Conversely, if G is generalized Ricci-recurrent, then

G(X, Y) = [omega](Z)S(X, Y) + [delta](Z)g(X, Y), (17)

where [omega](Z) and [delta](Z) are non-zero 1-forms.

Taking [omega](Z) = [[[alpha](Z)]/[[beta](Z)]] and [delta](Z) = [1/[beta](Z)]][m(Z) - [1/2]dr(Z) - K N(Z),

where [alpha], [beta], m, N are also non-zero 1-forms, and substituting in (17) we get

[beta](Z)G(X, Y) = [[alpha](Z)S(X, Y) + m(Z)g(X, Y)] + [- [1/2]dr(Z) - KN(Z)]g(X, Y). (18)

As the Riemannian manifold M is generalized Ricci-recurrent, using (5) and (7) in (18) we have

[beta](Z)T(X, Y) + N(Z)g(X, Y) = [1/K][([[nabla].sub.Z]S)(X, Y) - [1/2]dr(Z)g(X, Y)]. (19)

By virtue of covariant differentiation of (7) with respect to Z, (19) reduces to

([[nabla].sub.Z]T)(X, Y) = [beta](Z)T(X, Y) + N(Z)g(X, Y) (20)

which shows that T is generalized recurrent.

Theorem 3. In a 4-dimensional generalized Ricci recurrent Riemannian manifold with generalized recurrent energy momentum tensor which is without cosmological constant, an Einstein tensor is generalized Ricci-recurrent if and only if the manifold is semi-Einstein.

Proof. Further, if G is generalized Ricci-recurrent,then

G(X,Y) = ([[nabla].sub.Z]S)(X,Y). (21)

By virtue of covariant derivative of (7) with respect to Z we get

G(X,Y) = K([[nabla].sub.Z]T)(X,Y) + [1/2]dr(Z)g(X,Y). (22)

Using (2), (7) and (11) in (22) we obtain

KT(X,Y) = E(Z)g(X,Y), (23)

where E(Z) = [1/1-[psi](Z)][K[phi](Z) + [1/2]{rA(Z) + 4B(Z)}] provided [psi](Z)[not equal to]1.

From (7) and (23) we have

S(X,Y) = P(Z)g(X,Y), (24)

where P(Z) = E(Z) +[1/2]r, is a 1-form.

By virtue of definition (3), we assert that the manifold is semi-Einstein.

Theorem 4. A necessary and sufficient condition that the energy momentum tensor T with cosmological constant in a general relativistic space time is generalized recurrent is that space time is generalized Ricci-recurrent.

Proof. Differentiating equation (8) covariantly with respect to Z, we obtain

([[nabla].sub.Z]S)(X,Y)-[1/2]dr(Z)g(X,Y) + [d[lambda]](Z)g(X,Y) = K([[nabla].sub.Z]T)(X,Y) = K{[psi](Z)T(X,Y) + [phi](Z)g(X,Y)}. (25)

Using (2), (8) we get

([[nabla].sub.Z]S)(X,Y) = M(Z)S(X,Y) + N(Z)g(X,Y), (26)

where M and N are 1-forms such that

M(Z) = [psi](Z) and N(Z) = [1/2]dr(Z) + K[phi](Z) + ([lambda] - [1/2]r)[psi](Z) - [d[lambda]](Z).

Equation (26) shows that the space time is generalized Ricci-recurrent. Putting X = Y = [e.sub.i] in (26) we have

dr(Z) = rM(Z) + 4N(Z). (27)

Conversely, we assume that the space time is generalized Ricci-recurrent, then from (2), (25) and (26) we obtain

K([[nabla].sub.Z]T)(X,Y) = M(Z)S(X,Y) + {N(Z)-[1/2]dr(Z) + d[lambda](Z)}g(X,Y). (28)

From equations (8), (27) and (28) we have

([[nabla].sub.Z]T)(X,Y) = A(Z)T(X,Y)-[1/K]B(Z)g(X,Y), (29)

where A(Z) = M(Z) and B(Z) = {[lambda]M(Z) + N(Z)-d[lambda](Z)} and M, N are 1-forms. Equation (29) shows that T is generalized recurrent. Hence the proof.

References

(1) A. Bhattacharyya, On generalized Ricci-recurrent Kenmotsu manifold, Bull. Cal. Math. Soc., 95(6)(2003), 463-466.

(2) M. C. Chaki, S. Ray, Space time with covariant constant energy momentum tensor, Int. I. Theor. Phys, 35(1027), 1996.

(3) U. C. De, N. Guha, and D. Kamilya, On generalized Ricci-recurrent manifolds, Tensor. N. S., 56(1995), 312.

(4) L. P. Eisenhart, Riemannian geometry, Princaton Univ. press,(1947), 1925.

(5) Hanstephani, General relativity, an introduction to the theory of the gravitational field, Cambridge Univ. Press.

(6) A. Konar and B. Biswas, Space-timewith covariant recurrent energy momentum tensor, Jr.of Cal. Math. Soc., 1(2004), 75-90.

A. Bhattacharyya [dagger], T. De [double dagger], and D. Debnath [section]

Dept. of Mathematics, Jadavpur University, Kolkata-32, India

Received November 23, 2007, Accepted February 20, 2008.

* 2000 Mathematics Subject Classification. 83C47.

[dagger] E-mail:bhattachar1968@yahoo.co.in

[double dagger] E-mail:tapan97@indiatimes.com

[section] E-mail: dipankardebnath123@rediffmail.com

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Author: | Bhattacharyya, A.; De, T.; Debnath, D. |
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Publication: | Tamsui Oxford Journal of Mathematical Sciences |

Geographic Code: | 9TAIW |

Date: | Dec 1, 2009 |

Words: | 1894 |

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