Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices

(1)

Volume 2007, Article ID 20672,10pages doi:10.1155/2007/20672

Research Article

Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices

Rakotonirina Christian

Received 11 December 2006; Accepted 13 February 2007 Recommended by Howard E. Bell

We have expressed the tensor commutation matrixn⊗nas linear combination of the tensor products of the generalized Gell-Mann matrices. The tensor commutation matrices 3⊗2 and 2⊗3 have been expressed in terms of the classical Gell-Mann matrices and the Pauli matrices.

Copyright © 2007 Rakotonirina Christian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distri- bution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

When we had worked on Raoelina Andriambololona idea on the use of tensor product in Dirac equation [1,2], we had met the unitary matrix

U2⊗2=

⎛

⎜⎜

⎜⎝

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

⎞

⎟⎟

⎟⎠. (1.1)

This matrix is frequently found in quantum information theory [3–5] where one writes, by using the Pauli matrices [3–5],

U2⊗2=1

2I2⊗I2+1 2

3 i=1

σ_i⊗σ_i (1.2)

withI2the 2×2 unit matrix. We call this matrix a tensor commutation matrix 2⊗2. The tensor commutation matrix 3⊗3 is expressed by using the Gell-Mann matrices under

(2)

the following form [6]:

U3_⊗3=1

3I3⊗I3+1 2

8 i=1

λi⊗λi. (1.3)

We have to talk a bit about diﬀerent types of matrices because in the generalization of the above formulas, we will consider a commutation matrix as a matrix of fourth- order tensor and in expressing the commutation matricesU3⊗2,U2⊗3, at the last section, a commutation matrix will be considered as matrix of second-order tensor.

ᏹm×n(C) denotes the set ofm×nmatrices whose elements are complex numbers.

2. Tensor product of matrices

2.1. Matrices. If the elements of a matrix are considered as the components of a second- order tensor, we adopt the habitual notation for a matrix, without parentheses inside, whereas if the elements of the matrix are, for instance, considered as the components of sixth-order tensor, three times covariant and three times contravariant, then we represent the matrix of the following way, for example:

M=

⎛

⎜⎜

⎜⎝

⎛

⎜⎜

⎝ 1 0 1 1

1 1 3 2

0 0 0 0

1 1 1 1

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 1 0 1 2

7 8 9 0

3 4 5 6

9 8 7 6

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 1 1 1 1

0 0 3 2

4 5 1 6

1 7 8 9

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 5 4 3 2

1 0 1 2

3 4 5 6

7 8 9 0

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 1 2 3 4

9 8 7 6

5 6 7 8

5 4 3 2

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 9 8 7 6

5 4 3 2

1 0 1 2

3 4 5 6

⎞

⎟⎟

⎠

⎞

⎟⎟

⎟⎠ ,

M=

Mⁱ_j¹₁ⁱ_j²₂ⁱ³_j₃ i1i2i3=111, 112, 121, 122, 211, 212, 221, 222, 311, 312, 321, 322 row indices, j1j2j3=111, 112, 121, 122, 211, 212, 221, 222 column indices.

(2.1) The first indicesi1and j1are the indices of the outside parenthesis which we call the first-order parenthesis; the second indicesi2andj2are the indices of the next parentheses which we call the second-order parentheses; the third indicesi3and j3are the indices of the most interior parentheses, of this example, which we call third-order parentheses. So, for instance,M121³²¹=5.

(3)

If we delete the third-order parenthesis, then the elements of the matrixMare considered as the components of a fourth-order tensor, twice contravariant and twice covariant.

A matrix is a diagonal matrix if deleting the interior parentheses, we have a habitual diagonal matrix.

A matrix is a symmetric (resp., antisymmetric) matrix if deleting the interior parentheses, we have a habitual symmetric (resp., antisymmetric) matrix.

We identify one matrix to another matrix if after deleting the interior parentheses, they are the same matrices.

2.2. Tensor product of matrices

Definition 2.1. ConsiderA=(Aⁱ_j)∈ᏹm×n(C),B=(Bⁱ_j)∈ᏹp×r(C). The matrix defined by

A⊗B=

⎛

⎜⎜

⎝

A¹1B ... A¹_jB ... A¹_nB

... ... ...

Aⁱ1B ... Aⁱ_jB ... Aⁱ_nB

... ... ...

A^m1B ... A^m_jB ... A^m_nB

⎞

⎟⎟

⎠

(2.2)

is called the tensor product of the matrixAby the matrixB, A⊗B∈ᏹmp×nr(C), A⊗B=

Cⁱ_j¹1ⁱj²²

= Aⁱ_j¹1Bⁱ_j²2

, (2.3)

(cf., e.g., [3]) where,i1i2are row indices and j1j2are column indices.

3. Generalized Gell-Mann matrices

Let us fix n∈N,n≥2 for all continuations. The generalized Gell-Mann matrices or n×n-Gell-Mann matrices are the traceless Hermitian n×nmatricesΛ1,Λ2,...,Λn²−1

which satisfy the relationTr(ΛiΛj)=2δi j, for alli,j∈ {1, 2,...,n²−1}, whereδi j=δ^{i j}= δⁱ_jis the Kronecker symbol [7].

However, for the demonstration ofTheorem 4.3, denote, for 1≤i < j≤n, theC_n²= (n!/2!(n−2)!)n×n-Gell-Mann matrices which are symmetric with all elements 0 except theith row jth column and the jth rowith column which are equal to 1, byΛ^{(i j)}; the C²_n=(n!/2!(n−2)!)n×n-Gell-Mann matrices which are antisymmetric with all elements are 0 except theith rowjth column which is equal to−iand thejth rowith column which

(4)

is equal toi, byΛ^[^{i j}^]and byΛ⁽^d⁾, 1≤d≤n−1, the following (n−1)n×n-Gell-Mann matrices are diagonal:

Λ⁽¹⁾=

⎛

⎜⎜

⎝

1 0 ... 0

0 −1

0 ...

... . ..

. ..

0 ... 0

⎞

⎟⎟

⎠ ,

Λ⁽²⁾=_√1 3

⎛

⎜⎜

⎝

1 0 ... 0

0 1

−2 ...

... 0

. ..

0 ... 0

⎞

⎟⎟

⎠

,...,Λ⁽ⁿ⁻¹⁾=1 C²_n

⎛

⎜⎜

⎝

1 0 ... 0

0 1

1 ...

... . ..

1

0 ... −(n−1)

⎞

⎟⎟

⎠ .

(3.1) Forn=2, we have the Pauli matrices.

4. Tensor commutation matrices

Definition 4.1. Forp,q∈N,p≥2,q≥2, call the tensor commutation matrixp⊗qthe permutation matrixUp⊗q∈ᏹpq×pq(C) formed by 0 and 1, verifying the property

U_p⊗q·(a⊗b)=b⊗a (4.1)

for alla∈ᏹp×1(C),b∈ᏹq×1(C).

ConsideringUp⊗qas a matrix of a second-order tensor, one can construct it by using the following rule [6].

Rule 4.2. Let us start in putting 1 at first row and first column, after that let us pass into second column in going down at the rate ofprows and put 1 at this place, then pass into third column in going down at the rate ofprows and put 1, and so on until there are only for usp−1 rows for going down (then we have obtained number of 1 :q). Then pass into the next column which is the (q+ 1)th column, put 1 at the second row of this column and repeat the process until we have onlyp−2 rows for going down (then we have obtained number of 1 : 2q). After that pass into the next column which is the (2q+ 1)th column, put 1 at the third row of this column and repeat the process until we have onlyp−3 rows for going down (then we have obtained number of 1 : 3q). Continuing in this way, we will have that the element atp×qth row andp×qth column is 1. The other elements are 0.

(5)

Theorem 4.3. One has

Un⊗n= 1

nIn⊗In+1 2

n²−1 i=1

Λi⊗Λi. (4.2)

Proof. One has

I_n⊗I_n=

δⁱ_j¹₁ⁱ_j²₂= δⁱ_j¹₁δⁱ_j²₂, U_n⊗n=

δⁱ_j¹₂δⁱ_j²₁,

(4.3) where,i1i2are row indices and j1j2are column indices [3].

Consider at first theC_n²symmetricn×nGell-Mann matrices which can be written as Λ⁽^{i j}⁾=

Λ⁽^{i j}⁾^l_k1_≤l≤n, 1_≤k≤n

=

δ^ilδ_k^j1_≤l≤n, 1_≤k≤n+δ^jlδ_kⁱ1_≤l≤n, 1_≤k≤n

=

δ^ilδ_k^j+δ^jlδ_kⁱ1_≤l≤n, 1_≤k≤n.

(4.4)

Then

Λ⁽^{i j}⁾⊗Λ⁽^{i j}⁾=

Λ⁽^{i j}⁾⊗Λ⁽^{i j}⁾^l_k¹₁^l²_k₂=

δ^il¹δ_k^j₁+δ^jl¹δ_kⁱ₁δ^il²δ_k^j₂+δ^jl²δ_kⁱ₂, (4.5) wherel1l2are row indices andk1k2are column indices.

That is,

Λ^{(i j)}⊗Λ^{(i j)}^l_k¹₁^l²_k₂=δîl¹δ_k^j₁δîl²δ_k^j₂+δîl¹δ_k^j₁δ^jl²δ_kⁱ₂+δ^jl¹δ_kⁱ₁δîl²δ_k^j₂+δ^jl¹δ_kⁱ₁δ^jl²δⁱ_k₂. (4.6) TheC_n²antisymmetricn×nGell-Mann matrices can be written as

Λ^[^{i j}^]=

Λ^[^{i j}^]^l_k1_≤l≤n, 1_≤k≤n=

−iδ^ilδ_k^j+iδ^jlδ_kⁱ1_≤l≤n, 1_≤k≤n. (4.7) Then

Λ^[^{i j}^]⊗Λ^[^{i j}^]=

Λ^[^{i j}^]⊗Λ^[^{i j}^]^l_k¹₁^l²_k₂,

Λ^[^{i j}^]⊗Λ^[^{i j}^]^l_k¹₁^l²_k₂= −δîl¹δ_k^j₁δîl²δ_k^j₂+δîl¹δ_k^j₁δ^jl²δⁱ_k₂+δ^jl¹δⁱ_k₁δîl²δ_k^j₂−δ^jl¹δ_kⁱ₁δ^jl²δ_kⁱ₂,

1≤i< j≤n

Λ⁽^{i j}⁾⊗Λ⁽^{i j}⁾^l_k¹₁^l²_k₂+

1≤i< j≤n

Λ^[^{i j}^]⊗Λ^[^{i j}^]^l_k¹₁^l²_k₂

=2

1_≤i< j≤n

δ^il¹δ_k^j₁δ^jl²δ_kⁱ₂+δ^jl¹δ_kⁱ₁δ^il²δ_k^j₂=2

i /=j

δ^il¹δ_k^j₁δ^jl²δⁱ_k₂

(4.8)

is thel1l2th row,k1k2th column of the matrix

1≤i< j≤n

Λ⁽^{i j}⁾⊗Λ⁽^{i j}⁾+

1≤i< j≤n

Λ^[^{i j}^]⊗Λ^[^{i j}^]. (4.9)

(6)

Now, consider the diagonaln×nGell-Mann matrices. Letd∈N, 1≤d≤n−1, Λ⁽^d⁾= 1

C²_d₊₁ δ_k^l d p=1

δ_k^p−dδ_k^lδ_k^d+1

(4.10) and thel1l2th row,k1k2th of the matrixΛ⁽^d⁾⊗Λ⁽^d⁾is

Λ^(d)⊗Λ^(d)^l_k¹₁^l²_k₂= 1 C²_d+1δ_k^l¹₁δ_k^l²₂

d q=1

d p=1

δ^q_k₁δ_k^p₂

− 1

C_d+1² δ_k^l¹₁δ_k^l²₂ dδ_k^d₂⁺¹ d p=1

δ_k^p₁

− 1

C_d+1² δ^l_k¹₁δ_k^l²₂ dδ^d_k₁⁺¹ d p=1

δ_k^p₂

+ 1

C²_d+1δ_k^l¹₁δ_k^l²₂d²δ_k^d₁⁺¹δ^d_k₂⁺¹,

(4.11)

Λ⁽^d⁾⊗Λ⁽^d⁾is a diagonal matrix, so all that we have to do is to calculate the elements on the diagonal wherel1=k1andl2=k2. Then,

n−1 d=1

Λ⁽^d⁾⊗Λ⁽^d⁾^l_k¹₁^l_k²₂=

n−1 d=1

1 C_d²+1

d q=1

δ_k^q₁ d p=1

δ_k^p₂

−

n−1 d=1

1 C²_d+1

dδ_k^d₂⁺¹ d p=1

δ_k^p₁

−

n−1 d=1

1 C_d²+1

dδ^d_k₁⁺¹ d p=1

δ_k^p₂+

n−1 d=1

1 C²_d+1

d²δ_k^d₁⁺¹δ_k^d₂⁺¹

(4.12)

is thel1l2th row,k1k2th column of the diagonal matrixⁿ_d⁻₌¹₁Λ⁽^d⁾⊗Λ⁽^d⁾withl1=k1and l2=k2.

Let us distinguish two cases.

Case 1. k1=/1 ork2=/ 1.

Case 1.1. k1=/ k2. Ifk1< k2,

n−1 d=1

Λ⁽^d⁾⊗Λ⁽^d⁾^l_k¹₁^l_k²₂=

n−1 d=k2

1 C²_d+1

−k2−1 C²_k₂ ⁼2

_n₋₁

d=k2

1 d⁻

1 d+ 1

− 1 k2

= −2 n.

(4.13) Similarly, ifk1> k2,

n−1 d=1

Λ⁽^d⁾⊗Λ⁽^d⁾^l_k¹₁^l_k²₂= −2

n. (4.14)

Case 1.2. k1=k2=/1:

n−1 d=1

Λ⁽^d⁾⊗Λ⁽^d⁾^l_k¹₁^l_k²₂=ⁿ⁻ 1 d=k2

1 C²_d+1

+

k2−1² C²_k₂ ⁼

2 k2−2

n+

k2−1²

C²_k₂ ⁼2−2

n. (4.15)

(7)

Case 2. k1=k2=1:

n−1 d=1

Λ^(d)⊗Λ^(d)^l_k¹₁^l²_k₂=ⁿ

−1

d=1

1

C²_d₊₁⁼2−2

n. (4.16)

We can condense these cases in one formula as

n−1 d=1

Λ⁽^d⁾⊗Λ⁽^d⁾^l_k¹₁^l²_k₂= −2

nδ_k^l¹₁δ_k^l²₂+ 2 n i=1

δ^il¹δⁱ_k₁δ^il²δⁱ_k₂, (4.17)

which yields the diagonal of the diagonal matrixⁿ_d⁻₌¹₁Λ⁽^d⁾⊗Λ⁽^d⁾. For all then×nGell-Mann matrices, we have

1≤i< j≤n

Λ⁽^{i j}⁾⊗Λ⁽^{i j}⁾^l_k¹₁^l²_k₂+

1≤i< j≤n

Λ^[^{i j}^]⊗Λ^[^{i j}^]^l_k¹₁^l²_k₂+

n−1 d=1

Λ⁽^d⁾⊗Λ⁽^d⁾^l_k¹₁^l_k²₂

= −2

nδ_k^l¹₁δ_k^l²₂+ 2 n i=1

δ^il¹δ_kⁱ₁δ^il²δ_kⁱ₂+ 2

i /=j

δ^il¹δ_k^j₁δ^jl²δ_kⁱ₂

= −2

nδ_k^l¹₁δ_k^l²₂+ 2 n j=1

n i=1

δ^il¹δ_k^j₁δ^jl²δⁱ_k₂

= −2

nδ_k^l¹₁δ_k^l²₂+ 2δ^l_k¹₂δ_k^l²₁

(4.18)

for alll1,l2,k1,k2∈ {1, 2,...,n}. Hence, by using (4.3),

n²−1 i=1

Λi⊗Λi= −2

nI_n⊗I_n+ 2U_n⊗n (4.19)

and the theorem is proved.

5. Expression ofU3⊗2andU2⊗3

In this section, we derive formulas forU3_⊗2 andU2_⊗3, naturally in terms of the Pauli matrices

σ1= 0 1 1 0

, σ2= 0 −i i 0

, σ3= 1 0

0 −1

(5.1)

(8)

and the Gell-Mann matrices λ1=

⎛

⎜⎝

0 1 0

1 0 0

0 0 0

⎞

⎟⎠, λ2=

⎛

⎜⎝

0 −i 0

i 0 0

0 0 0

⎞

⎟⎠, λ3=

⎛

⎜⎝

1 0 0

0 −1 0

0 0 0

⎞

⎟⎠,

λ4=

⎛

⎜⎝

0 0 1

0 0 0

1 0 0

⎞

⎟⎠, λ5=

⎛

⎜⎝

0 0 −i

0 0 0

i 0 0

⎞

⎟⎠, λ6=

⎛

⎜⎝

0 0 0

0 0 1

0 1 0

⎞

⎟⎠,

λ7=

⎛

⎜⎝

0 0 0

0 0 −i

0 i 0

⎞

⎟⎠, λ8=_√1 3

⎛

⎜⎝

1 0 0

0 1 0

0 0 −2

⎞

⎟⎠.

(5.2)

Forr∈N^∗, defineE_{i j}⁽^r⁾as the elementaryr×rmatrix whose elements are zeros except theith row andjth column which is equal to 1. We constructU3_⊗2by usingRule 4.2, and then we have

U3⊗2=E⁽⁶⁾11 +E⁽⁶⁾23 +E35⁽⁶⁾+E⁽⁶⁾42 +E⁽⁶⁾54 +E⁽⁶⁾66. (5.3) Take

E11⁽⁶⁾=E⁽³⁾11 ⊗E11⁽²⁾. (5.4) Let

E⁽³⁾11 =α0I3+α3λ3+α8λ8 (5.5) withα0,α3,α8∈C, then

α0=1

3, α3=1

2, α8=

√3 6 , E⁽³⁾11 =1

3I3+1 2λ3+

√3 6 λ8.

(5.6)

Let

E⁽²⁾11 =β0I2+β3σ3 (5.7) withβ0,β3∈C, then

β0=1

2, β3=1 2, E⁽²⁾11 =1

2I2+1 2σ3.

(5.8)

So we have

E11⁽⁶⁾= 1

3I3+1 2λ3+

√3 6 λ8

⊗ 1

2I2+1 2σ3

. (5.9)

(9)

In a similar way, we have

E⁽⁶⁾23 = 1

2λ1+ i 2λ2

⊗ 1

2σ1− i 2σ2

, E⁽⁶⁾35 =

1 2λ6+ i

2λ7

⊗ 1

2I2+1 2σ3

, E⁽⁶⁾42 =

1 2λ1− i

2λ2

⊗ 1

2I2−1 2σ3

, E⁽⁶⁾54 =

1 2λ6− i

2λ7

⊗ 1

2σ1+ i 2σ2

, E⁽⁶⁾66 =

1 3I3−

√3 3 λ8

⊗ 1

2I2−1 2σ3

.

(5.10)

Hence U3_⊗2=

1 3I3+1

2λ3+

√3 6 λ8

⊗ 1

2I2+1 2σ3

+

1 2λ1+ i

2λ2

⊗ 1

2σ1− i 2σ2

+ 1

2λ6+ i 2λ7

⊗ 1

2I2+1 2σ3

+

1 2λ1− i

2λ2

⊗ 1

2I2−1 2σ3

+ 1

2λ6− i 2λ7

⊗1 2σ1+ i

2σ2

+

1 3I3−

√3 3 λ8

⊗1 2I2−1

2σ3

.

(5.11)

In an analogous way,

U2⊗3= 1

2I2+1 2σ3

⊗ 1

3I3+1 2λ3+

√3 6 λ8

+

1 2σ1+ i

2σ2

⊗ 1

2λ1−i 2λ2

+ 1

2I2+1 2σ3

⊗ 1

2λ6−i 2λ7

+

1 2I2−1

2σ3

⊗ 1

2λ1+ i 2λ2

+ 1

2σ1− i 2σ2

⊗ 1

2λ6+ i 2λ7

+

1 2I2−1

2σ3

⊗ 1

3I3−

√3 3 λ8

.

(5.12)

One can develop these formulas in employing the distributivity of the tensor product.

Acknowledgments

The author thanks the referee of an earlier manuscript for suggesting the topic. The author would like to thank Victor Razafinjato, Director of Civil Engineering Department of Institut Sup´erieur de Technologie d’Antananarivo(IST-T) and Ratsimbarison Mahasedra for encouragement and for critical reading of the manuscript.

(10)

References

[1] C. Rakotonirina, Thèse de Doctorat de Troisième Cycle de Physique Théorique, Université d’Antananarivo, Antananarivo, Madagascar, 2003, unpublished.

[2] R. P. Wang, “Varieties of Dirac equation and flavors of leptons and quarks,”http://arxiv.org/

abs/hep-ph/0107184.

[3] K. Fujii, “Introduction to coherent states and quantum information theory,” prepared for 10th Numazu Meeting on Integrable System, Noncommutative Geometry and Quantum theory, Nu- mazu, Shizuoka, Japan, May 2002,http://arxiv.org/abs/quant-ph/0112090.

[4] L. D. Faddev, “Algebraic aspects of the Bethe ansatz,” International Journal of Modern Physics A, vol. 10, no. 13, pp. 1845–1878, 1995.

[5] F. Verstraete, “A Study of Entanglement in Quantum Information Theory,” Th`ese de Doctorat, Katholieke Universiteit, Leuven, Belgium, 2002.

[6] C. Rakotonirina, “Tensor permutation matrices in finite dimensions,” http://arxiv.org/abs/

math.GM/0508053.

[7] S. Narison, Spectral Sum Rules, vol. 26 of World Scientific Lecture Notes in Physics, World Scien- tific, Singapore, 1989.

Rakotonirina Christian: Département du Génie Civil, Institut Supérieur de Technologie d’Antananarivo (IST-T), BP 8122, Madagascar

Email address:[email protected]

(11)

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Edson Denis Leonel,Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com