ON MIXED-TYPE REVERSE-ORDER LAWS FOR THE MOORE-PENROSE INVERSE OF A MATRIX PRODUCT

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1 IJMMS 2004:58, PII. S Hindawi Publishing Cop. ON MIXED-TYPE REVERSE-ORDER LWS FOR THE MOORE-PENROSE INVERSE OF MTRIX PRODUCT YONGGE TIN Received 13 Januay 2003 Some mixed-type evese-ode laws fo the Mooe-Penose invese of a matix poduct ae established. Necessay and sufficient conditions fo these laws to hold ae found by the matix ank method. Some applications and extensions of these evese-ode laws to the weighted Mooe-Penose invese ae also given Mathematics Subject Classification: 1503, If and ae a pai of invetible matices of the same size, then the poduct is nonsingula, too, and the invese of the poduct satisfies the evese-ode law ) 1 = 1 1. This law can be used to find the popeties of ) 1,aswellas to simplify vaious matix expessions that involve the invese of a matix poduct. Howeve, this fomula cannot tivially be extended to the Mooe-Penose invese of matix poducts. Fo a geneal m n complex matix, the Mooe-Penose invese of is the unique n m matix X that satisfies the following fou Penose equations: i) X =, ii) XX = X, iii) X) = X, iv) X) = X, whee ) denotes the conjugate tanspose of a complex matix. matix X is called a {1}-invese inne invese) of if it satisfies i) and is denoted by. Geneal popeties of the Mooe-Penose invese can be found in 2, 4, 16. Let and be a pai of matices such that exists. In many situations, one needs to find the Mooe-Penose invese of the poduct and its popeties. ecause,, and ae not necessaily identity matices, the elationship between ) and is quite complicated and the evese-ode law ) = does not necessaily hold. Theefoe, it is not easy to simplify matix expessions that involve the Mooe- Penose invese of matix poducts. Theoetically speaking, fo any matix poduct, the Mooe-Penose invese ) can be witten as ) = o ) = +X, 1) whee X is a esidue matix. Fo these two situations, one can conside the following two poblems: I) necessay and sufficient conditions fo ) = to hold,

2 3104 YONGGE TIN II) if ), find possible expessions of X in ) = + X, and then detemine necessay and sufficient conditions fo ) = +X to hold. The investigation of the Mooe-Penose invese of the poduct was stated in 1960s. Fo the standad situation ) =, a well-known esult due to Geville 9 assets that ) = R ) R), R ) R ), 2) whee R ) denotes the ange column space) of a matix. Many othe equivalent conditions fo ) = to hold can be found in 2, 4, 16, 26. Geneally speaking, the two ange inclusions in 2) ae stict conditions fo any pai of matices and to satisfy. Theefoe, it is necessay to seek vaious weake evese-ode laws fo ) to satisfy. In addition to 2), ) may satisfy some othe mixed-type evese-ode laws. Fo example, ) = ) ), ) = ), ) = ). 3) These evese-ode laws wee studied in 6, 8, 11, 26. lthough these matix equalities ae moe complicated than the law in 2), the conditions fo these equalities to hold ae weake than that fo 2) to hold. In fact, mixed-type evese-ode laws also stem fom vaious easonable opeations fo the Mooe-Penose invese of matix poducts see Remak 10). lthough ) can be witten as ) = + X in geneal, it is not easy to give an explicit expession fo the esidue matix X fo the given matices and. Some discussion fo the expession of X and its popeties wee given in 8. In the investigation of ), we obseve that a possible expession fo ) is ) = I n ) I n ). 4) diect motivation fo us to find out the esidue matix in 4) aises fom two diffeent decompositions of the following block matix: M = In 0 5) and its genealized inveses. In fact, it is easy to veify that M can be decomposed as the following two foms: In 0 In 0 In M = := P 1 N 1 Q 1, I m 0 0 I k In In ) T In 0 M = 0 I m 0 := P 2 N 2 Q 2, I k 6)

3 ON MIXED-TYPE REVERSE-ORDER LWS whee T = I n )I n ). Fom these two decompositions, one can find two {1}-inveses of M as follows: M = Q1 1 N 1 P 1 1 In In 0 In 0 = 0 I k 0 ) In ) = ) ), M = Q2 1 N 2 P 2 1 In 0 T In I n ) = I k 0 0 I m T T = T T. I m 7) These two {1}-inveses of M ae not necessaily equal. Theefoe, it is natual to conside unde what conditions the two {1}-inveses of M in 7) ae equal; o some blocks of them ae equal. The mixed-type evese-ode law 4) is noticed by compaing the lowe ight blocks of 7). ecause the ight-hand side of 4) involves complicated matix opeations, it is not easy to establish necessay and sufficient conditions fo 4) to hold by definitions, as well as vaious matix decompositions associated with,, and. In the investigation of vaious poblems on genealized inveses of matices, the pesent autho notices that the ank of matix is a simple and poweful method fo dealing with the elationship between any two matix expessions involving genealized inveses. In fact, any two matices and of the same size ae equal if and only if ) = 0, whee ) denotes the ank of a matix. If one can find some nontivial fomulas fo the ank of, then necessay and sufficient conditions fo = to hold can be deived fom these ank fomulas. This method can be used fo investigating the elations between any two matix expessions that involve genealized inveses. Seveal simple ank fomulas fo the diffeences of matices found by the pesent autho ae given below: k k) k + k, 2), ) 2 2 ), ), +) ) ), ),,, ), 0, 0 )+),, min ) ),+)+),, 8)

4 3106 YONGGE TIN whee, denotes a ow block matix; see Tian 20, 23, 25, 26. The significance of these simple ank fomulas is: they connect diffeent matix expessions though the ank of these matices. Fom these ank equalities, one can deive some basic popeties fo the matices on the left-hand sides. Fo instance, let the ight-hand sides of the above ank equalities be zeo and simplify by some elementay methods, one can immediately obtain necessay and sufficient conditions fo the matices on the left-hand sides to be zeo. In this pape, we establish a ank fomula associated with 4) and then deive fom the ank fomula a necessay and sufficient condition fo 4) to hold. The following ank fomulas ae well known:,)+ ) )+ ), 9) )+C)+ I ) I C C ) ; C 0 10) if R) R) and RC ) R ),then )+ D C ), 11) C D see Masaglia and Styan 12. Recall the equality ) = see Zlobec 28) and notice that R) = R ) and R ) =R ). y appealing to 11), the ank of the Schu complement D C is D C ) C ). 12) D This ank equality is quite useful in dealing with vaious matix expessions involving the Mooe-Penose invese. Rank fomulas fo the Schu complement D C, whee is a {1}-invese of, can be found in 23. The main esult of this pape is given below. Theoem 1. Let C m n and C n p, and let T = I n )I n ). Then, ) + T +, 2). 13) Hence, the evese-ode law 4) holds if and only if and satisfy the following two ange equalities: R ) =R), R ) =R ). 14) Poof. Fom 12), we fist obtain ) + T T TT T T ) T). 15)

5 pplying 10) to T yields T) ON MIXED-TYPE REVERSE-ORDER LWS In ) )= n+) ) ). 16) 0 It is easy to veify that T TT = I n )T I n ). Recall that elementay matix opeations and block elementay matix opeations do not change the ank of matix. Thus, we can find by 10) and block elementay matix opeations that T TT T T ) In ) T I n ) I n ) I n ) I n ) I n ) ) T In ) I n ) ) 0 ) ) 0 0 I n + ) 0 ) ) 0 0 I n ) ) 0 ) ) 0 0 I n ) ) ) ) ) 0 ) ) = n+ 0 ) ). 17) lso note that R{I n ) } R) ={0}. It follows that ) ) R R 0 ={0}. 18) Thus, ) ) 0 ) ) + 0 ) ) +. 19)

6 3108 YONGGE TIN pplying 9) gives ) ) ), ) ), ) ), ) ), ) ) ),. 20) Substituting 20) into19), and 19) into17), and then 16) and 17) into15) givesus 13). Let the ight-hand side of 13) be zeo and note that ) ) ). 21) Then the equivalence of 4) and14) follows. The establishment of 13) is not easy, because the matix expession on the left-hand side of 13) involves thee tems consisting of the Mooe-Penose invese and the ighthand side of 13) involve anks of block matices with the poducts,, and. Howeve, the two block matices on the ight-hand side of 13) and the two ange equalities in 14) ae easy to simplify when and satisfy some conditions. Fo example, if both and ae patial isometies, that is, = and =,then14) is satisfied. In this case, 4) becomes ) = ) I n ) I n ). 22) The most valuable consequence of 4) is concened with the Mooe-Penose invese of the poduct of two othogonal pojectos. Coollay 2. Let P and Q be a pai of othogonal pojectos of ode n. Then, the poduct PQ satisfies the following two identities: PQ) = QP Q Q P ) P, 23) PQ) 2 = PQ+PQ Q P ) PQ, 24) whee P = I n P and Q = I n Q. In paticula, PQ) = QP Q Q P ) P = 0, PQ) 2 = PQ PQ Q P ) 25) PQ= 0. Poof. Note that P 2 = P = P = P, Q 2 = Q = Q = Q fo any pai of othogonal pojectos P and Q. Thus, P and Q satisfy 14), and 4) is educed to 23). Pemultiplying and postmultiplying both sides of 23) bypq yield 24).

7 ON MIXED-TYPE REVERSE-ORDER LWS Recall that fo any pai of othogonal pojectos P and Q, PQ) = QP PQ) 2 = PQ PQ= QP, 26) see, fo example, 1. Thus, 25) could be egaded as two new equivalent statements fo the commutativity of two othogonal pojectos. Thee ae many esults in the liteatue on poducts of othogonal pojectos and elated topics; see, fo example, 1, 13. The two identities 23) and24) ae two fundamental esults fo the poduct of two othogonal pojectos. They can be used fo dealing with vaious matix expessions that involve poducts of two othogonal pojectos. Fo example, the poduct PQP satisfies the following identity: PQP) = PQ)PQ) = QP) PQ) = P I n P Q ) Q I n Q P ) P. 27) Moeove, it is easy to veify that PQ) 2 = PQ) QP) PQ), PQ) # = QP) PQ) QP), 28) whee PQ) # is the goup invese of PQ. Hence, one can also deive fom 23) twoidentities fo PQ) 2 and PQ) #. Fom 23), one can also deive some valuable expessions fo P ±Q) and PQ±QP) ;see5. Let denote the spectal nom of a matix, that is, the maximal singula value of. Fo a nonnull othogonal pojecto P, P =1. Fo any pai of othogonal pojectos P and Q with PQ 0, it can be deived fom 23) the following nom equality: PQ) In Q P ). 29) It was shown in 3, 15 that if P is idempotent with P 0 and P I, then I P = P. If P and Q ae two othogonal pojectos, then PQ) is idempotent; see 14. Note that I n P and I n Q ae othogonal pojectos. Hence, I n Q)I n P) is idempotent. Thus, if I n Q)I n P) 0, then I n Q P ) = Q P ). 30) pplying this equality to 29), we see that if PQ 0 and Q P 0, then PQ) Q P ). 31) Replacing P with I n Q and Q with I n P in 31) alsogives Q P ) PQ). 32) Hence, we have the following esult. Theoem 3. Let P and Q be a pai of othogonal pojectos with both PQ 0 and Q P 0. Then, PQ) = Q P ). 33)

8 3110 YONGGE TIN Identity 23) can be used to establish some identities fo the Mooe-Penose invese of C, whee,, and C ae thee othogonal pojectos. Theoem 4. Let,, and C be a tiple of othogonal pojectos of ode n. Then, C) = P ) C ) ), PC 34) C) = C I n C P ) ) ) C) I n PC ), 35) whee P ) = ) and P C = CC). Poof. Recall a simple esult in 7 that any matix poduct UV satisfies UV ) = U UV ) UVV ). 36) pplying this fomula to C = )C) gives 34). pplying 23) top ) C) and P C ) also gives P) C ) = CP) C C P ) P) ), ) PC = PC P C P C ) 37). Substituting these two esults into 34) yields35). lthough evese-ode laws fo the Mooe-Penose invese of matix poducts have many diffeent expessions, some of these evese-ode laws may be equivalent. simple example is ) = ) = ) = ; 38) see 21. When investigating 4), we also find that 4) is equivalent to the following two mixed-type evese-ode laws: ) = ), 39) ) = ). 40) The evese-ode law 39) was fist studied by Galpein and Waksman 8, and then by Izumino 11 fo a poduct of two linea opeatos. They showed that 39) holds if and only if R ) =R), R ) =R ). 41) This esult is also tue fo a complex matix poduct. ecause the Mooe-Penose inveses of and ae contained in the condition 41), it cannot be egaded as a satisfactoy necessay and sufficient condition fo 39) to hold. In fact, 41) is equivalent to 14) by noticing, ) ), ) ),,. 42)

9 Without much effot, one can show that ON MIXED-TYPE REVERSE-ORDER LWS ) ) +, 2); 43) see 26. Equality 43) is deived fom the following esult. Lemma Let X 1 and X 2 be a pai of oute inveses of a matix, that is, X 1 X 1 = X 1 and X 2 X 2 = X 2. Then, Hence, ) X1 X 1 X 2 + ) ) X 1,X 2 X1 X2. 44) X 2 X 1 = X 2 R X 1 ) =R X2 ), R X 1 ) =R X 2 ). 45) Obviously, the matix ) is an oute invese of by the definition of the Mooe- Penose invese. It is also easy to veify that ) is an oute invese of. Thus, it follows by 44) that ) ) ) + ), ) 46) ) ) ). Hence, one can deive fom 44) that39) holds if and only if R ) =R ) ), R{ } =R { ) }. 47) It is also easy to veify that ) ),, ), ), ) ) ). 48) Then, 43) follows. nothe ank equality elated to 39) is ) ) ) ) + ) ), 49) which is shown in 26. Two ank fomulas associated with 40) ae given below.

10 3112 YONGGE TIN Theoem 6. Let C m n and C n p. Then, ) ) ) ) ), ) + ) ), ) ) +, 2). 51) Hence, the following statements ae equivalent: a) 40) holds, b) R) =R ), R{) }=R{ ) }, c) 14) holds. Poof. Note that ) ) ) = ), that is, ) is an oute invese of. Thus, 50) is deived fom 44), and 51) is a simplification of 50). pplying 13), 43), and 51) tothetwopoducts and gives us the following esult. Theoem 7. Let C m n and C n p, and let T = I n )I n ). Then, ) + T ), 50) ) + T, ), ) ) ), ) ), ), ) ) ), 52) ) ), ). Hence, a) the following statements ae equivalent: i) ) = T, ii) ) = ), iii) ) = ), iv) R ) =R), b) the following statements ae equivalent: i) ) = T, ii) ) = ), iii) ) = ), iv) R ) =R).

11 ON MIXED-TYPE REVERSE-ORDER LWS The combination of 4), 14), 39), 40), 41), 47) and Theoems 6 and 7 givesusthe following esult. Theoem 8. Let C m n and C n p, and let T = I n )I n ). Then, the following statements ae equivalent: a) ) = ), b) ) = ), c) ) = ) = ), d) ) = ) and ) = ), e) ) = ) and ) = ), f) ) = ) and ) = ), g) ) = ), h) ) = ) ) ), i) ) = T, j) ) = T and ) = T, k) R) = R T 1 ) and R{) } = R T 1 ), whee T 1 =, l) R) = R T 2 ) and R{) } = R T 2 ), whee T 2 =, m) R ) =R) and R ) =R), n) R ) =R) and R ) =R). The esults given above can be extended to the weighted Mooe-Penose invese of a matix poduct. The weighted Mooe-Penose invese of a matix C m n with espect to a pai of Hemitian positive definite matices M C m m and N C n n is defined to be the unique n m matix that satisfies the following fou matix equations: i) X =, ii) XX = X, iii) MX) = MX, iv) NX) = NX, and this X is denoted as X = M,N. In paticula, when M = I m and N = I n, I m,i n is the standad Mooe-Penose invese of. Revese-ode laws fo the weighted Mooe- Penose invese of matix poducts have been studied; see 17, 24, 27. s is well known see, e.g., 2), the weighted Mooe-Penose invese M,N of can be ewitten as M,N = N 1/2 M 1/2 N 1/2) M 1/2, 53) whee M 1/2 and N 1/2 ae the positive definite squae oots of M and N, espectively. y appealing to 12), one can obtain the following basic ank fomula: ) D C M,N MN 1 M CN 1 ); 54) D see also 26. pplying Theoem 8 to M 1/2 N 1/2 ) in ) M,N = N 1/2 M 1/2 N 1/2) M 1/2, 55)

12 3114 YONGGE TIN and noting that M,I = M 1/2 ) M 1/2, I,N = N 1/2 N 1/2) 56) yields the following esult. Theoem 9. Let C m n and C n p, and let M C m m and N C p p be a pai of Hemitian positive definite matices. Then, the following statements ae equivalent: a) ) M,N = I,N M,I I,N) M,I, b) ) M,N = M,P) P,N M,P = P,N P,N) M,P, c) ) M,N = N 1 MN 1 ) M, d) ) M,N = I,N M,I I,NI n I,N)I n M,I) M,I, e) R M) =R) and RN 1 ) =R). Remak 10. Revese-ode laws can be established fom any easonable opeations fo the Mooe-Penose invese of matix poducts. Fo example, wite = = ) ) = ) ) := PNQ. 57) The evese-ode law PNQ) = Q N P is equivalent to ) = ) ) ). 58) The law ) = ) is obtained fom witing = ) ) ) := PNQ 59) and PNQ) = Q N P. On the othe hand, mixed-type evese-ode laws can be intoduced by compaing diffeent decompositions of a block matix and thei genealized inveses. The intoduction of 4) is such an example. In addition, one can conside some vaiations of 4). Fo instance, eplacing the Mooe-Penose inveses in 4) with {1}-inveses yields the following evese-ode law fo ) : ) = F E F ) E, 60) whee F = I n and E = I n. One can also establish some ank equalities associated with this evese-ode law and then deive necessay and sufficient conditions fo this law to hold. Fo a tiple matix poduct C, the Mooe-Penose invese C) can be witten as eithe C) = C o C) = C + X. The law C) = C was studied in 10, 19. Necessay and sufficient conditions fo this law to hold ae quite stict and complicated. Two easonable extensions of 4) toc) ae C) = C) ) C) P C P ) ), C) = C I p P C P ) I n P P ). 61)

13 ON MIXED-TYPE REVERSE-ORDER LWS Some easonable extensions of 39) and40) toc) ae C) = C CC ), C) = C CC ), 62) C) = C) ) CC) ). Vaious ank fomulas associated with these evese-ode laws can be established. Fo moe details, see 18. cknowledgments. The autho would like to expess his sincee thanks to Pofessos G. P. H. Styan, S. Duy, X.-W. Chang, and Y. Takane at McGill Univesity fo thei suppots duing the pepaation of this pape. Sincee thanks also ae due to the efeees fo the helpful comments and suggestions. Refeences 1 J. K. aksalay, lgebaic chaacteizations and statistical implications of the commutativity of othogonal pojectos, Poceedings of the Second Intenational Tampee Confeence in Statistics T. Pukkila and S. Puntanen, eds.), Univesity of Tampee, Tampee, 1987, pp en-IsaelandT.N.E.Geville,Genealized Inveses. Theoy and pplications, 2nded., CMS ooks in Mathematics, vol. 15, Spinge-Velag, New Yok, D. uckholtz, Hilbet space idempotents and involutions, Poc. me. Math. Soc ), no. 5, S. L. Campbell and C. D. Meye, J., Genealized Inveses of Linea Tansfomations, Dove Publications, New Yok, 1991, coected epint of the 1979 oiginal. 5 S. Cheng and Y. Tian, Mooe-Penose inveses of poducts and diffeences of othogonal pojectos, cta Sci. Math. Szeged) ), no. 3-4, R. E. Cline, Note on the genealized invese of the poduct of matices, SIM Rev ), R.E.ClineandT.N.E.Geville,n extension of the genealized invese of a matix, SIM J. ppl. Math ), M. Galpein and Z. Waksman, On pseudo-inveses of opeato poducts, Linea lgeba ppl ), T.N.E.Geville,Note on the genealized invese of a matix poduct, SIM Rev ), R. E. Hatwig, The evese ode law evisited, Linea lgeba ppl ), S. Izumino, The poduct of opeatos with closed ange and an extension of the evese ode law, Tôhoku Math. J. 2) ), no. 1, G. Masaglia and G. P. H. Styan, Equalities and inequalities fo anks of matices, Linea and Multilinea lgeba /75), T. Oikhbeg, Poducts of othogonal pojections, Poc. me. Math. Soc ), no. 12, R. Penose, genealized invese fo matices, Math. Poc. Cambidge Philos. Soc ), V. Rakočević, On the nom of idempotent opeatos in a Hilbet space, me. Math. Monthly ), no. 8, C. R. Rao and S. K. Mita, Genealized Invese of Matices and Its pplications, John Wiley & Sons, New Yok, W. Sun and Y. Wei, Invese ode ule fo weighted genealized invese,simj.matixnal. ppl ), no. 3,

14 3116 YONGGE TIN 18 Y. Tian, Some mixed-type evese ode laws fo the Mooe-Penose invese of a tiple poduct, to appea in Rocky Mountain J. Math. 19, Revese ode laws fo the genealized inveses of multiple matix poducts, Linea lgeba ppl ), , How to chaacteize equalities fo the Mooe-Penose invese of a matix, Kyungpook Math. J ), no. 1, , Poblem 29-10: Equivalence of thee evese-ode laws, Image The ulletin of the Intenational Linea lgeba Society) ), 35, Solutions by J. K. aksalay and O. M. aksalay, O. M. aksalay and K. Chylińska, S. Cheng, Y. Tian, and H. J. Wene, Image The ulletin of the Intenational Linea lgeba Society) ), , Rank equalities elated to oute inveses of matices and applications, Linea and Multilinea lgeba ), no. 4, , Uppe and lowe bounds fo anks of matix expessions using genealized inveses, Linea lgeba ppl ), , Revese ode laws fo the weighted Mooe-Penose invese of a tiple matix poduct with applications, Int. Math. J ), no. 1, , Rank equalities fo block matices and thei Mooe-Penose inveses, Houston J. Math ), , Using ank fomulas to chaacteize equalities fo Mooe-Penose inveses of matix poducts, ppl. Math. Comput ), no. 2, Y. Tian and S. Cheng, Some identities fo Mooe-Penose inveses of matix poducts, Linea and Multilinea lgeba, in pess. 28 S. Zlobec, n explicit fom of the Mooe-Penose invese of an abitay complex matix, SIM Rev ), Yongge Tian: Depatment of Mathematical and Statistical Sciences, Univesity of lbeta, Edmonton,, Canada T6G 2G1 addess: ytian@stat.ualbeta.ca

15 Mathematical Poblems in Engineeing Special Issue on Time-Dependent illiads Call fo Papes This subject has been extensively studied in the past yeas fo one-, two-, and thee-dimensional space. dditionally, such dynamical systems can exhibit a vey impotant and still unexplained phenomenon, called as the Femi acceleation phenomenon. asically, the phenomenon of Femi acceleation F) is a pocess in which a classical paticle can acquie unbounded enegy fom collisions with a heavy moving wall. This phenomenon was oiginally poposed by Enico Femi in 1949 as a possible explanation of the oigin of the lage enegies of the cosmic paticles. His oiginal model was then modified and consideed unde diffeent appoaches and using many vesions. Moeove, applications of F have been of a lage boad inteest in many diffeent fields of science including plasma physics, astophysics, atomic physics, optics, and time-dependent billiad poblems and they ae useful fo contolling chaos in Engineeing and dynamical systems exhibiting chaos both consevative and dissipative chaos). We intend to publish in this special issue papes epoting eseach on time-dependent billiads. The topic includes both consevative and dissipative dynamics. Papes discussing dynamical popeties, statistical and mathematical esults, stability investigation of the phase space stuctue, the phenomenon of Femi acceleation, conditions fo having suppession of Femi acceleation, and computational and numeical methods fo exploing these stuctues and applications ae welcome. To be acceptable fo publication in the special issue of Mathematical Poblems in Engineeing, papes must make significant, oiginal, and coect contibutions to one o moe of the topics above mentioned. Mathematical papes egading the topics above ae also welcome. uthos should follow the Mathematical Poblems in Engineeing manuscipt fomat descibed at Pospective authos should submit an electonic copy of thei complete manuscipt though the jounal Manuscipt Tacking System at mts.hindawi.com/ accoding to the following timetable: Guest Editos Edson Denis Leonel, Depatamento de Estatística, Matemática plicada e Computação, Instituto de Geociências e Ciências Exatas, Univesidade Estadual Paulista, venida 24, 1515 ela Vista, Rio Clao, SP, azil ; edleonel@c.unesp.b lexande Loskutov, Physics Faculty, Moscow State Univesity, Voob evy Goy, Moscow , Russia; loskutov@chaos.phys.msu.u Manuscipt Due Decembe 1, 2008 Fist Round of Reviews Mach 1, 2009 Publication Date June 1, 2009 Hindawi Publishing Copoation