Research Article The Group Involutory Matrix of the Combinations of Two Idempotent Matrices

Transcription

1 Hindwi Pulishing Corportion Journl of Applied Mthemtics Volume 2012, Article ID , 17 pges doi: /2012/ Reserch Article The Group Involutory Mtrix of the Comintions of Two Idempotent Mtrices Lingling Wu, 1, 2 Xioji Liu, 1 nd Yoming Yu 3 1 College of Mthemtics nd Computer Science, Gungxi University for Ntionlities, Nnning , Chin 2 College of Mthemtics nd Computer Science, Bijie University, Guizhou , Chin 3 School of Mthemticl Sciences, Monsh University, Clyton Cmus, VIC 3800, Austrli Correspondence should e ddressed to Xioji Liu, liuxioji.2003@yhoo.com.cn Received 19 Decemer 2011; Accepted 16 Mrch 2012 Acdemic Editor: Mehmet Sezer Copyright q 2012 Lingling Wu et l. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. We discuss the following prolem: when P Q cpq d ep f gp of idempotent mtrices P nd Q,where,, c, d, e, f, g C nd / 0,/ 0, is group involutory. 1. Introduction Throughout this pper C n n stnds for the set of n n complex mtrices. Let A C n n. A is sid to e idempotent if A 2 A. A is sid to e group invertile if there exists n X C n n such tht AXA A, XAX X, AX XA 1.1 hold. If such n X exists, then it is unique, denoted y A g, nd clled the group inverse of A. It is well known tht the group inverse of squre mtrix A exists if nd only if rnka 2 rnkasee, e.g., 1 for detils. Clerly, not every mtrix is group invertile. But the group inverse of every idempotent mtrix exists nd is this mtrix itself. Recll tht mtrix A with the group inverse is sid to e group involutory if A g A. A is the group involutory mtrix if nd only if it is tripotent, tht is, stisfies A 3 A see 2. Thus, for nonzero idempotent mtrix P nd nonzero sclr, P is group involutory mtrix if nd only if either 1or 1. Recently, some properties of liner comintions of idempotents or projections re widely discussed see, e.g., 3 12 nd the literture mentioned elow. In13, uthors

2 2 Journl of Applied Mthemtics estlished complete solution to the prolem of when liner comintion of two different projectors is lso projector. In 14, uthors considered the following prolem: when liner comintion of nonzero different idempotent mtrices is the group involutory mtrix. In 15, uthors provided the complete list of situtions in which liner comintion of two idempotent mtrices is the group involutory mtrix. In 16, uthors discussed the group inverse of P Q cpq d ep f gp of idempotent mtrices P nd Q, where,, c, d, e, f, g C with, / 0, deduced its explicit expressions, nd some necessry nd sufficient conditions for the existence of the group inverse of P Q cpq. In this pper, we will investigte the following prolem: when P Q cpq d ep f gp is group involutory. To this end, we need the results elow. Lemm 1.1 see 16, Theorems 2.1 nd 2.4. Let P, Q C n n e two different nonzero idempotent mtrices. Suppose PQ 2 2. Then for ny sclrs,, c, d, e, f, g, where, / 0 nd θ c d e f g, P Q cpq d ep fgpq 2 is group invertile, nd i if θ / 0, then P Q cpq d ep f gpq 2 g 1 PQ 1 P 1 1 Q 1 c 2 1 c d 2 2 c d cd e 2 P 1 d 1 2 cd e cd f θ c d P; cd f ii if θ 0, then P Q cpq d ep f gpq 2 g 1 P 1 1 Q 1 c 2 1 c d 2 2 c d cd e 2 cd e 2 1 PQ P 1 d 1 2 PQ 2. cd f 2 c d cd f Lemm 1.2 see 16, Theorem 3.1. Let P, Q C n n e two different nonzero idempotent mtrices. Suppose 2 0. Then for ny sclrs,, c, d, e, f, nd g,where, / 0, P Q cpq d ep f gpq 2 is group invertile, nd P Q cpq d ep f gpq 2 g 1 P 1 1 Q 1 c 1 PQ 1 d

3 Journl of Applied Mthemtics c d cd e 2 1 P 2 c d cd f c d g cd e ce cd f cf c2 d PQ Min Results In this section, we will reserch when some comintion of two nonzero idempotent mtrices is group involutory mtrix. First, we will discuss some situtions lying in the ctegory of PQ 2 2. Theorem 2.1. Let P, Q C n n e two different nonzero idempotent mtrices with PQ 2 2, nd let A e comintion of the form A P Q cpq d ep f gp, 2.1 where,, c, d, e, f, g C with, / 0. Denote θ c d e f g. Then the following list comprises chrcteristics of ll cses where A is the group involutory mtrix: the cses denoted y 1 3,inwhich PQ, 2.2 nd ny of the following sets of dditionl conditions hold: 1 either 1 or 1, eitherθ 1 or θ 1 or θ 0, nd Q PQ; 2 either 1 or 1, eitherθ 1 or θ 1 or θ 0, nd P PQ; 3 either 1 or 1, either 1 or 1, eitherθ 1 or θ 1 or θ 0 or PQ 0. the cses denoted y 1 6,inwhich PQ/, P, 2.3 nd ny of the following sets of dditionl conditions hold: 1 ±1, 1, eitherθ 1 or θ 1 or θ 0 or P 0; 2 ±1,c d 1,eitherθ 1 or θ 1 or θ 0 or P 0; 3 ±1,c 1, eitherθ 1 or θ 1 or θ 0, nd P; 4 ±1,d 1, eitherθ 1 or θ 1 or θ 0, nd PQ P; 5 ±1, c 1, nd 0; 6 ±1, d 1, nd PQ 0, c the cses denoted y c 1 c 18,inwhich P /, P P, 2.4

4 4 Journl of Applied Mthemtics nd ny of the following sets of dditionl conditions hold: c 1 ±1, 1,c d 2e ± cd ±1, eitherθ 1 or θ 1, nd P; c 2 e ±1,c d 1, eitherθ 1 or θ 1, nd P; c 3 ±1, 1,c d 2f cd 1, eitherθ 1 or θ 1, nd P P; c 4 f ±1,c d 1, eitherθ 1 or θ 1, nd P P; c 5 ±1, 1,c d 2e ± cd ±1,c d 2f cd 1, eitherg 1 or g 1; c 6 e f ±1,c d 1, eitherg 1 or g 3; c 7 ±1, 1,c d 2e ± cd ±1, nd 0; c 8 e ±1,c d 1, nd 0; c 9 ±1, 1,c d 2f cd 1, nd P 0; c 10 f ±1,c d 1, nd P 0; c 11 ±1, 1,c d 2e ± cd ±1,c d 2f cd 1, nd P 0; c 12 e f ±1,c d 1, nd P 0; c 13 ±1, 1, 2e c d ± cd ±1,θ 0, nd P; c 14 e ±1,c d 1,θ 0, nd P; c 15 ±1, 1, 2f c d cd 1,θ 0, nd P P; c 16 f ±1,c d 1,θ 0, nd P P; c 17 ±1, 1, 2e c d ± cd ±1, 2f c d cd 1,g 0; c 18 e f ±1,c d 1,g 2. Proof. Oviously, the condition 2.2 implies tht the group inverse of A exists nd is of the form 1.2 when θ / 0 or the form 1.3 when θ 0yLemm 1.1. So do the conditions 2.2, 2.3,nd2.4. We will strightforwrdly show tht mtrix A of the form 2.1 is the group involutory mtrix if nd only if A A g 0. Under the condition 2.2, A P Q μpq, where μ c d e f g. 1 If θ / 0, then A g 1 P 1 Q 1 θ 1 1 PQ, 2.5 nd so A A g 1 P 1 Q μ 1 θ 1 1 PQ Multiplying 2.6 y P nd Q, respectively, leds to 1 P 1 PQ μ 1 θ 1 1 PQ 0, 1 PQ 1 Q μ 1 θ 1 1 PQ 0, 2.7

5 Journl of Applied Mthemtics 5 nd then 1 P 1 PQ 1 PQ 1 Q. 2.8 Multiplying the ove eqution, respectively, y P nd y Q, weget 1 P PQ 0, 1 Q PQ Thus, since P / Q, we hve three situtions: P PQ nd 1 ; 1 nd Q PQ; 1 nd 1. When Q PQ nd 1, 2.6 ecomes θ θ 1 Q 0 nd then θ ±1. Therefore, we otin 1 except the sitution θ 0. Similrly, when 1 nd P PQ, we hve 2 except the sitution θ 0. When 1 nd 1, 2.6 ecomes θ θ 1 PQ 0 nd then θ ±1orPQ 0. Therefore, we otin 3 except the sitution θ 0. 2 If θ 0, then A g 1 P 1 Q 1 1 PQ, 2.10 nd then A A g 1 P 1 Q μ 1 1 PQ Anlogous to the process of reching 2.9 in 1, we hve 1 Q PQ 0, 1 P PQ Thus, we hve three situtions: P PQ nd 1 ; 1 nd Q PQ; 1 nd 1, since P / Q. Similr to the rgument in 1, sustituting them, respectively, into 2.11,we cn otin the sitution θ 0, respectively, in 1, 2,nd 3. Under the condition 2.3, A P Q cpq d νp, where ν e f g. 1 If θ / 0, then A g 1 P 1 1 Q 1 c 1 PQ 1 d 1 1 c d 1 P, θ 2.13

6 6 Journl of Applied Mthemtics nd so A A g 1 P 1 Q c 1 1 c PQ d 1 1 d ν 1 1 c d 1 θ P Multiplying the ove eqution, respectively, on the two sides y P yields P c 1 c 1 P d 1 d PQ ν d c 1 θ ν c d 1 θ P, P Multiplying 2.15 on the left sides y Q nd 2.16 on the right sides y Q,y2.3, we hve 1 c d ν 1 1 0, θ 1 PQ c d ν 1 1 0, θ 2.17 nd then 1 PQ0. Since / PQ, 1. Similrly, 1. Sustituting 1 inside 2.17 yields θθ 1 0 nd then θ θ 1 or 0. We will discuss the reminder for detil s follows: When 1, 1, 2.14 ecomes 0 c 1 1 c ν 1 1 c d 1 θ PQ d 1 1 d P, 2.18 i if 0, then nd so it follows from 2.18 tht c 1 1 c 0, d 1 1 d 0, 2.19 θ 1 P ν c d 1 P 0. θ θ 2.20 Therefore, either θ θ 1 or P 0 implies tht 2.18 holds, nmely, 2.14 holds. Thus, we hve 1 except the sitution θ 0.

7 Journl of Applied Mthemtics 7 ii if, then 2.18 ecomes 0 2c 2PQ 2d 2 2ν θ 1 P. θ 2.21 Multiplying the ove eqution, respectively, on the right side y P nd on the left side y Q, we hve 0 2c 2PQ ν d c 1 P, θ 0 2d 2 ν c d 1 P. θ So if θ θ 1, then the two equtions ove 2.22 nd 2.23 ecome, respectively, c PQ P 0, d P Or if P 0, then 2.22 nd 2.23 ecome, respectively, c PQ 0, d Since PQ/, it follows from 2.24 nd 2.25 tht we hve the six situtions: θ θ 1 nd c d ; θ θ 1, c nd P; θ θ 1, d, ndpq P; c nd 0; d nd PQ 0; c d nd P 0. Thus, we hve 2 4 except the sitution θ 0, nd 5 nd 6. 2 If θ 0, then A g 1 P 1 1 Q 1 c 1 PQ 1 d 1 1 c d P, 2.26 nd then A A g 1 P 1 Q c 1 1 c PQ d 1 1 d ν 1 1 c d P Anlogous to the process in 1,using2.27 we cn otin 1 1 P 0, 1 PQ 1 P

8 8 Journl of Applied Mthemtics Thus, since PQ/, PQ/ P nd/or / P nd then 1. Similrly, 1. Hence, ±. i If, then c 1 1 c 0, d 1 1 d 0, 2.29 ν 1 1 c d 2 0. Thus, 2.27 holds. Hence we hve the sitution θ 0in 1. ii If, then 2.27 ecomes c PQ d νp Multiplying the ove eqution on the left side, respectively, y P nd y Q, we hve c PQ P 0, d P Thus, c d ; c nd P; d nd PQ P. Hence, we hve the sitution θ 0, respectively, in 2, 3,nd 4. c Under the condition 2.4, A P Q cpq d ep fp gp If θ / 0, then A g 1 P 1 1 Q 1 c 1 PQ 1 d 2 1 c d cd e 2 1 P 2 c d 2 2 c d cd e cd f 1 P, 2 2 θ cd f nd so A A g 1 P 1 Q c 1 1 c e 2 1 c d cd e 2 P PQ d 1 1 d

9 Journl of Applied Mthemtics 9 f 1 2 c d cd f 2 g 2 2 c d cd e cd f 1 P θ 2.34 If PQ 0, then 0 P nd so it contrdicts 2.4.ThusPQ/ 0. Similrly, / 0. Multiplying 2.34 on the left side y yields 1 c 1 c d e f g c 1 P θ Multiplying the ove eqution, respectively, on the left side y P nd on the right side y PQ yields, y 2.4, P 1 θ 1 θ 1 θ 1 θ P, P Since P /, 1 y 2.36 nd Similrly, we cn gin 1. Sustituting 1 inside 2.36 yields θ θ 1 or P 0. i Consider the cse of 1, 1 nd θ θ 1. Sustituting 1, 1,ndθ θ 1 inside 2.35 yields Similrly, we hve c c P 0. d d P P If P P, then / P y the hypothesis P / nd so c c/ 0y2.38. Multiplying 2.34 on the right side y Q yields c d 2f cd P Thus, c d 2f cd/ 0 nd then 2.14 ecomes d d f 2 c d cd f e g θ P 0. cd f 2.41

10 10 Journl of Applied Mthemtics Multiplying the ove eqution on the right side y P yields d d P Assume PQ P. Then P P PQ P nd it contrdicts the hypothesis P /.Thus, d d/ 0. Similrly, if P, then we cn otin d d 0, cd2ecd/ 0, nd c c/ 0. Oviously, if / P nd / P, we hve d d/ 0, c c/ 0, c d 2e cd/ 0, nd c d 2f cd/ 0. Next, we clculte these sclrs. If 0, then c c/ 0 for ny c nd d d/ 0 for ny d, ndsoc, d, e re chosen to stisfy c d 2e cd/ 0. Similrly c, d, f re chosen to stisfy c d 2f cd/ 0. If, then c d, nde y solving c d 2e cd/ 0, nd f y solving c d 2f cd/ 0. Note tht c d 2e cd/ 0ndcd2f cd/ 0implyg θ. Hence, we hve c 1 c 6. ii Consider the cse of 1, 1,ndP 0. Multiplying 2.34, respectively, on the right side y nd on the left side y PQ yields c 1 1 c d 1 1 d 0, P If 0, then P / 0ndso d d/ 0nd2.34 ecomes 0 c 1 1 c PQ e 2 1 c d Multiplying 2.44 on right side y Q yields c 1 1 c PQ 0. Since PQ/ 0, c c/ 0 nd then 2.44 ecomes 2e c d cd P. cd e 2 P Thus, 2e c d cd/ 0. If P 0, then we, similrly, hve c c/ 0, d d/ 0, nd 2f c d cd/ 0.

11 Journl of Applied Mthemtics 11 If P / 0nd / 0, then, multiplying 2.34, on the right side y Q nd on the left side y P yields c c/ 0, nd multiplying 2.34 on the right side y P nd on the left side y Q yields d d/ 0. Thus, 2.34 ecomes e 2 1 c d cd e 2 P f 1 2 c d cd f Multiplying the eqution ove on the right side, respectively, y P nd y Q yields 2e c d cd 0, 2f c d cd As the rgument ove in i, we hve c 7 c If θ 0, then A g 1 P 1 1 Q 1 c 1 PQ 1 d 2 1 c d 2 2 c d cd e 2 cd e 2 1 P 2 c d cd f P, 2 cd f nd so A A g 1 P 1 Q c 1 1 c PQ d 1 1 d e 2 1 c d f 1 2 c d g 2 2 c d cd f 2 cd e 2 cd e 2 cd f P 0. 2 P 2.50 Anlogous to the process in c1, using2.50, we cn get 1 P P 0, 1 P

12 12 Journl of Applied Mthemtics Thus, since P /, P / P nd/or / P nd then 1. Similrly, 1. Therefore, multiplying 2.50 on the right side y Q nd on the left side y P yields c c PQ P Multiplying 2.50 on the right side y P nd on the left side y Q yields d d P Since PQ/ P nd / P, c c/ 0nd d d/ 0. Multiplying 2.50 on the left side, respectively, y P nd y Q yields 2e c d cd P P 0, 2f c d cd P Thus, we hve 2e c d cd/ 0nd P;2f c d cd/ 0nd P P;2e c d cd/ 0nd2f c d cd/ 0. Note tht 2e c d cd/ 0nd2f c d cd/ 0implyg y θ 0. As the rgument ove in c1, we hve c 13 c 18. Remrk 2.2. Clerly, 15, nd in Theorem re the specil cses in Theorem 2.1. Exmple 2.3. Let P, Q

13 Journl of Applied Mthemtics 13 Then they, oviously, re idempotent, nd PQ 2 2 ut P /. By Theorem 2.1c 5, A P Q 2PQ P 1 P is the group involutory mtrix, nmely, A A g,since / nd / By Theorem 2.1c 17, P Q PQ 2 2P 2.57 is group involutory since nd Next, we will study the sitution PQ 2 0or 2 0. Theorem 2.4. Let P, Q C n n e two different nonzero idempotent mtrices, nd let A e comintion of the form A P Q cpq d ep f gp, 2.58 where,, c, d, e, f, g C with, / 0. Suppose tht P/ 0, P 0, 2.59 nd ny of the following sets of dditionl conditions hold: d 1 ±1, c d 1, e f ±1, g 1; d 2 ±1, 1, 2e c d ± cd ±1, 2f c d cd 1. Then A is the group involutory mtrix. Proof. By Lemm 1.2, 0 A A g 1 P 1 Q c 1 1 c PQ d 1 1 d e 2 1 c d f 1 2 c d g 2 2 2c d g cd e 2 cd f 2 P cd e ce 2 cd f cf 2 c2 d PQ

14 14 Journl of Applied Mthemtics Since P/ 0, multiplying 2.60, respectively, on the right side nd on the right side y P yields 1 P 0, 1 P 0, 2.61 nd so 1 nd 1. Sustituting them inside 2.60, weget 0 c 1 1 c PQ d 1 1 d e 2 1 c d f 1 2 c d g 2 2 2c d g cd e 2 cd f 2 P cd e ce 2 cd f cf 2 c2 d P Multiplying 2.62 ontheleftsideyp yields c 1 1 c P 0, 2.63 nd then c 1 1 c So 2.62 ecomes 0 d 1 1 d e 2 1 c d f 1 2 c d g 2 2 2c d g cd f 2 cd e ce 2 cd e 2 cd f cf 2 P c2 d P Multiplying 2.65 ontheleftsideypq nd on the right side y P yields d 1 1 d P

15 Journl of Applied Mthemtics 15 Therefore, d 1 1 d Similrly, we cn otin 0 g e 2 1 c d 0 f 1 2 c d 2c d g By 2.64 nd 2.67, we cn otin cd e ce 2 cd e 2, cd f, 2 cd f cf 2 c2 d c d 2e cd 0, 1 c d 2f cd Since 1 nd 1, ±.If, then 2.64 holds for ny c, 2.67 holds for ny d, nd, for ny c, d, e, f stisfying 2.69 nd ny g, g 2 2 cd e ce cd f cf 2 2 c 2 d 2c d e f c e f c 2 d 2c d c d c 1 cd 0. 2c d g c2 d If, then, y , c d nd e f nd so g from Hence, we hve d 1 nd d 2. Exmple 2.5. Let P, Q Oviously they re idempotent, nd 2 0utPQ 2 / 0. By Theorem 2.4d 2, P Q 2PQ P 5 2P is group involutory since /22 2 1nd /

16 16 Journl of Applied Mthemtics Similrly, we hve the following result. Theorem 2.6. Let P, Q C n n e two different nonzero idempotent mtrices, nd let A e comintion of the form A P Q cpq d ep f hp, 2.73 where,, c, d, e, f, h C with, / 0. Suppose tht P / 0, P 0, 2.74 nd ny of the following sets of dditionl conditions hold: e 1 ±1, c d 1, e f ±1, h 1; e 2 ±1, 1, 2e c d ± cd ±1, 2f c d cd 1. Then A is the group involutory mtrix. Acknowledgment This work ws supported y the Ntionl Nturl Science Foundtion of Chin nd the Ministry of Eduction Science nf Technology Key Project nd Grnts HCIC of Gungxi Key Lorrory of Hyrid Computtionl nd IC Design Anlysis Open Fund. References 1 A. Ben-Isrel nd T. N. E. Greville, Generlized Inverses: Theory nd Applictions, Springer, New York, NY, USA, 2nd edition, R. Bru nd N. Thome, Group inverse nd group involutory mtrices, Liner nd Multiliner Alger, vol. 45, no. 2-3, pp , J. K. Bkslry nd O. M. Bkslry, On liner comintions of generlized projectors, Liner Alger nd its Applictions, vol. 388, pp , J. K. Bkslry, O. M. Bkslry, nd H. Özdemir, A note on liner comintions of commuting tripotent mtrices, Liner Alger nd its Applictions, vol. 388, pp , J. K. Bkslry, O. M. Bkslry, nd G. P. H. Styn, Idempotency of liner comintions of n idempotent mtrix nd tripotent mtrix, Liner Alger nd its Applictions, vol. 354, pp , 2002, Ninth specil issue on liner lger nd sttistics. 6 O. M. Bkslry nd J. Benítez, Idempotency of liner comintions of three idempotent mtrices, two of which re commuting, Liner Alger nd its Applictions, vol. 424, no. 1, pp , J. Benítez nd N. Thome, Idempotency of liner comintions of n idempotent mtrix nd t-potent mtrix tht commute, Liner Alger nd its Applictions, vol. 403, pp , J. Benítez nd N. Thome, Idempotency of liner comintions of n idempotent mtrix nd t-potent mtrix tht do not commute, Liner nd Multiliner Alger, vol. 56, no. 6, pp , Y. N. Chen nd H. K. Du, Idempotency of liner comintions of two idempotent opertors, Act Mthemtic Sinic, vol. 50, no. 5, pp , J. J. Kolih, V. Rkočević, nd I. Strškr, The difference nd sum of projectors, Liner Alger nd its Applictions, vol. 388, pp , 2004.

17 Journl of Applied Mthemtics H. Özdemir nd A. Y. Özn, On idempotency of liner comintions of idempotent mtrices, Applied Mthemtics nd Computtion, vol. 159, no. 2, pp , M. Srduvn nd H. Özdemir, On liner comintions of two tripotent, idempotent, nd involutive mtrices, Applied Mthemtics nd Computtion, vol. 200, no. 1, pp , J. K. Bkslry nd O. M. Bkslry, Idempotency of liner comintions of two idempotent mtrices, Liner Alger nd its Applictions, vol. 321, no. 1 3, pp. 3 7, 2000, Liner lger nd sttistics Fort Luderdle, FL, C. Coll nd N. Thome, Olique projectors nd group involutory mtrices, Applied Mthemtics nd Computtion, vol. 140, no. 2-3, pp , J. K. Bkslry nd O. M. Bkslry, When is liner comintion of two idempotent mtrices the group involutory mtrix? Liner nd Multiliner Alger, vol. 54, no. 6, pp , X. Liu, L. Wu, nd Y. Yu, The group inverse of the comintions of two idempotent mtrices, Liner nd Multiliner Alger, vol. 59, no. 1, pp , 2011.

18 Advnces in Opertions Reserch Hindwi Pulishing Corportion Advnces in Decision Sciences Hindwi Pulishing Corportion Journl of Applied Mthemtics Alger Hindwi Pulishing Corportion Hindwi Pulishing Corportion Journl of Proility nd Sttistics The Scientific World Journl Hindwi Pulishing Corportion Hindwi Pulishing Corportion Interntionl Journl of Differentil Equtions Hindwi Pulishing Corportion Sumit your mnuscripts t Interntionl Journl of Advnces in Comintorics Hindwi Pulishing Corportion Mthemticl Physics Hindwi Pulishing Corportion Journl of Complex Anlysis Hindwi Pulishing Corportion Interntionl Journl of Mthemtics nd Mthemticl Sciences Mthemticl Prolems in Engineering Journl of Mthemtics Hindwi Pulishing Corportion Hindwi Pulishing Corportion Hindwi Pulishing Corportion Discrete Mthemtics Journl of Hindwi Pulishing Corportion Discrete Dynmics in Nture nd Society Journl of Function Spces Hindwi Pulishing Corportion Astrct nd Applied Anlysis Hindwi Pulishing Corportion Hindwi Pulishing Corportion Interntionl Journl of Journl of Stochstic Anlysis Optimiztion Hindwi Pulishing Corportion Hindwi Pulishing Corportion