A new eigenvalue inclusion set for tensors with its applications

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1 COMPUTATIONAL SCIENCE RESEARCH ARTICLE A new egenvalue ncluson set for tensors wth ts applcatons Cal Sang 1 and Janxng Zhao 1 * Receved: 30 Deceber 2016 Accepted: 12 Aprl 2017 Frst Publshed: 20 Aprl 2017 *Correspondng author: Janxng Zhao, College of Data Scence and Inforaton Engneerng, Guzhou Mnzu Unversty, Guyang , P.R. Chna E-als: zjx810204@163.co, zhaojanxng@gzu.edu.cn Revewng edtor: Song Wang, Curtn Unversty, Australa Addtonal nforaton s avalable at the end of the artcle Abstract: In ths paper, we gve a new egenvalue localzaton set for tensors and show that the new set s tghter than those presented by Q (2005) and L et al. (2014). As applcatons, we gve a new suffcent condton of the postve (se-) defnteness for an even-order real syetrc tensor and new lower and upper bounds of the nu egenvalue for -tensors. Subjects: Scence; Matheatcs & Statstcs; Advanced Matheatcs; Algebra; Lnear & Multlnear Algebra; Nonlnear Algebra; Nuercal Algebra Keywords: -tensors; nonnegatve tensors; nu egenvalue; locaton set; postve (se-)defnte AMS subject classfcatons: 15A18; 15A69 1. Introducton For a postve nteger n, n 2, N denotes the set 1, 2,, n}. C (respectvely, R) denotes the set of all coplex (respectvely, real) nubers. We call =(a 1 ) a coplex (real) tensor of order denson n, denoted by C, n] (R, n] ), f a 1 C(R), where j N for j = 1, 2,,. An -order n-densonal tensor s called nonnegatve, f each entry s nonnegatve. A tensor of order denson n s called the unt tensor, denoted by, f ts entres are δ 1 for 1,, N, where ABOUT THE AUTHOR Janxng Zhao has obtaned PhD n appled atheatcs fro Yunnan Unversty. Currently, he s an assocate professor n College of Data Scence and Inforaton Engneerng, Guzhou Mnzu Unversty, Guyang, Chna. Hs an research nterests nclude crtera for H-tensors and ts applcatons, H(Z)-egenvalue ncluson set for general tensors wth ts applcatons, and estates of the nu egenvalue for -tensors. PUBLIC INTEREST STATEMENT One of any practcal applcatons of egenvalues of tensors s that one can dentfy the postve (se-)defnteness for an even-order real syetrc tensor by usng the sallest H-egenvalue of a tensor; consequently, one can dentfy the postve (se-)defnteness of the ultvarate hoogeneous polynoal deterned by ths tensor. However, t s not easy to copute the sallest H-egenvalue of tensors when the order and denson are very large, we always try to gve a set ncludng all egenvalues n the coplex. In partcular, f one of these sets for an even-order real syetrc tensor s n the rghthalf coplex plane, then we can conclude that the sallest H-egenvalue s postve, consequently, the correspondng tensor s postve defnte. Therefore, the an a of ths paper s to gve a new egenvalue ncluson set for tensors, and usng the set to obtan a weaer suffcent condton for the postve (se-)defnteness of an even-order real syetrc tensor The Author(s). Ths open access artcle s dstrbuted under a Creatve Coons Attrbuton (CC-BY) 4.0 lcense. Page 1 of 13

2 δ 1 = 1, f 1 = =, 0, otherwse. A real tensor =(a 1 ) s called syetrc (Q, 2005) f a 1 = a π(1 ), π Π, where Π s the perutaton group of ndces. A tensor =(a 1 2 ) s called reducble f there exsts a nonepty proper ndex subset J N such that a 1 2 = 0, 1 J, 2,, J. If s not reducble, then we call s rreducble (Chang, Zhang, & Pearson, 2008). Let =(a 2 ) be a nonnegatve tensor, G =(g j ) R n n, g j = a 2. s called wealy reducble f G s a j 2,, } reducble atrx. If s not wealy reducble, then t s called wealy rreducble; for detals, see Fredland, Gaubert, and Han (2013) and Zhang, Q, and Zhou (2014). For a general tensor =(a 1 ) C, n], Wang and We (2015) proved that f s rreducble, then s wealy rreducble, and for = 2, s rreducble f and only f s wealy rreducble. Gven a tensor =(a 1 ) C, n], f there are λ C and x =(x 1, x 2,, x n ) T C n 0} such that x 1 = λx 1], then λ s called an egenvalue of and x an egenvector of assocated wth λ, where x 1 s an n denson vector whose th coponent s ( x 1 ) = a 2 x 2 x, and 2,, N x 1] =(x 1 1, x 1 2,, x 1 n ) T. If λ and x are all real, then λ s called an H-egenvalue of and x an H-egenvector of assocated wth λ. Ths defnton was ntroduced by Q (2005) where he assued that R, n] s syetrc and s even. Independently, L (2015) gave such a defnton but restrcted x to be a real vector and λ to be a real nuber. Moreover, the spectral radus ρ of the tensor s defned as ρ =axλ:λ σ}, where σ s the spectru of,.e. σ =λ:λ s an egenvalue of } (see Chang et al., 2008; Yang & Yang, 2010). Let =(a 1 ) R, n]. s called a -tensor, f all of ts off-dagonal entres are non-postve, whch s equvalent to wrte = s, where s > 0 and s a nonnegatve tensor. A -tensor = s s an -tensor f s >ρ. Here, we denote by τ the nal value of the real part of all egenvalues of an -tensor, and note that f s a wealy rreducble -tensor, then τ > 0 s the unque egenvalue wth a postve egenvector; for detals, see Zhang et al. (2014) and Dng, Q, and We (2013). Page 2 of 13

3 Gven an even-order syetrc tensor =(a 1 ) R, n], the postve (se-)defnteness of s deterned by the sgn of ts sallest H-egenvalue, that s, f the sallest H-egenvalue s postve (nonnegatve), then s postve (se-)defnte. However, when and n are very large, t s not easy to copute the sallest H-egenvalue of. Then we can try to gve a set n the coplex whch ncludes all egenvalues of. If ths set s n the rght-half coplex plane, then we can conclude that the sallest H-egenvalue s postve, consequently, s postve defnte; for detals, see Q (2005), L, L, and Kong (2014), L and L (2016), L, Jao, and L (2016), L, Chen, and L (2015) and Huang, Wang, Xu, and Cu (2016). Therefore, one of the an as of ths paper s to gve a new egenvalue ncluson set for tensors, and use ths set to deterne postve (se-)defnteness of tensors. In Q (2005) generalzed Gerŝgorn egenvalue ncluson theore fro atrces to real supersyetrc tensors, whch can be easly extended to general tensors (L et al., 2014; Yang & Yang, 2010). Theore 1.1 (Q, 2005, Theore 6) Let =(a 1 ) C, n]. Then σ Γ = Γ, where Γ =z C:z r }, r = δ 2 =0 a 2. To get tghter egenvalue ncluson sets than Γ, L et al. (2014) extended the Brauer s egenvalue localzaton set of atrces (Varga, 2004) and proposed the followng Brauer-type egenvalue localzaton sets for tensors. Theore 1.2 (L et al., 2014, Theore 2.1) Let =(a 1 ) C, n]. Then σ = where,, j,, j =z C:(z r j )z a a r }, j j j j j r j = a 2 = r a j j. δ 2 = 0, δ j2 = 0 One of any applcatons of egenvalue ncluson sets s to bound the nu H-egenvalue of -tensors (He & Huang, 2014; Huang et al., 2016; Wang & We, 2015; Zhao & Sang, 2016). In He and Huang (2014) provded soe nequaltes on τ for an rreducble -tensor as follows. Theore 1.3 (He & Huang, 2014, Theore 2.1) Then Let =(a 1 ) R, n] be an rreducble -tensor. τ n a where R =, and n R 2,, N a 2. τ ax R, For the wealy rreducble -tensor, Wang and We (2015) obtaned the followng results on τ. Theore 1.4 (Wang and We, 2015, Lea 4.4) τ n a }. Let be a wealy rreducble -tensor. Then Page 3 of 13

4 In ths paper, we contnue ths research on the egenvalue ncluson sets for tensors and ts applcatons. We obtan a new egenvalue ncluson set for tensors and prove that the new set s tghter than Theores 1.1 and 1.2. As applcatons, we establsh a suffcent condton for the postve (se-)defnteness of tensors and gve new lower and upper bounds of the nu H-egenvalue for -tensors, whch are the correcton of Theore 4.5 n Wang and We (2015). 2. A new egenvalue ncluson set for tensors In ths secton, we propose a new egenvalue ncluson set for tensors and establsh the coparsons between ths new set wth those n Theores 1.1 and 1.2. Theore 2.1 Let =(a 1 ) C, n]. Then σ Ω =, Ω, j, where j Ω, j = z C: ( z ) } z r r j, r = a, r =r r. Proof For any λ σ, let x =(x 1,, x n ) T C n 0} be an egenvector correspondng to λ,.e. x 1 = λx 1]. (1) Let x p x q axx : N, p, q} (where the last ter above s defned to be zero f n = 2). Then, x p > 0. Fro (1), we have (λ a p p )x 1 p = δ p2 =0, ( 2 3 ) (jj j) a p2 x 2 x + j p a pj j x 1 j. Tang odulus n the above equaton and usng the trangle nequalty gve λ a p p x p 1 δ p2 =0, a p2 x 2 x + j p a pj j x j 1 ( 2 3 ) (jj j) δ p2 =0, ( ) (jj j) 2 3 a p2 x p 1 + a pj j x q 1 j p = r p x p 1 + r p x q 1, equvalently, (λ a p p )x p 1 r p x q 1. (2) If x q = 0, by x p > 0, we have λ a p p 0. Then for any j p, (λ a p p )λ 0 r p r j, whch ples that λ Ω p, j Ω. Otherwse, x q > 0. Slarly, fro (1), we can obtan λ a q q x q 1 r q x p 1. (3) Page 4 of 13

5 Multplyng (2) wth (3) and notng that x p 1 x q 1 > 0, we have (λ a p p )λ a q q r p r q, and λ Ω p, q Ω. Hence, σ Ω. Next, a coparson theore s gven for Theores 1.1, 1.2 and 2.1. Theore 2.2 Let =(a 1 ) C, n]. Then Ω Γ. (4) Proof Accordng to Theore 2.3 n L et al. (2014), Γ. Hence t suffces to show that Ω. Let z Ω, then there exst p, q N, p q such that z Ω p, q,.e. ( )z a q q r p r q. (5) The followng proof wll be dvded nto two cases accordng to a certan rule. (I) Suppose that r p r q =0. Then r p =0 or r q =0. () If r p =0, then a pq q = 0, r p =r q =r p p, and ( r q p )z a q q =( )z a q q r p r q =0 a pq q r q, whch ples that z p, q, consequently, Ω. () If r q =0, by r q r p p, we have ( r q p )z a q q ( )z a q q r p r q =0 a pq q r q, We can also have z p, q and Ω. (II) Suppose that r p r q > 0. Dvdng (5) by r p r q, we have z a q q 1, r p r q (6) whch ples 1 r p (7) or z a q q 1. r q (8) Let a =, b = r p, c = r p a pq q, d = a pq q. () When (7) holds and a pq q > 0, then fro Lea 2.2 n L and L (2016) and (6), we have r q p z a q q a pq q r q z a r p p p z a q q 1. r p r q Page 5 of 13

6 Furtherore, we have ( r q p )z a q q a pq q r q, whch ples Ω. When (7) holds and a pq q = 0, then r p + r p =r q p +a pq q,.e. r q p 0 = a pq q. Obvously, ( r q p )z a q q 0 = a pq q r q, whch also ples Ω. () When (8) holds, we only need to prove Ω under the condton of r p > 1,.e. r p > 1. If a qp p > 0, then fro Leas 2.2 and 2.3 n L and L (2016) and (6), we have z a q q r p q r p a qp p z a r p p p z a q q 1, r p r q whch leads to (z a q q r p q ) a qp p r p. Ths ples z Ω q, p Ω. Furtherore, Ω. If a qp p = 0, by (8), we can have z a q q r p q 0 = a qp p. Then (z a q q r p q ) 0 = a qp p r p, whch also ples Ω. Ths proof s copleted. In the followng, a nuercal exaple s gven to verfy Theore 2.2. Exaple 2.1 Let R 4, 2] wth entres be defned as follows: a 1111 = 14, a 1222 = 6, a 1333 = 9, a 2111 = 2, a 2222 = 15, a 2333 = 8, a 3111 = 3, a 3222 = 5, a 3333 = 17, Page 6 of 13

7 Fgure 1. Coparsons of Γ, and Ω and other a jl = 0. The egenvalue ncluson sets Γ,, Ω and the exact egenvalues are drawn n Fgure 1, where Γ, and Ω are represented by green boundary, blue boundary, and red boundary, respectvely. The exact egenvalues are plotted by blac +". It s easy to see σ Ω Γ,.e. Ω can capture all egenvalues of ore precsely than and Γ. 3. Deternng the postve defnteness for an even-order real syetrc tensor As shown n Q (2005), L et al. (2014), L and L 2016, L et al. (2016), L et al. (2015) and Huang et al. (2016), an egenvalue localzaton set can provde a suffcent condton for the postve defnteness and postve se-defnteness of tensors. As applcatons of the results n Secton 2, we n ths secton provde soe suffcent condtons for the postve defnteness and postve se-defnteness of tensors, respectvely. Theore 3.1 Let =(a 1 ) R, n] be an even-order syetrc tensor wth a > 0 for all N. If for any, j N, j, ( a ) a j j > r r j. then s postve defnte. Proof Let λ be an H-egenvalue of. Suppose that λ 0. By Theore 2.1, we have λ Ω, that s, there are soe, j N, j such that ( λ a ) λ r r j. Fro a > 0, N, we have ( λ a ) λ ( ) a j j > r r j. Ths s a contradcton. Hence, λ >0, and s postve defnte. The concluson follows. Slar to the proof of Theore 3.1, we can easly obtan the followng concluson: Let =(a 1 ) R, n] be an even-order syetrc tensor wth a 0 for all N. If for any, j N, j, Page 7 of 13

8 ( a ) a j j r r j. then s postve se-defnte. 4. New bounds for the nu egenvalue of -tensors In ths secton, new lower and upper bounds for the nu H-egenvalue of -tensors are gven, whch are the correcton and generalzaton of Theore 4.5 n Wang and We (2015). THEOREM 4.1 Let R, n] be a wealy rreducble -tensor. Then n τ ax, (9) where = 1 2 ( ) 2 2 r j. Proof Because τ s an egenvalue of, fro Theore 2.1, there are, j N, j, such that (τ )(τ a j j ) r r j. Fro Theore 2.1, we can get ( τ r )(a j j τ) r r j, equvalently, τ 2 ( )τ+a j j ( ) r r j 0. (10) Solvng for τ gves τ 1 2 = 1 2 n ( ) 2 2 4(a j j ( ) r r j ) ( ) 2 2 r j ] } 1 1 a 2 ( ) 2 2 r j. Next, we prove that the second nequalty n (9) holds. Suppose that x =(x 1,, x n ) T > 0 s an egenvalue of correspondng to τ,.e. x 1 = τx 1], and (11) x l x u nx : N, l, u}. Fro (11), we have (a u u τ)x 1 u = and δ u2 =0 a u2 x 2 x r u x 1 l, (12) Page 8 of 13

9 (a l l τ)x 1 l = δ =0 l2 ( ) (jj j) 2 3 a l2 x 2 x j l a lj j x 1 j r l x 1 l + r l x 1 u,.e. (a l l τ r l )x 1 l r l x 1 u. (13) Multplyng (12) wth (13) and notng that x l 1 x u 1 > 0, we have (a u u τ)(a l l τ r l ) r l r u. (14) Then solvng for τ(a) gves τ 1 2 ax a l l + a u u r l (a l l a u u r l ) r l r u ] } 1 1 a 2 ( ) 2 2 r j. Ths proof s copleted. Rear 4.1 Note that W j. Hence, the bounds (9) n Theore 4.1 are slghtly dfferent fro the bounds n Theore 4.5 of Wang and We (2015). In fact, the bounds (9) are the correcton of the bounds n Theore 4.5 of Wang and We (2015). Because the left (rght) nequalty of (4.2) n Theore 4.5 of Wang and We (2015) obtaned by solvng for τ(a) fro nequalty (4.2) ((14), respectvely); for detals, see the proof of Theore 4.5 n Wang and We (2015). However, solvng for τ by nequaltes (10) and (14) gves the bounds (9). In the followng, a counterexaple s gven to show that the result n Theore 4.5 n Wang and We (2015) s false. Consder the tensor =(a jl ) of order 4 denson 2 wth entres defned as follows: (:, :, 1, 1) =(30 2 2), (:, :, 2, 1) = ( 3 1 3), (:, :, 1, 2) =( 2 3 2), (:, :, 2, 2) =( ). By Theore 4.5 n Wang and We (2015), we have 11 τ 11. By Theore 2.1, we have τ In fact, τ = Next, we can extend the results of Theore 4.1 to a ore general case. Theore 4.2 Let R, n] be an -tensor. Then n τ ax. (15) Proof Because s an -tensor, by Theores 2 and 3 n Dng et al. (2013), there s x =(x 1,, x n ) T 0 such that x 1 > 0. Let δ = n x 1 }=n 2,, N a 2 x 2 x, x ax = ax x. Then x ax > 0. Let = 1, where = 1, 2,, and denote the tensor wth every entry beng 1. Then s an rreducble -tensor, and } s a onotoncally ncreasng sequence. Tang > n 1 x 1 ax δ ] + 1, then for any N, Page 9 of 13

10 2,, N ( a 2 1 ) x 2 x n 2,, N = δ n 1 x 1 ax a 2 x 2 > 0, x n 1 x 1 ax whch ples that x 1 > 0. Then, by Theores 2 and 3 n Dng et al. (2013), we can conclude that s an rreducble -tensor. By Theore 4.1 n He and Huang (2014), τ )} s a onotoncally ncreasng sequence wth upper bound τ, so τ ) has a lt, and let l τ( + )=λ τ. (16) By Theore 2.6 n Wang and We (2015), we see that τ ) s the egenvalue of wth a postve egenvector y (),.e. (y () ) 1 = τ )(y () ) 1]. As hoogeneous ultvarable polynoals, we can restrct y () on the unt ball y () = 1. Then y () } s a bounded sequence, so t has a convergent subsequence. Wthout loss of generalty, suppose that t s the sequence tself. Let y () y as +, we get y 0 and y = 1. Lettng +, we have y = λy 1] fro (y () ) 1 = τ )(y () ) 1]. So λ s an egenvalue of, furtherore, λ τ. Together wth (16) results n λ = τ, whch eans that l τ( )=τ. + Usng Theore 4.1 for, we have n ) τ ) ax ), (17) where )= r ( ) ( )) 2 2 )r j ), r )=r + n 1, r ( )= r + n 1, r )= r + n 1 (n 1). Lettng + n (17), we have that (15) holds. Slar to the proof of Theore 4.2, we can extend the results of Theore 1.3 to a ore general case. Theore 4.3 Let be an -tensor. Then τ n a, and n R τ ax R. Next, we copare the bounds n Theore 4.2 wth those n Theore 4.3. Theore 4.4 Let =(a 1 ) R, n] be an -tensor. Then n R n ax ax R. (18) Proof Slar to the proof of Theore 5 n Zhao and Sang (2016), we can obtan n easly. Next, we only prove that the last nequalty n (18) holds. (I) For any, j N, j, f R R j,.e. r a j j r j, then R n r +r j. Hence, Page 10 of 13

11 ] 2 r j ] r j ]r j = ] ]r j +4r j ] 2 = +2r j ] 2. When +2r j > 0, we have ( ) 2 r j ] r j ] = 2a j j 2r j = 2R j. And when +2r j 0,.e. a jj j 2r j, we have ( ) 2 r j ] 1 2 ( ) 2 ] 1 2 = = + ] = 2 2 r 2a j j 4r j 2a j j 2r j = 2R j. Therefore, = 1 2 whch ples ( ) 2 2 r j R j, ax ax R. j N j (II) For any, j N, j, f R j R,.e. a j j r j r, then r j a j j + r + r. Page 11 of 13

12 Slarly, we can obtan = 1 a 2 ( ) 2 2 r j R, whch ples ax,j N ax R. The concluson follows fro I and II. 5. Concluson In ths paper, a new egenvalue localzaton set for tensors s gven. It s proved that the new set s tghter than those n Q (2005) and L et al. (2014). As applcatons of the obtaned results, a new suffcent condton of the postve (se-)defnteness for an even-order real syetrc tensor, and new lower and upper bounds of the nu egenvalue for -tensors, whch are the correcton of the bounds n Wang and We (2015), are obtaned. Fnally, we extend Lea 4.4 n Wang and We (2015) and Theore 2.1 n He and Huang (2014) to a ore general case. Authors contrbutons All authors contrbuted equally to ths wor. All authors read and approved the fnal anuscrpt. Fundng Ths wor s supported by Natonal Natural Scence Foundatons of Chna grant nuber ], Foundaton of Guzhou Scence and Technology Departent grant nuber 2015]2073] and Natural Scence Progras of Educaton Departent of Guzhou Provnce grant nuber 2016]066]. Author detals Cal Sang 1 E-al: sangcl@126.co ORCID ID: Janxng Zhao 1 E-al: zjx810204@163.co, zhaojanxng@gzu.edu.cn ORCID ID: 1 College of Data Scence and Inforaton Engneerng, Guzhou Mnzu Unversty, Guyang , P.R. Chna. Ctaton nforaton Cte ths artcle as: A new egenvalue ncluson set for tensors wth ts applcatons, Cal Sang & Janxng Zhao, Cogent Matheatcs (2017), 4: References Chang, K. Q., Zhang, T., & Pearson, K. (2008). Perron-Frobenus theore for nonnegatve tensors. Councatons n Matheatcal Scences, 6, Dng, W. Y., Q, L. Q., & We, Y. M. (2013). M-tensors and nonsngular M-tensors. Lnear Algebra and ts Applcatons, 439, Fredland, S., Gaubert, S., & Han, L. (2013). Perron-Frobenus theore for nonnegatve ultlnear fors and extensons. Lnear Algebra and ts Applcatons, 438, He, J., & Huang, T. Z. (2014). Inequaltes for M-tensors. Journal of Inequaltes and Applcatons, 2014, 114. Huang, Z. G., Wang, L. G., Xu, Z., & Cu, J. J. (2016). a new S-type egenvalue ncluson set for tensors and ts applcatons. Journal of Inequaltes and Applcatons, 2016, 254. L, C. Q., Chen, Z., & L, Y. T. (2015). A new egenvalue ncluson set for tensors and ts applcatons. Lnear Algebra and ts Applcatons, 481, L, C. Q., Jao, A. Q., & L, Y. T. (2016). An S-type egenvalue locaton set for tensors. Lnear Algebra and ts Applcatons, 493, L, C. Q., & L, Y. T. (2016). An egenvalue localzato set for tensor wth applcatons to deterne the postve (se-) defntenss of tensors. Lnear Multlnear Algebra, 64, L, C. Q., L, Y. T., & Kong, X. (2014). New egenvalue ncluson sets for tensors. Nuercal Lnear Algebra wth Applcatons, 21, L, L. H. (2015). Sngular values and egenvalues of tensors: A varatonal approach. Proceedngs of the IEEE Internatonal Worshop on Coputatonal Advances n Mult-Sensor Adaptve Processng (CAMSAP05), 1, Q, L. Q. (2005). Egenvalues of a real supersyetrc tensor. Journal of Sybolc Coputaton, 40, Varga, R. S. (2004). Geršgorn and hs crcles. Berln Hedelberg: Sprnger-Verlag. Wang, X. Z., & We, Y. M. (2015). Bounds for egenvalues of nonsngular H-tensor. Electronc Journal of Lnear Algebra, 29, Yang, Y. N., & Yang, Q. Z. (2010). Further results for Perron- Frobenus Theore for nonnegatve tensors. SIAM Journal on Matrx Analyss and Applcatons, 31, Zhang, L. P., Q, L. Q., & Zhou, G. L. (2014). M-tensors and soe applcatons. SIAM Journal on Matrx Analyss and Applcatons, 35, Zhao, J. X., & Sang, C. L. (2016). Two new lower bounds for the nu egenvalue of M-tensors. Journal of Inequaltes and Applcatons, 2016, 268. Page 12 of 13

13 2017 The Author(s). Ths open access artcle s dstrbuted under a Creatve Coons Attrbuton (CC-BY) 4.0 lcense. You are free to: Share copy and redstrbute the ateral n any edu or forat Adapt rex, transfor, and buld upon the ateral for any purpose, even coercally. The lcensor cannot revoe these freedos as long as you follow the lcense ters. Under the followng ters: Attrbuton You ust gve approprate credt, provde a ln to the lcense, and ndcate f changes were ade. You ay do so n any reasonable anner, but not n any way that suggests the lcensor endorses you or your use. No addtonal restrctons You ay not apply legal ters or technologcal easures that legally restrct others fro dong anythng the lcense perts. Cogent Matheatcs (ISSN: ) s publshed by Cogent OA, part of Taylor & Francs Group. Publshng wth Cogent OA ensures: Iedate, unversal access to your artcle on publcaton Hgh vsblty and dscoverablty va the Cogent OA webste as well as Taylor & Francs Onlne Download and ctaton statstcs for your artcle Rapd onlne publcaton Input fro, and dalog wth, expert edtors and edtoral boards Retenton of full copyrght of your artcle Guaranteed legacy preservaton of your artcle Dscounts and wavers for authors n developng regons Subt your anuscrpt to a Cogent OA journal at Page 13 of 13