Axiomatic Quantification of Co-authors Relative Contributions

1 Axiomatic Quantification of Co-authors’ Relative Contributions Ge Wang 1 and Ji anshe ng Ya ng 2 1 VT-WFU School of Biomedical Engineering, Virginia Tech, Blacksburg, VA 2 4061, USA 2 LMAM, School of Mathema tics, Peking University, Beijing, 100871, P.R . China Over the past decades, the competition for academic resource s has gradually intensified, an d worsened with the current financial crisis. To optim ize the resou rce allo catio n, in dividu ali zed assessment of research results is being actively studied but the current indices, s uch as the number of papers, the number of citations, the h - factor and its va riants have limitations, especially their inability of dete rmining co-authors’ credit shar es fai rly. H ere w e estab li sh an axi oma tic system and quantify co-aut hors’ relative contributions. Our methodology av oids subjective assignment of co-authors’ credit s using the inflated, fractional or harmonic methods , and provides a quantitativ e tool for scientific management such as f unding and tenure decisions. Citation analysis | impa ct | co-authors’ relative contributions | axiomatic approach \body Introduction Because the number of publications and the number of co-authors have been rapidly increasing annually [1], there is a critical and immed iate need for individualized assessment of scientific productivity and impact [2-8]. A recent topic in bibliometrics is the use and extension of t he h -index (defined as the ma ximum h if h of a researcher’s papers have at least h citation s each) [4- 5] for measurement of his or her a cademic calibre. While the idea is insightful and widely used [9-13], the h -index is quite rough by definition [14] and subject to various biases [15-24]. A major obstacle to significant improvement of the h -index and other popular indices of thi s type has b een t he lack of a so und mech anis m for assessme nt o f co-autho rs’ individu al contribu tions [23, 25]. Current perception of a researcher’s qualification relies, to a great degree, on either inflated or fractional counting methods [26-27]. Wh ile the f ormer method gives the f ull c redit to an y co- auth or (f or ex ampl e, i t is only stated in a biography how many papers are published), the latter method distributes an equally divided recognition to each co-a uthor (as in so me bibli ome tri c anal yses ). N eith er of thes e met hods is ideal, because the order or r ank of co-authors and the corresponding authorship are almos t exclusively used to indicate co-authors’ r elative contributions. Generally speaking, the further down the list of co -authors for a publication, the less credit he or she receives. Often times, the first author and the corresponding author are considered the most p rominent. Now and then, a number of co-authors claim equal contribu tions. To quantify co-authors’ relative contributions, the harmonic counting me thod was proposed [27] in order to avoid the equal-share bias of the fractional counting method (a less sophisticated variant was also suggested [8]). While the harmonic counting method does permi t equ al ra nkin gs for s ubset s of co-aut hor s, witho ut loss of gen erali ty l et us ass ume th at th e orde r of co -aut hors is c onsis ten t with t hei r cr edit ranki ng, and that there are totally n co-aut hors on a public ati on whose shares are presented as a vector 12 (, , , ) n x xx x = G " ( 1 in ≤≤ ). Then, the k -th author’s harmonic credit k x is def ine d as 1 k x k α = , where 1 1 1 n j j α = = ∑ , 1 kn ≤≤ . (1) Despite its superiority to the fractional metho d, the harmonic method has not been practically used, due to its s ubjec t ive n atur e. E vide ntly , t here i s n o rati on ale behind the proportionality that the k- th author contributes 1/ k as much a s th e firs t aut hor’ s contribution. Realistically, there are ma ny possible ratios between the k -th and the first authors’ credits, which may be equal or ma y be rather small such as in the case s of d ata sharin g or tech nical assistance . Rigorous quantification of co-aut hors’ credits is a long overdue task. The Higher Education Funding Council for England (HEFCE) recently proposed the peer- review system “ Research Excellence Framework (REF) ” that will extensively utilize citation analyses (http://www.nature.com/news/2009/090923/full/new s.2 009.933.html). Nevertheless, HEFCE has ad mitted that bi bli omet rics is not " sufficiently ro bust " for assessment of research quality. Thus, it c ould be prone to misconducts if those bibliometric measures are administratively us ed for funding and t enure deci sions . Fo r exam ple, a popul ar Chi nes e we b foru m “ New Threads ” (http://www.xys.org/new.html) discusse d some cases in which th e numbe r of publications, the number of co-au thors, and even the h -indices were purposely manipulated and effectively inflated. In the USA, the National Institutes of Health recently adopted the enhanced review criteria (http://gran ts.nih.gov/grants/guide/no tice-files/NOT- OD-09-024.html), with mandatory quantification of an investigator’s qualification on a 9-point scale (revised from the initially plan ned 7- point sca le). How ever, the scoring has been largely subjective, still accommodating a substantial level of peer-review nois e. Results and Discussion Here we propose to use the axiomatic approach for quantification of co-au thors’ relative contributions. Assume that a publication has a total of n co-a utho rs who can be divided into m groups ( nm ≥ ) and that i c co-authors in the i -th group have t he same credit 2 12 ( , , , ) im x xx x x ∈= G " ( 1 im ≤≤ ). We postulate the follow ing axioms: Axiom 1 (Ranking Preference): 12 0 m xx x ≥≥ ≥ > " ; Axiom 2 (Credit Normalization) : 11 2 2 1 mm cx c x c x ++ = " ; Axiom 3 (Maximum Entropy): x G is uniformly dist ribu te d in t he do mai n d efin ed by Ax ioms 1 an d 2. The first axiom reflects the ranking process of co- authors’ relative con tributions, which happens during the production of a publication. In most cases, such a ranking determines the order of co- authors. More eff orts bey ond t his ran king to speci fy co -aut hors ’ credits may well be too complicated, highly controversial, and thus impractical and counter- productive. While a co-authors’ c ontribution statement has been encouraged by s ome journals, often times it cannot be directly translated into co-authors’ credit shares and disappears in the bibliometric measurement. Hence, we suggest that a ranking code be added to ea ch publication as shown in Figure 1, which will be the basis for further analysis. This straightforward ranking code is immediately superior to the inflated and fractional counting me thods, since it clearl y re pre sent s re lati ve im port ance of c o-aut hors ’ essential intelle ctual and techn ical contributions from their peers’ perspectives, and suppresses artifacts in terms of insignificant co-authors, un-qualified corresponding authors, and confusing weights associated with some particular co-authors’ positions on a pu blic atio n [28] . The second axiom ensures that the quantification of co-authors’ contributions is in a rela tive sense. The absolute value of a publica tion should be estimated independently, which can be the impact factor of a journal initially and the number of citations or its variants subsequently. The last axiom recognizes the impossibility of specifying exact relative contributions of co-authors on each and every publication, thereby asserting that all the cases permitted by Axioms 1 and 2 a re equally likel y, s ince there i s no gr ound for as sumi ng ot her wis e in the f iel ds of s cienc e and tech nolo gy as a whole . A co-author may have done his or her ultra best for academic excellence or may have only met a minimum requirement, and any scenario in between is quite poss ible. As in m any ar eas i nvol ving i nform ati on theoretic inference, the maximum e ntropy principle [29] in this bibliometric context requires that the distribution of the credit vector be uniform across the permissible domain. Nevertheless, in a specific area we could have m ore i nform at ion or a str ong er ass umpt ion. In such a ca se, our ge neric axio matic system can be adapted to ma ke use of available knowledge without any theoretical difficulty. Therefore, the fairest estimation of co-authors’ credit shares can be formulated as the expectation of all possibl e credit vector s. In other word s, the k- th set of co-a uthor s’ ind ivi dual cred it s hould be the e leme nta l mean, which is re ferred to as the a -index for its axio mati c f oundat io n and we hav e pro ved t o be 1 11 () m k j jk i i Ex m c = = = ⎛⎞ ⎜⎟ ⎝⎠ ∑ ∑ , 1 km ≤≤ . (2) It can be verified that 1 1 () 1 m kk k cE x m − = ∑ . In the spe cial case of unequal-contribution co-authors (no equal contributions are claimed by any sub-group of these co-authors), Eq. (2 ) becomes 11 () n k jk Ex n j = = ∑ , 1 kn ≤≤ , (3) as com puted in Tabl e 1 for n up to 10. Our axiomatic characte rization is significa ntly different from the existing credit countin g methods. As shown in Figure 2, the fractional measures are too rough compared to the harmonic and axiomatic mea sures. As far as the harmonic and axiomatic measures are concerned, the axiomatic method promo tes the first author’s share and dilutes the last author’s we ight more than the harmonic method doe s. It is interesting to note that this “Mathew effect” is not o nly generally desir able but als o axio mati call y just ifie d. Conclusion We anticipate our axiomatic system to be come a basis for development of academic assessment or peer- review systems [23]. It is hoped that our methodolo gy will be adopted by academic institutions and funding agencies, and help improve identification of productive and i nflue nti al inv esti gat ors an d ins tit uti ons. Furthermore, our w ork might be relevant in psychological, social and other contexts in wh ich ranking is fundamentally involved, such as subjective choice s and fuzzy re asoning . Materials and Methods Mathematically, our axiomatic quantification problem is to co mpute not onl y t he elem ent al m ean () k Ex as a co-autho r’s credit share but also the co rresponding standard deviation () k x σ ( 1 km ≤≤ ) for statistical testing. The formulas for the co-authors’ contributions and the corresponding standard deviations can be derived using either an algebraic or geometric appro ach . The deri vat ion pr oce sses are qu ite technical, and given in the SI text using the algebraic approach, leading to Eqs. (2) and (3) presented above. Acknowledgements : The authors thank Dr. Zhenjie Lin for discussion and help with implementing a Monte- Carlo algorithm for evaluation of co-authors’ credits in this axiomatic framework and Dr. Michae l Vannier for advice on the implication and refine ment of this work. 3 Refer enc es 1. Greene, M., The demise of the lone author. Nature, 2 007. 450(7173): p. 1165. 2. Foulkes, W. and N. Neylon, Redefining authorship. Relative contribution should be given after each author's name. BMJ, 1996. 312(7043): p. 1423. 3. Campbell, P., Policy on papers' contributors. Nature, 1999. 399(6735): p. 393. 4. Hirs ch, J.E ., An i ndex t o quant ify an indi vid ual' s sci enti fic resea rch output . P roc N atl Ac ad Sc i U S A, 2005. 102(46): p. 16569-72. 5. Hirsch, J.E., Does the H index have predict ive power? Proc Natl Acad Sci U S A, 2007. 104(49): p. 19193-8. 6. Anonymous, Who is accountable? Nature, 2007. 450(7166): p. 1. 7. Ball, P., A longer paper gathers more citations. 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Proceedings of the IEEE, 1982. 70(9): p. 939-952. 4 Figure Legends Fig. 1. Ranking code after the key words to remove any ambiguity in ranking co-authors. Fig. 2. Comparative visualization of co-authors’ rela tive contributions according to (A) the fraction al, (B) harmonic and (C) axiomatic measures respectively, for the nu mber of co-authors up to n=5. Table Legend Table 1. Axiomatic indices ( a -indices) for up to 10 unequal-contribution co-authors. Note that the sum of the rounding errors has been added to the first author’s share for n =4, 5, 8, 9 and 1 0 res pec tiv ely. Ranking code: 1, 2, 3, 3, 2 Fig. 1. Ranking code after the key words to remove any ambiguity in ranking co-autho rs. A n=1 n=2 n=3 n=4 n=5 A B C Fig. 2. Comparative visualizatio n of co-auth ors’ relative contribut ions according to (A) the fractional, (B) harmonic and (C) axiomatic measures respectively for the number of co authors up to n=5 Color Coding 5 th Author 1 st Author 2 nd Author 3 rd Author 4 th Author (B) harmonic and (C) axiomatic measures respectively , for the number of co - authors up to n=5 . 1 Table 1. Axiomatic indices ( a -indices) for up to 10 unequal-contribution co-authors. Note that the sum of the rounding errors has been added to the first author’s share for n =4, 5, 8, 9 an d 10 res pec tiv ely . Co-authors’ relative contributions 1 1.0000 2 0.7500 0.2500 3 0.6111 0.2778 0.1 111 4 0.5209 0.2708 0.1 458 0.0625 5 0.4566 0.2567 0.1 567 0.0900 0.0400 6 0.4083 0.2417 0.1 583 0.1028 0.0611 0.0278 7 0.3704 0.2276 0.1 561 0.1085 0.0728 0.0442 0.0204 8 0.3398 0.2147 0.1 522 0.1106 0.0793 0.0543 0.0335 0.0156 9 0.3145 0.2032 0.1 477 0.1106 0.0828 0.0606 0.0421 0.0262 0.0123 10 0.2928 0.1929 0.1429 0 .1096 0. 0846 0.0646 0.0479 0.0336 0.0211 0.0100 1 SI Methodology: Derivation of a-ind ices and associated deviations Ge Wang 1 and Jiansheng Yang 2 1 VT-WFU School of Biomedical Eng ineering, Virginia Tech, Blacksburg, VA 24 061, USA 2 LMAM, School of Mathematics, Peking University, Beijing, 100871, P .R. China As described in the main text, there are totally n co-authors on a public ation who can be divided into m groups ( nm ≥ ), and i c co-authors in the i -th group have the same credit 12 ( , , , ) im x xx x x ∈= G " ( 1 im ≤≤ ). Our axiomatic system co nsists of Axiom 1 (Ranking Prefere nce): 12 0 m xx x ≥≥ ≥ > " ; Axiom 2 (Credit Normalizat ion): 11 2 2 1 mm cx c x c x ++ = " ; Axiom 3 (Maximum Entropy): x G is unif orml y dis tri bute d in th e doma in def ine d by A xioms 1 and 2. Then, our problem is to compute no t only the elemental mean () k Ex as a co-author’s credit share but also the corresponding standard deviation () k x σ ( 1 km ≤≤ ) for stat istic al t estin g. For v is ualiz ati on of the key idea , the 3D case is illustra ted in Figu re S1. Figu re S1 . Domai n p ermit t ed by t he axi oma tic syst em i n the c ase of n= 3, wh ere t he dis tr ibut ion o f co-a uthor s’ c redit shar es is pos tul ated t o be t he mas s cen ter of the soli d red trian gle. Since 22 () ( ) () kk k x Ex Ex σ =− , we w ill need to find 2 () k E x ( 1 km ≤≤ ). For convenience of the induction to be used below, let us denote () k Ex and 2 () k E x ( 1 km ≤≤ ) by , mk R and , mk S respectiv ely. The sample space of the above problem is 21 2 2 1 1 {( ) ( , ) : 0 1 } m km m m i i i x mxx x x x c x c − = ⎛⎞ Ω= = ≤ ≤ ≤ ≤ ≤ − ⎜⎟ ⎝⎠ ∑ "" . (1) 2 Let () m m M dx m Ω = ∫ , (2) , () m mk k E xd x m Ω = ∫ , 1 km ≤≤ , (3) 2 , () m mk k F xd x m Ω = ∫ , 1 km ≤≤ , (4) wher e 1 2 1 1 1 m ii i x cx c = ⎛⎞ =− ⎜⎟ ⎝⎠ ∑ . (5) Clearly, we have , , mk mk m E R M = , 1 km ≤≤ , (6) , , mk mk m F S M = , 1 km ≤≤ . (7) To determine , mk R and , mk S ( 1 km ≤≤ ) in a recursive fashion, w e introduce the following fun ctions whose utilities will become evident later: 21 2 2 1 1 (, ) {( ) ( , ) : } m mm m m i i i ab x m x x b x x x a c x c − = ⎛⎞ Ω== ≤ ≤ ≤ ≤ ≤ − ⎜⎟ ⎝⎠ ∑ "" , (8) (, ) (, ) ( ) m m ab M ab d x m Ω = ∫ , (9) , (, ) (, ) ( ) m mk k ab Ea b x d x m Ω = ∫ , 1 km ≤≤ , (10) 2 , (, ) (, ) ( ) m mk k ab F ab x d x m Ω = ∫ , 1 km ≤≤ . (11) wher e 1 2 1 1 m ii i x ac x c = ⎛⎞ =− ⎜⎟ ⎝⎠ ∑ , (12) and a and b are c onst ant s wit h 0 a ≥ , 1 m i i ab c = ≥ ∑ . Then, we have the follow ing propositions. Proposition 1 (Equivalency) : 1 (, ) ( , 0 ) m mm i i Ma b Ma b c = =− ∑ ; Proposition 2 (Measurement) : 1 12 12 3 12 (, 0 ) (1 ) ! ( ) ( ) ( ) m m m a Ma mc c c c c c c c − = − + ++ +++ "" ; Proposition 3 (Re duction) : For 1 km ≤≤ , we have 3 1 1 ,, 1 12 12 3 12 (, ) (1 ) ! ( ) ( ) ( ) m m m i m i mk i mk i m ba b c Ea b a b c E mc c c c c c c c − = = ⎛⎞ − ⎜⎟ ⎛⎞ ⎝⎠ =− + ⎜⎟ − + ++ +++ ⎝⎠ ∑ ∑ "" ； 1 2 1 1 ,, , 11 12 123 12 (, ) 2 ; (1 ) ! ( ) ( ) ( ) m m mm i mm i mk i m k i mk ii m ba b c Fa b a b c F b a b c E mc c c c c c c c − + = == ⎛⎞ − ⎜⎟ ⎛⎞ ⎛⎞ ⎝⎠ =− + − + ⎜⎟ ⎜⎟ − + ++ +++ ⎝⎠ ⎝⎠ ∑ ∑∑ "" Proposition 4 (Substitut ion) : 2, 1 2 2,2 1 1 (1 ) R cR c =− , 2 2, 1 2 2,2 2 2,2 2 1 1 (1 2 ) Sc R c S c =− + . Proof of Proposition 1 : Making the variables transform ii yx b =− for 2 im ≤≤ , we have 1 (, ) ( , 0 ) 1 (, ) ( ) ( ) ( , 0 ) m mm i i m mm i ab a b c i Ma b d x m d y m Ma b c = ΩΩ − = == = − ∑ ∑ ∫∫ . (13) Proof of Proposition 2 : Let us proceed by induction w ith respect to m . For 2 m = , we have 12 2 22 (, 0 ) 0 12 (, 0 ) ( 2 ) a cc a a Ma d x d x cc + Ω == = + ∫∫ . (14) That is , the prop osition holds in this case. For 2 m > , by Prop osition 2 and the inductive hypothesis, we have (, 0 ) (, 0 ) ( ) m m a M ad x m Ω = ∫ ( ) 12 1 0( , ) (1 ) m mm m m a cc c m ac x x dx m dx − ++ + Ω− =− ∫∫ " 12 1 1 0( , 0 ) (1 ) m m mm i i a cc c m ax c dx m dx − = ++ + Ω− ⎛⎞ =− ⎜⎟ ⎜⎟ ∑ ⎝⎠ ∫∫ " 12 1 0 1 (, 0 ) m a m cc c mm i m i M ax c d x ++ + − = =− ∑ ∫ " 12 2 1 0 12 12 3 12 1 () (2 ) ! ( ) ( ) ( ) m m m a mi i cc c m m ax c dx mc c c c c c c c − = +++ − − = − + ++ ++ + ∑ ∫ " "" 1 12 12 3 12 (1 ) ! ( ) ( ) ( ) m m a mc c c c c c c c − = − + ++ ++ + "" . (15) Proof of Proposition 3 : For 1 km ≤≤ , we have 4 , (, ) (, ) (, ) ( ) ( ) ( ) mm mk k ab ab E a b x b dx m bdx m ΩΩ =− + ∫∫ (, ) () ( ) ( , ) m km ab x bd x m b M ab Ω =− + ∫ , (16) , (, ) mk Fa b 22 (, ) (, ) (, ) () ( ) 2 () ( ) ( ) mm m kk ab ab ab x b d xm b x b d xm b d xm ΩΩ Ω =− + − + ∫∫ ∫ 22 (, ) (, ) () ( ) 2 () ( ) ( , ) mm kk m ab ab x b d xm b x b d xm b M a b ΩΩ =− + − + ∫∫ . (17) Making th e variable s transform 1 i i m i i x b y ab c = − = − ∑ for 1 im ≤≤ , we have , (, ) 11 ( ) () () m m mm mm ki k i m k ab ii x bd x m a b c yd y m a b c E ΩΩ == ⎛⎞ ⎛ ⎞ −= − = − ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ ∑∑ ∫∫ , (18) 11 22 , (, ) 11 ( ) () () m m mm mm ki k i m k ab ii x bd x m a b c y d y m a b c F ++ ΩΩ == ⎛⎞ ⎛⎞ −= − = − ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ ∫∫ , (19) By Propositions 1 a nd 2 , we have 1 1 12 12 3 12 (, ) (1 ) ! ( ) ( ) ( ) m m i i m m ab c Ma b mc c c c c c c c − = ⎛⎞ − ⎜⎟ ⎝⎠ = − + ++ +++ ∑ "" . (20) Ins erti ng Eqs . (18 ), (19) and (2 0) int o Eq s. (16) an d (1 7) re spec tiv ely, we ob tain 1 1 ,, 1 12 12 3 12 (, ) (1 ) ! ( ) ( ) ( ) m m m i m i mk i mk i m ba b c Ea b a b c E mc c c c c c c c − = = ⎛⎞ − ⎜⎟ ⎛⎞ ⎝⎠ =− + ⎜⎟ −+ + + + + + ⎝⎠ ∑ ∑ "" , (21) 1 ,, , 11 (, ) 2 mm mm mk i mk i mk ii Fa b a b c F b a b c E + == ⎛⎞ ⎛⎞ =− + − + ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ 1 2 1 12 123 12 . (1 ) ! ( ) ( ) ( ) m m i i m ba b c mc c c c c c c c − = ⎛⎞ − ⎜⎟ ⎝⎠ + − + ++ +++ ∑ "" (22) Proof of Proposition 4 : For 2 m = , we have 11 2 2 1 cx c x += , (23) 12 2 1 1 (1 ) x cx c =− . (24) Hence, we have 5 2 , 1 1 22 22 11 11 () ( ( 1 ) ) ( 1 ) R Ex E c x E c x cc == − = − 22 2 2 , 2 11 11 (( 1 ) ( ) ) ( 1 ) E cE x cR cc =− = − , (25) and 222 2 , 1 1 22 22 22 11 11 () ( ( 1 ) ) ( ( 1 ) ) SE x E c x E c x cc == − = − 22 22 2 2 2 1 1 (( 1 ) 2 ( ) ( ) ) Ec E x c E x c =− + 2 22 , 2 2 2 , 2 2 1 1 (1 2 ) cR c S c =− + . (26) Theor em 1: , 1 11 m mk jk j R mc c = = ++ ∑ " , 1 km ≤≤ . Proof: By Propositions 1 and 2 , w e have ( ) 12 1 1 , 0( 1 , ) () ( 1 ) m mm m m m cc c mm m m m cx x E x dx m x dx m dx − ++ + ΩΩ − == − ∫∫ ∫ " 12 1 1 0 (1 , ) m cc c mm m m m m x Mc x x d x ++ + − =− ∫ " 12 1 1 0 1 (1 , 0 ) m m cc c mm m i m i x Mx c d x ++ + − = =− ∑ ∫ " 12 2 1 1 0 12 12 3 12 1 1 (2 ) ! ( ) ( ) ( ) m m m mm i i cc c m m xx c dx mc c c c c c c c − = +++ − ⎛⎞ − ⎜⎟ ⎝⎠ = − + ++ ++ + ∑ ∫ " "" 12 12 123 12 1 () ! ( ) ( ) () mm cc c m cc cc c cc c = ++ + + ++ ++ + "" " . (27) By Proposition 3 and Eq. (27) , for 11 km ≤≤ − we have ( ) 12 11 1 1 1 ,1 (,, ) 0 ( 1 , ,,, ) (, , ) ( ) ( 1 ) m mm m m m m m cc c mk m k k m cc c x x cc E c c x dx m x dx m dx −− ++ + ΩΩ − == − ∫∫ ∫ " "" " 12 1 1, 1 1 0 (1 , , , , ) m cc c mk m m m m m Ec x x c c d x ++ + −− =− ∫ " " 6 12 2 1 1 1 1, 1 2 1 0 1 12 12 3 12 1 (1 ) (1 ) ( , , , ) (2 ) ! ( ) ( ) ( ) m m m mm i m m i cc c mi m k m m i m xx c x cE c c c d x mc c c c c c c c − − = +++ −− = − ⎛ ⎞ − ⎜ ⎟ ⎜ ⎟ =− + −++ + + + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∑ ∑ ∫ " " "" 1, , 12 () mk mm m E E mc c c − =+ ++ + " . (28) By Proposition 2 , we have 12 12 3 12 1 (1 . 0 ) (1 ) ! ( ) ( ) ( ) mm m MM mc c c c c c c c == −+ + + + + + "" . (29) Hence, we have , , 1 1 () mm mm mm E R M mc c == ++ " , (30) and for 11 km ≤≤ − we have ,, 1 , , 12 () mk mm m k mk mm m m EE E R M Mm c c c M − == + +++ " 1, 1 , 12 () mk m mm mm RM R mc c c M −− =+ ++ + " ,1 , 1 mm m k m RR m − − =+ . (31) For 21 km ≤≤ − , repeatedly using Eq. (31) we have ,, 1 , 1 2 , 12 () 1 mk mm m m m k mm RR R R mm −− − −− =+ + − ,1 , 1 2 , 12 mm m m m k mm RR R mm −− − −− =+ + ,1 , 1 2 , 2 , 12 mm m m m m k k mm k RR R R mm m −− − − −− =+ + + + " 11 1 1 2 1 11 1 1 () ( ) ( ) ( ) mm m k mc c mc c mc c mc c −− =+ + + ++ ++ ++ ++ " "" " " 1 11 m jk j mc c = = ++ ∑ " . (32) For 1 k = , repeatedly using Eq. (31) we have , 1 , 1 , 1 2, 2 3, 3 2, 1 12 3 2 mm m m m m m mm RR R R R R mm m m −− − − −− =+ + + + + " 2, 1 3 1 11 2 m j j R mc c m = =+ ++ ∑ " . (33) 7 Inserting 2 2, 1 2 2,2 11 1 2 11 (1 ) (1 ) 2( ) c Rc R cc c c =− =− + into Eq. (33) , we hav e 2 ,1 31 11 1 2 1 11 2 11 (1 ) 2( ) mm m jj jj c R mc cm c c c mc c == =+ − = ++ + ++ ∑∑ "" . (34) Combining Eqs. (30 ), (32) and (34), w e have , 1 11 m mk jk j R mc c = = ++ ∑ " , 1 km ≤≤ . (35) Theor em 2: , 11 21 (1 ) ( ) ( ) mk ki jm ji S m m cc cc ≤≤ ≤ = ++ + + + ∑ "" , 1 km ≤≤ . Proof: By Propositions 1 and 2 , w e have ( ) 12 1 1 22 , 0( 1 , ) () ( 1 ) m mm m m m cc c mm m m m cx x Fx d x m x d x m d x − ++ + ΩΩ − == − ∫∫ ∫ " 12 1 2 1 0 (1 , ) m cc c mm m m m m x Mc x x d x ++ + − =− ∫ " 12 1 2 1 0 1 (1 , 0 ) m m cc c mm m i m i x Mx c d x ++ + − = =− ∑ ∫ " 12 2 2 1 1 0 12 12 3 12 1 1 (2 ) ! ( ) ( ) ( ) m m m mm i i cc c m m xx c dx mc c c c c c c c − = +++ − ⎛⎞ − ⎜⎟ ⎝⎠ = − + ++ ++ + ∑ ∫ " "" 2 12 12 12 3 12 2 () ( 1 ) ! ( ) ( ) () mm cc c m cc cc c cc c = ++ + + + ++ ++ + "" " . (36) For 11 km ≤≤ − , ut ilizing P roposition 3 and Eq. (36), w e have ( ) 12 1 1 22 , 0( 1 , ) () ( 1 ) m mm m m m cc c mk k k m cx x F xd x m xd x m d x − ++ + ΩΩ − == − ∫∫ ∫ " 12 1 1, 0 (1 , ) m cc c mk m m m m Fc x x d x ++ + − =− ∫ " 12 1 1 1, 1, 0 11 12 1 m mm mm cc c mi m k m mi m km ii x cF x x c E d x − ++ + −− == ⎛⎞ ⎛⎞ ⎛⎞ =− + − ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ∑∑ ∫ " 12 2 2 1 1 0 12 123 12 1 1 (2 ) ! ( ) ( ) ( ) m m m mm i i cc c m m xx c dx mc c c c c c c c − = ++ + − ⎛⎞ − ⎜⎟ ⎝⎠ + −++ + + + + ∑ ∫ " "" 1, 1, , 2 11 2 (1 ) ( ) (1 ) ( ) mk mk mm mm FE F mc c m mc c −− =+ + ++ + ++ + "" , (37) 8 Therefore, we have , , 2 1 2 (1 ) ( ) mm mm mm F S M mm c c == ++ + " , (38) and for 11 km ≤≤ − we have ,1 , 1 , , 2 11 2 (1 ) ( ) (1 ) ( ) mm m k m k mk mm m m m FF E S M mc c M m mc c M −− =+ + ++ + ++ + "" ,1 , 1 , 1 12 ( 1 ) 1( 1 ) ( ) m m mk mk m mm SS R mm m c c −− −− =+ + ++ + + " . (39) For 21 km ≤≤ − , repeatedly using (39) we have ,, 1 , 1 , 1 2( 1 ) 1 (1 ) ( ) 1 mk mm m k m k m mm SS R S mm c c m −− −− =+ + ++ + + " ,1 , 1 2( 1 ) (1 ) ( ) mm m k m m SR mm c c − − =+ ++ + " 1, 1 2 , 2 , 11 12 ( 2 ) 2 1( 1 ) ( ) m m mk mk m mm m SR S mm m c c m −− − − − ⎛⎞ −− − ++ + ⎜⎟ +− + + ⎝⎠ " ,1 , 1 1 , 2 , 11 1 12 ( 1 ) 2 ( 2 ) 1( 1 ) ( ) ( 1 ) ( ) m m mm mk m k mm mm m SS R R m m mc c m mc c −− − − − −− − =+ + + ++ + + + + + "" 2, (1 ) ( 2 ) (1 ) mk mm S mm − −− + + ,1 , 1 2 , 2 , 1 ( 1) ( 2 ) ( 1) 1( 1 ) ( 1 ) mm m m m m k k mm m k k SS S S mm m m m −− − − −− − + =+ + + + ++ + " 1, 2 , , 11 1 1 1 2( 1 ) 2( 2) 2 (1 ) ( ) (1 ) ( ) (1 ) ( ) mk m k k k mm k mm k RR R mm c c mm c c mm c c −− −+ −− ++ + + ++ + ++ + ++ + " "" " 2 11 1 21 1 (1 ) ( ) ( ) ( ) m jk ki jm jj i m m cc cc cc =≤ < ≤ ⎛⎞ =+ ⎜⎟ ⎜⎟ ++ + + + + + ⎝⎠ ∑∑ "" " 11 21 (1 ) ( ) ( ) ki jm ji m m cc cc ≤≤ ≤ = ++ + + + ∑ "" . (40) For 1 k = , repeatedly using Eq. (39) we have , 1 , 1 , 1 2, 2 3,3 2, 1 1( 1 ) ( 2 ) 4 3 3 2 1( 1 ) ( 1 ) ( 1 ) mm m m m m m mm m SS S S S S mm m m m m m −− − − −− − × × =+ + + + + ++ + + " 1,1 2 ,1 2 ,1 11 1 1 2 3 2( 1 ) 2( 2) 2 2 (1 ) ( ) (1 ) ( ) (1 ) ( ) mm mm mm RR R m m cc m m cc m m c c c −− − −− × ++ + + ++ + ++ + ++ + " "" 9 2, 1 2 31 11 1 1 1 2 21 1 1 3 (1 ) ( ) ( ) ( ) ( ) m ji j m jj i S m m cc cc cc c c c =≤ < ≤ ⎛⎞ =+ + − ⎜⎟ ⎜⎟ ++ + + + + + + ⎝⎠ ∑∑ "" " . (41) Since 2 2 22 2, 1 2 2, 2 2 2 ,2 22 2 1 1 12 12 22 11 (1 2 ) (1 ) 2( ) 2 3( ) cc Sc R c S c c cc cc =− + =− + +× + 22 22 1 2 2 2 22 2 2 2 2 2 11 1 2 1 1 2 1 1 2 1 1 2 1 () 3 () () 3 () cc c c c c c cc c cc c cc c cc c +− =− + = + ++++ 2 2 22 11 2 1 1 2 1 () 3 () c cc c c c c =+ ++ , (42) we have 22 22 1 1 2 2, 1 22 22 11 2 1 1 2 11 2 1 1 2 2( ) 12 3 () () () () cc c c c S cc c c c c cc c c c c ++ −= += +++ + 22 12 1 22 2 2 11 2 1 1 2 () 11 () () cc c cc c c c c ++ == + ++ . (43) Inserting Eq . (43) into Eq. (41), we have ,1 2 11 11 1 21 1 (1 ) ( ) ( ) ( ) m m ji j m jj i S m m cc cc cc =≤ < ≤ ⎛⎞ =+ ⎜⎟ ⎜⎟ ++ + + + + + ⎝⎠ ∑∑ "" " . (44) Combining Eqs. (36 ), (40) and (44), w e obtain , 11 21 (1 ) ( ) ( ) mk ki jm ji S m m cc cc ≤≤ ≤ = ++ + + + ∑ "" , for 1 km ≤≤ . (45) Finally, w e have Theor em 3: For 1 km ≤≤ , 2 11 1 11 1 2 1 () 1( ) 1 ( ) ( ) m k jk ki jm j ji m x mm c c m c c c c σ =≤ < ≤ − =− ++ + + + + + + ∑∑ "" " . Proof: For 1 km ≤≤ , we have 22 2 ,, () ( ) () kk k m k m k xE x E x S R σ =− = − 2 11 1 11 1 2 1 1( ) 1 ( ) ( ) m jk ki jm j ji m mm c c m c c c c =≤ < ≤ − =− + ++ + ++ ++ ∑∑ "" " . (46) Remark: In the case of mn = , 12 1 m cc c === = " , we have 10 11 () n k jk Ex nj = = ∑ , 2 11 1 2 1 () 11 n k jk ki jn n x nn j n i j σ =≤ < ≤ − =− ++ ⋅ ∑∑ , 1 kn ≤≤ . (47)