Mlysi Jourl of Librry & Iformtio Sciece, Vol. 9, o. 3, 04: 4-49 A geerl theory of miiml icremets for Hirsch-type idices d pplictios to the mthemticl chrcteriztio of Kosmulski-idices L. Egghe Uiversiteit Hsselt (UHsselt), Cmpus Diepebeek, Agorl, B-3590 Diepebeek, BELGIUM Uiversiteit Atwerpe (UA), IBW,Stdscmpus, Veusstrt 35, B-000 Atwerpe, BELGIUM e-mil: leo.egghe@uhsselt.be ABSTRACT For geerl fuctio f ( ) ( =,,... ) chrcterize the first icremet I ( ) ( ) f ( ) f ( ) icremet I ( ) I ( ) I ( ), defiig geerl Hirsch-type idices, we c = s well s the secod =. A pplictio is give by presetig mthemticl chrcteriztios of Kosmulski-idices. Keywords: Icremet; Hirsch-type idex;kosmulski-idex. INTRODUCTION Let us hve set of ppers where the i th pper hs c i cittios. We ssume tht ppers re rrged i decresig order of received cittios (i.e. ci c j if d oly ifi j). The most geerl Hirsch-type idex c be defied s follows. Let f ( ) (,,3,... ) = be geerl icresig fuctio. The the Hirsch-type idex (bsed o f) for this set of ppers d f cittios is the highest rk such tht ll the ppers o rks,..., hve t lest ( ) cittios. Exmples re f ( ) = for the Hirsch-idex (Hirsch (005)), f ( ) = ( > 0) for the geerl Wu-idex (Egghe (0) d Wu (00)for = 0) d f ( ) = ( > 0) for the geerl Kosmulski-idex (Egghe (0) d Kosmulski (006) for = ). Note tht the geerl Wu- d Kosmulski-idices reduce to the Hirsch-idex for =. Permet ddress Pge 4
Egghe, L. Give such fuctio f ( ) ( =,,3,... ) the miimum situtio to hve idex equl to is to hve ppers with exctly f ( ) cittios ech d where the other ppers hve zero cittios. I this cse we hve totl of f ( ) cittios. The miimum situtio to hve idex equl to is ppers ech hvig f ( ) cittios d where the other ppers hve zero cittios. Here we hve ( ) f ( ) cittios i totl, hece icrese of ( ) f ( ) f ( ) cittios. We defie (see lso Egghe (03,b)) the geerl icremet of order s, for every =,,3,... The geerl icremet of order is defied s ( ) ( ) ( ) ( ) I = f f () ( ) ( ) ( ) I = I I () which is equl to, by () ( ) ( ) ( ) ( ) ( ) ( ) I = f f f (3) for ll =,,3,... Exmples (see lso egghe (03, b)):. For the geerl Wu-idex ( f ( ) = ) we hve ( ) ( ) I = (4) ( ) I = (5) for ll. This gives for the Hirsch-idex ( = ) : ( ) I = (6) for ll. I ( ) = (7). For the geerl Kosmulski-idex ( f ( ) for ll. = ) we hve ( ) ( ) I = (8) ( ) ( ) ( ) I = (9) Pge 4
A Geerl Theory of Miiml Icremets for Hirsch-Type Idices 3. For the threshold idex ( f ( ) for ll. = C, costt) we hve I ( ) = C (0) I ( ) = 0 () I Egghe (03, b) we chrcterized the geerl Wu-idex (hece lso the Hirsch-idex) I. I the preset pper we I d ( ) will chrcterize geerl Hirsch-type-idices (give fuctio f ( )) by mes of their first d secod icremets give certi fuctio I ( ) = ϕ ( ) d I ( ) = ψ ( ). This will d the threshold idex usig the icremets ( ) be doe i the ext sectio. I the third sectio we pply this geerl theory to the chrcteriztio of the Kosmulski-idices by mes of the icremets (8) d (9). The pper closes with coclusios d suggestios for further reserch. CHARACTERIZATIONS OF GENERAL HIRSCH-TYPE-INDICES USING THE INCREMENTS I () AND I () Let us hve geerl Hirsch-type-idex defied by geerl icresig fuctio f ( ). For geerl fuctio ϕ ( ) we hve: Theorem : The followig ssertios re equivlet: I = ϕ () (i) ( ) ( ) for ll =,,3,... (ii) f ( ) ϕ f ( ) = for ll =,,3,... (3) Proof: (i) => (ii) Sice I ( ) ϕ ( ) Hece, from (4), =, for ll, we hve by () ( ) f ( ) f ( ) ϕ ( ) = (4) ( ) ϕ f ( ) = f ( ) (5) Pge 43
Egghe, L. Choosig the free prmeter f ( ) > 0 we hece hve from (5) f f ( 3) f ( ) ( ) f ( ) ( ) ϕ = (6) ( ) ϕ ( ) ϕ = (7) 3 3 (usig lso (6)). So we hve show tht (3) is true for =,,3. Complete iductio supposes (3) to be true for d we hve to show (3) for replced by : By (5) d (3) (for ) ( ) f which is (3) for replced by. (ii) => (i) By (3)(pplied to d ) we hve Hece I ( ) ϕ ( ) theorem. ( ) f ϕ ϕ = ϕ f ( ) = f ( ) ( ) = ( ) ( ) ( ) I f f ( ) ϕ ϕ = ( ) f ( ) f ( ) = s is redily see. Hece we proved (i), completig the proof of this Usig the secod icremet yields other chrcteriztio of geerl Hirsch-type-idices. ψ we hve: For geerl fuctio ( ) Theorem : The followig ssertios re equivlet: (i) I ( ) ψ ( ) = (8) for ll =,,3,... Pge 44
A Geerl Theory of Miiml Icremets for Hirsch-Type Idices (ii) f ( ) = ( ) f ( ) ( ) f ( ) ( i ) ψ Proof: (i) => (ii) for ll =,,3,... From (8) d (3) we hve for ll. Hece (9) ( ) ( ) ( ) ( ) ( ) ( ) ψ ( ) I = f f f = (0) ( ) ( ) ψ f ( ) = f ( ) f ( ) () So we choose two free prmeters f ( ) f ( ) f ( )) d this gives, usig (): > 0 (to obti icresig fuctio ( ) 4 ψ f ( 3) = f ( ) f ( ) () 3 3 3 ( ) 6 ψ f ( 4) = f ( ) f ( ) ψ ( ) (3) 4 4 4 4 usig lso (). So we hve tht (9) is true for =,,3,4 (defiig 0 = = 0 ). Complete iductio supposes (9) to be true for d d we hve to prove (9) for. By () we hve: ( ) f ( ) = f ( ) ( ) f ( ) ( ) ψ ( ) ψ ( ) f ( ) ( ) f ( ) ( i ) ψ = 4 f ( ) ( ) f ( ) ( ) ψ ( ) f ( ) ( ) f ( ) ( i ) ψ ψ ( ) = ( ) f ( ) f ( ) ψ ( ) ( i) ψ ( i ) ψ ψ ( ) = ( ) f ( ) f ( ) ( i ) ψ Pge 45
Egghe, L. which is (9) with replced by. (ii) => (i) By (9), pplied to, d we hve, by (3): I ( ) = ( ) f ( ) f ( ) ( i ) ψ ( ) ( ) f ( ) ( ) f ( ) ( ) ψ = ψ ( ) f ( ) ( ) f ( ) ( i ) ψ s is redily see. This completes the proof of (ii) => (i) d hece the proof of theorem. Bsed o this theory we will, i the ext sectio, give two chrcteriztios of the geerl Kosmulski-idices. CHARACTERIZATIONS OF THE GENERAL KOSMULSKI-INDICES For I ( ) s i (8) we hve the followig theorem. Theorem 3: The followig ssertios re equivlet: I = (4) (i) ( ) ( ) (ii) f ( ) for ll =,,3,... ( ) f = ( ) (5) Proof: This follows from Theorem d the fct tht ψ = ( ) ( ) ( ) =... The ext corollry gives chrcteriztio of the geerl Kosmulski-idices usig the first icremet. Pge 46
A Geerl Theory of Miiml Icremets for Hirsch-Type Idices Corollry 4: The followig ssertios re equivlet (i) f ( ) = d ( ) ( ) = I (ii) f ( ) for ll =,,3,... = for ll =,,3,..., hece the geerl Kosmulski-idices. Proof: This follows redily from theorem 3. For I ( ) s i (9) we hve the followig theorem: Theorem 5: The followig ssertios re equivlet: I = (6) (i) ( ) ( ) ( ) for ll =,,3,... (ii) f [ f f ( ) = ( ) ( ) ( ) ( ) ( )( ) ( ). 3 ] (7) Proof: (9) trsforms ito ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f = f f i i i i whichis (7) sice ll coefficiets of 3,4,...,( ) re zero which is redily see by evlutio of ll the Σ s. So Theorem 5 follows from Theorem. T The ext corollry gives chrcteriztio of the geerl Kosmulski-idices usig the secod icremet. Corollry 6: The followig ssertios re equivlet: (i) f ( ) =, f ( ) = d ( ) ( ) ( ) for ll =,,3,... I = Pge 47
Egghe, L. (ii) f ( ) = for ll =,,3,..., hece the geerl Kosmulski-idices. Proof: This follows redily from Theorem 5 d the fct tht f ( ) = ( ) ( ) ( )(. ) ( 3 ) (( ) ( ) 3 ) ( ) ( = ) = Remrk It is cler from Theorems 3 d 5 d Corollries 4 d 6 tht the icremetl idetities (4) d (6) yield impct mesures tht re geerliztio of the Kosmulski-idices due f to the fct tht they coti prmeter f ( ) d prmeters f ( ) d ( ) respectively. This is remrkble fct but is i lie with the results obtied i Egghe (03,b) o the Wu- d Hirsch-idices. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH We hve chrcterized geerl Hirsch-type idices by mes of their first d secod icremets. They idicte wht effort is ecessry to icrese such impct mesure from to, for every =,,3,.... We lso chrcterized the geerl Kosmulski-idices by mes of their icremets of first d secod order. Sice we treted the most geerl Hirsch-type idices, this fiishes this type of study but leves ope the similr tretmet of impct mesures which re ot of Hirsch-type such s e.g. the impct fctor. I this cotext, impct mesures such s the g-idex (Egghe (006)) or the R-idex (Ji et l. (007)) fll i the ctegory of h-type idices sice, by usig f of icremets ( ) cittios. I d I ( ), we oly cosider ppers with equl umber ( ) ACKNOWLEDGEMENT This reserch received o specific grt from y fudig gecy i the public, commercil, or ot-for-profit sectors. Pge 48
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