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2 O Sly of Mulvre Polyols 9 E Rodrguez-Ageles Mexco Se Auooous Uversy Egeerg Fculy Mexco Iroduco I he uvre polyol cse here re oly wo oos of sly: Hurwz sly for couous polyols d Schur sly for dscree polyols However he ulvre polyol cse here exss ore coplex suo sce here re ore clsses of sly: Wde Sese Sle (WSS Scerg Hurwz Sle (SHS d Src Sese Sle (SSS for couous polyols (Fewes & Bsu 987 d Wde Sese Schur Sle (WSSS Scerg Schur (SS d Src Sese Schur Sle (SSSS for dscree polyols (Bsu & Fewes 987 hese clsses hve dffere properes for exple soe clsses reduce o he Hurwz or Schur uvre oo d soe polyols fro soe clsses y lose her sly propery he presece of rrry sll coeffce vros Besdes ewee hese clsses hs o ee possle o eslsh slr reloshp s does for Hurwz d Schur uvre polyols y he Moeus rsforo (Bose 98 For log e SSS d SSSS polyols hve ee eployed o o ey properes of sly d rous sly her ow do ecuse hey hve ore cocde chrcerscs wh Hurwz d Schur uvre oos h he oher ulvre clsses hve (Bsu & Fewes 987; Fewes & Bsu 987 Despe of hs hs wor he eres s focused wo dffere oos of sly: Sle clss for he couous cse (Khroov & orres-muñoz 999 d Schur Sle clss for he dscree cse (orres- Muñoz e l 6 he reso s wofold: frsly oh clsses hve he propery of eg he lrges clsses preservg sly whe fced o rrry sll coeffce vros d secodly hs ee recely show h y eer of he Sle clss s ssoced y ler rsforo o oe eer of he Schur Sle clss he se wy h Hurwz d Schur uvre polyols re reled y he Moeus rsforo (orres-muñoz e l 6 Besdes oh clsses re he url exeso of her uvre couerpr: Hurwz d Schur uvre clsses I geerl he lyss d corol of y syse s por o hve effce fro he copuol po of vew crer o es he sly of s chrcersc polyol For he uvre cse here s g vrey of well-ow effce lgorhs o del wh he Hurwz d Schur sles (Bre 983; Prs & Hh 99; Bhchryy 995 However he ulvre cse hs prole s ore coplex: he -vre ( > cse here re few lgorhs repored d hey hve he prole of her effcecy (Bose 98 Despe of hs he vre ( cse here re lo of lgorhs o del wh he Schur Sle vre ssue d soe of he re effce (Aderso & Jury 973;

3 9 Syses Srucure d Corol Mr & Fhy 973; Sl 975; Bose 977; Jury 988; Yg & Uehue 998; Bsrz ; Xu e l 4; Durescu 6 I cors he couous vre cse he repored lgorhs re devoed o he SSS clss (Zehe & Wlch 98; Bose 98 e here re o repored lgorhs delg wh he Sle vre clss I hs wor ep s de o gve sple d effce crero o he Sle vre clss I he uvre cse he fc h he Moeus rsforo of y Hurwz polyol gves Schur polyol d vcevers hs llowed exedg sly d rous sly resuls fro oe do o he oher (Prs & Hh 99 I hs wor s used slr fc ewee he Sle d Schur Sle vre clsses he followg wy: frsly s o he dscree couerpr of gve couous vre polyol ex s Schur sly s proved d flly he Schur sly of hs polyol ples he sly of he couous polyol For hs ew Schur Sle vre es s developed y cosrucg reduced order polyol rry for uvre Schur polyols wh lerl coeffces such wy h he Schur sly of hese polyols ogeher wh specfc coeffce codos ples he sly of he orgl polyol hs wor egs wh he roduco of soe prelres oos d oo of ulvre polyols d sury of soe ey properes of he Sle d Schur Sle clsses Nex he prole s clerly defed followg wh he preseo of he Schur Sle d Sle ess Flly soe exples d coclusos rers re gve Prelres of ulvre polyols (Bose 98 A ulvre polyol he vrle vecor ( s s of he for p ( s s fe su s s s s ( where s re he depede vrles of prl degree deg { p( s } he coeffces re gve rel (or coplex uers Oe y defe followg he lexcogrphc order of he dces he coeffce vecor ( I he lyss of ulvre polyols s very useful o wre he polyol p ( s s uvre polyol wh polyol coeffces e ( ( ( s s s s s ps s ( s for d where he coeffces ( ( s s s s s re ( -vre polyols I hs cse he free d he polyol coeffces wh respec o he vrle s correspod o d s s s respecvely s s

4 O Sly of Mulvre Polyols 93 A roo of p ( s s vecor s ( s s s such h p ( s If ( s s3 s ( s s3 s re fxed o soe rrry vlue he p ( s s s s uvre polyol he vrle s of degree I cocluso d cors wh he uvre cse ulvre p ( s hs fe uer of roo folds -desol coplex spce Besdes cors wh he uvre cse wo ulvre polyols y e copre u possessg coo roos (Khroov & orres-muñoz 999 Le us deoe he se of cos degree -vre polyols y where ( P { p( s deg{ ( s } } p N s he vecor of cos prl degrees Slr defos wll hold for uvre polyols I he lyss of he couous ulvre polyols s ofe used he oo of he couge polyol he couge polyol of p ( s wh respec o he vrle s usg p ( s s he decoposo ( wh respec o he vrle s s gve y where ( ( s s p * ( ( s p( s s s ( s s3 s ( s (3 3 s es h ll coeffces d vrles s s 3 s re chged y her coplex couges Clerly he couge polyol c e e fro oe ul * vrles Herefer wll e cosdered uless oherwse sed he couge p ( s wh respec o he vrle s o dsgush he dscree polyols fro he couous cse d for rdo dscree ulvre polyol s oed s ( z vecor used re ( z z z d ( z q he vrle vecor d he coeffce respecvely Besdes he srucure of dscree polyol s he se s ( d s lso possle o wre s he decoposo ( I he lyss of he dscree ulvre polyols s ofe used he oo of he recprocl couge polyol he recprocl couge polyol of q ( z wh respec o he vrle z usg q ( z s he decoposo ( wh respec o he vrle z s gve y where ( ( z z q ( ( z q( z z z z ( z z3 z z z (4 3 z es h ll coeffces d vrles z z 3 z re chged y her coplex couges Clerly he recprocl couge polyol c e e fro oe ul vrles Herefer wll e cosdered uless oherwse sed he couge q ( z wh respec o he vrle z Sle ulvre polyols I he couous cse cosder he followg polydo

5 94 Syses Srucure d Corol ogeher wh s essel oudry Γ ( {( s s s Re( s } ( {( s s s Re( s } Ω Defo : A ulvre polyol p( s P s clled Sle f ssfes he followg codos ( p( s s Γ -vre polyols ( ( s s s s s polyols ccordg o hs defo wh degree ( M ( for re Sle A polyol ssfyg us codo s clled Src Sese Sle (SSS Such clss of polyols hs plyed por role sly d rous sly lyss (Feewes & Bsu 987 hs clss reduces o he sdrd oo of Hurwz sly he uvre cse u y lose s sly propery he presece of sll rrry coeffces peruros (Khroov & orres-muñoz 999 hs frgly s very udesrle whe oe sudes he rousess ssue Noe h he sle clss s proper suclss of he SSS clss s s he lrges ulvre clss preservg sly uder sll coeffce vros (Khroov & orres-muñoz 999 Besdes he sle clss reduces o he rdol Hurwz clss he uvre cse so he sle clss lso preserves severl useful properes of uvre Hurwz polyols oo (Khroov & orres-muñoz 999; Khroov & orres-muñoz A sury of soe of properes of he sle ulvre clss s he followg Le : Le p( s P e sle ulvre polyol Le s e fxed soe vlue s such h Re( s he he ( -vre polyol p ( s s s ( 3 s sle polyol of degree Le : Le p( s P e sle ulvre polyol Assue h > he he polyol p ˆ( s p( s s s s s sle ulvre polyol of he se degree s p ( s Noce h y successve plco Le d Le c e e fro oe ul vrles Nex resul s he exeso of he Lucs heore for he Hurwz uvre polyols (Mrde 949 e shows he vrce of he sly propery uder dffereo h c e e y successve pplco fro oe ul vrles heore 3: Le p( s P e sle ulvre polyol Assue h he he ( s s sle ulvre polyol of degree ( polyol ~ ps s s p( s > Le 4: Le p( s P e sle ulvre polyol he ll ( coeffces ( ( s s s s s for s for re sle polyols of degree ( -vre polyol he decoposo ( wh respec o he vrle Nex propery s he exeso of he clsscl Sodol s Codo for Hurwz uvre polyols (Gcher 959

6 O Sly of Mulvre Polyols 95 heore 5: Le p( s P e sle ulvre polyol wh rel coeffces he ll coeffces of he polyol hve he se sg: eher ll of he re posve or ll of he re egve Le 6: Le p( s P e sle ulvre polyol he he coeffce s o zero Proof: For he cse he see s ovous For cosder sle vre polyol p ( s s of degree ( By Defo s s he decoposo ( wh respec o he vrle uvre polyol coeffce ( ( s s Hurwz sle polyol of degree Assue h he see s rue for ( e coeffce -vre polyols d cosder he -vre cse Gve sle ulvre polyol p ( s of degree ( s ( -vre polyol coeffce ( ( s s3 s s s sle ( -vre polyol of degree ( 3 respec o he vrle duco hypohess coeffce fro Defo follows h he decoposo ( wh he y he Nex resuls show h for sle ulvre polyols he rous sly c e cosdered whou srucurl resrcos o ucer preers d h sle ulvre polyol hs o roos close o he essel oudry heore 7: Le p( s P e sle ulvre polyol he here lwys exss ε > such h every ulvre polyol wh coeffce vecor lyg he ε-eghourhood of he coeffce vecor of p ( s s sle oo heore 8: Le ( s P p e sle ulvre polyol he here lwys exss ε > such h hs o roos he ε-eghourhood of he essel oudry Schur Sle ulvre polyols I he dscree do cosder he polydsc gve y d s essel oudry gve y U ( ( z z z ( Ω { z } { z } ( ( z z z ( Defo : A ulvre polyol q( z P s clled Schur Sle f q( z z U he so-clled Src Sese Schur Sle (SSSS clss s ofe eployed he lerure d cosders q( z P SSSS f ( z z z z z z (Bsu & q for ll ( { } Feewes 987 Despe of SSSS clss preserves sly uder sll coeffce vros reduces o he sdrd Schur polyols oo he uvre cse so soe ey properes of Schur uvre polyols c o e exeded o he ulvre cse s he vrce of he Schur sly propery uder dffereo (orres-muñoz e l 6 he Schur sle clss he sese of Defo s lso used he lerure (Hug 97; Kczore 985 Acully s he recprocl clss of he SSSS clss d lso preserves sly uder sll coeffce vros Besdes he Schur sle clss reduces o he

7 96 Syses Srucure d Corol sdrd Schur clss he uvre cse so severl useful properes y e exeded fro he uvre Schur polyols o he ulvre cse (orres-muñoz e l 6 A sury of soe of he properes of he Schur sle ulvre clss s he followg Le 9: Le q( z P e Schur sle ulvre polyol Le z e fxed soe vlue z such h z he he ( -vre polyol q ( z z z s Schur sle polyol of degree ( 3 qz P e Schur sle ulvre polyol Assue h he he Le : Le ( polyol q ~ ( q( z z z q ( z heore : Le ( z P z s Schur sle ulvre polyol of he se degree s q e Schur sle ulvre polyol Assue h he he ( polyol ~ q z z z q( z s Schur sle ulvre polyol of degree z ( Noce h Le 9 Le d heore re he dscree verso of Le Le d heore 3 respecvely he c e lso e fro oe ul vrles z P -vre Le : Le q( e Schur sle ulvre polyol he he ( polyol coeffces ( ( z z z z z for z re Schur sle polyols of degree ( > > he decoposo ( wh respec o he vrle I s worh o eo h oher polyol coeffces dffere fro he polyol coeffces he decoposo ( re o ecessrly Schur sle (orres- Muñoz e l 6 Nex propery s he exeso of he clsscl coeffce codo for Schur uvre polyols (Bhchryy 995 Le 3: Le q( z P e Schur sle ulvre polyol he he coeffce codo < s hold Corollry 4: Le ( s o zero qz P e Schur sle ulvre polyol he he coeffce Proof: I drecly follows fro Le 3 d he fc h Nex resuls show h he Schur sle ulvre clss s sule o sudy he rousess ssue heore 5: Le q( z P e Schur sle ulvre polyol he here lwys exss ε > such h every ulvre polyol wh coeffce vecor lyg he ε-eghourhood of he coeffce vecor of q ( z s Schur sle oo qz P e Schur sle ulvre polyol he here lwys exss ε > heore 6: Le ( such h hs o roos he ε-eghourhood of he essel oudry 3 Prole see ( Fro prccl po of vew s essel o dspose of copuolly fesle polyol sly crer For he uvre cse here re soe very well-ow effce sly

8 O Sly of Mulvre Polyols 97 crer (Bre 983; Prs & Hh 99; Bhchryy 995 u for he -vre cse here re However here re soe crer for he vre cse (Jury 988; Bsrz ; Xu e l 4; Durescu 6 u her pleeo he ulvre cse s o esy I hs wor he gol s o cle he followg Prole: Gve couous vre polyol p ( s s fd effce polyol coeffces depede crero llowg o coclude wheher or o elogs o he sle clss he sese of Defo hs crero us e lso poelly sule for s exeso o he ulvre cse A frs glce y ure of he couous sle clss ryg o o o-recursve crer gh e hrd s hs corss wh he dscree Schur cse where reserch effors leded o relle lgorhs llowg o lyze sly depedg o he polyol coeffces fe uer of seps I he uvre polyol cse Hurwz sly ples Schur sly d vcevers hs correspodece hs llowed o rsle sly resuls ewee couous d dscree dos For sce rslo of Rouh-Hurwz sly crero spred he develope of coeffce-sed lgorhs for Schur sly (Prs & Hh 99 I such ve he elef h SSS vre polyols re src equvlece wh Schur sle vre polyols ws he ceer of erler eps o develope vre sly heory I hese eps were used dffere rsforo of rsforo (5 Uforuely he erly cocluso ws oly SSS sly s pled y Schur sly d o he reverse sese (Bose 98 he se cocluso s oed usg rsforo (5: cosder he SSS polyol ps ( ss s urs ou h he rsfored dscree polyol usg rsforo (5 ( zz z z Schur sle s hs he roo ( ( U qz s o 3 (orres-muñoz e l 6 herefore here s o wy o fer sly resuls ewee SSS d Schur sle clsses However recely ws show h he ulvre sle clss he sese of Defo s he couerpr of he ulvre Schur sle clss he sese of Defo heore 7: (orres-muñoz e l 6 he polyol p ( s of degree ( s sle f d oly f he polyol z z z qz (5 ( ( ( ( z z z p z z z s Schur sle polyol of degree ( Oserve h hs rsforo s he url exeso of he Moeus uvre rsforo hs resul ws used s rdge o rsle properes d sly resuls fro he couous do o he dscree oe d vcevers (orres-muñoz e l 6 4 A drec crero for couous vre polyol sly O he ss of heore 7 drec vre couous sly lgorh c e sed s follows: Gve couous vre polyol ( ps s s

9 98 Syses Srucure d Corol Cosruc he dscree polyol ( z se degree s ( s q usg he rsforo (5 If q ( z hs he p he coue If s o he cse he he polyol p ( s s o sle Apply y Schur vre sly es o q ( z 3 If q ( z s Schur sle he he polyol p ( s s sle If s o he cse he he polyol p ( s s o sle I s worh ocg h here exss vrey of crer for he Schur vre sly cse (Jury 988; Bsrz h gh e poelly dped o he cse of couous vre sle polyols he sep However ew sple Schur sly es ws roduced recely s oe lerve wy o cle he prole of gvg relle crero for he couous sle clss (Rodrguez-Ageles e l 7 he uderlyg phlosophy s spred o he uvre cse o ry o fd rry of reduced degree polyols whose sly wll ply sly of he orgl polyol (Bhchryy e l 995 heore 8: (Rodrguez-Ageles e l 7 he polyol q( z z P s Schur sle f d oly f q ( z z for ll z d for fxed z z such h z Gve he followg polyol sequece { [ ] } ( ( ( ( ( ( ( ( q z ( z z q z z z q ( z z (6 z he followg equly holds ( ( ( z ( > z (7 for ll z z such h z where ( ( z z q( z z of he h polyol ( ( z z q d q for ( ( z s he -h coeffce he Schur vre sly lgorh sed o heore 8 c e sed s follows: Gve dscree vre polyol he decoposo ( wh respec o he vrle z ( ( z z ( z qz Verfy f he uvre polyol ( z q s Schur sle If s Schur sle he coue If s o he cse he he vre polyol q ( z s o Schur sle Verfy sep y sep f he equly holds for ll θ [ π ] ( θ ( ( θ e > ( e d for re oed fro sequece (6 wh polyol ( z z d where coeffces ( θ ( d ( θ ( e z e e θ If ll equles hold he he vre q s Schur sle If oe of he coeffce codos fls he sop d he vre polyol q ( z s o Schur sle

10 O Sly of Mulvre Polyols 99 Acully he sep c e pleeed uercl d grphcl wy provdg sple es for Schur vre sly Besdes oce h he grphcl esg s us eeded ouded ervl depedely of he polyol vecor degree Exple : Deere he sly of he couous vre polyol ( ( 75 s ( ( 5s s 3s s 5 3s 75s s p s Accordg o heore 7 he sly of p ( s s equvle o he Schur sly of he rsfored polyol q ( z gve y ( ( 5z ( ( 5z 5z z 5 5z z z q z Hece q ( z hs he se degree s ( s Followg he lgorh for Schur vre sly oe y verfy he sep he le us cle he sep he frs polyol of he sequece (6 s q p oe hs o chec he Schur sly of ( z ( θ θ θ θ θ θ ( z ( ( ( e 5e 5e 5e z 5 5e e z Fro Fgure oe c see h he equly ( ( ( ( θ e e θ he le us coue wh he es q > holds for ll θ [ π ] 8 6 ( (e θ ( (e θ θ Fgure Coeffce codo for he s polyol ( θ q ( z e he secod polyol of he sequece (6 s q of he polyol sequece ( ( ( ( θ θ θ θ θ θ θ θ z e 5 5e 5e 5e 5 65e 65e 5e 5e z Fro Fgure oe c see h he equly ( ( ( ( θ e e θ > holds for ll θ [ π ] he he dscree vre polyol q ( z s Schur sle s s repored severl ppers

11 Syses Srucure d Corol (Hug 97; Jury 988; Bsrz herefore he couous vre polyol p s s sle ( 3 5 ( (e θ ( (e θ θ Fgure Coeffce codo for he d polyol ( θ q ( z e of he polyol sequece 5 Nuercl exples he s o show he poel pplcly of he drec lgorh preseed he prevous seco whe delg wh vre polyols of relvely hgh degree Fro copuol po of vew s sruel o e o ccou he reloshp ewee he coeffces of couous ulvre polyol p ( s d hose of s dscree q Acully he coeffces of he vre polyols p ( s s d couerpr ( z q ( z z re reled y ler rsforo s s expressed he followg heore 9: Le ( p s s s e gve vre polyol of degree ( s rsfored dscree vre polyol Cosder he q ( z c e expressed s z z qz (8 ( ( z ( z p z z ( qz z z where he coeffces d re reled s follows r r ( φ l l l l l l

12 O Sly of Mulvre Polyols where ( ( ( ( l l r r φ Corollry : Le ( s p d ( z q wo vre polyols reled s (8 wh coeffce vecors d respecvely he coeffce vecors re reled hrough he rx equo hs reloshp c e expressed s where [ ] [ ] d s cos osgulr rx gve y where ( ( ( ( ( ( ( ( ( wh ( ( r p r q l q l p q l p φ where ( ( ( ( q p l l r r l φ Noce h prevous sees y e deduced y srghforwrd rx clculos fro he rsforo (8

13 Syses Srucure d Corol Besdes for he copuol pleeo s useful o wre polyol ( ps s s rx for e s s s for d where [ ] Exple : Chec he sly of he (6 for (9 wh ( s s As p (9 A -degree couous polyol p ( s expressed he A Applyg heore 9 or Corollry s possle o fd s dscree couerpr ( z z Bz q wh B

14 O Sly of Mulvre Polyols 3 Hece ( z q hs he se degree s ( s p oe hs o chec he Schur sly of ( z q Followg he Schur vre sly lgorh s esy o chec h sep holds he le us proceed o chec sep of he lgorh Acully he sequece of polyols (6 d he coeffce codos (7 of he Schur vre sly lgorh y e esly pleeed uercl wy Ideed oe y geere sep y sep sequece of grphcs llowg o decde f such codos re ssfed I hs exple we hve he followg grphcs 4 35 ( (e θ ( (e θ θ Fgure 3 Coeffce codo for he s polyol ( ( θ q z e of he polyol sequece 8 ( (e θ ( (e θ θ Fgure 4 Coeffce codo for he d polyol ( θ q ( z e of he polyol sequece

15 4 Syses Srucure d Corol 5 9 ( (e θ ( (e θ θ Fgure 5 Coeffce codo for he 3 rd polyol ( θ q ( z e of he polyol sequece 5 8 (3 (e θ (3 (e θ θ Fgure 6 Coeffce codo for he 4 h polyol ( θ q ( z e 3 of he polyol sequece he frs grphcs Fgure 3 Fgure 4 d Fgure 5 show h codo (7 s hold However he ls grphc Fgure 6 shows h codo (7 s o respeced Cerly polyol q ( z s o Schur sle d y cosequece polyol p ( s s o sle

16 O Sly of Mulvre Polyols 5 7 Coclusos I hs wor ufed ulvre polyol sly heory ws cosdered d s sed o he Sle d he Schur sle ulvre clsses for couous d dscree dos respecvely he focus ws o gve fesle crer o deere wheher or o couous vre polyol elogs o he Sle clss I drec pproch he recursve ure of he couous Sle clss poses he eedg of checg Hurwz sly of he wo uvre polyol coeffces where prl degree preservo s requred s well d he SSS sly of he orgl polyol o chec he frs es oe c use every of he well-ow uvre crer d o chec he SSS sly here re soe crer h c e used u he hve soe effce proles I drec pproch he sly of couous vre polyol s deduced y lyzg he Schur sly of s dscree vre polyol couerpr Frsly he ehod preseed hs wor requres of checg Schur sly of cos coeffces uvre polyol d secodly checg Schur sly of uvre polyol wh lerl coeffces o chec he frs e here re o prole o chec he ls e s ecessry he fulfle of sequece of coeffce codos of he for (7 he fe frequecy ervl θ [ π ] If hese wo es re ssfed he he couous polyol elogs o he Sle clss Becuse of s splcy coeffce codos re fesle grphcl er d y cosruco he coplexy of he lgorh s depede of he polyol degree I fuure wor he exeso of he proposed drec vre lgorh o he ulvre cse c e lyzed d here re oher wy o use he reloshp ewee Sle d Schur sle ulvre polyols: o drec couous sly crero y rslg hrough he relo (5 exsg Schur sle es 8 Refereces Aderso BDO & Jury EI (973 Sly es for wo-desol Recursve Flers IEEE rscos o Audo d Elecrocouscs Vol No4 Augus 973 pp ISSN Bre S (983 Polyols d Ler Corol Syses Mrcel Deer Ic ISBN New Yor Bsu S & Fewes A (987 New Resuls o Sle Muldesol Polyols Pr II: Dscree Cse IEEE rscos o Crcus d Syses Vol34 No Noveer 987 pp64-74 ISSN Bhchryy SP; Chpell H & Keel LH (995 Rous Corol: he Prerc Approch Prece Hll ISBN X Upper Sddle Rver Bsrz Y ( O esg Sly of -D Dscree Syses y Fe Colleco of -D Sly ess IEEE rscos o Crcus d Syses-I Vol49 No Noveer pp ISSN 57-7 Bose NK (977 Ipleeo of New Sly es for wo-desol Flers IEEE rscos o Acouscs Speech d Sgl Processg Vol5 No Aprl 977 pp7- ISSN Bose NK (98 Appled Muldesol Syses heory V Nosrd Rehold Copy ISBN New Yor

17 6 Syses Srucure d Corol Durescu B (6 Sly es of Muldesol Dscree-e Syses v Su-of- Squres Decoposo IEEE rscos o Crcus d Syses-I Vol53 No4 Aprl 6 pp ISSN 57-7 Fewes A & Bsu S (987 New Resuls o Sle Muldesol Polyols Pr I: Couous Cse IEEE rscos o Crcus d Syses Vol 34 No Ocoer 987 pp-3 ISSN Gcher FR (959 he heory of Mrces Vol II Chelse Pulshg Copy ISBN New Yor Hug S (97 Sly of wo-desol Recursve Dgl Flers IEEE rscos o Audo d Elecrocouscs Vol No Jue 97 pp58-63 ISSN Jury EI (988 Modfed Sly le for -D Dgl Flers IEEE rscos o Crcus d Syses Vol35 No Jury 988 pp6-9 ISSN Kczore (985 wo-desol Ler Syses Lecure Noes Corol d Iforo Sceces No68 Sprger-Verlg ISBN Khroov VL & orres-muñoz JA (999 Rous Sly of Mulvre Polyols Pr I: Sll Coeffce Peruros Muldesol Syses d Sgl Processg Vol No Jury 999 pp7- ISSN Khroov VL & orres-muñoz JA ( Rece Resuls o he Rous Sly of Mulvre Polyols IEEE rscos o Crcus d Syses-I Vol49 No6 Jue pp75-74 ISSN 57-7 Mrde M (949 he Geoery of Zeros of Polyol Coplex Vrle Aerc Mhecl Socey ISBN 8853 New Yor Mr GA & Fhy MM (973 O he Sly of wo-desol Dgl Flers IEEE rscos o Audo d Elecrocouscs Vol No5 Ocoer 973 pp47-47 ISSN Prs PC & Hh V (99 Sly heory Prece Hll ISBN Rodrguez-Ageles E; orres-muñoz JA & Medez-Brros CF (7 A Sple es for Sly of Couous Bvre Polyols 3 rd IFAC Syposu o Syse Srucure d Corol Foz do Igussu (Brzl Ocoer 7 Sl DD (975 Sly Crer for wo-vrle Polyols IEEE rscos o Crcus d Syses Vol No3 Mrch 975 pp85-89 ISSN orres-muñoz JA; Rodrguez-Ageles E & Khroov VL (6 O Schur Sle Mulvre Polyols IEEE rscos o Crcus d Syses-I Vol53 No5 My 6 pp66-73 ISSN 57-7 Xu L; Yd M; L Z; So O & Azw Y (4 Furher Iprovees o Bose s D Sly es Ierol Jourl of Corol Auoo d Syses Vol No3 Sepeer 4 pp39-33 ISSN Yg X & Uehue R (998 New Sly es Algorh for wo-desol Dgl Flers IEEE rscos o Crcus d Syses-I Vol45 No7 July 998 pp ISSN 57-7 Zehe E & Wlch E (98 Zero Ses of Mulpreer Fucos d Sly of Muldesol Syses IEEE rscos o Acouscs Speech d Sgl Processg Vol9 No Aprl 98 pp97-6 ISSN

18 Syses Srucure d Corol Eded y Per Huse ISBN Hrd cover 48 pges Pulsher Iech Pulshed ole Augus 8 Pulshed pr edo Augus 8 he le of he oo Syse Srucure d Corol ecopsses rod feld of heory d pplcos of y dffere corol pproches ppled o dffere clsses of dyc syses Oupu d se feedc corol clude og ohers rous corol opl corol or ellge corol ehods such s fuzzy or eurl ewor pproch dyc syses re eg ler or oler wh or whou e dely fxed or ucer oedesol or uldesol he pplcos cover ll rches of hu cves cludg y d of dusry ecoocs ology socl sceces ec How o referece I order o correcly referece hs scholrly wor feel free o copy d pse he followg: E Rodrguez-Ageles (8 O Sly of Mulvre Polyols Syses Srucure d Corol Per Huse (Ed ISBN: Iech Avlle fro: hp:///oos/syses_srucure_d_corol/o_sly_of_ulvre_polyols Iech Europe Uversy Cpus SeP R Slv Kruze 83/A 5 Re Cro Phoe: 385 ( Fx: 385 ( Iech Ch U 45 Offce Bloc Hoel Equorl Shgh No65 Y A Rod (Wes Shgh 4 Ch Phoe: Fx:

19 8 he Auhor(s Lcesee IechOpe hs chper s dsrued uder he ers of he Creve Coos Aruo-NoCoercl- ShreAle-3 Lcese whch pers use dsruo d reproduco for o-coercl purposes provded he orgl s properly ced d dervve wors uldg o hs coe re dsrued uder he se lcese