Journal of Mathematics and System Science

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1 oural of Mathematcs ad System Scece Volume 4 Number February 04 Seral Number 4 Davd Publshg Davd Publshg Comay

2 Publcato Iformato: oural of Mathematcs ad System Scece s ublshed mothly hard coy ad ole Prt: ISSN 59-59; Ole: ISSN by Davd Publshg Comay located at 670 ast ohso Drve Cty of Idustry C 9745 US. ms ad Scoe: oural of Mathematcs ad System Scece a mothly rofessoal academc ad eer-revewed oural artcularly emhaszes ew research results theory ad methodology realm of ure mathematcs aled mathematcs comutatoal mathematcs system theory system cotrol system egeerg system bology oeratos research ad maagemet robablty theory statstcs formato rocessg etc.. rtcles terretg ractcal alcato of u-to-date techology are also welcome. dtoral Board Members: Wllam P. Fo US Wattaavadee Srwattaaogse Thalad Zoubr Dahma lgera Claudo Cuevas Brazl lsa Fracomao Italy Baha ŞN Turey ssa Guezae-Laoud lgera leader Nolaevch Raov Russa Shelly SHN Cha Mauscrts ad corresodece are vted for ublcato. You ca submt your aers va web submsso or -mal to Submsso gudeles ad web submsso system are avalable at htt:// or htt:// dtoral Offce: 670 ast ohso Drve Cty of Idustry C 9745 US Tel: Fa: mal: Coyrght 03 by Davd Publshg Comay ad dvdual cotrbutors. ll rghts reserved. Davd Publshg Comay holds the eclusve coyrght of all the cotets of ths oural. I accordace wth the teratoal coveto o art of ths oural may be reroduced or trasmtted by ay meda or ublshg orgas cludg varous webstes wthout the wrtte ermsso of the coyrght holder. Otherwse ay coduct would be cosdered as the volato of the coyrght. The cotets of ths oural are avalable for ay ctato. However all the ctatos should be clearly dcated wth the ttle of ths oural seral umber ad the ame of the author. bstracted / Ideed : Database of BSCO Massachusetts US Ide Coercus Polad Google Scholar Ulrch s Perodcals Drectory CS Techology Research Database Norwega Socal Scece Data Servces NSD Database for Statstcs o Hgher ducato DBH Norway Uverse Dgtal Lbrary S/B Prouest Summo Serals Solutos Chese Database of CPS rt Ic. & OCLC Chese Scetfc ourals Database VIP Cororato Chogg P. R. Cha Subscrto Iformato: Prce er year: Prt $50; Ole $300; Prt ad Ole $560 Davd Publshg Comay 670 ast ohso Drve Cty of Idustry C 9745 US Tel: ; Fa: mal: Dgtal Cooeratve Comay: D DV I D PUBLISHING Davd Publshg Comay

3 oural of Mathematcs ad System Scece Volume 4 Number February 04 Seral Number 4 Cotets 69 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet Guocheg Wag Ye Ta ad Pg L 83 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems Fay M. Gozález Barros ad rue Castro Martíez 93 From Dgtal alogs Through Recursve Maches to Quatum Comuter M.B. Igatyev Yu.. Shey.. Ltov 99 Iterolated Tesor Products of oetal Tye Vectors of Ubouded Oerators Mara Dmytryshy 05 Measuremet of Surlus Labor Vet Nam grculture Quyet Dag Pham ad Hoa Huu Vo stmato of Demograhc Statstcs the Camboda Soco-coomc Survey CSS Pe Socheat ad ders Holmberg 6 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg Kodwo a Brya Bafll ad Stehae Broc 7 Develomet ad lcato of Statstcal Busess Regster Gudeles frca Coutres Mchael. Colledge ad Besa Muwele 33 ffect of Heteroscedastc Varace Covarace Matrces o Two Grous Lear Classfcato Techues Frday Zzedoff Owou ad bdul Rahma Othma

4 oural of Mathematcs ad System Scece D DVID PUBLISHING Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet Guocheg Wag Ye Ta ad Pg L. Isttute of Quattatve & Techcal coomcs Chese cademy of Socal Sceces CSS Beg 0073 Cha. Deartmet of Idustral coomcs Graduate School of Chese cademy of Socal Sceces Beg 0488 Cha Receved: November / cceted: December 03 / Publshed: February bstract: The utlty of ublc goods vary wth the behavors of staeholders layers ad t s arorate to study effectve suly ad maagemet of ublc goods wth game modelg ad aalyss. The comarso effect s the ey ssue of ublc good rovso both theoretcal aalyss ad ractce. Oe maor cotrbuto of the aer s the eteso of Clare-Groves mechasm to acheve whch strategc behavor aalyss s aled through the aalyss ad the comarso effect amog varous staeholders dfferet stages s created ad hghly emhaszed. I the frst secto of ths aer the defto of tegrated water resources maagemet IWRM the mortace of staeholder artcato as well as some models ad methods that have bee aled are llustrated. Followg ths the framewor of aalyss s elaborated whch the scearo ad ams are show ad t s clamed that game theory s the ma aroach whch cludes both cooeratve games ad o-cooeratve games. To acheve the ams of the ublc roect fve aroaches from game theory are able to cover the etre rocess of the roect ad the fourth aroach o terest comesato mechasm s the hghlght of the research. fter ths the terest comesato mechasm s demostrated the model secto ad s roved to be a cetve comatble mechasm that maes each staeholder choose to behave accordace wth the terest of the etre roect. The Clare-Groves mechasm s aled ad eteded establshg the model ad the utlty chage by the comarso amog staeholders defed as the comarso effect s volved. I the alcato secto a water roect s aalyzed cosderato of varous staeholders ad other ossble alcatos are also dcated. Keywords: Game modelg strategc behavor aalyss tegrated water resources maagemet IWRM terest comesato mechasm the Clare-Groves mechasm.. Itroducto It s a arsg ad actve area to study ublc goods rovso wth game modelg ad eermetal game methods game theory ad ublc maagemet. Water maagemet that deeds o teractg or strategc behavor of staeholders layers s a tye of ublc goods as well ad t s more arorate to aalyze them wth game modelg ad aalyss. Issues related to water resources have attracted creasgly more atteto as the roblems of water shortage ad Corresodg author: Ye Ta Ph.D. research feld: dustral ecoomcs techcal ecoomcs. -mal: water olluto hder ecoomc growth ad affect socal rogress. s a result voces callg for tegrated water resources maagemet IWRM have bee heard dfferet laces all over the world. I the rocess of tegrated water resources maagemet staeholder volvemet ublc artcato ad mult-staeholder aalyss are usually emhaszed may maagemet domas. s for the cotet of tegrated water resources maagemet t s defed dfferet ways after ts emergece. most commoly cted defto was gve by Global Water Partersh whch ca ofte be read lterature related to IWRM. It was defed that IWRM s a rocess whch romotes the co-ordated

5 70 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet develomet ad maagemet of water lad ad related resources order to mamze the resultat ecoomc ad socal welfare a eutable maer wthout comromsg the sustaablty of vtal ecosystems. [] Ths defto was well elaborated by Bswas [] ad aother relatvely comrehesve defto was cted asers s artcle cf. Va Hofwege & asers 999 [3]. Beg a wdely used tool for mult-staeholder aalyss the cocet of mult-staeholder latform s ofte see lterature ad Warer uaced ad aalyzed that. Meawhle Warer suosed that mult-staeholder rocesses mult-staeholder artershs mult-staeholder dalogues mult-staeholder fora ad mult-staeholder roudtables ca also be used as mult-staeholder latforms f the focus s wde eough [4].. The Imortace of Staeholder Partcato Staeholder aalyss s wdely used may felds ad staeholder artcato ca hel to buld a better relatosh amog staeholders as well as a more harmoous relatosh betwee huma begs ad ature. Ostrom cosdered the role of dfferet eole atural resources solvg the roblem of the tragedy of the commos whch was frstly roosed by Hard [5 6]. Later Ostrom demostrated that strog temtatos of short-ru self-terest ca be overcome wth the hel of recrocty reutato ad trust [7]. Wth regard to water Moss Dowg ad Roucher argued that evtably where water s cocered cocers of staeholders reflect or egeder socal coflct [8]. Therefore solvg coflcts water resources maagemet has always bee a heated toc for research. For a log tme techologcal fes have bee a effcet way solvg urget evrometal roblems but dssatsfacto ad much eedture reures ublc oo whch s also aroused by evrometal awareess [9]. I asers s artcle t was stated that staeholder artcato has become a crucal ssue ad water resources lag wthout staeholder artcato s hghly effectve [3]. I addto Mostert et al. foud that artcatory rocesses ca lead to chages rver-bas maagemet ad beeft all staeholders as well as the evromet. These rocesses creased the uderstadg of ey ssues maagemet heled to buld trust ad mrove relatos establshed ad develoed ew orgazatos [8]. Hemmat cocluded the beefts of Mult-Staeholder Processes the smlar cocet to mult-staeholder latforms MSPs: The ualty of oo-formg ad decso-mag s mroved. Credblty ad moral authorty ca be gaed f doe a eutable trasaret ad democratc way. 3 Peole s commtmet to the outcomes ad mlemetato ca be ehaced. 4 Mutual resect ad tolerace socety are creased ad coflct o cotetous ssues ca be solved more easly [0].. Models ad Methods led Lterature fter may became aware of the sgfcace of staeholder artcato varous models ad methods were aled staeholder artcato ad mult-staeholder aalyss. s early as 985 Hederso ad Schllg used decso suort systems DSS the ublc sector for a commuty metal health system as a eerece ad ths DSS ca also suort other decso rocess the ublc sector []. The DSS ecomasses a multle obectve allocato model as well as a multle arty decso rocess. Moss Dowg ad Roucher aled smulato modelg for water demad olcy ad resose cosderato of staeholder artcato []. I addto by usg The Itegrated Systems for Kowledge Maagemet ISKM as framewor collaboratve learg ad formato sharg rocess was coducted [3]. I the research coducted by Borsu et al. may meas were aled such as lterature searches hoe tervews ersoal tervews ublc meetgs

6 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet 7 wrtte surveys or uestoares [4]. They reled o a robablstc model whch s called robablty etwor ad ths dects robablstc relatoshs amog ucerta varables. Hamalae et al. used modelg aroaches to grou decso suort based o Multle Crtera Decso Mag MCDM ad value tree aalyss Pareto-otmalty ad cosesus seeg were aled ad role layg eermets were coducted to evaluate ad mrove the models [5]. Besdes Turcotte ad Pasuero troduced the method of Mult-staeholder Collaboratve Roudtable MCR ad cosdered the outcomes three asects: cosesus learg ad roblem solvg caactes [6]. Pahl-Wostl metoed that the focus grou a mortat ad ovel method tegrated evrometal assessmet s wdely used ublc oo research ad maretg [7]. Pahl-Wostl also troduced aget-based models BMs [7]. Wag et al. utlzed a cooeratve game aroach water allocato ad acheved effcet use of water through water trasfers by reallocato of water [8]. I the case of South frca Smugwe eamed the cetral ssues of Catchmet Maagemet Forums CMFs whch are oe of the ew sttutoal forms beg referred to as MSPs. Desto research secodary data collecto formal survey tervews artcat observato staeholder aalyss ad worshos were used as maor methods ad techues [9]. These methods rovde remarable sghts for the alcato of staeholder artcato varous stuatos esecally for water resources maagemet. Some meas ca be adoted the research but at the same tme alcato hghly reles o the secfc scearo whch reures modfcato ad adatato utlzg them ad ovato for further develomet.. The Framewor of alyss The ma scearo ths aer s a water roect for eamle a water dverso roect or a desalato roect whch s oe maor doma water resources maagemet ad s esecally mortat for Cha may areas of whch water shortage has become a severe roblem. The etre roect dscussed here ca be dvded to three arts: rearato for the roect costructo of the roect ad oerato mateace ad reovato after comleto. I the water roect ams two asects are to be fulflled. Oe s eole s harmoous relatosh wth the evromet a basc rcle mag the roect more sustaable the future. fter logtme develomet t s commoly acceted that ature s a dsesable foudato for huma develomet ad ay d of large roect should cosder the mact mosed o the evromet. If the roect ca ot get alog well wth the surroudgs t ca ot be sustaable the log ru. other s the ecoomc ad socal asect. Publc artcato s creasgly mortat decso mag ad the smle but dffcult target s to let all staeholders o the decso rocess ad fally ga from the roect at least ot worse off. Oe reaso for ths s that more artcats may geerate more deas ad otetally erches rocess substace [0]. I addto wllgess as well as accetace of the staeholders heavly flueces the costructo ad mateace of the roect from the mcro ersectve ad the socal welfare ad socal stablty a much wder rage. I a more comrehesve way Global Water Partersh llustrated rcles for effectve water goverace whch reure aroaches to be oe ad trasaret clusve ad commucatve coheret ad tegratve eutable ad ethcal; erformace ad oerato should be accoutable effcet resosve ad sustaable []. These should be well tae to accout water roects. I ths aer the emhass s maly ut o the ecoomc asect. The aalyss of staeholders water resources maagemet for ths aer hghly emhaszes o the strategc behavors of staeholders. lthough

7 7 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet commoly used methods MSPs are ecessary to be adoted such as wrtte surveys or uestoares tervews collaboratve learg ublc meetgs formal or formal vsts ad formato sharg the most mortat techue the aalyss s the use of both cooeratve games ad o-cooeratve games. Frst of all water resources maagemet volves may staeholders from dfferet sectors whch maes t more dffcult to fd eulbrum for mult-staeholder aalyss cosderg comrehesvely the terests of varous staeholders. ust wth the frst level of staeholders ths secfc scearo there are staeholders from the govermet resdets dustres ad other ossble staeholders ot to meto the secodary level or eve the tertary level. I ths sese t s clearer ad more coveet to use game theory aroaches to aalyze strategc behavors of staeholders. Besdes cooeratve games ad o-cooeratve games are both aled. The overall roect s a comlcated ad log-lastg tas the aalyss of whch s dversfed for varous asects durg the rocess. Hece both cooeratve games ad o-cooeratve games are to be cosdered. I addto water resources maagemet ofte lasts for a log tme ad stuatos ofte chage after more formato s dsclosed ad artcats get to ow each other better. Therefore mult-stage aalyss s eeded so as to reflect the dyamc chages as tme goes o. O the whole the framewor of the aalyss s a mult-stage mult-staeholder mult-doma mult-factor ad mult-target game aalyss rocess. Wth the etre rocess fve maor aroaches are of great sgfcace thus eedg elaato. Before elag the fve maor aroaches detal some geeral deftos are to be llustrated: The set of layers s llustrated by N ad N { : } where dvdual N. Ths shows that there are staeholders the model. They rereset dfferet staeholders from the govermet sector resdets dfferet dustres ad other ossble ettes. It s to be oted that govermet ageces are volved as staeholders ths model to rereset the admstratve factors. lthough some of them stll hel to coordate varous staeholders ths roect ther ow beefts ad losses are tae to accout the model. For each staeholder the set of strategc behavors s S ad oe strategc behavor s s S. The etre sace of strategc behavors s S S S S. 3 The sace of formato s Θ { }. s secfcally ow by dvdual for ad caot be drectly revealed to other staeholders. I the followg fve maor stages wth the strategc aalyss are gve ad a bref llustrato of the fve stages as well as mechasms ad ossble models that may be aled ca be foud Table. Dsclosure of formato. Wthout owg the actual formato of the staeholders t s hard to determe how much each staeholder should ay for the roect. Hece uderstadg how to reveal the rvate formato of each staeholder s ecessary. I ths rocess the obectve s to mae each staeholder tell the truth ad reveal hs or her real reveue ad loss. Table Fve Stages of Staeholder alyss. Stage Mechasm aled Possble model Dsclosure of formato Mechasm for tellg the truth The revelato rcle Negotato mechasm Negotato mechasm Nash egotato model Icetve mechasm ad regulatory Prcal-get model cetve Icetve/ealty/regulatory mechasm mechasm mechasm desg otmal cotract Iterest comesato mechasm Clare-Groves mechasm Free-rdg model ublc goods terest comesato Trust ad sustaable develomet Sustaable ad cooerato mechasm Trust games dyamc/reeated games evolutoary games reutato model

8 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet 73 By dog so t becomes ossble to let staeholders bear ther costs a relatvely far way ad fally ga beefts from the oerato of the roect. Ths belogs to o-cooeratve games ad ca be realzed maly by the revelato rcle. Negotato mechasm. Durg the rearato erod of the roect geeral cosesus should be made before costructo. I the costructo erod ad mateace stage whe roblems arse egotato mechasm ca also be effectve. I ractce meetgs ad worshos ca be held to get staeholders together ad dscuss roblems that aear the roect. Durg ths rocess formato ad deas are echaged through commucato ad eole wll adust ther eectatos order to reach agreemet. Several rouds of egotatos may be volved as rogress s made ste by ste. s was cocluded by Ravborg ad Westerma ot learg rovdes a crucal ad ofte essetal bass for solvg or ameloratg roblems ad thrd arty facltato lays a mortat role staeholder detfcato brgg coflcts ad terdeedeces to oe ad facltatg egotatos []. Moreover the uderstadg of the goverace ad cultural systems ad the way they are structured ad maaged s also oe ot to be coscous of [3]. I egotato mechasm collaborato s uderled ad the egotato model ca be aled as the bass. 3 Icetve mechasm ad regulatory mechasm. The cetve mechasm ca hel to realze that what the staeholder wats to behave s ust what the roect or the socety wshes to see. By desgg a roer mechasm the goal of the roect ca be acheved automatcally by the dvdual otmzato of each staeholder although t s ofte a secod best soluto as a whole. I the aalyss dvdual ratoalty ad cetve comatblty should be tae to cosderato. Idvdual ratoalty esures that the artcats are better off ad each oe s wllg to o the roect. Icetve comatblty s the oe that maes the dvdual volutarly chooses what the socety wats to see. But whe reachg ths outcome dvduals ca get etra reveue due to ther rvate formato. Ths s why the best soluto for the ublc s almost mossble to be obtaed wth the cetve mechasm. O the other sde regulatory mechasm wll force staeholders to ee ther romses ad mae the egotato rocess more effectve. By suervso costrats ad ushmet set by the ublc are good sulemet to cetve mechasm. They lmt the behavors of staeholders ad s coducve to reachg the ot where socal beefts are attaed at the hghest ossble level. 4 Iterest comesato mechasm. Iterest comesato mechasm s the creatve ot ad the essetal toc ths aer whch s the ey to the success of a roect. It wll be elaborated a model the followg secto. Based o the aroaches show above ad uder certa assumtos t s to be roved that as log as the total reveue of the roect s more tha the total cost t s ossble to fd a soluto whch s acceted by all staeholders. I ths way Pareto mrovemet ca be acheved wth oe roud or several rouds of terest comesato. ll staeholders wll ether be better off or stll hold ther reservato utlty. 5 Trust ad sustaable develomet. I the oerato mateace ad reovato stages after the comleto of the roect trust amog staeholders s the most mortat factor for a sustaable develomet. Wth o trust thgs ofte get worse tha they ca be whe eole truly trust each other. eamle that s usually metoed s the moetary olcy [4]. Whe the authorty betrays the ublc ad chooses hgh flato t loses ts authorty eeg flato at a low rate ad s less beleved by the ublc. t ths tme oe moetary olcy that ams to stmulate the ecoomy may be effectve ad trggers hgh flato mag the stuato eve worse. Reutato model ca be used solvg the roblem ad dyamc game s troduced. The rocess volves adustmet of fath whch s determed by revous behavors

9 74 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet coducted by each staeholder. I addto to rewards ushmet s also ecessary to ehace trust ad mae the roect more sustaable after comleto. mechasm ought to be desged the way that staeholders are aware that eeg romses s better tha volatg what s agreed o thus reachg a better soluto ad buldg trust amog eole. These fve aroaches are closely coected to each other ad cover the etre rocess of the roect from the rearato stage to the oerato ad mateace stage after comleto. From the techue ersectve both cooeratve games ad o-cooeratve games are utlzed ad oe aroach may rovde roer techue for aother. For stace the terest comesato mechasm s the core elemet the aalyss whch has to be based o the frst three aroaches esecally the dsclosure of formato the cetve mechasm ad the regulatory mechasm. Trust ad sustaable develomet s formed uo the results of rearato ad costructo stages of the roect ad s asssted by rewards ad ushmet methods derved from the regulatory mechasm. 3. Model for Comesato Mechasm The model corresods to the fourth aroach the last secto ad maly ales ad eteds the Clare-Groves mechasm whch rovdes a mechasm that maes staeholders tell the truth mag decsos for ublc goods [5-8]. I establshg the model some dffereces are to be cosdered. I Cha both admstratve factors ad ecoomc mechasms have sgfcat fluece o the strategc behavors of the staeholders ad ure ecoomc model wthout cosderg the secfc stuatos of Cha may ot be feasble sometmes. The mechasm of formato dsclosure cetve mechasm ad egotato mechasm metoed the revous secto ca be aled solvg roblems of deedet dvduals whereas staeholders from Cha have artcular characterstcs resulted by the two factors thereby eedg further dscusso a ew scearo. Hece terest comesato mechasm wth comarso effect of strategc behavor amog staeholders s the ey method the aalyss of staeholders ad the behavors of govermet sectors are volved whe both admstratve factors ad ecoomc mechasms are tae to accout. d the other mechasms ca lay a role o the bass of the terest comesato mechasm. I addto revous theores ofte aalyze staeholders a relatvely deedet way ad do ot cosder the effects mosed by the amout of trasfer gve to the other staeholders. Whe oe dvdual ows the amout of trasfer receved by the others ths dvdual may comare that amout to hs or her ow. lthough sometmes the trasfer s large eough after oe owg that what he or she receves s less tha what aother erso gets ths erso wll be dssatsfed. I ths sese the utlty fucto s ot oly determed by the utlty of the roect ad the absolute amout of the trasfer but t s also affected by the comarso amog dfferet dvduals. Ths d of effect s defed as comarso effect the aer ad wll be added the model volvg comarso effects. 3. Deftos I the model for comesato mechasm besdes those deftos llustrated above that ca be used all fve stages a few deftos are to be gve for ths model: The fal la of the ublc roect s whch have may alteratves ad s determed by the strategc behavors of each staeholder s for. The utlty of staeholder s U ad t s assumed to tae a uas-lear form. The utlty fucto s eressed as U e t v + e + t 3. for where s the tye of staeholder ;

10 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet 75 v s the utlty gaed from the roect ad s determed by the fal la ad the tye of the staeholder ; e s the tal edowmet of dvdual or the reservato utlty whch ca be thought as the utlty of holdg a certa amout of currecy or the utlty gaed from rvate goods bought by the same amout of currecy ad e ca be see as a costat whch wll ot have ay fluece o the mamzato rocess; t s the trasfer from the roect to staeholder ad the trasfer ca be ostve egatve or ust eual to zero but the total trasfer caot be ostve sce bas are also cosdered as staeholders ad t s assumed that there s o eteral source to rase fuds ad s deoted as t 0. I the aalyss t s assumed that t Model aalyss For the etre roect the total reveue s the sum of the beefts gaed by each dvdual who s better off from the roect. s for the total cost besdes the costructo ad oerato costs of the roect losses caused by the roect to those who are worse off should also be cosdered. Staeholders beefts ad losses are rereseted by v for as was metoed the defto art above. The goal of the roect s to mae everyoe volved or affected become better off or at least do ot get worse off mata ther reservato utlty amely a Pareto mrovemet stuato. I ths sese a basc codto s that the total reveue of the roect must eceed the total cost of t but ths codto s ot suffcet sce those who eoy the beefts ad those who burde the costs are ot the same staeholders. Therefore moetary trasfer s eeded amog staeholders. However the uesto s how much the trasfer should be for each staeholder. Hece a mechasm for terest comesato s reured as a way to aswerg the uesto. It s suosed that f the tye of each staeholder were ow t would be much easer to solve the roblem accordg to v. Whereas realty the tye of staeholders for or the formato s et by each dvdual ad t s lely for them to sea out a utrue tye to the ublc order to ga more from the roect ad ths s deoted as s v. I the followg aalyss t s to be roved that there ests a mechasm that drves staeholders to behave the same way as reured for achevg socal mamzato. For the etre roect the total reveue mus the total cost ca be eressed as v C 3. where v cotas both reveues ad losses caused by the roect to varous staeholders ad C s the costructo ad oerato costs of the roect. uato 3. s the ayoff fucto of the etre roect the mamzato of whch dcates the socal otmzato. However as the ublc does ot ow v ad oly ows s the mamzato has to be doe wth s C 3.3 The ey to the mechasm s the establshmet of the trasfer fucto: t s C + rs 3.4 for where r s s a fucto that ca be foud a secfc crcumstace to mae the trasfer fucto sutable for the mechasm ad the reuremet s that ts chages have o relevace wth v or s oly deedg o the strateges of staeholders ecet dvdual. Therefore the utlty fucto of staeholder becomes derved from uato 3. ad 3.4 U e t v + e + t v + e + s C + rs 3.5 I the rght sde of the euato e ad C are costat s ad r s are decded by other dvduals rather tha staeholder. Hece the oly

11 76 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet elemet that deeds o dvdual s choce s v. Loog bac at uato 3.3 the mamzato rocess that s coducted by the ublc f the dvdual wats to mamze ts ow utlty fucto the roer strategy should be tellg the truth that s s v. The utlty fucto ca be chaged to U e t e + s C + rs 3.6 I ths euato staeholder s decso ad e ad r s do ot deed o s C s mamzed by the ublc. s a result tellg the truth s chose by staeholder to mamze ts utlty fucto. I ths way each staeholder for chooses the strategy tellg the truth ad uato 3. s acheved rovg that ths mechasm s able to solve the roblem. 3.3 Model Ivolvg Comarso ffects Based o the model above a small chage ests after the comarso effects of staeholders are troduced whch ca show the utlty chage due to the comarso of trasfer wth the other staeholders esecally those dvduals who are ale. The ew utlty fucto s eressed as U e t t v + e + t + com t t 3.7 Where com t t reresets the comarso effects ad vares from erso to erso. I geeral f staeholder fds the trasfer t s smaller tha the trasfer of the other staeholders who have smlar characterstcs ad fds the trasfer s ot suffcet after comarso ths dvdual wll get egatve utlty comarso effects that s com t t < 0; vce versa com t t > 0. Net cosder the etre roect. Sce the comarso effects chage the utlty of dvduals the total reveue mus the total cost becomes v + com t t C 3.8 The stuato s smlar that the ublc does ot ow v ad com t t. The ublc oly ows the strateges chose by each staeholder. The strategy does ot oly clude cosderato of v but t also reflects com t t. The strategy chose by dvdual s ow amed as s ' the the mamzato of the ublc has to be doe wth ' s C 3.9 The trasfer fucto s chose as t s C r s 3.0 ' ' ' + ' ' for where r s s a fucto that ca be foud a secfc crcumstace to mae the trasfer fucto sutable for the mechasm ad the reuremet s that ts chages have o relevace wth ' v com t t ad s oly deedg o the strateges of other staeholders. Therefore the utlty fucto of staeholder becomes derved from uato 3.7 ad 3.0 ' ' ' C + r s 3. U e t t v e com t t s I the rght sde of the euato e ad C are ' costat s ' ' ad r s are decded by other dvduals rather tha staeholder. The elemets that deed o dvdual s choce are v ad com t t. Loog bac at uato 3.9 the mamzato rocess coducted by the ublc f the dvdual wats to mamze ts ow utlty fucto the roer strategy should be tellg the truth that s s ' v + com t t. The utlty fucto ca be chaged to ' ' ' U e t e + s C + r s 3. I ths euato e ad do ot deed o ' ' r s staeholder s decso ad ' s C s mamzed by the ublc. It s roved that ths mechasm s also cetve comatble whe comarso effects are volved.

12 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet Polcy Recommedato from the Comarso ffects From the sychologcal sde whe eerecg the same amout of chages dfferet drectos a erso s utlty chage s ofte larger the crcumstace of losses tha gas. For eamle the degree of sadess caused by a loss of 00 uts of currecy s usually larger tha the degree of haess wth a ga of 00 uts of currecy. s for comarso effects t may be atural to reach the same cocluso. I the scearo of two eole who are eactly the same characterstcs erso receves more trasfer tha erso B. fter comarso the crease of erso s utlty by the comarso effect s smaller tha the decrease of erso B s utlty by the comarso effect ad the total utlty chage caused by comarso effects s egatve. Ths ca be eteded whe more eole are cosdered. I coseuece com t t < 0 cosdered uato 3.8 f eualty ests. s the degree of eualty creases the value of com t t becomes smaller thus decreasg the total utlty of the etre roect. I cocluso whe comarso effects are cosdered eualty terest comesato amog dfferet staeholders reduces the total welfare of the roect ad eualty should be avoded as much as ossble. I ractce comarso effects are more obvous amog resdets tha the other staeholders ad ths wll be aalyzed the alcato later. 3.5 Cocluso of the Models From the aalyss above t s roved that the terest comesato mechasm roosed s a cetve comatble mechasm that maes each staeholder volutarly choose to behave accordace wth the terest of the etre roect whe they are actually mamzg ther ow utlty fucto. It should be oted that wthout owg the tye of each staeholder the ublc roect s fally chose as f the refereces of all staeholders were ucovered ad the roect were determed for socal mamzato as a whole. I addto durg the etre rocess the most mortat techue s the establshmet of the trasfer fucto whch s reured ot to be deedet o the strategy of staeholder. Meawhle the utlty fucto ad the socal mamzato target are somewhat overlag whch artly results a cetve comatble stuato of the dvduals. Wth the models the admstratve factors ca be aalyzed by the volvemet of govermet sectors ad the comarso effects are cosdered the secod model. 4. lcatos 4. I a Water Proect Now cosder a water roect for eamle a water dverso roect. Staeholders ca be dvded to four categores: the govermet sector resdets dustres ad other ossble staeholders. I each category there are dfferet staeholders the secodary level. Classfcato ad behavoral characterstcs wll be dscussed brefly ad a rough llustrato of classfcato of staeholders s show Table. 4.. Govermets The frst rmary category of staeholders s govermets. For the govermet sector t s classfed to e staeholders the secodary level: cetral govermet mucal govermet relevat dstrct ad couty govermets rver bas maagemet agecy Mucal Bureau of Water Resources Mucal Bureau of vrometal Protecto legal ettes of water resources roects ettes for roect route maagemet ad water suly comaes. The cetral govermet s resosble for mag olces laws ad regulatos for macro level water resources maagemet ad the mucal govermet has the smlar dutes a more mcro level. lso relevat dstrct ad couty govermets have eve more mcro resosbltes. Ther utlty fuctos are ale asects of facal come ad eedture

13 78 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet Table Classfcato of Staeholders. Prmary staeholders Govermets Resdets Idustres Other ossble staeholders Secodary staeholders Cetral govermet mucal govermet relevat dstrct ad couty govermets rver bas maagemet agecy Mucal Bureau of Water Resources Mucal Bureau of vrometal Protecto legal ettes of water resources roects ettes for roect route maagemet ad water suly comaes Urba resdets rural resdets as water users ad mmgrats caused by buldg the roect The frst dustry the secod dustry ad the thrd dustry Bas meteorologcal deartmets meteorologcal deartmets research sttutes ad o govermet orgazatos NGOs shares ower satsfacto of resdets ad eterrses as well as socal harmoy ad socal stablty. The mucal govermet ad relevat dstrct ad couty govermets are also flueced by aroval or crtc from the uer level govermets. The utlty fuctos of other govermet ageces that are to be aalyzed are also deedet o the evaluato from the uer level govermet sectors dfferet degrees ad ths ot wll ot be reeated the followg aalyss. The rver bas maagemet agecy s the reresetatve agecy of the Mstry of Water Resources. It mlemets water maagemet for maor water bass but has o rght to eact laws. Its target s to coordate dfferet staeholders ad follow the structos of the cetral govermet. Mucal Bureau of Water Resources tegrates three maor sectors water rrgato water suly ad draage allocates water resources ad acheves ufed maagemet of the cty s water affars. Besdes Mucal Bureau of vrometal Protecto s resosble for lag mlemetato maagemet ad cotrol of evrometal olluto ad atural resources coservato. These two ageces ay more atteto o the harmoous relatosh betwee humas ad ature. Legal ettes of water resources roects are set u for the costructo erod ad are resosble for roect ualty safety schedule fudg ad the use of fuds. ttes for roect route maagemet s resosble for maagemet surveyg cludg surveyg ad mag desg suervso ad costructo etc. The last cosdered agecy s water suly comaes. They are resosble for costructo oerato ad maagemet of water suortg roects ad rovde water suly to users. t the same tme they ca cover the costs by chargg water users ad may receve trasfer from the local govermets. Here the water comaes are suosed to be govermet cotrolled comaes for the coveece of aalyss but more ad more water comaes are becomg rvate-owed eterrses. For govermet sectors v s the ma cocer ad s determed by ther refereces to the outcomes of the water roect. For stace the cetral govermet may gve the satsfacto of eole ad socal stablty hgh weghts. t may be govermet trasfers betwee dfferet govermet sectors or may be trasfers to resdets or dustres. t ca also be gas from secfc taes or ealtes o some staeholders. 4.. Resdets Resdets are roughly dvded to urba resdets rural resdets as water users ad mmgrats caused by buldg the roect accordg to ther dfferet behavors geeral. lthough water users are dfferet eve wth the urba area or the rural area they are ot further dvded for the coveece of aalyss. If t s aalyzed a more secfc way more tyes of resdets ca be formed whch wll mae the aalyss more le the realty. Urba resdets may actually ay more taes due to the costructo of the water roect sce large frastructure roects ofte get subsdes from the govermet ad ths s essetally ad by ta ayers. Urba resdets mght also have to ay more for hgher water rce but they ca beeft from better water trasort more stable water suly hgher water ualty ad flood cotrol

14 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet 79 etc. ll of these cosderatos ca be ut to the utlty fucto affectg v ad t. Rural resdets may suffer from taes ad creased costs from hgher water rce but beeft from the mrovemet of rural water suly whch wll add to ther utltes. Immgrats that are forced to leave ther home lose utlty for both ecoomc ad metal reasos but they should be comesated wth moetary trasfer ad other comesatos whch ca be calculated currecy form such as ew houses or aartmets a better lace for lvg ew worg oortutes etc. Comarso effects are esecally obvous resdets ad ought to be focused uo emrcal studes. Frst of all varous resdets may have dfferet fucto tyes for com t t. Some may feel etremely aoyed by eualty whe ther trasfer s less tha that of the others ad com t t taes a mortat role the utlty fucto; whereas some may ot be affected severely by the comarso effects. Therefore emrcal aalyss methods from eermets ad surveys may be utlzed fdg dfferet tyes of staeholders ad ther roorto a certa area. I cosderato of the three tyes of resdets mmgrats caused by buldg the roect eed the most cocer. They are more sestve to eualty ad may reure hgher terest comesato f the roblem of eualty s serous. O oe had hgher terest comesato maes the roect harder to realze ad oerate. O the other had some socal roblems ad coflcts ofte arse whe the roblem of eualty s ot treated a rght way Idustres Idustres ca be roughly classfed to the frst dustry the secod dustry ad the thrd dustry a less detaled way. Wthout suffcet water farmlads may suffer from droughts whe rafall s small or uevely dstrbuted dfferet tme of the year or areas. But whe a relable water suly system s establshed t wll hel the agrcultural dustry tmes of droughts. lthough roducto costs may rse due to hgher water rce as well as some costructo costs for water facltes f t s ecessary staeholders wth the frst dustry beeft from eased water ressure ad the overall ecoomc beefts are ofte mroved. s for the secod dustry ad the thrd dustry they also ga from eased water ressure ad are able to roduce roducts ad servces a more relable way deste ossble rse water rce. For dustres the comarso effect may ot be as sgfcat as that of the resdets. They ay more atteto o the roducto rocess ad roft mamzato ad com t t does ot tae u a hgh ercetage ther utlty fucto. For dfferet dustres ad dfferet sectors wth a dustry varous rces should be charged accordg to ther amout of water cosumto ad eteralty as well as the cocet of water savg are to be emhaszed Other ossble staeholders Other ossble staeholders may clude bas that offer loas ad other facal servces for the water roect meteorologcal deartmets geograhcal deartmets research sttutes ad o govermet orgazatos NGOs cocerg dfferet domas for eamle olluto reveto evrometal chages amal rotecto or socal euty. Bas get reveues from terests of loas ad cosultg fees for servces but also tae rss for ledg moey to the water roect. Meteorologcal deartmets geograhcal deartmets ad research sttutes rovde hel the arasal of the water roect ad set rgd costrats wth the hel of relevat techology ad ther advce s good for rs cotrol ad may reduce costs ad allevate ossble coflcts for the roect. No govermet orgazatos NGOs are becomg a dsesble art for roblem solvg socal lfe. They ca be see as thrd artes that ofte suervse the rocess uhold the terests of a certa grou ad fd roblems a more deedet ersectve. I the etre rocess of the water roect fve aroaches troduced above ca be used for dfferet stages of the roect from the rearato erod of the

15 80 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet roect the costructo rocess to the oerato mateace ad reovato stages after the comleto of the water roect. I the terest comesato asect the most crtcal cocer s the trasfer fucto of each staeholder whch ought to be set the way demostrated the model secto ad s deedet from the staeholder s ow strateges. For some staeholders comarso effects have a relatvely hgh weght ad should be ad much atteto o. 4. Other Possble lcatos art from the water roect le the water dverso roect or a desalato roect the fve aroaches dscussed the aer esecally the terest comesato mechasm ca be tae to ractce may areas related to ublc or uas-ublc goods ad servces. For stace frastructure roects are smlar to the water roect aalyzed the aer ad ths ca be used the same way such as electrcty or eergy lats ad etwors commucato facltes hghways ad ralways cty ublc trasortato systems etc. Besdes evrometal rotecto ad atural resources rotecto ca also get hts from the mechasms. Wth a certa rage staeholders eed to ay for costs of rotecto ad cofe ther ow behavors but wll beeft from better evromet ad the use of the mroved evromet ad atural resources. I ths rocess the terest comesato mechasm s ut to ractce ad gets everyoe better off or at least ot harmed from the rogram. I addto other ossble alcatos are basc cultural ad educatoal servces scetfc ad techologcal develomet the goverace of maret dsorders rotecto of tellectual roertes at-corruto [9] as well as other domas related to eteraltes ad formato asymmetry. lthough the rage of alcato s vast t should be oted that due to tme costs ad effort costrats the alcato of the roosed fve aroaches has t bee ut to ractce for the tme beg ad may ecouter dffcultes ractce thereby eedg further emrcal research ad aalyss the future. 5. Cocluso s s llustrated the troducto secto the role of tegrated water resources maagemet ad staeholder artcato s of great mortace. fter gettg sghts from some lterature o ths ssue some methods ca be aled strategc behavor aalyss of staeholders water resources maagemet. Besdes the alcato of game theory ca be see as a alteratve or a sulemet. I the scearo of a water roect two ams are to be accomlshed. Oe s eole s harmoous relatosh wth the evromet whch s a basc rcle mag the roect more sustaable the future. The other am s cosdered the ecoomc ad socal asect whch emhaszes ublc artcato ad wats to mae all staeholders o the decso rocess ad fally ga from the roect or are at least ot worse off. I order to acheve these two ams fve aroaches from game theory are reseted ad the fourth aroach whch s o terest comesato mechasm s elaborated a model. There are two categores wth the rovso of ublc goods: oe s the secfc tems or roects such as street lghts ad ars; ad aother s the oe ublc goods whch ca be usecfc thgs or materals le evromet ad sttutos the border ssues of whch such as the beefcares utltes costructo costs ad evaluato crtera are vague. Meawhle these may also vary wth chages of artcats or staeholders as well as ther orgazatoal forms ad the margal utlty may crease as more artcats are volved. I ths aer the ssues of IWRM are more cled to have features of the latter tye ad the comesated mechasms desged for IWRM ca better solve the effectve suly roblem of ublc goods by attractg more artcats ad arousg the ethusasm of varous staeholders to the mamum etet.

16 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet 8 By alyg the Clare-Groves mechasm oe terest comesato mechasm for ublc roect s foud that maes each staeholder tell the truth ad fally realzes socal otmzato. The ey elemet s the establshmet of the trasfer fucto whch motvates the staeholders to behave accordace wth the terest of the etre roect or the socety. Sce the admstratve factors are mortat esecally Cha these ca be aalyzed by the volvemet of govermet sectors wth the models. Meawhle a model cludg the comarso effects are cosdered whch hghlghts the utlty chage caused by staeholders comarso the amout of terest comesato. Thus the framewor of the aalyss s a mult-stage mult-staeholder mult-doma mult-factor ad mult-target game aalyss rocess ad s eteded the comarso betwee staeholders. s for the alcato of the mechasms a water roect s aalyzed ad the classfcato as well as strategc characterstcs s stated. Meawhle the fve aroaches esecally the terest comesato mechasm ca be used other domas relevat to ublc or uas-ublc goods ad servces. But t should be oted that the alcato has t bee ut to ractce for the tme beg ad may ecouter dffcultes ractce thereby eedg further emrcal research ad aalyss the future. cowledgmets We tha the Natoal Basc Research Program of Cha 973 Program Seral No. 0CB95580 ad Cha-Netherlads ot Scetfc Thematc Research Programme STP alyss of Water Resource Maagemet Mechasms o Socal-coomc ad vrometal Regoal Develomet-- Comaratve Case Study Research of Ta Cty Cha ad Rotterdam Cty The Netherlads. Refereces [] Global Water Partersh Itegrated water resources maagemet Global Water Partersh GWP Techcal dvsory Commttee Bacgroud Paer No []. K. Bswas Itegrated water resources maagemet: a reassessmet Water Iteratoal [3] F. G. W. asers Isttutoal arragemets for tegrated rver bas maagemet Water Polcy [4]. F. Warer Mult-staeholder latforms: tegratg socety water resource maagemet? mbete & Socedade [5]. Ostrom Goverg the commos: the evoluto of sttutos for collectve acto Cambrdge Uversty Press 990. [6] G. Hard The tragedy of the commos Scece [7]. Ostrom behavoral aroach to the ratoal choce theory of collectve acto: resdetal address merca Poltcal Scece ssocato 997 merca Poltcal Scece Revew [8]. Mostert C. Pahl-Wostl Y. Rees B. Searle D. Tàbara. Tett Socal learg uroea rver-bas maagemet: barrers ad fosterg mechasms from 0 rver bass cology ad Socety [9] C Pahl-Wostl Towards sustaablty the water sector--the mortace of huma actors ad rocesses of socal learg uatc Sceces 64 00a [0] M. Hemmat The world commsso o dams as a mult-staeholder rocess: some future challeges Poltcs ad the Lfe Sceces []. C. Hederso D.. Schllg Desg ad mlemetato of decso suort systems the ublc sector MIS Quarterly [] S. Moss T.. Dowg. Roucher Demostratg the role of staeholder artcato: a aget based socal smulato model of water demad olcy ad resose Cetre for Polcy Modellg Dscusso Paers CPM Reort No [3] W.. lle O.. H. Bosch M.. Klvgto D. Harley I. Brow Motorg ad adatve maagemet: addressg socal ad orgasatoal ssues to mrove formato sharg Natural Resources Forum [4] M. Borsu R. Cleme L. Magure K. Rechow Staeholder values ad scetfc modelg the Neuse Rver watershed Grou Decso ad Negotato [5] R. Hamalae. Kettue M. Marttue H. htamo valuatg a framewor for mult-staeholder decso suort water resources maagemet Grou Decso ad Negotato [6] M.-F. Turcotte. Pasuero The arado of multstaeholder collaboratve roudtables The oural of led Behavoral Scece [7] C. Pahl-Wostl Partcatve ad staeholder-based olcy desg evaluato ad modelg rocesses Itegrated

17 8 Game Modelg ad Strategc Behavor alyss Publc Goods Provso: vdece From Water Resources Maagemet ssessmet 3 00b 3-4. [8] L. Z. Wag L. Fag K. W. Hel Water resources allocato: a cooeratve game theoretc aroach oural of vrometal Iformatcs [9]. Smugwe Water staeholders ad commo groud: challeges for mult-staeholder latforms water resource maagemet South frca Dssertato at Wagege Uversty Wagege The Netherlads 006. [0]. delebos. H. Kl Maagg staeholder volvemet decso mag: a comaratve aalyss of s teractve rocesses the Netherlads oural of Publc dmstrato Research ad Theory [] Global Water Partersh ffectve water goverace Global Water Partersh GWP Techcal Commttee TC Bacgroud Paer No [] H. M. Ravborg O. Westerma Uderstadg terdeedeces: staeholder detfcato ad egotato for collectve atural resource maagemet grcultural Systems [3] C. Pahl-Wostl. Mostert D. Tàbara The growg mortace of socal learg water resources maagemet ad sustaablty scece cology ad Socety [4] R. Barro Reutato a model of moetary olcy wth comlete formato oural of Moetary coomcs [5]. G. Clare Multart rcg of ublc goods Publc Choce Fall [6] T. Groves Icetves teams coometrca [7]. Mas-Colell M. D. Whsto. R. Gree Mcroecoomc theory Oford Uversty Press 995. [8] G. D. Myles Publc ecoomcs Cambrdge Uversty Press 995. [9] G. Wag O the game mechasm maagemet & suly of ublc goods Coferece aer at WCC&CMGT 004 Qgdao Cha ug

18 oural of Mathematcs ad System Scece D DVID PUBLISHING Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems Fay M. Gozález Barros ad rue Castro Martíez Deartmet Ddactc of Mathematcs Uversty of Graada 807 Sa Receved: November / cceted: December 6 03 / Publshed: February bstract: We vestgated the use of dagrams multlcatve comarso word roblems. The dagrams have bee cosdered as oe of the effectve heurstc strateges or solvg math roblems. However how studets use durg ther school ad the degree develomet that shows ther erformace whe aled to secfc felds of owledge s a tas to be elucdated. We lace our study the school stage whch t maes the trasto from arthmetc to algebra ad arthmetc roblems we focus o the uderlyg multlcatve comarso scheme. I ths aer we aalyzed the resoses of hgh school studets to the traslato of multlcatve comarso word roblems to reresetato grahs. We have used the resoses of -4 year old studets freshma year of secodary school to rereset multlcatve comarso word roblems to detfy ad categorze the studets resoses whch allowed us detfy categores for each tye of reresetato ad hyothesze rorty order ad subordato betwee the categores. Results show that studets are ot famlar wth buldg dagrams that tegrate estg relatos word roblems. Most of the studets do ot use all the uattatve formato cotaed the word roblem therefore draw dagrams referrg to the subect or cotet of the roblem wthout relatg to the data t. We descrbe detal the uattatve dagram tyes roduced by these studets. We have detfed four ds of uattatve dagrams that the studets used to rereset the multlcatve comarso roblems wth cosstet statemets ad these dagrams corresod to the four strateges for taclg the costructo of the dagram. Key words: Reresetatos dagrams multlcatve comarso word roblems.. Itroducto The research commuty the feld of teachg ad learg mathematcs recogzes the usefuless of usg dfferet ds of reresetatos for mathematcal thg ad roblem solvg. Several researchers roose that smultaeous wor wth dfferet tyes of reresetatos allows better cocetual costructo of the mathematcal obects ad mroved roblem solvg []. Ths leaves may oe uestos cocerg the role ad relatosh of the dfferet reresetatos the md of the roblem solver. Dagrams are used varous actvtes of cogtve character such as learg comreheso roblem solvg ad decso mag. Our study focuses o dagrams as a way of Corresodg author: Fay M. Gozález Barros researcher feld research: Numercal Thg. -mal: reresetg the mathematcal relatoshs ad the way whch the studets coceve them whe they solve word roblems verbally. dagram s a vsual reresetato that resets the formato satally []. Dagrams are cosdered to be structural reresetatos whch the suerfcal detals are ot mortat. I roblem solvg a dagram may serve to rereset the structure of a roblem ad t ca thus be a useful tool for uderstadg that roblem. We have stressed the mortace of dagrams the hase of comrehedg or reresetg arthmetc word roblems sce dagrams ca be used to hel uac the structure of a roblem ad thus establsh. The bass for solvg t. Wthout dagrams roblems ca be more dffcult to solve [3-5]. The use of dagrams has bee detfed as oe of the roblem solvg strateges that has bee

19 84 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems cororated as a methodologcal elemet roosals for mrovg the effcecy of solvg mathematcs roblems [6]. Dagrams have also bee used as facltators the roblem-solvg rocess. The resece of dagrams does ot crease studets geeral erformace the solvg of oroute roblems [7]. s a mlcato of the coclusos obtaed ther study suggests that teachers ca gve studets oortutes ot oly to use dagrams but also to vet or fd ther ow strategy for a soluto cludg the actve costructo ad use of dagrams as tools solvg roblems. Varous studes have show that dagrams are more useful whe the artcats vet them tha whe the dagrams are rovded. Studets the frst year of secodary school use a wde varety of grahc strateges sotaeously solvg mathematcs roblems [8 9]. There s ot uamous agreemet o the use of dagrams however sce some study results coclude that sotaeous dagrams geerated by studets are ot always effectve [0] whle other studes dcate that the dagrams costructed by the studets themselves are heurstcs wth great ossbltes for helg stuatos that volve roblem solvg []. Pror studes have ad cosderable atteto to the mact of rovdg studets wth dagrams as a ad ror to the rocess of solvg mathematcs roblems. Studes have also show that dagrams are more useful whe the roblem solvers use them sotaeously tha whe the dagrams are rovded for them. However lttle research has robed what ds of dagrams the studets themselves costruct whe they use them how they use them ad why they decde to use a dagram. We beleve that the use or ot of dagrams roblem solvg s codtoed by the d of roblem tacled. We aalyze ths dea usg word roblems volvg multlcatve comarso wth cosstet statemets []. Multlcatve comarsos word roblems descrbe statc relatoshs betwee uattes that are aouced usg terms such as tmes as may as ad volve three uattes: referet uatty comared uatty ad scalar. y oe of the three uattes may be the uow the roblem. For roblems whch the referet s the uow we say that the word roblem s cosstet whereas roblems whch the comared s uow are called cosstet []. Comarso roblems are cosdered to be oe of the most dffcult roblems for studets [3] a d whch the solvers ted to commt ersstet errors [4]. Because Sa these roblems are ot usually treated elctly as school tass they are ot flueced by the structo receved mag them useful for detectg actos of the studets that are ucoected to ror structo.. Schematc ad Pctoral Reresetatos Use a drawg or a grah to solve a roblem sometmes hel ad other dffcult to get your soluto [0 5]. The usefuless of the reresetato deeds o the ature of the roblem ad the cometece ad slls of the solvers to grah roblems. Grahcal reresetatos of a roblem are ot the same ad deed o the dvdual who erforms t. The lterature detfes two dstct ways of reresetg a roblem schematc ad ctoral reresetato [5-7]. Schematc reresetatos are those that clude relevat formato roblem cotrbutg to the soluto of the same [6]. They cota a schematc drawg of the relatos betwee the amouts rereseted. You ca clude detals but the detals rereset a comoet of the roblem such as the uattes. Note that the schematc drawgs are ot ecessarly smlstc drawgs. Rather the schematc drawgs are a tye of grahcal reresetato showg the amouts may as the relatoshs establshed betwee them o the ssue. O the other had ctoral drawgs ot schematc reresetatos are those that clude elemets or

20 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems 85 surface wthout reflectg roblem aecdotal teral structure mlct relatoshs there. The ctoral drawg focuses o ualtatve ad uattatve asects vsually reflects that are ecessary to solve the roblem. The detals that focus o drawg are ecessary for solvg the math roblem [5 6]. Pctoral reresetatos may drect atteto to roblem sgfcat detals later dverted studet cosderato of the ey elemets of the roblem [8]. There are studets who are slled the schematc reresetatos reresetatos that show the satal relatoshs betwee the obects metoed the roblem. s for the vestgato of the relatosh betwee the tye of vsual reresetato schematc agast ctoral ad successful roblem solvg foud that use of schematc vsual reresetatos assocated wth the successful resoluto of roblems [5]. lso they studed the use of schematc ad ctoral reresetatos a samle of studets wth ad wthout learg dsabltes. They foud success the tas s ostvely correlated wth the use of schematc reresetatos ad a egatve correlato wth the use of ctoral reresetatos [7]. Pror research has eamed the way elemetary studets grahcally rereset mathematcal formato order to solve a roblem. The study sought to determe f studets sotaeously geerated vsual dagrammatc or ctoral maly whe ased use a drawg to hel solve the roblem. Studets usg sotaeously schematc vsual reresetatos were able to solve roblems more successfully tha those who rereset ctorally the elemets reset the roblem statemet verbally. The results dcated studets usg schematc vsual reresetatos were able to solve roblems more successfully tha ctorally reresetg the elemets of the roblem. They foud a sgfcat relatosh betwee the use of schematc vsual reresetatos ad roblem solvg [6]. The results of ths vestgato are agreemet wth the results obtaed revous research [5-7] whch the schematc reresetatos are ostvely related to solvg the roblem.. Str Dagrams ad Word Problems arthmetc word roblem ca be classfed accordg of the tme-dmeso statc or dyamc. Research oted out that dyamc roblems tur out to be easer for chldre tha statc oes [9]. Lear models shaed bad are a artcular case of schematc dagrams ad are beg used currculum develomet dfferet coutres [3]. The statc ature occurs arthmetc word roblem that cotag the art-whole relatosh art-whole scheme or comarso relatosh comarso scheme. These two categores of roblems are modeled wth str dagrams that ow dstctve characterstcs whch reflect ths dstcto from the dagrams that llustrate the tets of Sgaore [3]. Statc roblems of art-whole tye are modeled by a rectagle dagram reresetg the whole whch tur s dvded to arts. For eamle the roblem Mary made 686 bscuts. She sold some of them. If 98 were left over how may bscuts dd she sell? [3]. It s assocated wth the dagram Fg.. I Fg. the umber 686 s the whole ad the art umber s 98. Ths tye of reresetato resects the dmesos assocated wth each uatty. Comarso roblems are rereseted by two arallel rectagular lear bads oe reresets the grou comared to the other the referece grou both for roblems as for dscrete cotuous. s they aear varous documets [3 0 ]. For eamle the followg roblem: Fg. ssocated dagrams to art-whole relatosh a word roblem.

21 86 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems Table Problems ad uestos used ths study. Fg. ssocated dagrams to comarso relatoshs a word roblem. farmer has 7 ducs. He has 5 tmes as may chces as ducs.how may more chces tha ducs does he have? The roblem s rereseted together wth a dagram smlar to Fg.. We ote that ths tye of reresetato of the two comared uattes are laced arallel the frst art of the dagram show the referece uatty ad the secod the uow uatty that s to say the comared uatty. The goal of ths study s to aalyze the models or tye of dagrams emloyg studets solvg word roblems volvg multlcatve comarso wth cosstet statemets. We wll focus o the followg uestos: a Do studets use some d of dagrammatc reresetato to solve comarso roblems? b What ds of dagram do they costruct? c What dfferet ds of uattatve dagrams do the roblem solvers use?. ermetal Secto. Methods total of 89 studets from the frst year of the secodary school two ublc hgh schools the cty of Graada artcated the study. The studets were -4 years old. They had receved o secfc ror structo o ether the costructo of dagrams or solvg comarso roblems. We osed two wrtte uestos to the studets a ad b for two multlcatve comarso roblems wth cosstet statemets see Table. The studets were gve o structos o what d of dagram to mae. The roblems were aled ad solved dvdually a test usg aer ad ecl durg the usual mathematcs class tme. The test was admstered by the researcher wth the teacher of the corresodg class also reset. s for the tye of tass reseted ths study we foud coectos wth wors related to the toc of fractos as those of Ref. [] whch they use word roblems smlar to ours ad they ca be related to the dea of artto sce these authors use two roblem-solvg rocesses tegratg the ut fed from the drve to all ad the decomosto of uty arts movg from whole to arts. 3. Results ad Dscusso 3. Results The 78 aswers 74 % correct ad 6 % correct gve by the studets for secto a of roblems ad were roduced symbolcally. That s they used o dagram or other d of fgure to solve the comarso roblems. Studets have ot used elctly grahc strateges to solve roblems but we are terested owg f they ca grah the multlcatve comarso ad what level of structural comlety acheved ther reresetatos. Thus the followg we reset detal the results obtaed for the secod uesto osed each roblem that s secto b. I secto b of roblems ad the studets were ased to draw a dagram of a roblem stated words. To aalyze the subects roductos resose to ths traslato betwee verbal ad grahc reresetato we establshed ror crtera based o whch we categorzed the aswers. Secfcally the crtera for aalyzg the dagrams roduced by the studets were: a whether the resose cluded a

22 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems 87 drawg or ot; b the degree to whch the studet roductos reflect the data of the roblem ad the relatoshs betwee the uattes. ccordg to these crtera we have observed studet roductos resoses of the followg ds: Wthout drawg Left bla. Verbal reformulato of the roblem statemet. I the form of a schematc eresso. Wth drawg Qualtatve drawg. Quattatve drawg: ad wth ths level whether the drawgs: - Rereset oly the uattes comared or - Rereset the relatosh betwee the uattes. Based o these tyes of resoses we have establshed a classfcato of resoses to fve categores. 3.. Descrto of the Categores We wll ow descrbe the categores establshed beforehad accordg to the studets roductos ths study ad rovde eamles for each of them ecet for the categores wthout drawgs. The descrto of the categores we have made sce less elaborate resoses to those that reflect a greater mathematcal cometece the use of dagrams. Category C : wthout aswer. We have cluded t whch cota o formato drawgs ot aear or left bla. Category C : reformulato of the statemet. I them the studet does ot draw but mae otes or rewrte the roblem statemet. Category C 3 : schematc eresso. The studet wrtes sythetc telegrahc summary form setches the formato that s wrtes hrases that gve a dea of schematc reresetatos. Some of these are smlar to schematc reresetatos for roortoalty. They are vsual ad smle caturg art or all of the formato the roblem statemet. Category C 4 : ualtatve drawg. I ths category we have cluded resoses of the studets whch we see drawgs of eole or obects that refer to the toc or cotet of the statemet. Por eemlo e la Fg. 3 se observa ue el estudate rereseta e forma aralela los dos vehículos mecoados e las catdades de la comaracó tetado mostrar e la mage el tamaño relatvo de las catdades ero o establece la relacó multlcatva etre ellas. Category C 5 : uattatve drawg. These are more comleted or develoed dagrams. They show a drawg that reresets the two uattes cluded the comarso scheme referet ad comared uatty as well as the multlcatve relato betwee them. These uattatve drawgs gve a coheret reresetato of the structure of the roblem. ccordg to the hases of word roblem solvg [3] the roblem solvers have erformed a rocess of traslato ad tegrato of the roblem we therefore call these uattatve dagrams. For eamle the resose show Fg. 4 shows how to use a crcle to rereset the comared the crcle s dvded to as may arts as dcated by the scalar ad arallel comarso shows the referet. I ths case the studet reresetato adats to ow characterstcs of art-whole scheme the detfes as comared uatty to a whole the crcle ad the referece uatty to a art of that crcle. Fg. 3 Qualtatve dagram roduced by a studet for Problem. Fg. 4 Quattatve dagram roduced by a studet for Problem.

23 88 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems dagrame more cosstet wth comarso scheme show fgure 5. Here ether the referece or comared are assocated wth a whole or a defed ut ad shows the ualtatve relatosh betwee the comared ad referece uattes. Therefore ths resose cotas all the features or reuremets of the comaratve scheme. 3.. Freueces of Categores The frst three categores do ot cota llustratos ad ther freuecy s lower tha the other two categores therefore we have groued the freueces a sgle bloc we deomate wthout drawg. Table shows that the subects used uattatve drawgs less freuetly tha ualtatve oes ad that the uattatve drawgs occurred eve less freuetly tha resoses wthout a drawg Relato Betwee Such Processes ad Dagrams We study two tyes of relatoshs. I rcle we aalyze the ablty of studets the costructo of a dagram from a multlcatve comarso roblem statemet verbally relato to success or falure ts resoluto. I the secod art we aalyzed the assocato betwee the effcecy solvg multlcatve comarso word roblems ad the tye of dagram roduced. Table 3 shows together the freueces of the success rate solvg multlcatve comarso word roblems ad buldg dagrams to them. We have obtaed that o sgfcat assocato betwee the varables that mae u the rows ad colums of table 3 that s to say betwee the success rate for solvg the multlcatve comarso word roblems ad the ablty of studets to draw dagrams although the degree of assocato betwee varables s ot very hgh Ph 0.7. Not beg the two deedet varables ad shows the dvdual effect of each varable t s ecessary to stress the ot effects of the two varables I total values see Table 3 shows that oly 3 of studets costruct uattatve dagrams whle 3 Table Freueces for each category. Table 3 Problem solvg versus draw a dagrams stated roblems solved correctly that s to say studets are more successful solvg roblems detfed verbally the costructo of a sutable dagram. The artal assocatos see Table 3 allow a more detaled aalyss whch we ote that there are oly 3 of studets successfully solve roblems ad mae a dagram set rght. Furthermore we observed that 0 of studets are successful solvg the roblems metoed ad do dagrams. Fally 46 of studets fal both to costruct a dagram as to solve the word roblem Tyes of Quattatve Dagrams We aalyzed the uattatve dagrams roduced by the studets to determe the theoretcal models to whch the studets drawgs beloged as well as the strateges emloyed ther costructo. The studets used rectagles ad crcles as basc fgures see Table 4 vsual reresetatos some of whch cota a asect of statstcal dagrams bar ad e charts ad other assocated wth fractoal uattes art ad whole. Table 4 cludes the dfferet theoretcal models that sythesze the dagrams draw by the studets the frst year of secodary educato resose to the reuest that they draw a dagram for the multlcatve comarso roblems. We wll ow descrbe sythetcally the strateges used costructg dagrams D D D3 ad D4

24 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems 89 Table 4 Quattatve dagrams roduced by the studets Fg. 5 dagram D roduced by the studets. whch were foud the artcats aswers. D-dagram: The startg ot that the roblem solvers used to mae ths model s the referece. These dagrams clude two drawgs. The frst reresets the referet uatty as the startg ot ad the secod arses from the frst reresetg the comared uatty where we ca observe the comared uatty a dvso to arts that reflects the scalar. D-dagram: The studets costructed ths dagram model by tag the comared uatty as ther startg ot ad reresetg t. I aother arallel drawg they the made a reresetato that s the euvalet of the drawg of the comared uatty but usg the referece as may tmes as the scalar dcates. D3: Ths d of dagram roduced by the study artcats hghlghts the hel of a aulary eteral measuremet strumet ths case a as smlar to the as of Cartesa coordates whch serves to establsh the vertcal measuremets of the drawgs of both the referet ad the comared uatty. D4-dagram: I ths dagram the studets rereset the referet uatty ad the comared uatty a sgle fgure as arts of a whole whether as a art-whole or a art-art relatosh. Table 5 shows the freueces of the tyes of uattatve dagram tass b ad b. ccordg to the results obtaed the table we see that the studets used lear reresetato more freuetly Fg. 6 dagram D roduced by the studets. Fg. 7 dagram D3 roduced by the studets. Fg. 8 dagram D4 roduced by the studets. Table 5 Freueces of tyes of dagrams.

25 90 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems tha crcular ad that the dagrams that we labeled D ad D3 occur wth greater freuecy tha D ad D4. 3. Dscusso Our frst goal ths aer s to study whether secodary school studets that have dffcultes uderstadg the multlcatve comarso roblems tur to used a dagram. Ths s the urose of the uesto that aears secto a solvg word roblems volvg multlcatve comarso. We eected that some studets used dagrams or models that wll facltate the uderstadg esecally those studets wth correct aswers. Thus try to evaluate ther cometece regardg the ossblty of buldg some d of grahcs. Sce studets have ot bult ay llustrato to hel them solve the multlcatve comarso roblems that we have roosed we beleve that such roblems do ot elct the costructo of dagrams. Studets solve them drectly from verbal reresetato wrtg drectly the mathematcal oerato. It became clear to studets of smlar age [8] whe these studets have reseted other roblems that do elct the use of grahcs; studets sotaeously have bult a varety of grahc strateges roblem solvg. other elcato s that studets have a lac the costructo of grahcal models. The secod ssue that we roosed to the subects of the study was teded to elucdate ths dchotomy ad to demostrate ther level of cometece develog multlcatve dagrams for comarso. To do ths we have ased aragrah b of the uestoare a dagram reresetg the comarso roblems revously roosed. s a Result a thrd of the studets have ot draw aythg so they do ot aear to be cometet to assocate a vsual model to the multlcatve comarso. The remag two thrds f they mae a drawg but the aswers are ot homogeeous. The ualty of the resoses ad ts adatato to a schema comarso are very dfferet. The classfcato of the dagrams roduced by subects led us to dstgush betwee them reflectg o ualtatve asects of the statemet agast whch clude comaratve relatosh o your dagram. The dstcto we have made betwee ualtatve ad uattatve dagrams other authors call ctoral ad schematc ad has rove adeuate for categorzg the reresetatos made by the subects [5-7]. Hghlght schematc reresetatos roduced are two tyes of models: the art-whole whch the referece s made to cocde wth a art of a greater whole see Fg. ad the comarso model whch the referece ad the comarso are draw arallel ad deedet of each other see Fg.. Deste ot havg receved formal structo studets seem to ft the models or rectagular bad dagrams that are used other coutres [3]. 4. Coclusos Oe urose of ths study s to detect whether or ot studets use dagrams solvg arthmetc roblems multlcatve comarso ad what d of reresetatos or dagrams assocated wth multlcatve comarso. I resose to the frst uesto we have obtaed that the studets dd ot use dagrams resose to the reuest that they solve the comarso roblems. Ths result leads us to coclude that multlcatve comarso does ot elct the use of vsual reresetato from the studets. More terestg s what we have obtaed for the secod uesto. The resoses obtaed to the reuest to draw a dagram show the studets lac of vsual terretato of the multlcatve comarso. romately 33% of the resoses do ot cota drawgs 39% are ualtatve drawgs ad oly 8% are uattatve dagrams that show the relatoshs betwee the data the roblem. The latter are the most terestg sce they reflect a reresetato of the relatosh comared wth schematc dagrams or uattatve. Wth the 8% of resoses that cluded a uattatve o schematc dagram the studets oly use rectagular ad crcular forms ad

26 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems 9 they use rectagular forms more ofte tha crcular. These dagrams rereset the comarso models that studets have adated sotaeously wthout havg receved trag about. We beleve that ths result s due to the fact that the studets use the owledge acured revously mathematcs class about fractos ad statstcal grahs to mae the dagrams. Ths would coclude that studets are ot famlar wth the costructo of dagrams that tegrate the relatoshs the statemet ad that they therefore etraolate ther owledge to aother area of mathematcs to rereset the comarso roblems we osed. The aalyss of the uattatve dagrams has led us to detect four ossble strateges for costructg dagrams: Some studets beg the costructo of the dagram by tag the drawg of the referece as ther startg ot ad the a arallel fgure draw the comared uatty whch the value of the scalar s reflected; some studets follow a secod strategy whch they frst draw the comared uatty ad the aother arallel fgure draw the referet uatty as may tmes as the scalar dcates; a thrd strategy cossts of usg a aulary umercal scale for the costructo of the comared uatty ad the referet; fally the fourth strategy followed by studets s to draw the comared uatty ad the referet uatty a sgle fgure. These strateges are reflected the four uattatve dagrams whch we call D D D3 ad D4 resectvely. Refereces []. Cuoco F. R. Curco The Roles of Reresetato School Mathematcs Yearboo 00 NCTM Resto V 00. [] C. M. Dezma L. D. glsh Promotg the use of dagrams as tools for thg : The Roles of Reresetato School Mathematcs NCTM Resto V [3] S. Becma Solvg lgebra ad Other Story Problems wth Smle Dagrams: a Method Demostrated Grade 4-6 Tets Used Sgaore The Mathematcs ducator [4]. B. Lews Trag studets to rereset arthmetc word roblems oural of ducatoal Psychology [5] Y. Uesaa. Maalo S. Ichawa What ds of ercetos ad daly learg behavours romote studets use of dagrams mathematcs roblem solvg? Learg ad Istructo [6] D. Va Gardere Teachg Studets wth LD to Use Dagrams to solve Mathematcal Word Problems oural of Learg Dsabltes [7] M. Patzara. Gagatss I. la Usg dagrams as tool for the soluto of o-route mathematcal roblems ducatoal Studes Mathematcs [8]. Castro N. Morcllo. Castro Reresetatos Produced by Secodary ducato Puls Mathematcal Problem Solvg : Proceedgs of the Twety Frst ual Meetg of the North merca Chater of the Iteratoal Grou for the Psychology of Mathematcs ducato. RIC Clearghouse for Scece Mathematcs ad vrometal ducato Columbus OH Vol [9] D. De Boc L. Verschaffel D. asses W. Va Doore K. Claes Do realstc cotets ad grahcal reresetatos always have a beefcal mact o studets erformace? Negatve evdece from a study o modellg o-lear geometry roblems Learg ad Istructo [0] G. Va sse C. Hamaer Usg self-geerated drawgs to solve arthmetc word roblems oural of ducatoal Research [] K. R. Koedger.. Terao Cogtve tas aalyss of usg ctures to suort re-algebrac reasog : Proceedgs of the Twety-Fourth ual Coferece of the Cogtve Scece Socety Lawrece rlbaum ssocates Mahwah N []. B. Lews R.. Mayer Studets' mscomreheso of relatoal statemets arthmetc word roblems oural of ducatoal Psychology [3]. Ster What maes certa arthmetc word roblems volvg the comarso of sets so dffcult for chldre? oural of ducatoal Psychology [4]. Castro L. Rco. Castro Choce of structure ad terretato of relato multlcatve comare roblems : Proceedgs of the steeth PM Coferece. Uversty of New Hamshre. Durham NH Vol [5] M. Hegarty M. Kozhevov Tyes of vsual-satal reresetatos ad mathematcal roblem solvg oural of ducatoal Psychology [6] K. des. Potter How studets "uac" the structure of a word roblem: Grahc reresetatos ad roblem solvg School Scece ad Mathematcs

27 9 Dagrams Produced by Secodary Studets Multlcatve Comarso Word Problems [7] D. Va Gardere M. Motague Vsual-satal reresetato mathematcal roblem solvg ad studets of varyg abltes Learg Dsabltes Research ad Practce [8] N. C. Presmeg Vsualzato hgh school mathematcs For the Learg of Mathematcs [9] P. Nesher The Stereotyed Nature of Word Problems For the Learg of Mathematcs [0] Ch. Forste Ste-by-ste model drawg: solvg word roblems. The Sgaore Way Crystal Srgs Boos Peterborough NH 00. [] L. Waler Model drawg for challegg word roblems. Fdg Solutos the Sgaore Way Crystal Srgs Boos Peterborough NH 00. []. Wls. Norto S. Boyce Valdatg a wrtte strumet for assessg studets fractos schemes ad oeratos The Mathematcs ducator [3] R.. Mayer Mathematcal ablty : Huma abltes: formato rocessg aroach. Freema New Yor

28 oural of Mathematcs ad System Scece D DVID PUBLISHING From Dgtal alogs Through Recursve Maches to Quatum Comuter M.B. Igatyev Yu.. Shey.. Ltov Sat -Petersburg State Uversty of erosace Istrumetato Receved: November 0 03 / cceted: December 6 03 / Publshed: February bstract: The reort eames the evoluto of comuters from dgtal aalogs through o-vo Neuma maches to uatum comuters whch are also dgtal aalogs. I the 60 years of dgtal aalogs successfully develoed at the Isttute of lectromechacs of the USSR Legrad. mortat stage the develomet of o-classcal multrocessor mache erformace ad relablty has bee the develomet of recursve maches whch was carred out at the Isttute of Cyberetcs led V.M.Glushov ad the Legrad Isttute of vato Istrumetato. The geeral aroach to the sythess s carred out through lguo- combatoral modelg wth structured ucertaty. Keywords: aflos comutato dgtal aalog recursve maches lguo-combatoral smulato of atoms structural ucertaty. Itroducto fter the orgazato of the Deartmet of Comuter Systems ad Networs at the Legrad Isttute of vato Istrumetato LIP - ow t's St. Petersburg State Uversty of erosace Istrumetato GUP 97 ecet for robotcs mortat asect of ts actvtes was eabled develog o-tradtoal comutg systems archtecture. To uderstad the logc of ths decso you eed to tell the world about the state of comuter techology the early sevetes. t ths tme the comay was domated by IBM flagrat volato of the laws of the moooles ad coductg ltgato may states the U.S. ad other coutres. Ths moooly was mafested a comuter lterature - t descrbed the IBM mache ad almost othg was sad about the maches of other comaes such as Cotrol Data Cororato Burroughs ad others who were rvals IBM. The comuters of the comay IBM mlemeted a classcal vo Neuma archtecture whch could ot Corresodg author: M.B. Igatyev Sat-Petersburg State Uversty of erosace Istrumetato. -mal: satsfy the cosumers. I the Sovet Uo there was a struggle betwee two tedeces - betwee the tedecy to develo ther ow desg such as BSM the Urals ad others ad the tedecy to coy foreg eerece esecally IBM coy mache. I ths stuato our youg deartmet released from the Deartmet of geerg Cyberetcs LIP February 97 decded to develo o-covetoal mult-rocessor systems whch the log term for hgh erformace ad relablty. For Dr. M.B.Igatev the head of ths deartmet the decso was a cotuato of hs wor the feld of dgtal dfferetal aalyzers whch were secalzed multrocessor recursve structures wth feedbac hgh erformace ad relablty by troducg redudacy redudat varables method that was revously develoed by them [ 4-9]. mortat ste was tae by our lecturer V.. Torgashev [35] who roosed to eted ad develo these rcles to maframe comuters. I the ed was bor the cocet of recursve maches whch receved the suort of the State Commttee for Scece ad Techology Moscow ad the Isttute of Cyberetcs headed by cademca V.M. Glushov Kev. There was the

29 94 From Dgtal alogs Through Recursve Maches to Quatum Comuter grou of muscovtes who rereseted the V.. Myasov the grou of Kev whch was rereseted by V.M. Glushov ad Legrad wth a commo ceter LIP. I the most vbrat form ths cocet was reseted at the Iteratoal Cogress of IFIP Stocholm 974 our reort [0]. Frst Sovet comuter develomet was aouced o the teratoal scee that has draw atteto from all sdes. The result of ths acto was frstly the cluso the rogram of wor ad the allocato of the SCS & T face to create eermetal rototye mache recursve secod a agreemet wth the frm Cotrol Data to create a recursve mache o the bass of our archtectural decsos ad thrdly to rovde the best for the tme elemet base ad debuggg tools. M.B. Igatev became the head of the worg grou o cooerato wth the frm Cotrol Data Cororato ad as a roect develoed by the recursve mache ad other roects amog whch was the urchase of the suercomuter Cyber for Legrad Scetfc Ceter of the USSR o the bass of ths mache frst orgazg the Legrad Research Scetfc Comutg ceter ad the the Legrad Isttute for Iformatcs ad utomato cademy of Sceces of the USSR. It should be oted t was a tme of a warmg of US-Sovet relatos t s the tme the roect was mlemeted Soyuz- ollo. Thus as a result of a cofluece of favorable crcumstaces we were able to tate wor o the actual establshmet of the recursve mache. Wor started whch was atteded by may members of our deartmet - V.. Torgashev V.I. Shrtl S.V. Gorbachev VB Smrov V.M. Kselov.M. Lual Yu.. Shey ad may others. s a result by 979 may uts were maufactured maches ad autum 979 a eermetal model of recursve mache was reseted to the state commsso headed by academca.. Dorodtsy. I the secal Decree of the USSR ad the SCST of the Presdetal Commttee of the Coucl of Msters o g for umber 47/76 oted that the lauch of the world's frst eermetal model of a mult- recursve mache erformace ad relablty s the achevemet of a world level. Plas have bee made for further develomet of ths wor but December 979 Sovet troos vaded fghasta ad the U.S. govermet severed all scetfc ad techcal tes wth the Sovet Uo cludg o -le frm Cotrol Data whch caused us a lot of damage. But the wor wet o eve though our team was dvded - some staff auary 980 led by V.. Torgashev moved to Legrad Research Scetfc Comutg Ceter cademy of Sceces of the USSR whle others cotued to wor our deartmet to create dfferet versos of multrocessor systems. t the Isttute of Cyberetcs Kev was to create Deartmet of recursve maches. These are the eteral cotours of ths oeerg wor.. The Prcles of Orgazato Recursve Maches ad Systems I mathematcs there s a large secto - a recursve fucto [3]. For a log tme the term "recurso" was used by mathematcas ot beg clearly defed. Hs rough tutve sese ca be descrbed as follows. The value of the desred fucto F at a ot a ot meas a set of argumets s defed geeral through the values of the same fucto other arts of H whch some sese recede H. The word "recurso" meas the retur [7]. Recursve fuctos - s comutable fuctos. ssetally all comutable fuctos o comuters - s a recursve fucto but dfferet o dfferet comuter archtectures are comutatoal rocesses. The better the structure corresodg to the structure of the comuter roblems the lower the cost of memory ad tme. So whe we tal about recursve maches we are talg about maches ad structures accordg to the tass as well as the tass are dfferet the structure of the mache should be fleble to adat to the structure of the roblem. Math beg mmersed rogrammg ad dstrbuted rogrammg recursve

30 From Dgtal alogs Through Recursve Maches to Quatum Comuter 95 oeratos. Comuter acts as a meas of materalzg the logcal- mathematcal trasformatos. Comuter s a llustrato of the cocet of otetal feasblty as the absece of restrctos o the tme ad memory caacty of comuters ay codto to ay calculato. The secfc flow s show oly comutg rocesses at the eterrse level chage formato secfc regsters are utlzed swtches rocessors data les a secfc order ad combato etc.. From ths ot of vew "comuter archtecture" - that s ts structure the rocess of the algorthm t s le a amated teture. The hlosohcal bass of ths reresetato s the theory of reflecto revealg a ma of categores ad heomea of the same ature umbers algorthms to the obects of a dfferet ature hyscal elemets sgals. Moreover ths mag s ambguous - a algorthm ca match may archtectures {} ad bac - archtecture does ot drectly corresod to ay algorthm a. The secfcty of teracto {a} ad {} reveals the uderlyg roertes of the dalectcal rocess of develomet of mathematcs ad comuter scece as a artcular case of the teracto of abstract ad cocrete. s the S Yaovsa "face mechazed mathematcs s becomg creasgly deedet o the develomet of hlosohcal ad logcal foudatos of mathematcs" [3]. It s ot ossble to dslay a cosstet formalzato of {a} {} because of ts ambguty. Therefore to costruct the corresodg aomatc theory of comuter desg s ot feasble [4]. Whe we formulated the rcles of recursve maches we started from the eeds of the develomet of comuters ad systems have receved may certfcates of authorsh [3-5 etc.] t was a fu ad creatve rocess from the ot of vew of relablty made bac years. Our reort o the IFIP Cogress Stocholm [ 0] the tet of the reort s reroduced full Ref. [34] cotaed a aalyss of the shortcomgs of tradtoal archtecture maches secto rcles vo Neuma archtecture rcles recursve maches the ma features of the laguage of recursve maches fragmetary descrto of the recursve mache. s a llustrato of a recursve structure of the system ca cause 3M - modular mcrorocessor system [8]. 3M system s bult from modules of three tyes - oeratoal commucato ad terface. Oeratg modules erform most of the wor o data rocessg mathematcal obects of the memory rocesses determe the avalablty ad mlemetato of the rogram oerators the teral laguage. The commucato module s desged to mlemet the commucato system - establshg a logcal coecto betwee the modules the echage of formato betwee modules search the resource system of the tye reuested. The terface modules are coected to eteral devces for ts ut-outut ut. Questos to echage formato wth the outsde world are essetal to sgfcatly multrocessor systems have a sgfcat mact o ther actual erformace. Dfferet classes of roblems reure dfferet testes of echage wth eteral devces. The comuter system should rovde the costructo of ts cofguratos for each artcular alcato whch would have the otmum for ths alcato secfcatos for ut-outut. 3M system rovdes a cremetal buld-u of comutg ower to ay desred value by addg addtoal blocs wthout mag chages to the estg system ad ts software as a system develomet hase ad the course of ts oerato. Methodology for the desg ad mlemetato of a system based o a revew of 3M's comuter system as a herarchy of vrtual maches. 3M system s recursvely orgazed a mult-level structure. Recursve structure s that the structure of ay modfcato of set recursve defto. Dyamcally chagg the course of vrtual comutg rocesses reure costat dyamc recofgurato of ls betwee modules. Now mlemeted systems cotag thousads or mllos of rocessors.

31 96 From Dgtal alogs Through Recursve Maches to Quatum Comuter 3. Prosects for the Develomet of Comutg Systems. I coecto wth the above would cosder the roblems of comutg. Comutg maches are desged to solve roblems. The geeral scheme of solvg the roblems of the form L mal -> L math -> L r -> L mache -> L result where L ma - formulato of the roblem atural laguage L math - formulato of the roblem the laguage of the basc relatos L r - formulato of the roblem a rogrammg laguage L mache - formulato of the roblem mache laguage L result - formulato of the roblem the laguage of the result the form of grahs tables mages tets souds etc. ach of these formulatos has ts ow meag. Ufortuately for most uroses there s oly the wordg atural laguage most of the tass oorly formalzed. So s the actual trasto from the descrto atural laguage to the laguage of the basc relatos lgustc- mture modelg s a way of formalzg []. s a result of the formalzato of recursve structures geerated wth structured ucertaty. Thus the recursve structure of the maches should clude three comoets the words the heomeo of meags ad structured ucertates whch are reset ay tas. terestg drecto the develomet of hardware comoets - a uatum comuter a hyothetcal comutg devce that by erformg uatum algorthms essetally uses whe worg uatum mechacal effects such as uatum arallelsm ad uatum etaglemet. Quatum arallelsm ca be realzed by usg the redudat varables ad uatum etaglemet structure realzed by arbtrary coeffcets. Lguo-combmatoral modelg allows you to buld such a system based o the use of ey words ey cocets revalg the subect feld. The model cossts of three grous of varables - characterstcs of the basc cocets meag of these characterstcs ad structured ucertaty euvalet euatos whch ca be used to adat cotrol ad calculato. You ca buld a lgustc-combatoral model of atoms whch ca be used as a uatum comuter. Cosder as a eamle a hydroge atom ad as eywords tae the word "atom" "roto" "electro" the the tal hrase s of the form tom + Proto + lectro I atural laguage we deote the word ad the meag s mled. There s a eed to troduce the otato costructve sese such as * + * + 3*3 0 3 ths euato - characterstcs of the hydroge atom - the meag of the hydroge atom -characterstcs of the roto - the meag of the roto of the hydroge atom 3 - characterstcs of the electro 3 - the meag of the electro of a hydroge atom. Ths s the algebrac sematc comutg model of hydroge t ca be used as a calculator. I order to solve the euato 3 or the varables or the varables are coductg a thrd grou of varables - the arbtrary coeffcets Us the the structure of euvalet euatos wll have the form U * + U * 3 - U * + U3 * U* U3* or U* + U*3 - U * + U3 * U * - U3 * where U U U3 - the arbtrary coeffcets. Substtutg euato 4 or 5 euato 3 t s detcally zero for all U. ll calculatos are made by us the algebrac rgs t's lguo- combatoral algebrac models. For smulato of deuterum usg the ey word "atom" "roto" "electro" "eutro" tom + roto + electro + eutro 6 fter the oerato the olarzato * + * + 3*3 + 4*4 0 7 ad the euvalet euatos are

32 From Dgtal alogs Through Recursve Maches to Quatum Comuter 97 U * + U * 3 + U3 * 4 - U * + U4 * 3 + U5 * U * - U4 * + U6 * U3 * - U5 * - U6 * 3 where U U U3 U4 U5 U6 - the arbtrary coeffcets whch may be the wave fuctos; -characterstcs of the deuterum atom - the meag of the deuterum atom; - characterstcs of the roto deuterum atom; - the meag of the roto deuterum atom; 3 - characterstc of a electro of a atom of deuterum 3 - the sese of a electro of a atom of deuterum; 4 - characterstcs of the eutro of the deuterum atom; 4 - meag a eutro of the deuterum atom. I mosg a further restrcto o the varables of the system * + * + 3*3 + 4 * uvalet euatos have the form 0 U * D3 + U * D4 + U3 * D34 - U * D3 - U * D4 + U4 * D U * D3 - U3 * D34 - U4 * D34 4 U * D4 + U3 * D43 + U4 * D43 where U U U3 U4 - arbtrary coeffcets D3 * 3-3 * etc. Smlarly the ossble costructo of lguo-combatoral models of all the elemets of the erodc table ad varous molecules. From the structure of these models mles the estece of a cotrol ut whch ca hadle arbtrary coeffcets e our model of the atom - a model of the atom wth the cotrol ut [] the develomet of whch oes u the ossblty to carry out formatoal fluece o the atoms whch s mortat for ther use as calculators. Ths s aother way for comuter modelg of hyscal ad chemcal reactos. Ths verfcato s ecessary to solve the roblem of such models for secfc systems. Basele characterstcs of uatum comuters theory allow them to overcome some of the lmtatos that arse whe worg wth classcal comuters. uatum comuter - a d of dgtal aalog dgtal devce aalog ature[]. Model of Fluorum atom ca be the model of cty model of Carboeum atom ca be he model of orgasm[30334]. The S. Llod boo [37] the uatum comuters ad ths cosderato ales to the etre uverse whle t uses the Shao's defto of formato the classcal laws of thermodyamcs ad the cocet of etroy whch the meag s ot defed. Ths meas that the S. Llod buldg ot be used for the costructo of calculators they ca be used oly for the aalyss of meagless hyscal systems. We cosdered aroach ca be used to costruct meagful tellget sesble comutg structures based o atomc ad molecular systems. 4. Cocluso Basele characterstcs of uatum comuters theory allow them to overcome some of the lmtatos that arse whe worg wth classcal comuters. uatum comuter - a d of dgtal aalog dgtal devce aalog ature. Lguo-combaoral aroach ermt o create both the model of atoms ad moleculs ad the model of comle systems cty orgasm etc. whch ca smulate each other. Today uatum comuters are the coy of the dgtal uversal comuter but the dgtal aalogs of uatum comuters have he terestg ossbltes. Referece [].. Voroov.R. Garbuzov B.L. rmlov M.B. Igatyev G.N. Soolov Yag X Zhe "Dgtal aalogues for automatc cotrol systems" zd. USSR cademy of Sceces 960 traslated to Chese 963. [] M.B. Igatyev "Holoomc automatc systems" zd. N USSR L - M 963. [3].I. Mal'tsev "lgorthms ad recursve fuctos" Moscow 965. [4] M.B. Igatyev "O the ot use of the rcles of redudacy ad feedbac to buld ultrastablty systems" Proceedgs of the III ll-uo Coferece o utomatc Cotrol Volume zd. cademy of Sceces of the USSR 968. [5] M.B. Igatyev "The method of redudat varables for the fucto of codg dgtal maches Sb." utomata Theory 4 Kev 969 [6] M.B.Igatyev "O the lgustc aroach to the aalyss ad sythess of comle systems" Proceedgs Iteruversty Scetfc Coferece "Techcal

33 98 From Dgtal alogs Through Recursve Maches to Quatum Comuter Cyberetcs" M [7] M.B. Igatyev "Redudacy the mult-urose systems" Proceedgs of the IV Symosum o the roblem of redudacy L [8] M.B. Igatyev F.M. Kulaov.M. Porovsy "Cotrol algorthms robotc maulators" zd. Mashostroee 97 the U.S. edto of Vrga Press 973 thrd edto 977. [9] G.S. Brtov M.B. Igatyev L.. Mroovsy Yu.M. Smrov "Cotrol of comuter rocesses zd. LGU 973. [0] V. Glushov M. Igatyev V. Masov V. Torgashev "Recursve maches ad comutg techology" Proceedgs IFIP- 74 comuter hardware ad archtecture Stocholm ugust [] M.B. Igatyev V.. Myasov.M.Porovsy " Program cotrol of eumets" zd. Mashostroee 974 secod edto 984. [] M.B. Igatyev V.. Myasov V.. Torgashev "Recursve comuters" Prert umber ITM ad CM cademy of Sceces of the USSR 977. [3] M.B. Igateyv V.M. Kselov V.. Torgashev V.B. Smrov " ssocatve memory" Certfcate umber Iveto Bullet umber [4].. Beasova S.V. Gorbachev M.B. Igatyev V.. Myasov V..Torgashev "CPU wth frmware cotrol" certfcate umber 848 wth rorty from [5] S.V. Gorbachev M.B. Igatyev V.M. Kselov V.. Myasov V.. Torgashev Multrocessor system" Certfcate umber wth a rorty date of 7 ugust 974 the Ivetos Bullet umber [6] M.B. Igatyev V.. Myasov V.V. Flchaov "The orgazato of the comutatoal rocess for solvg aled roblems o a multrocessor system wth a recursve orgazato" oural of Cyberetcs cademy of Sceces of the USSR [7] "Recurso " artcle the Mathematcal cycloeda Vol. 4 ages M 984. [8] M.B. Igatyev.. Koch V.. Myasov Yu.. Shey" Mcrorocessors - system rogrammg ad debuggg" tomeergozdat M [9] S.V. Gorbachev M.B. Igatyev Yu.. Shey "Recursve maches for wde alcato" Theses of ll-uo coferece o Comuter Scece zd. rmea SSR Yereva 987 [0] M.B. Igatyev Ya. Komora" geeralzed arametrc model of the locally - recursve structures three-dmesoal tegrated crcuts" Reorts of the USSR 99 tom.30 5 str [] M.B. Igatyev V.V. Flchaov L.G. Osovetsy "ctve methods to esure the relablty of algorthms ad rograms" zd.polteha 99. [] M.B. Igatyev "Lguo-combatoral modelg badly formalzed systems" Iformato ad Cotrol Systems [3] S.. Yaovsaya "Methodologcal roblems of scece" M 97. [4] P.G. Suvorova "Dalectcs of the abstract ad the cocrete the cocet of comuter archtecture" Sb "New deas the hlosohy of scece ad techcal owledge."m. 00. [5] M.B. Igatyev Yu.. Shey "5 years sce the creato of recursve comuter erformace ad relablty ad the roblem of arallel comutg" Reort to the leary sesso of the 3rd Iteratoal Coferece o the comuter scece ad sustaable develomet St. Petersburg house of Scetsts Russa cademy of Sceces rl [6] M.B. Igatyev Yu.. Shey "Recursve Comutg" Iformatzato ad Commucato [7] [8] R. Perouz "The ath to realty or the laws that gover the Uverse" Moscow 007 [9] Igatev M.B. "Smulato of datatoal Mamm Pheomeo Develog Systems" Proceedgs of The SIMTC' Iteratoal Smulato Techology Coferece Sa Fracsco US [30] Igatyev M.B. D.M. Maa N.N. Petrschev I.V. Polaov.V. Ulrch.V. Gub "Global model of orgasm for decso mag suort" Proceedgs of the Hgh Performace Comutg Symosum - HPC 000 d.. Teter 000 dvaced Smulato Techologes Coferece Washgto DC US [3] Igatyev M.B. "Lguo-combatoral method for comle systems smulato" Proceedgs of the 6th World Multcoferece o Systemcs Cyberetcs ad Iformatcs vol. XI Comuter scece II Orlado US [3] R. Bader "toms Molecules" M zd. Mr 00. [33] Igatyev M.B."Iformato techology ad mcro - aoad otoelectrocs " zd. GUP St. Petersburg 008. [34] Igatyev M.B. "Cyberetc world cture. The theory of comle systems" St. Petersburg 0 [35] Torgashev V" utomato etwors ad comuters: hstory ad curret state of develomet" / Hstory of comuter scece ad comuter scece at St. Petersburg Legrad. Issue 3. Uder the geeral edtorsh of R.M.Yusuov St. Petersburg 0. [36] M.Igatyev The lguo-combatoral smulato moder hyscs merca oural of Moder Physcs US 0 vol. No. 7-. [37] S.Llod Programmg Uverse US 006

34 oural of Mathematcs ad System Scece D DVID PUBLISHING Iterolated Tesor Products of oetal Tye Vectors of Ubouded Oerators Mara Dmytryshy Deartmet of Mathematcs ad Iformatcs Precaratha Natoal Uversty 57 Shevcheo str Ivao-Fravs Urae Receved: November 7 03 / cceted: December 8 03 / Publshed: February bstract: I ths aer we defe the tesor roducts of saces of eoetal tye vectors of closed ubouded oerators Baach saces. Usg the real method of terolato K-fuctoal we rove the terolato theorems that ermt to characterze of tesor roducts of saces of eoetal tye vectors. We show a alcato of abstract results to the theory of regular elltc oerators o bouded domas. For such oerators the eoetal tye vectors are root vectors. Thus we descrbe the tesor roducts of root vectors of regular elltc oerators o bouded domas. Key words: oetal tye vector tesor roduct terolato sace regular elltc oerator. Itroducto The cocet of etre eoetal tye vector of ubouded oerator Baach sace s defed Ref. []. The saces of such vectors are used coecto wth the roblems of oerator calculus theory of dfferetal euatos aromato theory. I Ref. [] s costructed the oerator calculus o the eoetal tye vectors of the oerator wth dscrete sectrum. Notce that for such oerator the subsace of eoetal tye vectors cocdes wth the lear sa of all ts sectral subsaces. The roblem of aromato wth dfferet classes of smooth vectors of closed oerator has bee studed Ref. [3]. characterzato of some classes of ftely dfferetable vectors of a ormal oerator Hlbert sace s gve terms of the rate wth whch the best aromato of the vectors by etre eoetal tye vectors ted to zero. Usg the K-fuctoal stead of module of cotuty the drect ad verse theorems roblems o the aromato Corresodg author: Mara Dmytryshy Ph.D. rofessor research feld: fuctoal aalyss oerator theory terolato theory aromato theory. -mal: of fte degree vectors are studed Ref. [4]. The aalogues of classcal Berste ad acso theorems the oclasscal cotet of the stratfed grous are establshed Ref. [5]. I stratfed grou s defed the class of fuctos that lays a role aalogous to etre eoetal tye fuctos. The codtos o a closed oerator Baach sace whch are ecessary ad suffcet for the estece of solutos of a dfferetal euato y t y t t [ 0; the classes of etre vector-fuctos wth gve order of growth ad tye are establshed Ref. [6]. The saces of ultrasmooth vectors of some elltc dfferetal oerators are descrbed Ref. [7]. Some terolato roertes of saces of eoetal tye vectors of the ubouded oerators Baach saces are reseted Ref. [8]. The motvato of ths aer s to establsh ew terolato roertes of tesor roducts of eoetal tye vectors of ubouded oerators Baach saces Theorems. The alcato of the obtaed abstract results for the regular elltc oerators o bouded domas s also show Theorem 3. I Theorem 4 we show that the tesor roduct of

35 00 Iterolated Tesor Products of oetal Tye Vectors of Ubouded Oerators the saces of all eoetal tye vectors for the regular elltc oerators wth costat coeffcets L Ω s somorhc to the tesor roduct of the subsaces of etre eoetal tye fuctos that ther restrctos o Ω belog to L Ω.. Ma Results Let X. X be a fte collecto of Baach comle saces. I the sace X we cosder a closed ubouded lear oerator wth a dese doma D. For ay umbers 0 < < we defe the sace of eoetal tye vectors of : : < D : 0 > 0 X / X < : su. Proosto. The sace s comlete. Proof. Let us use the eualty / wth. It follows that f X s a Cauchy seuece the sace the ad / are Cauchy seueces the sace X for all Z+. The comleteess of X mles that there est y X such that ad / y by orm of X. The grah of s closed X X therefore y. It s true for all Z+ so / by orm of X for all / Z +. For ay ε > 0 there ests ε N such that ε for all m. It follows that m < ε / for all m ε ad Z+. m < X ε So for all Z+ there ests mε ε such that / < ε / ad m X / < ε / for m m ε. Hece from the eualty / / X ε X / + / + t follows for all Z+ m ε X / / + ε / X ε X m m X X. We may use that the scalar seueces ξ / ad ε Baach sace obta ξ + ζ l ζ belog to the. From the revous eualty we X l ξ l + ζ l Hece. Moreover ε / / m X ε / + mε X X + 4ε. where ths eualty all seueces by belog to l. We obta < 4 ε ε. So s comlete. Let : be a tesor roduct wth the roectve orm w N f w : where f taes o all reresetato of w as the sum N w wth fte N ad :. Comleteess of o ths orm we deote by. Let 0 < < r 0 < < ad 0 < t <. Usg the real method of terolato K-fuctoal the otato of [9] we defe the

36 Iterolated Tesor Products of oetal Tye Vectors of Ubouded Oerators 0 terolato sace r wth the orm [ ] r t dt t K t / 0 ; : where f : ; r t t K 0 r. For we also cosder the terolato sace r wth the orm [ ] r t dt w K t t w / 0 ; : where + + f : ; r v u w v t u w t K u r v. Theorem. Let < < 0 0 < < ad. The the followg eualty holds wth euvalet orms. Proof. Notce that the eualty s vald for < < 0 ad. The for < < τ 0 ad t τ we obta N v u N v v N u u v u w u v v t u u w K t f f f f ; N v v t u +. ; ; f f N v u N v u K τ K τ v τ u v τ u Usg the Youg s eualty we have [ ] [ ] [ ] [ ]. ; ; ; ; ; N N τ dτ K τ τ τ dτ K τ τ τ dτ K τ K τ τ t dt w K t t Thus we have the eualty N w where :. Taes f o all reresetato of elemet w as the fte sum N w we obta w w whch shows the embeddg. Usg the terolato eualty c we obta

37 Iterolated Tesor Products of oetal Tye Vectors of Ubouded Oerators 0 N N w N w N w w c w K t t c τ τ c c w ; f f f + for all N w such that. Sce the sace s dese the the eualty w c w s vald for all w. From ths eualty we have. 3 The desred result follows from ad 3. Theorem. Let < < < 0 η 0 < < ad. The the followg eualty holds η 4 wth euvalet orms. Proof. The sace s sometrc to the sace of seueces { } : : 0 l wth the orm l. Let us relace σ σ σ η 0 whch the codto η turs to eualty σ σ σ 0 +. Moreover the followg euvaleces are vald X σ σ t l l t K 0 su m ; 0 m ; X σ σ t l l t K. We use reasog wth Theorem.8. [9]. For l l ad σ σ 0 > we have m su η l X σ σ X σ σ σ σ σ σ l l c + + whch shows the embeddg η l l l. 5 Usg the Holder s eualty for η l we have X σ σ σ σ σ σ l l m η l X η c whch shows the embeddg η l l l. 6 From 5 ad 6 we have η η l l l l l l l l. Thus we roved the eualty η. 7 The desred result follows from ad Regular lltc Oerators Let R Ω be a oe bouded doma wth the ftely smooth boudary C ad the system of oerators Ω C a u D a u α m α α α

38 Iterolated Tesor Products of oetal Tye Vectors of Ubouded Oerators 03 α Bu b α D u b α C Ω α m m s regular elltc see e.g. [9] Secto 5... I the comle sace L Ω r < we cosder the closed oerator W r < wth doma m { u Wr Ω : Bu 0 m} m r B Ω Ω r m Ω where W s the Sobolev sace. Suose that the resolvet set of ow [9] Secto { } s oemty.e. ρ. s s has a dscrete sectrum σ λ ad lm λ. Deote by R a root subsace of λ. Theorem 3. Let that corresods to the egevalue 0 < < < η 0 < <. The the followg eualty holds sa{ R : λ < η} wth euvalet orms. Proof. I [ Theorem.] t s rove that for 0 < the followg eualty holds < { R λ < } sa :. 8 From the eualtes 7 ad 8 we obta sa{ R : λ < η }. It remas to use the eualty 4. Suose that the coeffcets a α of are costats. Deote by [ L r Ω] the sace of etre eoetal tye fuctos that ther restrctos o Ω belog to L Ω ad B r [ Lr Ω]: { u [ Lr Ω] B u Ω : 0 m Z For ay 0 < the embeddg < holds. The we ca defe the sace : : lm d ad r > 0 r tesor roduct : r r r r + }. wth roectve locally cove toology see e.g. [0] Chater III 6. Comleteess of r o ths toology we deote by. r Theorem 4. The followg toologcal somorhsm holds L Ω. r Proof. Cosder the sace B [ r ] α { u C Ω : D u L Ω α } r D : r Z + edowed wth the orm / r r r α u : D D u. r L Ω 0 α r Show that the sace D : D cocdes r wth the sace [ L r Ω] > 0 r. For smlcty we ut 0 Ω. If l > / r ad u D the Sobolev s embeddg theorem yelds Ω α su D u c ma 0 r l { } u c D r where a costats c 0c are deedet of. It follows that u α y + y D u c e 0 α y 9! for all Ω ad y R where a costat c s deedet of. Hece u has etre eteso oto C of a eoetal tye. Coversely let a etre fucto u satsfes 9. α The the eualty all D u c e holds for R ad α Z+. Va boudedess of Ω α we have D u с 3 u 4 r α L Ω r D because α D u r L Ω r. It follows that r r 4 0 α Thus u D ad we have r α r D u L Ω α r su. [ L Ω] D. 0 r r Now we rove the eualty

39 04 Iterolated Tesor Products of oetal Tye Vectors of Ubouded Oerators r { D : B r u Ω m Z From Theorem [9] for ay ostve umbers c c such that u m Ω u c u W m Ω W Ω r Lr r c + 0 }. N there est u D. It follows the eualtes 0 c m c 0 α r mr Thus the embeddg u α D u r L Ω r r L Ω r c u r D r. τ { D : B u 0 m Z+ } r wth Ω m τ c holds. Coversely let u r. The r r c 0 α r α u D u. It follows that r wth r σ c r L Ω r σ { D : B u 0 m Z + }. Ω The desred result follows from 0 ad. Refereces [] Ya. V. Radyo Vectors of eoetal tye oerators r calculus ad dfferetal euatos. Dfferetal [] M.I. Dmytryshy O.V. Loushasy Oerator calculus o the eoetal tye vectors of the oerator wth ot sectrum. Chater boo "Geeral Toology Baach Saces" Nova Sc. Publ. Hutgto New Yor [3] M.L. Gorbachu V.I. Gorbachu O aromato of smooth vectors of a closed oerator by etre vectors of eoetal tye Ur. Math [4] G.V. Radzevs Drect ad verse theorems roblems o the aromato of fte degree vectors Mat. Sb [5] S. Gul romato ad Besov saces o stratfled grous Proc. mer. Math. Soc [6] M.L. Gorbachu O aalytc solutos of dfferetal-oerator euatos Ur. Math [7] M.I. Dmytryshy Ultrasmooth vectors of sgular dfferetal oerators oural of Mathematcal Sceces [8] M.I. Dmytryshy O.V. Loushasy Iterolated subsaces of eoetal tye vectors of the ubouded oerators Baach saces Demostrato Mathematca [9] H. Trebel Iterolato theory. Fucto saces. Dfferetal oerators Srger Berl-Hedelberg-New Yor-Toyo 995. [0] H. Schaefer Toologcal vector saces The Macmlla Comay New Yor 966.

40 oural of Mathematcs ad System Scece D DVID PUBLISHING Measuremet of Surlus Labor Vet Nam grculture Quyet Dag Pham ad Hoa Huu Vo. Isttute of Statstcal Scece Geeral Statstcs Offce Vet Nam. Duy Ta Uversty Da Nag Vet Nam Receved: uly / cceted: ugust 03 / Publshed: February bstract: The large shft of surlus labor from agrculture to dustry ad servces s see may coutres aroud the world as well as Vetam the rocess of dustralzato ad moderzato of the coutry that has set uestos about the sustaablty of rural surlus labor: s there stll a source of surlus labor rural areas? If so how large s the source of surlus labor ad how log t ca be lastg? These uestos were hotly debated the lterature abroad. But Vetam there s very lttle or hardly echaged oos about the cocets ad methods of measuremet of surlus labor geeral ad surlus labor agrculture artcular. Ths artcle refers to the measuremet aroach of surlus labor agrculture Vetam. Key words: Labor force agrcultural labor surlus labor surlus labor agrculture.. Itroducto It s o cocdece that the theme of the World Develomet Reort 008 s related to agrculture: "Stregtheg grculture for Develomet". "I the st cetury agrculture cotues to be a fudametal tool for sustaable develomet ad overty reducto. Three uarters of the oor the develog coutres lve rural areas where. bllo eole lve o less tha $ er day ad 880 mllo eole lve o less tha $ er day ad most tae agrculture as ther lvelhood". For Vetam a coutry wth early 70% of the oulato lvg rural areas ad agrculture cotues to be the ma lvelhood of mllos of rural labor the develomet of ths sector lays a mortat role strategy for soco-ecoomc develomet of the coutry. I rural areas agrculture remas the ma occuato but the ablty to actually create ew obs of the agrcultural sector s ute low. grcultural roducto methods stll bascally heavy tradtoal ad Corresodg author: Quyet Dag Pham Ph.D. research feld: statstcs ad labor research. -mal: Preface cover of World Develomet Reort 008 of the World Ba Stregtheg grculture for Develomet ublshed 0/007 fragmetato whle the agrculture s also rs sector. Besdes the ecoomc value of the agrcultural roducts always belog to the low grou comared wth may other commodtes that mae socal labor roductvty of the agrcultural sector s far dstat from other dustres. Ths fact maes more rural labor surlus ad laborers whch wat to stc wth agrculture are declg esecally amog youg worers. The large shft of surlus labor from agrculture to dustry ad servces s see may coutres aroud the world as well as Vetam the rocess of dustralzato ad moderzato of the coutry that has set uestos about the sustaablty of rural surlus labor: s there stll a source of surlus labor rural areas? If so how large s the source of surlus labor ad how log t ca be lastg? These uestos were hotly debated the lterature abroad. But Vetam there s very lttle or hardly echaged oos about the cocets ad methods of measuremet of surlus labor geeral ad surlus labor agrculture artcular. Ths artcle refers to the measuremet aroach of surlus labor agrculture Vetam. Surlus labor as defed by most ecoomsts s the

41 06 Measuremet of Surlus Labor Vet Nam grculture codto ests whe a orto of the labor force ca be removed wthout causg a reducto outut. Surlus labor techcally seag s that there s too much labor comared to the demad to roduce the same outut as t s the curret oe. The assumto of zero margal roductvty whch suggests that the margal roductvty of labor agrculture relevat develog coutres s very low s most useful as a devce to facltate clarty aalyss. Ths assumto offers a coveet measure of how the margal roduct of labor s creasg develog coutres by comarg the tred of the margal roduct over tme [4]. Surlus labor s a cocet of low usg labor that was dscussed ecoomc develomet but rarely measured. sde from the uesto of whether there s a ecess suly of labor the maret the form of uemloyed or uderemloyed eole who are ready to resod to ew emloymet oortutes as they emerge there s the uesto of whether some sectors smly have surlus labor the sese of havg too may worers relato to the umber techcally reured to roduce curret levels of outut. The olcy mlcato here s that f there s surlus labor of ths d t reresets a hdde savg otetal: surlus eole could be removed from ther reset actvty wthout affectg outut ad ut to wor o develometal roects of varous ds Cosderato amout of lterature ublshed o measurg surlus labor shows three broad aroaches have bee used to measure the sze of the surlus labor agrculture the re-990s [4]. They are: erece method; stmato method ad Labor orm method. The eerece method s the most tradtoal aroach to estmate the average labor-hour reuremets for agrcultural roducto rural areas. The amout of surlus s calculated by comarg these estmates wth actual labor hours. It's ute smle method to aly. Surlus agrcultural labor ca be uderstood as the dfferece betwee the total agrcultural labor suly comared to the actual demad for agrcultural labor by adotg more advaced agrcultural roducto ad maagemet techologes. I the study by Wag ad Dg 006 the agrcultural roducto fucto s descrbed as: Y FTKD Where T K D ad dcate the worg days catal vestmet the lad areas ad techology resectvely. The the demad for wordays roducto of a mamum amout outut Y' s calculated as follows: T F Y' KD ssumg that there are L agrcultural labor the umber of wordays rovded by oe worer oe year s calculated as follows: t T / L 3 Ths reflects the actual worload of oe farmer. The a ratoal worload for oe farmer must be set.e. a farmer s ratoal umber of worg days oe year. Scholars geerally agree that a farmer s umber of worg days er year should be 70 days Che 99. Therefore the actual demad for agrcultural labor ca be calculated as follows: L' T / 70 4 From euato 3 ad 4 the rato of agrcultural labor demad to the suly s calculated as follows: d L'/ L t/ 70 5 Thus the rato of agrcultural surlus labor to the total agrcultural labor s calculated as follows: r t/ 70 6 Thus there s o eed to ow the formato o agrcultural outut lad areas umber of lvestoc rased total worg days etc.... but oly a farmer s wor load t.. The estmato method sees to determe the labor reuremets based o the lad-labor rato desged for a artcular year. The a comarso ca be made betwee the actual labor ad labor orms assocated wth the bechmar year order to derve the amout of surlus labor. eamle of ths method was foud Che 004. Che argued that

42 Measuremet of Surlus Labor Vet Nam grculture 07 uder curret atural socal ecoomc ad techologcal crcumstace agrcultural resources roducto methodologes ad govermet olces regardg agrculture have mortat effects o the demad of agrcultural labor. mog these factors the agrcultural resources esecally arable lad are decsve factors. Che cosdered 95 a year wth o surlus labor ad thus fed the rato of arable lad to labor at the level of that 95. Che estmated agrcultural surlus labor usg the followg formula: SL L S / M 7 t t t t Where SL t s surlus labor to be estmated L t s the real labor force agrcultural labor suly S t s the actual area of arable lad ad M t s the cultvated area er cata. Furthermore M t s eressed as follows: M β t 95 t + 8 Where reresets the average area uder cultvato er cata from 949 to 957 ut: hectares ad β s the rate of chage agrcultural maagemet due to advaces agrcultural roducto techology. Che 004 set β through comutato. 3. The labor orm method. Istead of selectg a base year for effcet labor use ths techue calculates the total labor reured ad derves the surlus by subtractg the reured labor from actual labor used. Total labor reuremets ca be calculated four dfferet ways Wag 994 L La Z Q La D 9 X 300 a L/ D Where D L : demad for agrcultural labor La: total arable lad X: er laborer arable lad Z: sum of arable lad; Q: value of agrcultural outut a: er cata agrcultural outut L: aual wor days for each laborer D: umber of worg days reured er hectare ad : rural labor force. Oe materal of the ILO [6] has troduced two aroaches of measuremet of surlus labor agrculture accordg to the metoed above methods. The usual aroach to measuremet of such surlus labor the case of agrculture geerally as follows: The removable agrcultural surlus labor labor-hours s defed as the dfferece betwee the labor avalable ad labor reured where the labor avalable cossts of the total ecoomcally actve oulato the agrculture multled by the umber of full worg days avalable for agrculture the erod allowg for days off o weeeds ad holdays! multled aga by the umber of hours er day ormally wored ad labor reured to roduce a gve agrcultural outut s calculated by alyg the labor coeffcets to outut or acreage fgures. The ma roblem wth a eercse of ths d cocers the orms o whch t s based. However a tycal hectare that s selected as the bass for calculatg labor coeffcets varatos from t cro m lad ualty farm sze agro-clmatc zoe techology system etc wll greatly affect demad for labor er hectare farm o dvdual farms ad thus the aggregate. alteratve aroach - aroach based o Mehra 966 whch avods the eed to set techcal orms essetally comares the use of labor o farms whch emloy wage labor wth ts use o farms whch do ot. The bass of the method s the assumto that farms whch emloy wage worers do ot have ay surlus famly worers avalable otherwse they would ot eed to emloy wage labor. So f famly farms are usg more worers er hectare tha labor farms smlar every other resect they are usg more tha they eed. The assumto as wth Se s that surlus labor agrculture taes the form ot of the eedture of more labor-hours or labor-days tha ecessary but of the sreadg of the ecessary umber of labor-hours or days more thly tha ecessary over the umber of famly worers that s avalable. I formal terms the amout of surlus labor a famly farm of a gve tye.e. gve sze cro m agro-clmatc zoe fertlzer use rrgato tye etc. wll be calculated as follows. Sce emloyg wage farms have o surlus labor where N R w w 0

43 08 Measuremet of Surlus Labor Vet Nam grculture Rw s the reured umber of worers er hectare farms emloyg wage labor; Nw s the umber of worers actually emloyed farms emloyg wage labor. The umber of worers reured o a famly farm s the umber that they would use f ther worers wored the same umber of hours er day as worers o wage emloyg farms of a smlar tye..e. f where L f w R L f N w L f s total umber of labor-hours of labor eeded er hectare er year o famly farms; R f s the reured umber of worers er hectare o famly farms; Lw s total umber of labor-hours of labor eeded er hectare er year o farms emloyg wage labor; the ad f R f N w. L Lw f S f N f - R f N f - N w. L Lw 3 where S f s the umber of surlus labor er hectare famly farms. So far the coutry does ot have ay research o comrehesve ad systematc measuremet of surlus labor to be able to motor surlus labor the coutry. The Isttute of Statstcal Scece ISS has made the research toc " Study ad test measuremet method of surlus labor Vet Nam agrculture" to detfy ad roose methods measurg surlus labor agrculture to aswer the uesto whether or o amout of surlus labor rural areas? If so how large s ths worforce ad for how log t ca survve? From there we wll mae the rght decso to resolve the challege ad develomet tred of the labor maret geeral ad labor marets agrculture ad dustry artcular.. ermetal Secto The research toc receves the cocet of surlus labor the sese that f a umber of agrcultural labor ca be removed wthout causg a reducto outut ths art of the labor force s called surlus labor. The amout of removable agrcultural surlus labor labor-hours s defed as the dfferece betwee the labor avalable ad labor reured to roduce a certa outut of agrculture. The research toc otced measuremet aroach of surlus labor gve by the ILO the method of comarg the use of labor o farm wth emloy wage labor wth ts use o farms seems more comrehesve systematc ad feasble alteratve method. So the toc roosed use ILO methodology to test the measuremet of surlus labor agrculture Vetam. I order to gather the formato eeded to calculate the agrcultural surlus labor uder the ILO methodology descrbed above the toc has desged a survey o surlus labor rural agrcultural areas Ha Duog rovce wth the samle sze of 900 rural households 0 farms s commues of Ha Duog rovce 0. The uestoare was desged to collect formato o ecess farm labor suly. Household uestoare s dvded to 5 sectos: Secto I. Demograhcs of households/farms. Secto II. Labor ad worg tme volved the ast moths. Secto III. Lad use ad farmg of households/farms. Secto IV. Results of farm roducto. Secto V. mloyg wage labor ad catal of vestmet of households/farms for agrcultural roducto. The ey uesto ths survey s to collect eactly the umber of self-emloyed ad emloyg wage labor by households as well as wor hours wor days the households ad umber of wor hours wor days of emloyg wage labor samles. The uestoare was desged more detal for ths uesto. The toc cocetrated o rocessg ad aalyss of data from household uestoare. Here are the results of calculato of surlus labor agrculture Ha Duog.

44 Measuremet of Surlus Labor Vet Nam grculture 09 I total 90 households/farms surveyed there are 69 farm households 75.% 8 o-agrcultural households 4.8% wth 4 households emloyg wage labor 45.9% ad 498 households ot emloyg wage labor 54.% The total surveyed household demograhcs s 794 eole of whch total umber of ersos 5 years of age or older s 8 eole. I total 8 ersos 5 years of age or older there are 988 eole wor agrculture cultvato eole lvestoc - eole accoutg for 46.4% 797 of o-agrcultural emloymet accoutg for 37.5% ad 343 eole ot worg accoutg for 6.%. The survey results show that the average lad area er household s 96 m 0. ha of whch er emloyg wage labor household s 00 m 0. ha er household ot emloyg wage labor s 905 m 0.9 ha. I geeral the average umber of worg moths er laborer s moths of whch er cultvato labor s 0.3 moths er lvestoc labor s. moths er o-agrcultural worer s 6 moths. verage umber of worg days the year er labor s 5 day of whch er cultvato laborer s 66 days er lvestoc worers s 38 days ad er off-farm labor s 64 days. verage umber of worg hours er laborer er day for a year s 6. hours of whch er cultvato laborer s 5. hours er lvestoc worers s 4.8 hours ad er off-farm labor s 7.6 hours Table shows how calculate the total umber of labor-hours of labor used er hectare for a year of emloyg wage labor households. Where N w s the umber of laborers actually emloyed the households emloyg wage labor N w 496; L w s the total umber of labor-hours of labor eeded er hectare er year o households L w 7773 labor-hours. Table shows how calculate the umber of agrcultural surlus labor the samle households Where N f s the umber of laborers actually emloyed households N f 987; L f s total umber of labor-hours of labor eeded er hectare er year o households L f R f s the reured umber of laborers er hectare o households calculated usg the formula : L f R f N w. 80 ersos Lw S f s the umber of surlus labor er hectare households calculated usg the formula 3: S f N f - R f ersos Calculato results showed that there are 85 surlus labor of total 987 agrcultural laborers the samle survey Ha Duog rovce. S f Surlus labor rato r surlus %. N f Ths result s useful evdece for the study of agrcultural surlus labor Ha Duog rovce where arable lad s shrg ad declg due to urbazato rate radly creased. Table Calculato of labor-hours of labor used er hectare for a year of emloyg wage labor households. No. Idcators Numbers Number of laborers ast moths of wage emloyg households N w 496 verage umber of wor moths er labor ast moths of wage emloyg households.4 3 verage umber of wor days er labor ast moths of wage emloyg households 73 4 verage umber of wor er labor ast moths of wage emloyg households Total labor-hours of wage emloyg households Total labor-hours er hectare of wage emloyg households 60 7 Total wor hours of emloyg wage labor Total umber of wor hours of household labor lus emloyg wage labor Total umber of wor hours of household labor lus emloyg wage labor er hectare L w 7773

45 0 Measuremet of Surlus Labor Vet Nam grculture Table Calculato of surlus agrcultural labor No. Idcators Numbers Number of laborers ast moths of households N f 987 verage umber of wor moths er laborer ast moths of households verage umber of wor days er laborer ast moths of households 75 4 verage umber of wor hours er laborer ast moths of households Total labor-hours of households Total labor-hours er hectare of households L f The coeffcet of labor demad er hectare.6 8 Total labor demad R f 80 9 Number of redudat laborers S f 85 09: Surlus labor rato Results ad Dscusso The results of the research toc are ust the result of eermetal research the sze of survey s stoed oly of oe rovce. The research toc has ot uestoed o easo ad geeralzato samle to estmate the overall scale of surlus labor rural agrculture at atoal level. Ths lmtato should be addressed further studes. I the future tme a module of survey o surlus labor would be roceeded to stall the labor force survey or samle survey of agrculture ad rural aually. Results of ths samle survey ad estmato of surlus labor wll aswer uestos about the sustaablty of the labor maret: whether or ot the amout of surlus labor rural areas? If so how large s ths worforce ad how log t ca be lastg. From there the rght decsos wll be made ad solve challegg develomet tred of the geeral labor maret the labor maret the dustry ad agrculture sectors artcular. I order to ead the research results we suggest to set u a roect volvg research collaborato ot oly of the researcher the Isttute of Statstcal Sceces statstca of Geeral Statstcs Offce Vet Nam but all eerts scetsts abroad. It s ossble through ths tal study we troduce to teratoal orgazatos such as the Iteratoal Labour Orgazato ILO Statstcal Isttute for sa - Pacfc SIP Iteratoal Statstcal Isttute ISI... such a roect o measurg labor surlus Vet Nam. Refereces [] maresh Dubey Rchard Palmer oes ad Kua Se. 004: Surlus Labor Socal Structure ad Rural to Urba Mgrato: vdece from Ida Data School of Develomet Studes Uversty of ast gla Norwch. [] Che Xayu 004 The methodologcal research of measurg rural resdual labor forces Statstcal Research : Chese. [3] Che X-ag 99 alyss of Iut-Occuato-Outut of Urba ad Tow coomcs Cha. Beg: Scece Press Chese. [4] Fug Kwa 008: grcultural labour ad the cdece of surlus labour: eerece from Cha durg reform Uversty of Nottgham ublee Camus Wollato Road Nottgham. [5] Furog ad Keu Lee 009: Surlus Labor Oeess ad the Urba-Rural Ieualty Cha Deartmet of coomcs Seoul Natoal Uversty. [6] ILO/SMT 998: Maual o labour maret aalyss ad olcy Bago. [7] Lews W coomc Develomet wth Ulmted Suly of Labour The Machester School : [8] Mehra S. 966 Surlus Labour Ida grculture Ida coomc Revew. [9] Wag agu ad Shouha Dg 006 re-estmato of Cha s agrcultural surlus labor The demostrato ad modfcato of three revalet methods Froters of coomcs Cha : 7-8.

46 oural of Mathematcs ad System Scece D DVID PUBLISHING stmato of Demograhc Statstcs the Camboda Soco-coomc Survey CSS Pe Socheat ad ders Holmberg. Natoal Isttute of Statstcs Demograhc Statstcs Cesus ad Survey Deartmet #386 Preah Movog Blvd Boeug Keg Kog Chamarmor Phom PehCamboda. Statstcs Swede Deartmet of Research ad Develomet Örebro Swede Receved: Setember 03 / cceted: Octorber 3 03 / Publshed: February bstract: Sce 007 the CSS s the oly source that every year has rovded estmates of Camboda demograhc statstcs such as oulato ad household szes by geograhcal rego se martal status ad ethcty. Ths wor descrbes the methodology used to mae the estmates cosstet wth Cesus roectos tag to accout chage household szes. The methods are based o assumtos ad they wll adust the weghts of the survey. The assumtos eed to be evaluated cotuously ad ths resetato s the frst documetato ad evaluato. Usg tme seres of some ey statstcs the wor also resets ad dscusses the demograhc develomet Camboda. future decso f the methodology the CSS has to be comletely chaged or ust udated must be made whe the Iter-cesual result from 03 are avalable. Key words: Cosstecy cesus roectos weghtg method. Nomeclature t Tme dcator years oetal de couter 0 for 008 The total oulato POPof Camboda year t The umber of Households HHs Camboda year t The Cesus estmate of the Camboda oulato adusted for udercout for ormal households HHs The estmated oulato of Camboda from CSS adusted wth calbrated weghts The estmated average HH Sze of year t where X Y M The oulato chage sce 008 accordg to Cesus roectos. The chage average household sze sce 008 accordg to cesus roecto Costat based o the assumto of the same δ aual chage the average household durg a te year erod. Gree Letters [] Delta Corresodg author: Pe Socheat Natoal Isttute of Statstcs Camboda. -mal: Hotmal.com.. Itroducto Sce Camboda s 008 Cesus the offcal demograhc statstcs are based o yearly roectos of the Cesus results. The statstcs ad the uderlyg methodology of the roectos are descrbed Refs. [] ad []. The Natoal Isttute of Statstcs Camboda has wth the assstace of Statstcs Swede ad the facal suort of the Swedsh Iteratoal Develomet cooerato gecy coducted the Camboda Soco-coomc Survey CSS every year sce 007. mortat art of the CSS s ts estmates of demograhy ad mgrato. They are used as bacgroud formato ad for statstcal comarsos of all the other subect matter cotet the CSS. The CSS cota modules of coutry wde samle of households ad household members about housg codtos educato ecoomc actvtes household roducto ad come household level ad structure of cosumto health vctmzato etc. There are also uestos related to eole the labour force e.g. labour force artcato.

47 stmato of Demograhc Statstcs the Camboda Soco-coomc Survey CSS CSS was coducted termttetly the erod 993 to 004 but sce 007 the survey s aual. The 004 ad 009 were large samle surveys 000 households whereas the years betwee have small samles about 3600 households. The data from the CSS rovde mortat formato about lvg codtos Camboda ad have a wde rage of use. Results from CSS are used for motorg the Natoal Strategc Develomet Pla NSDP ad rogress towards the Mlleum Develomet Goals. It s mortat that the demograhc estmates of the CSS ad other Camboda surveys to a certa degree are cosstet wth the offcal Cesus statstcs. Not rmarly because the roectos from the Cesus are artcularly accurate a gve doma of study but rather to ee Camboda offcal statstcs harmozed. Ths aer descrbes the method used u to ow ad dscusses ts roertes ad future develomets. The outle s the followg. fter tally defg ecessary otato ad arameters I wll descrbe the methods that are used to estmate the oulato growth ad the chages household comostos. Thereafter I wll reset demograhc estmates ad commet.. Bacgroud of the Camboda Soco- coomc Survey The CSS s a multstage samle survey of Camboda ormal households ad ersos. Presetly the samlg desg cossts of three stages. The rmary samlg uts come from atowde frame of vllages. These are dvded to a rural ad a urba stratum ad ordered by geograhcal locato. From each stratum a systematc samle wth robabltes roortoal to sze umber of households er vllage s selected. The secodary samlg uts are eumerato areas the vllages selected by smle radom samlg ad the fal Normal households are all household ecet homeless Isttutoal household boat oulato traset oulato stage te households are selected by systematc samlg. Data are collected every moth face to face tervews. The two ma arameter of terest cosdered ths aer are : : stmates of s gve by : The Cesus estmate of the Camboda oulato adusted for udercout for ormal households HHs ad : The estmated oulato of Camboda from CSS adusted wth Calbrated weghts. Furthermore defe : as the estmated average HH Sze of year t where. Whe we estmate demograhc characterstcs of Camboda we eed to tae to accout two dfferet treds over tme. Frst the growth umber of ersos ad secod the chage HHs comosto. The household szes are decreasg. I wll beg by descrbg method used for the oulato growth. The method has bee used sce CSS 00 ad t uses the cesus roecto ad the cesus estmate from 008 as a base. 3. Methods ad ssumtos to stmate Demograhc Chage the CSS Poulato Growth Let be the oulato chage sce 008 accordg to Cesus roectos. The the yearly estmate of chage from March 008 to March 009 for the whole Camboda s gve by.0 accordg to Poulato Cesus of Camboda 008. Sce the CSS s stratfed by Urba/Rural smlar estmates are comuted for Urba/Rural searately. Ths wll gve us a seres of estmates of chage from Table stmate of demograhc chage sce 008 CSS % 3.7% 5.3% 6.9%

48 stmato of Demograhc Statstcs the Camboda Soco-coomc Survey CSS From the cesus roectos some adustmets must be made for the umber of ersos lvg ormal HHs or ot ad adustmet also eeds to be made for admstratve chages haeg the rovces sce the cesus. 4. Chage HHs Sze The other tred that s ecessary to estmate s the chage HHs comosto. Defe : as the chage average household sze sce 008 accordg to cesus roecto. For the chage household sze the CSS estmate have bee based o a crude model usg the observed chage betwee 998 cesus ad 008 cesus. I 998 the average household sze was 5.8 ersos er household ad 008 the household sze had decreased to 4.66 accordg to cesus data. Ths meas a aual decrease household sze by aromately %. Ths s used for a roecto model of the aual chage average household sze based o the assumto of the same aual chage every year the erod 998 to 008 that s δ X X Hece a decrease by aromately ercet every year s assumed however household sze caot cotue to decrease learly so the yearly roecto formula aled to the chage the average household sze s 0.9 where 0 for 008 for 009 ad so o. Therefore gves the roecto 4.73 for ormal household sze 009. The roected umber of ormal households are the gve by the followg comutato..g for 009 we get / Tag to accout the admstratve chages ad the two treds creasg oulato ad decreasg household szes we ca comute a seres of estmated total umber of households er year usg the above formula. 5. stmato the CSS ad Calbrato stmate of the Poulato Sze. The CSS estmate of the oulato sze s calbrated wth weghts to satsfy the euato. Theory about calbrato ca be foud Ref. [3]. The formato used the estmato s the roectos from the Cesus. Sce the CSS has ractcally o oresose the estmator s of the form wth beg a geeral otato of a study varable whch whe coutg the oulato here s eual to oe for each ut. However t ca also be a doma dcator coutg the oulato a e.g. a rovce. The weght cossts of two arts s the desg weght from the samlg desg ad s the calbratg factor determed as to satsfy. I our case t gves us a smle easo estmator wth. The calbrato techue guaratees that the CSS estmates of the oulato sze wll be eual to the forecasts from the 008 Cesus ad ths assures cosstet results the Camboda oulato statstcs. The weghts that are comuted wll also fluece other CSS estmates. ossble roblem s that the accuracy of the cesus oulato forecasts are decreasg as tme asses. Comaratve studes of Camboda oulato forecasts ad results from later cesuses ad a Iter-Cesal survey show ute dsturbg cossteces See Refs. [] ad [4]. It s mortat that the CSS s calbrated agast formato that s as relable as ossble; otherwse the calbrated weghts may troduce a uecessary source of error to the survey results. Cotug to use the above method based o the roectos from the 008 cesus s ot a good alteratve. The Camboda 03 ter-cesal survey wll rovde ew formato that ca be used for the 03 CSS results ad owards utl the et laed Cesus Demograhc stmates I ths secto we llustrate the demograhcs

49 4 stmato of Demograhc Statstcs the Camboda Soco-coomc Survey CSS estmated recet CSS comared wth Geeral Poulato Cesus 008. The fgure shows the oulato crease Camboda. The umbers for Cesus for estmated oulato by se thousad reseted here dffer a lttle bt from those reseted the CSS ad 0 reorts. The umbers reseted the CSS reorts do ot tae to accout chages the defto of urba area betwee 998 ad 008 ad for 004 ad 009 also dd ot adust for areas ucovered the 998 cesus see Ref. [5]. revew of the estmato rocedure for 009 revealed that the rocedure gave a slght uward bas. The rocedure has coseuetly bee adusted ad the 009 estmates have bee udated. The oulato of Camboda dstrbuted by se s show Fg. above. The se rato me relato to wome has creased sgfcatly betwee the two cesuses but recet years there seems to be o chage but t mght be descrbed as a ormal at the atoal level that stll deotes a ecess of wome tha me varyg over a rather arrow rage from about to 4000 thousad for both sees. Household sze demograhcs Oe of the terestg arts of the CSS s the household sze demograhcs. Fg. below shows roected average household sze based o curretly used model. I tred the roect average household sze s gog to decrease every year etre Camboda both urba ad rural. The estmated household sze for 009 s greater tha those roected for 00 0 ad 0 cludg Fg. Poulato develomet Camboda 998 to 0 Fg. Proected average household sze based o curretly used model

50 stmato of Demograhc Statstcs the Camboda Soco-coomc Survey CSS Fal Commets ad Cocluso Fg3 verage household sze s CSS ad Cesus 008 urba ad rural see Fg. However Fg. 3 below shows a dfferet cture wth estmates from the 00 ad 0 CSS llustratg a steeer decrease average household sze tha the modelled assumtos suggest. We also have dcatos from the mgrato statstcs CSS 00 that the oulato Camboda was movg to Phom Peh ad other abroad. Most for ob oortuty ad some cases Marres Reatrato or retur after dslacemet Isecurty Trasfer of wor lace. tc.. I these cases the household sze s decreasg. Fg. 3 shows the estmated mea household sze the CSS surveys ad the mea household sze ormal households accordg to cesus 008 see Ref. [6]. The cesus fgure s recorded after adustg for the udercout. Sce we beleve that systematc measuremets bas s small recordg the CSS household szes we susect from comarg Fgs. -3 that the assumtos about the chage household szes eeds to be revsed. There are sgs that the decrease s faster tha revously assumed. The CSS 0 ad the Iter-Cesal survey 03 wll sread ew lght o ths. For a coutry t s of outmost mortace that the oulato statstcs are relable. other reuremet that sometmes aears s cosstecy the umbers ad estmates hece that may sources rovde f ot the same but at least a smlar cture of realty. The Camboda 008 cesus ad ts oulato roectos have u tll today bee the basele for mrovg the estmates of the CSS. Ths aer descrbes ths estmato methodology ad t also rases some uestos ad doubts whch are the ey fdgs of ths aer. For eamle the cesus results are gettg older ad there are sgs that the accuracy s decreasg. Moreover the methodology used to roect the household szes does also show sgs of beg off the mar. The future las are to comare these fdgs wth the results from the ogog Camboda Iter-Cesal survey. It wll be used to assess the ualty of the cesus roectos ad to draw coclusos of how to mrove future demograhc estmates of the CSS. Refereces [] rraga Poulato alyss wth Mcrocomuters US Bureau of the Cesus New Yor [] Natoal Isttute of Statstcs Camboda. 0 Geeral Poulato Cesus of Camboda 008 alyss of the Cesus Result Reort Poulato Proecto of Camboda. auary 0. [3] Särdal C-. ad Ludström S. 005 stmato Surveys wth Noresose. New Yor Wley. [4] Natoal Isttute of Statstcs Camboda 004 Frst revsos of Poulato Proectos for Camboda ue 004. [5] Natoal Isttute of Statstcs Camboda. 00. alyss of the Cesus Results Reort Famly ad Households. NIS ue 00. [6] Pettersso H. 00. verage household szes the CSS-surveys teral reort Natoal Isttute of Statstcs Camboda.

51 oural of Mathematcs ad System Scece D DVID PUBLISHING Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg Kodwo a Brya Bafll ad Stehae Broc. Deartmet of Mathematcs & Comuter Scece Mot State Uversty Mot ND Deartmet of Scece Mot State Uversty Mot ND Receved: December 8 03 / cceted: auary 3 03 / Publshed: February bstract: lesy s a severe eurologcal dsorder clcally detfed by hyer-ectablty ad/or hyer-sychroy the corte ad other subcortcal regos of the bra. To regulate such ectablty ad sychroy Hodg ad Huley model has bee deloyed wth ether PUF or calcum bufferg couled wth TP modulate eurotrasmtter release. We formulate ad aalyze a system of dfferetal euatos that descrbe the effects of PUF TP ad calcum bufferg regulatg euroal hyer-ectablty ad hyer-sychroy eletc atets. We observed that PUF had dverse effects o the gatg varables. Secfcally there was a sgfcat reducto the hbtory otecy of PUF o the m -gates whch may cause a drect hbto of the voltage-gated Na+ chaels ad thus reduce euroal ectablty eletc atets. lso the actvato of the otassum chaels by PUF drectly lmted the euroal hyer-ectablty whle a small chage voltage otetal couled wth PUF restrat actvated the voltage deedet o chaels whch aded lowerg eletc ectablty atets. I addto hgher TP buffer levels the resece of PUF caused a sgfcat hyerolarzato whch may decrease euroal ectablty whle lower TP level tated euro deolarzato. These results clearly suggest that PUF couled wth calcum ad TP bufferg could be used to modulate euroal ectablty ecessve sychroy eletc atets. Keywords: PUF Calcum & TP bufferg mathematcal model euroal ectablty.. Itroducto lesy s a severe eurologcal dsorder characterzed by esodes of sotaeous recurret sezures. The clcal mafestatos of the sezures are detfed by hyerectablty ad/or hyersychroy the corte ad other subcortcal regos of the bra. ve though there are may factors led to the etology the o chaels cocet rooted the semal wor of Hodg ad Huley [] have become a cetral aradgm to descrbe ad regulate such ectablty ad sychroy of euros [-8]. The uderlyg dea behd the Hodg ad Huley model s that the oeg of voltage-gated sodum Na chaels Corresodg author: Kodwo a Ph.D. assstat rofessor research felds: edemologcal modelg bomedcal. -mal: creases ectablty whle oeg of voltage-gated otassum K + chaels reduces ectablty. Thus several ga-of-fucto mutatos Na + chaels [] as well as loss-of-fucto mutatos both delayed rectfer K ad -tye V K K chaels [7-8] are V 3 led wth elesy. Studes have show that hghly sychrozed euroal actvty durg eletogeess s suressed by the K V 3 chael dedrtes whch therefore may fucto as a rotectve mechasm agast hyerectablty [9-]. I elesy sce the K V 3 curret may ot be strog eough to comesate for the ectablty chages ateletc substaces such as the olyusaturated fatty acds PUF have bee used suressg cellular resoses to sychrocty durg etogec det to hel revet eletc sezures

52 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg [3-5]. It s beleved that PUF targets ad teracts wth a wde rage of o chaels cludg Na + ad K + chaels whch lead to modfed voltage deedece of the chaels actvato ad actvato [6-7]. I addto earler classc wor of Hodg ad Huley cluded calcum os Ca + whch studes have show to regulates eurotrasmtter release ad oc chael ermeablty [8-]. t the euroal level Ca + deolarzes as the sodum curret but also acts o the slower tmescale of the otassum curret. For eamle research has show that the cocetrato of calret actg as a fast calcum buffer cotrols the ectablty of cerebellar graule cells ad through the actvato of hgh-coductace voltage- ad Ca + actvated K + [0-]. Furthermore Roussel et al. 006 have dcated that chages the buffer cocetrato ca dramatcally affect the electrcal dscharge atter of cerebellar graule cells hece allowg trastos betwee regular euroal frg ad dfferet tyes of burstg []. Therefore the basc cocet of euroal Ca + s that the flu of Ca + through voltage-oerated chaels causes trasmtter release. mog them are the fter-hyerolarzato HPs ad the deolarzg fter-potetals DPs whch modfy euroal actvty by suressg or romotg hyerectablty resectvely. Thus to cotrol the Ca + -deedet o chaels resosble for the varatos the membrae otetal Ca + eterg through the voltage-oerated chaels or Ca + released from the teral stores eeds to be regulated. To modulate euroal ectablty secfcally calcum homeostass modulator CLHM have bee show to duce cato currets ad elevate cytolasmc Ca + cocetrato [0-]. deose-5-trhoshate TP deedet otassum chael K TP s aother substace show to be otet regulators of euroal ectablty. Closg ad oeg of K TP deeds o the tracellular cocetrato of TP. Whe the tracellular 7 TP/adeose dhoshate rato creases K TP closes ad vce versa. So closg of K TP leads to membrae deolarzato ad thus creases the ectablty whle oeg of K TP hyerolarzes the membrae ad s hbtory. Hece actvato/ oeg of K TP otetally rotects agast ectotocty. The am of ths aer s to use coductace-based comutatoal model to vestgate the etet to whch PUF calcum bufferg ad K TP regulate euroal ectablty ad sychrosm eletc atets. The model allows us to vestgate the effect of varatos the cocetrato of PUF calcum ad K TP euroal actvty durg eletogeess. The model formulato s reseted secto followed by results ad dscusso secto 3. We fally mae our cocluso secto 4.. Model Formulato We adat the coductace-based model roosed by Bscho et al. 0 for euroal ectablty ad calcum bdg rotes [3]. The modfcato of the model cludes calcum homeostass modulator ad deose-5-trhoshate chaels. I addto we modfy the model to clude PUF modulated cellular resosble for alterg the actvty of the voltage gated o chaels. We focus o the o chaels I I I Na KV K ad I V3 TP. The curret coservato euato goverg the membrae otetal dyamcs s gve by dv Cm INa IK I V K I V3 Ca dt ICaK ITP Ilea + Iut where C m ad V are the membrae caactace ad otetal of the euro resectvely I Na s a fast traset Na + curret I K v s a fast delayed rectfer 3 otassum curret I K v a slow delayed rectfer otassum curret I CaK the Ca + actvated K + curret I Ca the calcum curret I lea the assve lea curret I TP the TP-deedet otassum o curret ad I ut a eteral aled curret. The oc currets are gve by

53 8 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg 3 INa gnam hv VNa I g V V 4 Kv Kv K I g V V Kv3 Kv3 3 K ICaK gcak V VK ICa gcaa V VCa ITP gtp V VTP Il gl V Vl. I. h ad m are the actvato ad actvato gatg varables of the I Na curret resectvely a 3 are resectvely the actvato varables of ICa IKCa IK I v K currets. Based o the v3 a clcal membrae caactace values betwee 5 30 F wth membrae resstace 400 M Ω ad restg membrae otetal of 70 mv the arameter values for s reseted Table.. Modulato of PUF PUF modulato of the Na + chael I Na s mlemeted as a hyerolarzg shft of the steady state actvato curve gve as 9 mv [ PUF] Na where [ PUF ] Na s altered the terval betwee zero ad oe [4 5]. Ths leads to the reducto of the sodum chael [4]. The mlemetato of PUF modulato o the slow delayed rectfer otassum curret I K V s a hyerolarzg shft of the steady state actvato curve by 8 mv [ PUF] K V that lead to a crease the otassum curret [6]. Smlarly we assume that the fast delayed rectfer otassum o curret I K V 3 s also affected by PUF wth the same value 8 mv [ PUF] K V 3. Furthermore we mlemet PUF modulato of TP deedet otassum chael I TP as a shft from the restg membrae otetal by 5 mv [ PUF] TP. I our model PUF shfted actvato ad actvato wll be deoted by a PUF ad PUF resectvely whle shftg both actvato ad actvato s rereseted by a PUF.. Modelg gatg Varables To model gated voltage chaels each gate ca be oe or closed wth trasto robablty from oe to close α V ad from close to oe β V. Thus the fracto of oe chaels dyamcs of the gatg varables y h 3 follow [7] dy αy V y βy V y. 3 dt Itally the restg values of y s α y 0 y 0. 4 αy0 + βy0 However as the voltage s clamed to a dfferet voltage V the steady state values become α y V y V. 5 αy V + βy V Soluto to 3 s a eoetal fucto of the form t y t y V [ y V y 0]e 6 τ Where the tme costat τ gve by τ y V ad the steady-state gatg varables αy V + βy V y deed o the membrae otetal V. I artcular Table The model arameters used to smulate euato. Parameter Value Referece Membrae caactace C 30 F m [3] Calcum coductace g Ca 30 S [3] Ca + actvated K + coductace g CaK S [8] K V otassum coductace g K V S [3] K V 3 otassum coductace g K V 300 S [3] 3 Lea coductace g lea.5 S [3] Sodum coductace g Na 700 S [3] Calcum reversal otetal V Ca 80 mv [] Potassum reversal otetal V K 90 mv [9] Lea reversal otetal V lea 68 mv [3] Sodum reversal otetal V Na 74 mv [9]

54 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg 9 sce the actvatos varables a ad m are cosdered fast as comared to the other gatg varables we set ther steady state values as a a V ad m m V resectvely wth a ad m defed as 6 V αm a + e m αm + βm Wth the etcs of the a varable defed by Roussel et al. 006 as Ref. [] V τ a + + e[ 0.07 V 5] e[0. V + 8.9] 8 The K varable for the Ca + actvated K + chaels do ot deed o voltage ad ca be modeled accordg to Goldberg et al. 009 as Ref. [8] + [ Ca ] τ + [ Ca ] τ + + KCaK + [ Ca ] KCaK + [ Ca ] d dt 9 off CaK where KCaK wth off CaK 0. ms ad o CaK the Ca + bdg rate o CaK 0.4 µ M ms..3 Calcum Io Curret Currets through voltage-actvated Ca + chaels lay a crtcal double role. Frst ward Ca + flu I Ca deolarzes the cell ad thus cotrbutes to acto otetal formato. Oce reachg the tracellular comartmet however Ca + s also a ey secod messeger cotrollg a broad rage of euroal fuctos resosble for eurotrasmtter release whch cludes oeg of Ca + -actvated K + chaels I CaK. The basc cocet of euroal Ca + s the Ca + cocetrato chage ad calcum bdg rotes. Chages of Ca + cocetrato [ Ca + ] have bee observed to modulate euroal ectablty. For eamle CLHM eresso has bee show to duce cato currets euro cell ad elevate cytolasmc Ca + cocetrato [ Ca + ] resose to removal of etracellular Ca +. Calcum bdg rotes such as the Parvalbum PV s a member of the calcum bdg rotes grou that has two med + Ca Mg + bdg stes. Sce the bdg of Ca + s determed by the slow Mg + off- rate the dyamc mechasm of calcum the resece of PV s modeled as + d[ Ca ] dt ICa + + d[ PVCa] [ Ca ] [ Ca ] rest df dt 0 d[ PVCa] + o Ca[ Ca ][ PV ] off Ca[ PVCa] dt d[ PVMg] + o Mg[ Mg ][ PV ] off Mg[ PVMg] dt where [ Ca + ] ad [ PV ] rereset the free tracellular Ca + ad free PV cocetratos resectvely [ PVCa ] ad [ PVMg ] are the cocetrato of PV boud to Ca + ad Mg + wth the total cocetrato of PV [ PV ] T [ PV ] + [ PVCa] + [ PVMg] The ward flu term s I Ca where F s the Faraday costat ad s df the cell surface area of a shell wth thcess + + d 0. µ m. The term [ Ca ] [ Ca ] rest s the clearace mechasm assocated wth Ca + flues across the membrae where ms ad [ Ca + ] rest µ M I addto PV mght also act as a fast buffer [34] ad so we cosder both fast ad slow buffers by deotg [ B ] ad [ BCa ] resectvely as the cocetrato of free ad boud buffer for slow or fast buffer wth total buffer [ B] T [ B] + [ BCa]. Therefore we ca coule the calcum dyamcs as + d[ Ca ] dt ICa + + d[ BCa] [ Ca ] [ Ca ] rest 3 df dt

55 0 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg d[ BCa] dt Ca B BCa 4 + o[ ][ ] off [ ]. 3. Numercal Results ad Dscussos We umercally smulate the s. -4 wth model arameter values stated Tables ad by usg a fourth-order Ruge-Kutta method ad MTLB software tool. Gve a dfferetal euato y t f t yt wth tal codto the method determes the et ot o the curve by usg the revous ot ad the weghted averages of the four cremets.e. hf t y h hf t + y + h 3 hf t + y + hf t + h y The frst cremet s smly uler cremet. It taes to cosderato the sloe of the taget of the revous ot wth the ste sze h. The secod cremet s based o the sloe of the taget at the ceter of the ste. Whle the thrd ad fourth cremets are the sloe the mddle wth resect to the secod cremet ad at the ed of the ste resectvely. 3. PUF ffects o Gatg Varables Fg. characterzes the otecy of PUF o voltage -gated sodum chaels. The actvato m -gates ad the actvato h -gate curves ad PUF effects o both gated varables are lotted agast tme. There s a sgfcat reducto the hbtory otecy of PUF o the m-gates whle the h -gate became almost comletely resstat to PUF. The strog but traset hbto of the m-gates may drectly decrease the sodum curret ad thus reduce euroal membrae ectablty. I addto sce PUF also shft the actvato of the sodum chael [5] t may have a drect hbto o voltage-gated sodum o chael ad thus act to lmt hyerectablty actvty. The actvato of otassum chaels such as the slow ad fast delayed rectfer otassum currets K V ad K V 3 wth ther PUF effects are llustrated Fg.. The K V shows strog but ersstet otassum currets wth delayed oset ad slow actvato wth PUF whle K V 3 dects wea but fast oset ad fast actvato of otassum curret wth PUF. Therefore a hghly sychrozed euroal actvty may be suressed by the K V 3 chael ad may fucto as a rotectve mechasm agast hyerectablty. Hece PUF may act drectly to lmt hyerectablty ad sezure actvty by actvatg the otassum chaels. Table The model arameters used to smulate euatos 0-4. Parameter Value Referece Cell surface area 3000 µ m [3] Shell thcess d 0. µ m [3] Ca + etruso rate ms [3] Restg Ca + cocetrato [ Ca + ] rest 0.07 µ M [3] Mg + cocetrato [ Mg + ] 500 µ M [30] Ca + bdg rate to PV K o Ca 0.µ M ms [3] PV affty for Ca + K 0.0µ M D Ca [3] Fast Ca + buffer bdg rate o 0.µ M ms [3] Slow Ca + buffer bdg rate o 0.µ M ms [33] Fast buffer affty for Ca + K 0.0µ M D [3] Slow buffer affty for Ca + K D 0.µ M [33] PV affty for Mg + D Mg 3µ M [3] Mg + ubdg rate from PV K off Mg 0.05 ms [3] Mg + bdg rate from PV K o Mg µ M ms [3]

56 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg The ffects of PUF o m ad h Gatg Varables m Wthout PUF m Wth PUF h wthout PUF h wth PUF Gatg Varables Tme ms Fg. The sestvty of the actvato ad actvato gatg varables h ad m to PUFs wth resect to tme s llustrated. PUFs caused a sgfcat reducto the actvato m -gates whle there s early o effects o the actvato h -gate. Ths may cause a drect hbto of the voltage-gated Na + chaels ad thus reduce euroal ectablty The ffects of PUF o Kv ad Kv3 Gatg Varables Kv Wthout PUF Kv Wth PUF Kv3 wthout PUF Kv3 wth PUF Gatg Varables Tme ms Fg. The delayed rectfers of the otassum currets are dslayed. Whle there s strog otassum curret of the slow delayed rectfer there est a wea fast delayed rectfer wth or wthout PUF. However both cases there s a sgfcat actvato crease of otassum currets wth PUF whch may suress hyerectablty ad/or hyersychroy of euros. Fg. 3 shows the effect of PUF o the actvato of calcum ad the calcum-actvated otassum chaels. s ca be see there s a crease steeess as well as the shft of the calcum actvato whe PUF s added. The rad tal actvato of the a -gatg varable may affects calcum ward curret whch

57 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg The ffects of PUF o a- ad - Gatg Varables Gatg Varables a Wthout PUF a Wth PUF wthout PUF wth PUF Tme ms Fg. 3 PUF radly actvates ad shfts the a -gatg varable curve alog the gatg as tally the slow dow afterwards whle o chage s ecoutered the Ca + -actvated K + chaels. The slow ad fast actvato of the a -gatg varable leads to calcum bufferg whch may result burstg. may cause fast calcum bufferg the system. The slowdow the actvato of a-gatg later o dcates a slower Ca + dyamcs. Ideed the trasto to burstg arsg wth a creased calcum bufferg caacty may be the drect coseuece of the o-puf effects the actvato of a-gatg -actvated K + chaels due to the slower a -gatg dyamcs. The coestece of the fast ad slow gatg varables whch lead to burstg s caused by the resece of PUF cocetrato. I addto the resece of PUFs cause eglgble coductace of Ca + -deedet K + chaels that are ow to coule voltage-gated Ca + chaels. 3. PUF ffects o Voltage Potetal PUF also has a otetal of alterg the actvty of voltage deedece chael. Wthout PUF Studes have show that all euros are electrcally ectable matag voltage gradets across ther membraes. From Fg. 4 a small chage the voltage otetal results a large effect o euros ectablty the absece of PUF. The stee voltage chage alters the fuctos of the voltage-deedet otassum chael whch s the rego that matters for eletc reettve frg. However alyg PUF sgfcatly chages the stee curvature ad shae of the voltage otetal curve. PUF duced a shft of early ad fal voltage deedece ad learzed trasto whch may strogly suorts the hyothess that PUF acts drectly o the voltage deedece. Hece may suggest that PUF has maor effects o early ad fal voltage deedece trastos that lower eletc ectablty atets. 3.3 ffects of TP Level Couled wth PUF To evaluate the role TP lays the resece of PUF we assume the oto that PUF geerally creases TP level ad vared the TP levels above ad below the stadard TP.e. mm. Hgher TP level 5 mm causes a small traset outward curret voltage mode ad oes the TP-sestve otassum K TP chael see Fg. 5. Sce the K TP

58 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg Voltage Chage er Tme wth ad wthout PUF Wthout PUF Wth PUF 60 Voltage mv Tme ms Fg. 4 The effect of PUF o a small chage voltage otetal s lotted agast tme. small voltage chage couled wth PUF moderates the voltage-gated o chaels by shftg the steady-state actvato curve egatve drecto alog the voltage as Voltage Chage er Tme for Hgher TP Value Wthout PUF Wth PUF 60 Voltage mv Tme ms Fg. 5 PUF couled wth sgfcat or hgh TP cocetrato level causes small traset outward curret ad may decrease euroal ectablty eletc atets. The TP amout used was 5 mm whch s sgfcatly hgher tha the stadard TP of mm. chael s oted to be a ecellet medator for ectablty we suggest that PUF couled wth suffcet or hgh TP cocetrato level wll cause euros to sgfcatly hyerolarze themselves va drect TP release. Ths hyerolarzato may decrease euroal ectablty ad subseuetly

59 4 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg Voltage Chage er Tme for lower TP Value Wthout PUF Wth PUF 60 Voltage mv Tme ms Fg. 6 PUF couled wth sgfcatly low TP value of 0.5 mm causes a large traset ward curret ad therefore resultg euro deolarzato. reduces eletc actvty see Fg 5. Cotrary Fg. 6 shows that reducg TP cocetrato level to 0.5 mm causes a large but traset ward curret voltage mode. Therefore the K chael whch reresets a tye of ward TP rectfyg otassum chael s actvated whe TP cocetrato level s low ad thus deolarzes the euros. 4. Cocluso I cocluso we have geerated a system of dfferetal euatos that descrbe the role PUF lays regulatg euroal hyer-ectablty ad hyer-sychroy eletc atets by moderatg calcum ad TP Bufferg. Frst we observed that PUF has dverse effects creasg ad decreasg o the gatg varables. There s a sgfcat reducto the hbtory otecy of PUF o the m-gates whle the h -gate became almost comletely resstat to PUF. Ths observato may cause a drect hbto of the voltage-gated Na + chaels ad thus reduce euroal ectablty eletc atets. I addto the actvato of otassum chaels the slow ad fast delayed rectfer otassum currets K V ad K V 3 by PUF drectly lmted the euroal hyer-ectablty. Furthermore a small chage voltage otetal couled wth PUF moderato actvated the voltage deedet o chaels whch aded lowerg eletc ectablty atets. We the altered the TP bufferg below ad above the stadard value of mm ad observed that PUF couled wth hgher TP level caused a sgfcat hyerolarzato whch may decrease euroal ectablty whle low TP level caused euro deolarzato. These results clearly suggest that PUF couled wth calcum ad TP bufferg could modulate euroal ectablty eletc atets. Referece [] Hodg Huley 95 uattatve descrto of membrae curret ad ts alcato to coducto ad ectato erve.. Physol. 7: [] Mesler MH Kearey 005 Sodum chael mutatos elesy ad other eurologcal dsorders. Cl Ivest 5: [3] Hah Neubauer B 009 Sodum ad otassum chael dysfuctos rare ad commo doathc elesy sydromes. Bra Dev 3:

60 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg 5 [4] Sgh N Charler C Stauffer D DuPot BR Leach R et al. 998 ovel otassum chael gee KCNQ s mutated a herted elesy of ewbors. Nat Geet 8: 5 9. [5] Charler C Sgh N Rya SG Lews TB Reus B et al. 998 ore mutato a ovel KQT-le otassum chael gee a doathc elesy famly. Nat Geet 8: [6] Bervert C Schroeder BC Kubsch C Berovc SF Prog P et al. 998 otassum chael mutato eoatal huma elesy. Scece 79: [7] Browe DL Gacher ST Nutt G Brut R Smth et al. 994 sodc ataa/myoyma sydrome s assocated wth ot mutatos the huma otassum chael gee KCN. Nat Geet 8: [8] Sgh B Ogwara I Kaeda M Tooam N Maza et al. 006 Kv4. trucato mutato a atet wth temoral lobe elesy. Neurobol Ds 4: [9] Brag Mody I Wlso CL gel 00 Local geerato of fast rles eletc bra. Neurosc : 0 0. [0] Lasztocz B tal K Nyos L mr Z Kardos 004 Hgh-freuecy syatc ut cotrbutes to sezure tato the low-[mg+] model of elesy. ur Neurosc 9: [] Frasé Tgerholm 00 Role of -tye otassum currets ectablty etwor sychrocty ad elesy. Hocamus 0: [] Gasar S Mglore M Magee C 004 O the tato ad roagato of dedrtc ses C yramdal euros. Neurosc 4: [3] Leaf Xao YF Kag X Bllma G 005 Membrae effects of the -3 fsh ol fatty acds whch revet fatal vetrcular arrhythmas. Membr Bol 06: [4] Taha Y Burham WM uv S 00 Polyusaturated fatty acds ad elesy. lesa 5: [5] Xu XP rchse D Böresso SI Dahl M Åmar P et al. 008 Polyusaturated fatty acds ad cerebrosal flud from chldre o the etogec det oe a voltage-gated K chael: utatve mechasm of atsezure acto. lesy Res 80: [6] Böresso SI Parar T Hammarström S lder F 00 lectrostatc Tug of Cellular ctablty. Bohys 98: [7] Böresso SI lder F 0 electrostatc otassum chael oeer targetg the fal voltage sesor trasto. Ge Physol 37: [8] Schwaller B The cotug dsaearace of ure Ca+ buffers. Cell. Mol. Lfe Sc [9] Lambers T. T. Maheu F. Oacea. Hoofd L. de Lage F. Meseam. R. Voets T. Nlus B. Claham D.. Hoedero. G. ad Bdels R Calbd-d8 dyamcally cotrols trv5-medated Ca+ trasort. MBO [0] Gall D. Roussel C. Neus T. Chero G. Servas L. D gelo. ad Schffma S. N Role of calcum bdg rotes the cotrol of cerebellar graule cell euroal ectablty: eermetal ad modelg studes. Prog. Bra Res [] Bearzatto B. Servas L. Roussel C. Gall D. Baba-ïssa F. Schurmas S. de Kerchove d aerde. Chero G. ad Schffma S. N Targeted calret eresso graule cells of calret-ull mce restores ormal cerebellar fuctos. FSB [] Roussel C. reu T. Schffma S. N. adgall D Modulato of euroal ectablty by tracellular calcum bufferg: from sg to burstg. Cell Calcum [3] Bscho DP Orduz D Lambot L Schffma SN. Gall D. 0 Cotrol of euroal ectablty by calcum bdg rotes : a ew mathematcal model for stratal fast-sg tereuros Froters Molecular Neuroscece Vol 5:78 DOI0.3389/fmol [4] Vreugdehl M bruehl C Vosuyl R Kag X Leaf et al. 996 Polyusaturated fatty acds modulate sodum ad calcum currets C euros. Proc Natl cad Sc US 93: [5] Leaf Xao YF Kag X Bllma G 003 Preveto of sudde cardac death by -3 olyusaturated fatty acds. Pharmacology & Theraeutcs 98: [6] Xu XP rchse D Böresso SI Dahl M Åmar P et al. 008 Polyusaturated fatty acds ad cerebrosal flud from chldre o the etogec det oe a voltage-gated K chael: utatve mechasm of atsezure acto. lesy Res 80: [7] Maclla. G. Lews T.. Pto D.. Rzel. ad Coors B. W Sychrozato of electrcally couled ars of hbtory tereuros eocorte.. Neurosc [8] Goldberg.. Teagarde M.. Foehrg R. C. ad Wlso C Noeulbrum calcum dyamcs regulate the autoomous frg atter of rat stratal cholergc tereuros.. Neurosc [9] olvet R. Lews T.. ad Gerster W Geeralzed tegrate-ad-fre models of euroal actvty aromate se tras of a detaled model to a hgh degree of accuracy.. Neurohysol [30] L-Smer Y. Levta. S. ad ohso. W. 00. Free tracellular Mg+ cocetrato ad hbto of NMD resoses cultured rat euros.. Physol. Lod. 533Pt [3] Lee S. H. Schwaller B. ad Neher Ketcs

61 6 Regulatg Neuroal Hyer-ctablty ad Hyer-Sychroy letc Patets by Usg PUF Calcum ad TP Bufferg of Ca+ bdg to arvalbum bove chromaff cells: mlcatos for [Ca+] trasets of euroal dedrtes.. Physol. Lod. 55Pt [3] berhard M. ad re P Calcum ad magesum bdg to rat arvalbum. ur.. Bochem. 6. [33] Schwaller B. Meyer M. ad Schffma S. 00. New fuctos for old rotes: the role of the calcum-bdg rotes calbd d-8 calret ad arvalbum cerebellar hysology studes wth ocout mce. Cerebellum [34] ggerma. ad oas P. 0. How the slow Ca+ buffer arvalbum affects trasmtter release aodoma-coulg regmes. Nat. Neurosc. 5 0.

62 oural of Mathematcs ad System Scece D DVID PUBLISHING Develomet ad lcato of Statstcal Busess Regster Gudeles frca Coutres Mchael. Colledge ad Besa Muwele. Mchael Colledge Statstcal Cosultg Caberra ustrala. frca Develomet Ba Pretora South frca Receved: Setember / cceted: Octorber 8 03 / Publshed: February bstract: Wth the framewor of ts Statstcal Caacty Buldg Program the frca Develomet Ba fdb s suortg develomet ad mrovemet of statstcal busess regsters SBRs frca coutres. s a frst ste the fdb reared a documet ettled Gudeles for Buldg Statstcal Busess Regsters frca whch descrbes SBR desg costructo troducto use ad mateace. To suort dssemato terretato ad effectve use of the Gudeles the fdb s ow sosorg a rogramme of revew ad recommedatos for ehacemets to SBRs selected frca atoal statstcal offces. The aer outles the cotet of the Gudeles ad eereces ther alcato. The vews eressed are those of the authors ad do ot ecessarly reflect a offcal osto of the fdb. Keywords: dmstratve data ecoomc statstcs roflg statstcal caacty buldg statstcal tegrato. Itroducto. Motvato for Gudeles Coordato of the dvdual surveys ad admstratve collectos that costtute the ecoomc statstcs rogram of a atoal statstcal offce NSO s vtal. It deeds uo the use of a commo cocetual framewor cludg artcular the System of Natoal ccouts SN ad the Iteratoal Stadard Idustral Classfcato of ll coomc ctvtes ISIC. crucal reuremet mag ths framewor oeratoal s to esure that the frames for the dvdual surveys are roerly harmozed. u-to-date survey frame meag a lst of uts ad formato about those uts eeded for the survey s reured for each reetto of a regularly coducted survey. It s more effectve ad effcet to mata a frame so that t ca suort the seuece of reettos of a survey tha t s to create the frame afresh wth Corresodg author: Mchael. Colledge Mchael Colledge Statstcal Cosultg Pty Ltd e-mal: each reetto. Survey frame mateace s best acheved through the develomet of a sgle statstcal busess regster SBR ad ts use as the source of frames for all ecoomc surveys. The adectve statstcal s added to the usual term busess regster to emhasze that the regster s develoed by a NSO for statstcal uroses. The desg develomet ad troducto or ehacemet of a SBR s cosdered so mortat that t aears as a core goal the fve-year atoal strateges of most f ot all frca NSOs. I resose to reuests from NSOs the frca Develomet Ba fdb establshed a roect to reare ad romote the documet ettled Gudeles for Buldg Statstcal Busess Regsters frca abbrevated SBR Gudeles.. Obectves of SBR Gudeles The obectves of the Gudeles are to rovde: a geeral bacgroud o the eed for a SBR ad the cocets o whch t s based; a detaled descrto of the fuctos of a SBR ad ts uts ad oututs; detaled formato o the develomet ad

63 8 Develomet ad lcato of Statstcal Busess Regster Gudeles frca Coutres mlemetato of a SBR; ad a startg ot for harmozato of SBRs across frca NSOs. The Gudeles are eected to be useful to: SBR maagers ad staff by detalg SBR cocets SBR creato ad mateace rocedures ad SBR ualty ad erformace measures; ecoomc survey maagers ad staff by rovdg the basc cocets o whch a SBR s based by descrbg the geerato of survey frames from the SBR ad by dscussg the ossblty of ublshg SBR data; staff resosble for resodet relatos by defg ad eablg calculato of dvdual ad cumulatve resodet burde mosed by ecoomc surveys; staff resosble for laso wth other orgazatos the atoal statstcal system ad wth teratoal orgazatos by rovdg the basc cocets; ad seor maagers outlg the basc cocets ad rovdg ualty ad erformace measures ad suggestos for ualty mrovemets..3 stg Documetato o Busess Regsters The startg ot for develomet of the SBR Gudeles was estg teratoal documetato o SBRs. Ths cludes: aers from 3 meetgs of the Wesbade Grou o Busess Regsters formerly Iteratoal Roud Table o Busess Survey Frames whch rovdes a forum for the echage of vews o develomet mateace ad use of busess regsters; aers from ot UNC/UROSTT/OCD meetgs of the ert Grou o Busess Regsters; the Busess Regsters Recommedato Maual for uroea Uo U coutres; dscussos of the uro Grous Regster Proect a etwor of SBRs U coutres focusg o multatoal eterrse grous; U regulatos - Regulato 77/008 establshg a commo framewor for busess regsters for statstcal uroses ad Regulato 696/93 o the statstcal uts for the observato ad aalyss of roducto systems. By ad large ths documetato s too sohstcated to rovde the basc gudace reured by develog frca NSOs some of whom do ot have a worg SBR or have oe wth oly a very lmted fuctoalty. The Wesbade Grou recogzed the eed for teratoal SBR gudeles ad a gudeles develomet roect was subseuetly set u by the Bureau of the uroea Coferece of Statstcas. However the resultg documet wll go through several teratos whch wll tae tme. lso t wll be amed at all coutres artcular NSOs wth well develoed SBRs ad thus wll lely address wth the sort of comle ssues that the Grou members are curretly taclg..4 Develomet of SBR Gudeles Tag to accout relevat teratoal stadards estg SBR documetato ad curret status of SBRs Souther frca coutres the fdb reared a relmary draft of the SBR Gudeles auary 0.The draft was dscussed at a ert Grou Meetg GM held Pretora May 0 ad atteded by SBR ut heads from NSOs belogg to Souther frca Develomet Commuty SDC ad Commo Maret for aster ad Souther frca COMS coutres. Commets ad eamles from GM were used rearato of the frst verso of the Gudeles October 0..5 lcato of SBR Gudeles To romulgate ad romote the SBR Gudeles the fdb set u a rogram of mssos to selected SDC ad COMS coutres. The rogram ams are: to revew the estg regsters f ay at each NSO vsted ad to reare a comrehesve sute of recommedatos for regster mrovemet cororatg deas from SBR Gudeles but talorg them to each artcular coutry s secfc reuremets; to detfy addtos ad chages to the SBR

64 Develomet ad lcato of Statstcal Busess Regster Gudeles frca Coutres 9 Gudeles to mae them more useful.. Cotet of SBR Gudeles. SBR Gudeles Part I: Uderlyg Cocets ad Methods The ams of Part I are frst to descrbe the evromet wth whch a SBR oerates ad to dcate the role of the SBR secod to detal the uderlyg cocets ad methods cludg broadly acceted rcles ad ractces o the bass of whch a SBR should be desged develoed ad mlemeted. Chater summarzes the cocetual framewor for ecoomc statstcs rovded by the System of Natoal ccouts 008 SN008 so far as t s relevat to the SBR. It defes what s meat by ecoomc roducto ad by eterrse. It elas the eed to rofle dvde large comle eterrses to smaller uts such as establshmets for data collecto uroses ad t troduces the Iteratoal Stadard Idustral Classfcato of ll coomc ctvtes ISIC Rev 4. Chater 3 descrbes the tyes of uts legal admstratve ad statstcal uts that are mortat the cotet of survey frames ad how they relate to oe aother. Chater 4 dscusses the elemets of a ecoomc statstcs rogram ad the eed for ad use of a frame for each survey ad t detals the cotets of a survey frame. Chater 5 elas the reasos for a SBR ts rmary fucto rovdg survey frames ts other ossble fuctos measurg busess resodet burde lg busess statstcs databases ad as a stad-aloe source of busess statstcs.. SBR Gudeles Part II: SBR Desg Part II focuses o desg of SBR coverage uts ad cotet data tems ad the ut sources ad fuctos by whch coverage ad cotet are created ad mataed. Chater 6 resets a framewor for secfcato of a ecoomc uts model ad the coverage ad cotet of the SBR. Chater 7 rovdes detals of the admstratve sources of SBR data ad how they are used combato to costruct ad mata the SBR. Chater 8 descrbes addtoal sources of SBR data volvg drect data collecto cludg roflg of large busesses SBR mrovemet surveys ad use of survey feedbac. Chater 9 dcates how the varous sources are used combato to udate the SBR as busesses are created trasformed ad dsaear over tme. Chater 0 sets out a SBR mateace strategy ad outles udate rocedures. Part IIB focuses o desg of the oututs ad outut fuctos of the SBR. Chater dscusses the rmary outut fucto of the SBR amely the roducto of survey frames also the coordato of samle selecto across surveys ad the creato of survey cotrol fles. Chater deals wth the other outut fuctos relatg to resodet maagemet busess statstcs ad lage of data across sources. It descrbes how the reortg commtmets of dvdual busess resodets ca be detfed ad overall resodet burde ca be comled. It dscusses the roducto of busess statstcs drectly from the SBR ad the role of the SBR brgg together data from surveys ad admstratve sources. Part IIC focuses o the orgazato of the SBR ad the system that suorts t. Chater 3 dscusses SBR orgazato ad oeratos. Chater 4 deals wth SBR systems.e. alcato rograms ad database. Chater 5 focuses o SBR ualty ad erformace maagemet ad evaluato..3 SBR Gudeles Part III: SBR Imlemetato Part III detals the stes the mlemetg or ehacg a SBR. Chater 6 resets a broad level la for frst-tme desg develomet ad troducto of a SBR.

65 30 Develomet ad lcato of Statstcal Busess Regster Gudeles frca Coutres Chater 7 resets a broad level la for revew of a estg SBR for determato of the etet of the chages reured reegeerg maor ehacemet or cotuous mrovemet ad for mlemetato of these chages..4 Summary Dagrams Fgures ad summarse the SBR fuctos uts oututs ad data model. 3. ereces lcato of the SBR Gudeles 3. NSOs ad Coutres Vsted SBRs have bee revewed ad recommedatos for mrovemets made fve coutres to date: Statstcs Botswaa the Cetral Statstcal Offce CSO Zamba the Natoal Bureau of Statstcs NBS Tazaa Fgure : Summary of SBR Fuctos Iuts ad Oututs. Fgure : Summary model of orgazatoal admstratve ad statstcal uts.

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