Human

KyushuUniversity

21stCenturyCOEProgram

DevelopmentofDynamicMathematicswith

HighFunctionality

Humanfingerveinimagesarediverseanditspatternsareusefulforpersonalidentification

T.Yanagawa,S.Aoki

T.Ohyama

󰉹

MHF2007-12

(ReceivedApril5,2007)

FacultyofMathematicsKyushuUniversityFukuoka,JAPAN

Humanfingerveinimagesarediverseanditspatterns

areusefulforpersonalidentification

TakashiYanagawa∗,

SatoshiAoki†,

and

TetsujiOhyama‡

Abstract

Diversityofhumanfingerveinimagesisshownbyextractingtheirpatternsfromtherightandleftindexfingersandmiddlefingersof506personandtheusefulnessoffingerveinspatternsforpersonalidentificationisshownbyevaluatingthefalseacceptanceandfalserejectionrates.

Keywords:beta-binomialdistribution,biometry,falseacceptancerate,falserejectionrate,veinpatterns

1Introduction

Ourfingerveinsarenotvisibleand,forexample,wedonotknowwhethertheimageofmyrightindexfingerveinissimilartothatofmyfriends,oreventothatofmymiddlefinger,ormyleftindexfinger.Infact,nothinghavebeenknownaboutnumbers,locationsandlengthsoffingerveins.However,recenttechnologiesmadeitpossibletogetsharpveinimages.Kounoetal.(2000)developedfeatureextractionoffingerveinpatternsbasedonrepeatedlinetrackingandFan(2004)usedthermalimagesofvein-patterns.AmongthemwedealwithinthispaperistheonedevelopedbytheCentralResearchLaboratory,Hitachi,Ltd(Miuraetal.,2002,2004a,2004b).Themethodextractsfingerveinimagesasfollows(Miuraetal.,2002).Firstanewideaoflinetrackingisintroduced.Secondthelinetrackingisconductedbystartingatseveralrandompositionstotakeintoaccountunknownnumbers,locationandlengthofveins.Thirdweobtainthefingerveinpatternsastheoneshavinghighfrequencyinthetotalnumberofpixelstobetracked.

Figure1showstheresultofextractedfingerveinpatterns,representedfromMiuraetal.(2002).ReadersshouldrefertoMiuraetal.(2002)fordetails.Weextractedveinpatternsfromtherightindexandmiddlefingersandthecorrespondingpatternsfromtheleftfingersofabout500person.Hereweshowdiversityofhumanfingerveinimagesanddiscusstheusefulnessoffingerveinsforpersonalidentification.Theargumentsinthispaperaremainlybasedontheevaluationofthefalseacceptancerateandfalserejectionrate.

BiostatisticsCenter,KurumeUniversity,Fukuoka,830-0011,Japan

DepartmentofMathematicsandComputerScience,GraduateSchoolofScienceandEngineering,KagoshimaUniversity,Kagoshima0-0065,Japan‡

BiostatisticsCenter,KurumeUniversity,Fukuoka,830-0011,Japan

†∗

1

Figure1:Extractedfingerveinimage(left)andpatterns(right)

2Diversityoffingerveinpatterns

Thepixelsthatconsistofanextractedveinpatternareclassifiedintothreecategories;VEIN,AMBIGUOUS,andBACKGROUND.Twoveinpatternsareoverlappedandarecomparedpixelbypixel.IfapixelthatisclassifiedintoVEINoverlapswithapixelclassifiedintoBACKGROUND,thepairofthepixelissaidtobemismatched.Notethatitisnotsymmetric,i.e.,suppose,forexample,thattherearetwofingerpatterns,sayRandL,whichconsistofthreepixelsclassifiedinto(AMBIGUOUS,VEIN,BACKGROUND)and(AMBIGUOUS,VEIN,VEIN),thenthereisnomismatchingfromRtoL,butonemismatchingfromLtoR.Themismatchrate(MMR)isdefinedas

MMR=

totalnumberofmismatchedpairs

.

totalnumberofpixelsclassifiedintoVEINinthetwofingerpatterns

TheMMRiscomputedfrom506person(405male,101female),whoseagedistributionisgiveninTable1.

Table1:Agedistributionofpersonwhoaretested

Agenumbers

10-191

20-29149

33-39220

40-4991

50-5941

refusedtotell

4

total506

Figure2showshistogramsoftheMMRcomputedfrom1,012(=506person×2)pairsofidenticalrightindexfingersand255,530(=506×505)pairsofunrelatedrightindexfingers.Thefigureshowsthattwohistogramsareseparated,indicatingthesignificantdifferenceofveinpatternsoftherightindexfingerbetweenindividuals.Figure3showshistogramsoftheMMRcomputedfrom255,530pairsofrightmiddlefingerandrightindexfingerfromidenticalperson(dark)and255,530pairsofunrelatedrightindexfingers(light).Thefigureshowsthattwohistogramsarealmostcompletelyoverlapped;thatoftwofingersoftheidenticalpersonshiftedtoleftslightly.Thisindicatesthatthedifferenceoftheveinpatternsoftherightmiddlefingerandrightindexfingersofidenticalpersonissimilartothedifferenceoftheveinpatternsoftherightindexfingersofdifferentperson.Figure4showshistogramsoftheMMRfrom255,530pairsoftherightindexfingersandleftindexfingersofidenticalperson(dark)andthatfrom255,530pairsofunrelatedrightindexfingers(light).Againtwohistogramsalmostcompletelyoverlap,showingthatthedifferenceoftheveinpatternsoftherightindexfingerandleftindexfingerfromanidenticalpersonissimilartothedifferenceoftheveinpatternsoftherightindexfingers

2

㪇㪅㪈㪍㪇㪅㪈㪋㪇㪅㪈㪉㪇㪅㪈㪇㪅㪇㪏㪇㪅㪇㪍㪇㪅㪇㪋㪇㪅㪇㪉㪇㪇㪅㪇㪈Identical right index fingers Unrelated right index fingers 㪝㫉㪼㫈㫌㪼㫅㪺㫐㪇㪅㪉㪈㪤㫀㫊㫄㪸㪺㪿㩷㫉㪸㫋㪼㪇㪅㪋㪈㪇㪅㪍㪈 Figure2:Histogramsofmismatchratescomputed(MMR)from1,078pairsofidenticalrightindexfingerand254,012pairsofunrelatedindexfinger.

㪇㪅㪊㪇㪅㪊㪝㫉㪼㫈㫌㪼㫅㪺㫐㪝㫉㪼㫈㫌㪼㫅㪺㫐㪇㪅㪉㪇㪅㪉㪇㪅㪈㪇㪅㪈㪇㪇㪅㪊㪍㪇㪅㪋㪍㪤㫀㫊㫄㪸㫋㪺㪿㩷㫉㪸㫋㪼㪇㪅㪌㪍㪇㪇㪅㪊㪍㪇㪅㪋㪍㪤㫀㫊㫄㪸㫋㪺㪿㩷㫉㪸㫋㫀㫆㪇㪅㪌㪍Figure3:HistogramsofmismatchratesFigure4:Histogramsofmismatchrates

(MMR)computedfrom254,012pairsoftherightindexfingersandmiddlefingersofidenticalperson(dark)and254,012pairsofunrelatedindividuals(light).

(MMR)computedfrom254,012pairsoftherightindexfingersandleftindexfingersofidenticalperson(dark)and254,012pairsofunrelatedindividuals(light).

ofdifferentindividuals.Theseobservationsindicatethattwofingersareidenticalifandonlyiftheyarethesamefingerinthesamehandofthesameperson,andalltheothercasescanbetreatedsimplyasunrelated.

ThereproducibilityofthedistributionsoftheMMRisfairlygood.Forexample,Table2showsthemeansandstandarddeviationsofthedistributionsoftheMMRofunrelatedrightindexfingersthatareobtainedbymeasuring506individuals10times;ineachtimethemeanandthestandarddeviationarecomputedfrom255,530pairsofunrelatedindexfingers.Thetableshowsthatthevariabilitiesofthemeanands.d.throughoutthose10examinationsarenegligible.

3Personalidentification

Supposethatafingerveinpatternofanindividualisregisteredandanewfingerpatternisarrived.Figure2suggeststhatwemayjudgethatthenewarrivalisunrelatedifMMR>C,andidenticaltotheregisteredifMMR≤Cforsomecut-offpointC.Thegoodnessofthispersonalidentificationisevaluatedbythefalserejectionrate(FRR)

3

Table2:Themeansandstandarddeviation(s.d.)ofthedistributionsofthemismatchrates(MMR)in10repeatedexaminations:ineachexaminationthemeanands.d.arecomputedfrom255,530unrelatedpairsoftherightindexfinger.

1

means.d.

0.48490.0309

20.48540.0308

30.48590.0309

40.48560.0308

50.48630.0308

50.48580.0307

70.48600.0310

80.48620.0309

90.48630.0309

100.48610.0308

pooled0.48590.0308

㪇㪅㪈㪍㪇㪅㪈㪋㪇㪅㪈㪉㪇㪅㪈㪇㪅㪇㪏㪇㪅㪇㪍㪇㪅㪇㪋㪇㪅㪇㪉㪇㪇㪅㪇㪉㪇㪅㪇㪎㪇㪅㪈㪉㪤㫀㫊㫄㪸㫋㪺㪿㩷㫉㪸㫋㪼㪇㪅㪈㪋㪝㫉㪼㫈㫌㪼㫅㪺㫐㪜㫄㫇㫀㫉㫀㪺㪸㫃㪙㪙㪃㩷㫄㪔㪋㪇㪇㪃㩷㪸㫃㫇㪿㪸㪔㪏㪅㪋㪐㪃㪹㪼㫋㪸㪔㪐㪋㪅㪈㪐㪝㫉㪼㫈㫌㪼㫅㪺㫐㪇㪅㪈㪉㪇㪅㪈㪇㪅㪇㪏㪇㪅㪇㪍㪇㪅㪇㪋㪇㪅㪇㪉㪥㫆㫉㫄㪸㫃㪜㫄㫇㫀㫉㫀㪺㪸㫃㪇㪅㪈㪎㪇㪇㪅㪉㪐㪇㪅㪊㪐㪇㪅㪋㪐㪤㫀㫊㫄㪸㫋㪺㪿㩷㫉㪸㫋㪼㪇㪅㪌㪐Figure5:HistogramsofmismatchratesFigure6:Histogramsofmismatchrates

(MMR)computedfrom1,012pairsofiden-ticalrightindexfinger(empirical)andBeta-Binomialdistributionwithm=400,󰐋=8.49and󰐌=94.19.

(MMR)computedfrom2,540,120unre-latedpairsofrightindexfinger(empirical)andnormaldistributionwithmean=0.4859ands.d.=0.0308.

andfalseacceptancerate(FAR);definedbyFRR=Pr(MMR>C|identicalfingers)andFAR=Pr(MMR≤C|unrelatedfingers).

WeintroduceamathematicalmodeltoevaluatetheFRRandFARfromthedata.LetmbethetotalnumberofpixelsclassifiedintoVEINinthetwoveinpatterns,andXibeabinaryvariabletaking1withprobabilityPiftheithpairofthepixelsismismatchedand0otherwise,wherePrepresentsthemismatchrate.ThemismatchratePisestimatedby

m󰀅1¯=Xi.X

mi=1Sinceaveinisconnected,X1,...,Xmwouldnotbestatisticallyindependent.Further-moremisunknownanddependsonthelengthoftheveins.Thesecharacteristicsmustbetakenintoaccountinthemodeling.AssumethatPisarandomvariablefollowingabetadistributionofparametersαandβ,andthat,forgivenP=p,X1,...,XmareconditionallyindependentandidenticallydistributedwithaBernoullidistributionwithPr(Xi=1|P=p)=p,wheremisanunknownconstant.Itfollowsundertheseassump-¯followsabeta-binomialdistributionandthatthemean,varianceandthetionsthatmX

¯aregivenbythirdcentralmomentofX

󰀂󰀃

¯)=π,V(X¯)=π(1−π)1+(m−1)φE(X,

m1−φ

󰀃󰀂

󰀆󰀁3󰀆󰀁2π(1−π)¯−E(X¯)=EXφ3(π−1)+2φ(π−2)m+3(π−φ−1)m+2π−1,

m2(1+φ)(1+2φ)

4

where

π=

α

,α+β

φ=

1

.α+β

3.1EstimatingtheFRR

Parametersα,βandmareestimatedtobe8.49,94.19and400byapplyingthemethodofmoment(see,forexampleBickelandDoksom,1976,p.92)totheMMRdatafrom1,012(=506person×2)pairsofidenticalrightindexfingers.Figure5showsthehistogramandfittedbeta-binomialdistribution;showingthefittingisfairlygood.ThesecondcolumnofTable3givestheestimatedFRRfromthebeta-binomialdistributionforselectedvaluesofthecut-offpoints.

3.2EstimatingtheFAR

Themeansands.d.ofthe10examinationsinTable2issoclosethatwepooledthosedata,

ˆ=391.116andmandobtainedestimatesαˆ=369.593,βˆ=400from2,540,120unrelated

pairsofrightindexfingeraltogether.Whenα+βislarge,thebeta-binomialdistributionisapproximatedbyanormaldistribution;inparticular,theapproximationisgoodwhenα/(α+β)iscloseto0.5.Figure6showsthehistogramfrom2,540,120unrelatedpairs(empirical)andfittednormaldistributionN(0.4859,0.03082),showingthatthefittingisprettygood.ThethirdcolumnofTable2givestheFARestimatedfromthefittednormaldistribution.95%confidenceintervals(c.i.)oftheFARareobtainedfrom10,000bootstrapsamplesconsistedof1,000samplesdrawnfromeachofthepopulationinTable2.

Table3:Estimatedfalseacceptancerate(FRR)andfalserejectionrate(FAR).

Cut-offpoint

0.2700.2750.2800.2850.2900.2950.3000.3050.310

FRR3.16E-062.03E-061.30E-068.23E-075.20E-073.27E-072.04E-071.27E-077.86E-08

FAR1.31E-124.10E-121.25E-113.73E-111.08E-103.07E-108.47E-102.28E-095.97E-09

95%c.i.ofFAR6.32E-132.07E-126.41E-122.00E-115.82E-111.74E-104.84E-101.35E-093.69E-09

2.56E-127.80E-122.45E-116.96E-111.94E-105.49E-101.46E-093.85E-099.81E-09

3.3Riskofthepersonalidentification

Thevalueofthecut-offpointCmustbedecidedfromFRRandFARwhenweusetheabovedeviceforpersonalidentification.SimilarlytothetypeIandtypeIIerrorsinthetheoryofthestatisticaltestinghypothesis,FRRandFARcompete,i.e.,whenthecut-offpointCincreasesFRRdecreases,whereasFARincreases,andviceversa.TheFARshouldbemoreimportantinpracticethanFRRforthepurposeofpreservingsecurity.Inadditionweshowherethatitisdirectlyrelatedtotheriskforpersonalidentification.

5

Supposethatapersonregistershis/herfingerpatterntothedevice,andthatanewpersonarrivesonsomedayslaterwhoisunknowntobewhetheridenticaltotheregisteredpersonornot.Thenthemainconcernoftheuserofthedevicewouldbetheriskthatifthepersonpassesthedeviceandyethe/sheisnotidenticaltotheregistered,i.e.,cheatercouldimpersonatetheregisteredpersonforsomemaliciouspurpose.Theriskmayberepresentedbymeansofprobability,Pr(different|passed),andbeestimatedasfollows.LetPr(identical)andPr(different)betheprobabilitiesthatthenewpersonisidentical/differenttotheregisteredperson,wherePr(identical)+Pr(different)=1.ThenitfollowsfromtheBayestheoremthat

FAR·Pr(different)

Pr(different|passed)=.

FAR·Pr(different)+(1−FRR)Pr(identical)IfwehavenoinformationonPr(identical)andPr(different),itwouldbenaturaltosetPr(identical)=Pr(different)=0.5.Thenwehave

FAR

Pr(different|passed)=󰀇FAR,

FAR+(1−FRR)sinceFAR󰀆FRR󰀆1.Similarly,wemayshowthatPr(identical|notpassed)󰀇FRRwhenPr(identical)=Pr(different)=0.5.Butthisprobabilitywillbelessimportantthantheaboveprobabilityfromtheviewpointofpreservingsecurity.

3.4Universalunequalnessofthehumanfingervein

AlsoFARisusefultoassessuniversalunequalnessofthehumanfingervein.SupposeNbethesizeofpopulationthatweconsider.Nmaybe,forexample,thesizeofthepopulationsinTokyo(N∼12millions),thepopulationsinJapan(N∼120millions)orthepopulationsintheworld(N∼6.5billions).Supposethatonepersonischosenrandomlyfromthepopulationandregistershis/herfingerveintothisdevice.Then”howmanypersonsarethereinthepopulationwhoaregoingtobejudgedas“identical”tothisperson?”wouldbeaninterestingquestion.Thequestionisansweredasfollows.Supposethatwenumberallthepersonsofthepopulationas1,2,...,N,andsupposetheperson1registershisfingerveinwithoutlossofgenerality.LetYibetherandomvariabledefinedas

󰀄

1,personiisjudgedas“identical”totheperson1,

Yi=

0,otherwise,fori=2,...,NandY=

N󰀅i=2

Yi.Thentheexpectedvalueofthenumberofthepersons

N󰀅i=2

whoissupposedtobejudgedas“identical”totheperson1is

E(Y)=

E(Yi)=(N−1)FAR󰀇N·FAR.

Table4liststhisexpectedvaluesforsomeselectedvaluesofCandN.Wemayesteem

thosevaluesinthetableastheindexofuniversalunequalnessofpersonalidentificationbymeansoffingerveins.Fromthetable,forexample,theuniversalunequalnessofthehumanfingerveinwouldbestatisticallyguaranteedinJapan,whenwesetCasC≤0.310.Ontheotherhand,ifwewanttoensuretheuniversalunequalnessofthefingerveinintheworld,wewouldhavetosetCasC≤0.290.

6

Table4:Expectedvalueofthenumberofpersonswhojudgedas“identical”tosomespecificpersonintheNpopulations

Cut-offpointc

0.2700.2750.2800.2850.2900.2950.3000.3050.310

N=1million0.000000.000000.000010.000040.000110.000310.000850.002280.00597

N=10millions

0.000010.000040.000130.000370.001080.003070.008470.022800.05970

N=100millions

0.000130.000410.001250.003730.010800.030700.084700.228000.59700

4Discussion

Weinvestigatedthediversityofhumanfingerveinpatternsbycomparingtherightandleftindexfingersandmiddlefingersofabout500personsandconsidereditsusefulnessforpersonalidentification.Thevalidityofourpersonalidentificationisevaluatedbytwoprobabilitiesinherenttothedevice,thefalserejectionrate(FRR)andthefalseacceptancerate(FAR).FromFRRandFAR,wecanestimatethereliabilityofthepersonalidentificationbythehumanfingervein.

Thepersonalidentificationsarealsoreportedbythehumanirispatterns.Forexample,DaugmanandDowning(2001)gavethroughinvestigationontherandomnessofunrelatedhumanirispatternsof2.3milliondifferentpairsofeyeimagesandconsidereditsuseforpersonalidentifications.TheymeasuredtheHammingdistancebetweenunrelatedirispatternsandobservedthatthedistributionoftheHammingdistanceiswellapproximatedbythebinomialdistribution.Inthepresentmanuscript,ontheotherhand,weobservedthatthedistributionofthemismatchrateofthefingerveinscannotbeapproximatedfairlybybinomialdistribution.Instead,weconsiderthebeta-binomialdistributionforthemismatchrate,reflectingthefeatureoftheover-dispersion.

ComparingthedistributionofourmismatchratebetweenunrelatedfingerveinstothatoftheHammingdistanceofirispatterngiveninDaugmanandDowning(2001),theestimatedstandarddeviationis0.0308inourstudy,whileitis0.032inDaugmanandDowning(2001).Thisresultimpliesthatthediversityamongthedifferentpersonsofthehumanfingerveinissimilartothatoftheirispattern,andourmethodalsomaybeutilizedforthepersonalidentification.

Acknowledgments

WegreatlyacknowledgeDr.TakafumiMiyatake,theChiefResearcher,theCentralResearchLaboratory,Hitachi,Ltd,forhiscooperationofobtainingthedatainthestudyfromwhichFRRandFARarecomputed.ThisresearchwasconductedundertheauspicesoftheJapanTransdisciplinaryFederationofScienceandTechnology.Theirsupportforthisresearcharealsogreatlyacknowledged.

7

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8

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MHF2006-9ToruKOMATSU

GenericsexticpolynomialrelatedtothesubfieldproblemofacubicpolynomialMHF2006-10ShuTEZUKA&AnargyrosPAPAGEORGIOU

ExactcubatureforaclassoffunctionsofmaximumeffectivedimensionMHF2006-11ShuTEZUKA

Onhigh-discrepancysequences

¯MHF2006-12RaimundasVIDUNAS

DetectingpersistentregimesintheNorthAtlanticOscillationtimeseriesMHF2006-13ToruKOMATSU

TamelyEisensteinfieldwithprimepowerdiscriminant

MHF2006-14NaliniJOSHI,KenjiKAJIWARA&MartaMAZZOCCO

GeneratingfunctionassociatedwiththeHankeldeterminantformulaforthesolutionsofthePainlev´eIVequation¯MHF2006-15RaimundasVIDUNAS

DarbouxevaluationsofalgebraicGausshypergeometricfunctionsMHF2006-16MasatoKIMURA&IsaoWAKANO

NewmathematicalapproachtotheenergyreleaserateincrackextensionMHF2006-17ToruKOMATSU

ArithmeticofthesplittingfieldofAlexanderpolynomialMHF2006-18HirokiMASUDA

LikelihoodestimationofstableL´evyprocessesfromdiscretedata

¨MHF2006-19HiroshiKAWABI&MichaelROCKNER

Essentialself-adjointnessofDirichletoperatorsonapathspacewithGibbsmeasuresviaanSPDEapproachMHF2006-20MasahisaTABATA

Energystablefiniteelementschemesandtheirapplicationstotwo-fluidflowproblemsMHF2006-21YuzuruINAHAMA&HiroshiKAWABI

AsymptoticexpansionsfortheLaplaceapproximationsforItˆofunctionalsofBrownianroughpathsMHF2006-22YoshiyukiKAGEI

ResolventestimatesforthelinearizedcompressibleNavier-Stokesequationinaninfinitelayer

MHF2006-23YoshiyukiKAGEI

AsymptoticbehaviorofthesemigroupassociatedwiththelinearizedcompressibleNavier-StokesequationinaninfinitelayerMHF2006-24AkihiroMIKODA,ShuichiINOKUCHI,YoshihiroMIZOGUCHI&Mitsuhiko

FUJIO

Thenumberoforbitsofbox-ballsystemsMHF2006-25ToruFUJII&SadanoriKONISHI

Multi-classlogisticdiscriminationviawavelet-basedfunctionalizationandmodelselectioncriteriaMHF2006-26TaroHAMAMOTO,KenjiKAJIWARA&NicholasS.WITTE

Hypergeometricsolutionstotheq-Painlev´eequationoftype(A1+A󰀁1)(1)MHF2006-27HiroshiKAWABI&TomohiroMIYOKAWA

TheLittlewood-Paley-SteininequalityfordiffusionprocessesongeneralmetricspacesMHF2006-28HirokiMASUDA

Notesonestimatinginverse-Gaussianandgammasubordinatorsunderhigh-frequencysamplingMHF2006-29SetsuoTANIGUCHI

Theheatsemigroupandkernelassociatedwithcertainnon-commutativeharmonicoscillatorsMHF2006-30SetsuoTANIGUCHI

StochasticanalysisandtheKdVequation

MHF2006-31MasatoKIMURA,HidekiKOMURA,MasayasuMIMURA,HidenoriMIYOSHI,

TakeshiTAKAISHI&DaishinUEYAMA

QuantitativestudyofadaptivemeshFEMwithlocalizationindexofpatternMHF2007-1TaroHAMAMOTO&KenjiKAJIWARA

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Hypergeometricsolutionstotheq-Painlev´eequationoftypeA4

MHF2007-2KoujiHASHIMOTO,KentaKOBAYASHI&MitsuhiroT.NAKAO

VerifiednumericalcomputationofsolutionsforthestationaryNavier-StokesequationinnonconvexpolygonaldomainsMHF2007-3KenjiKAJIWARA,MartaMAZZOCCO&YasuhiroOHTA

AremarkontheHankeldeterminantformulaforsolutionsoftheTodaequationMHF2007-4Jun-ichiSATO&HidefumiKAWASAKI

DiscretefixedpointtheoremsandtheirapplicationtoNashequilibriumMHF2007-5MitsuhiroT.NAKAO&KoujiHASHIMOTO

Constructiveerrorestimatesoffiniteelementapproximationsfornon-coerciveellipticproblemsanditsapplications

MHF2007-6KoujiHASHIMOTO

Apreconditionedmethodforsaddlepointproblems

MHF2007-7ChristopherMALON,SeiichiUCHIDA&MasakazuSUZUKI

MathematicalsymbolrecognitionwithsupportvectormachinesMHF2007-8KentaKOBAYASHI

OntheglobaluniquenessofStokes’waveofextremeform

MHF2007-9KentaKOBAYASHI

Aconstructiveapriorierrorestimationforfiniteelementdiscretizationsinanon-convexdomainusingsingularfunctionsMHF2007-10MyoungnyounKIM,MitsuhiroT.NAKAO,YoshitakaWATANABE&Takaaki

NISHIDA

Anumericalverificationmethodofbifurcatingsolutionsfor3-dimensionalRayleigh-B´enardproblemsMHF2007-11YoshiyukiKAGEI

LargetimebehaviorofsolutionstothecompressibleNavier-StokesequationinaninfinitelayerMHF2007-12TakashiYANAGAWA,SatoshiAOKIandTetsujiOHYAMA

Humanfingerveinimagesarediverseanditspatternsareusefulforpersonalidentification