Ee?anueeee oi?aa?neoao

?iai? Oa?ana Oaa/aiea

Oea?aanueea Ieueaa TH???aia

OAeE 519.716.35

Eeane ooieoe?e,

aecia/aieo ia?aoai?thaa/aie

iaae iaaeneiaaie

01.01.08 – iaoaiaoe/ia eia?ea, oai??y aeai?eoi?a

? aeene?aoia iaoaiaoeea

AAOI?AOA?AO AeENA?OAOe??

ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Ee?a-1999

Aeena?oaoe??th ? ?oeiien.

?iaioo aeeiiaii ia eaoaae?? oai?aoe/ii? e?aa?iaoeee Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea

Iaoeiaee ea??aiee: – aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani?, E?NIAEE Eaii?ae Iao?iae/,

eaoaae?a oai??? i?ia?aioaaiiy

Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea.

Io?oe?ei? iiiiaioe: – aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani? EAI?OIIIAA THe?y Aieiaeeie??aia

?inoeooo e?aa?iaoeee IAI Oe?a?ie,

caa?aeoth/a a?aeae?eii oai??? oeeoi?aeo aaoiiao?a;

– eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe,

aeioeaio I?AOeUeIAEOEE Ieeiea Iao?iae/,

caa?aeoth/ee eaoaae?ith aeui? iaoaiaoeee

Iaoe?iiaeueiiai Iaaeaaia?/iiai Oi?aa?neoaoo

?iai? I.I. Ae?aaiiaiiaa.

I?ia?aeia onoaiiaa: ?inoeooo iaoaiaoeee IAI Oe?a?ie

Caoeno a?aeaoaeaoueny “31” o?aaiy 1999 ?ieo i 14 aiae.

ia can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.001.18 i?e
Ee?anueeiio oi?aa?neoao? ?iai? Oa?ana Oaa/aiea ca aae?anith:

252127, i.Ee?a — 127, i?iniaeo aeaae. Aeooeiaa,6 ,Ee?anueeee oi?aa?neoao
?iai? Oa?ana Oaa/aiea, iaoai?ei-iaoaiaoe/iee oaeoeueoao

C aeena?oaoe??th iiaeia iciaeiieoeny a a?ae?ioaoe? Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea (aoe. Aieiaeeie?nueea, 58)

Aaoi?aoa?ao ?ic?neaiee “ 27” ea?oiy 1999 ?ieo

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee
A.I.Iao?aa/oe

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

O ?iaio? ?icaeyaeathoueny eeane ia?aoai?thaa/?a iaae iane?i/aiieie
neiaieueieie cia?aaeaiiyie ae?enieo /enae. Aeine?aeaeothoueny
ia/enethaaeuei? iiaeeeaino? oeeo ia?aoai?thaa/?a ye cania?a caaeaiiy
iaia?a?aieo ae?enieo ooieoe?e ? o?aeoaeueieo iiiaeei.

Aeooaeuei?noue oaie. Iaoeie O’th??iaa, cie?aia ne?i/aii? aaoiiaoe I?e?,
iiaeia ?icaeyaeaoe ye can?a caaeaiiy ae?enieo ooieoe?e. C oe??? oi/ee
ci?o ne?i/aii? aaoiiaoe I?e? ? ocaaaeueiai? iine?aeiai?ni? iaoeie
?icaeyaeaeeny N. Aeeaiaa?aii. Aeey aecia/aiiy cia/aiiy f(x) aaoiiao
iine?aeiaii ia?a?iaeya ao?aeiee ia’?eo – ?iceeaae ae?eniiai /enea x o
aea?eeia?e (aeanyoeia?e) nenoai? /eneaiiy – o aeo?aeiee ia’?eo –
a?aeiia?aeiee ?iceeaae /enea f(x).

Iiiyooy R-ia?aoai?thaa/a, aaaaeaia E. I. E?niaeeii, ocaaaeueith?
i?aeo?ae Aeeaiaa?aa. Ne?i/aii? R-ia?aoai?thaa/?, ia a?aei?io a?ae
aaoiiao?a I?e?, iiaeooue aooe aneio?iiieie ? caaeaaaoe a?aeia?aaeaiiy
aea?eeiaeo ?iceeaae?a o oae caai? aea?eeia? ?iceeaaee c ia?aiiaiaiiyie,
oiaoi o?, ui aeiioneathoue iiyao neiaieo 2 o cia?aaeaii? /enea.
Noia?iiceoe?? ooieoe?e, ye? caaeathoueny R-ia?aoai?thaa/aie,
aeine?aeaeoaaeeny E?niaeeii ? I.A. Ae?iaeoaaei. I?ci?oa iiiyooy
R-ia?aoai?thaa/a aoei ocaaaeueiaia aei iiiyooy Rnm-ia?aoai?thaa/a,
eio?ee ia? m ao?aeieo ? n aeo?aeieo no??/ie, oa R(s,t)-ia?aoai?thaa/a,
eio?ee ia?a?iaey? s-eia? cia?aaeaiiy ae?enieo /enae o aeo?aei? t-eia?
cia?aaeaiiy. Ua a?eueo caaaeueieie ia’?eoaie ? RA[0, 1)-ia?aoai?thaa/?,
eio?? ia?iaeythoue niaoe?aeuei? iane?i/aii? cia?aaeaiiy ae?enieo /enae.

Oaeee i?aeo?ae aei caaeaiiy ae?enieo ooieoe?e ?, c iaeiiai aieo,
eiino?oeoeaiei, c ?ioiai – a?ae??ciy?oueny a?ae o?aaeeoe?eieo oai??e
eiino?oeoeaiiai aiae?co, ine?eueee ia ao?aei? aeai? ia iaeeaaea?oueny
oiiaa eiino?oeoeaiino? (aoaeoeaii? ia/enethaaiino?), a eiino?oeoeai?
aeeni? ooieoe?? (EAeO) ca cae/ae caaeathoueny ia eiino?oeoeaieo ae?enieo
/eneao (EAe*). A caeaaeiino? a?ae iiooaeiino? oa no?oeoo?e iai’yo?,
ia?aoai?thaa/? iiaeooue aooe ne?i/aiieie, iaaaceiieie, noaeiaeie,
C-iaoeiaie, aai aooe ia?aoai?thaa/aie caaaeueiiai aeaeyaeo c? ce?/aiiith
iiiaeeiith noai?a. sse iieacaa E?niaee, R-ia?aoai?thaa/aie caaaeueiiai
aeaeyaeo caaeathoueny an? iaia?a?ai? ooieoe?? oa ooieoe??, ui iathoue
?ic?eae ia?oiai ?iaeo o aea?eeiai-?aoe?iiaeueieo oi/eao. O ca’yceo ?c
oeei aeieea? iiaee e?eoa??e aeey eeaneo?eaoe?? ae?enieo ooieoe?e – ca
oeiii iai’yo? R-ia?aoai?thaa/a, eio?ei aecia/a?oueny caaeaia ooieoe?y.

C i?aeoe/ii? oi/ee ci?o ia?aoai?thaa/? oyaeythoue niaith aeai?eoie,
eio?? c aeia?eueiith oi/i?noth iiaeooue ia/enethaaoe cia/aiiy ooieoe?? o
caaeaii?e oi/oe?. Ia?aoai?thaa/? oaeiae iiaeooue ?icaeyaeaoeny ye caniae
caaeaiiy o?aeoaeueieo iiiaeei ?aeo?neaii? i?e?iaee ? ooieoe?e ia oaeeo
iiiaeeiao.

Iaoa ? caaea/? aeine?aeaeaiue. Iaoa aeaii? aeena?oaoe?eii? ?iaioe –
iiaoaeoaaoe ca’ycee i?ae eeanaie ae?enieo ooieoe?e, aecia/aieo ??cieie
oeiaie ia?aoai?thaa/?a iaae iaaeneiaaie, ? eeanaie ae?enieo ooieoe?e,
iienoaaieo iaoiaeaie eeane/iiai aiae?co – aeeoa?aioe?eiaieie ooieoe?yie,
ao?iieie ia?aoai?aiyie, oa?aeoa?enoe/ieie ooieoe?yie o?aeoaeueieo
iiiaeei.

Iaoiaee aeine?aeaeaiue. Iniiaiei “no?aoaa?/iei” iaoiaeii aeine?aeaeaiue
noaei aenoaaiiy a?iioac ?c iiaeaeueoei ?o aiae?cii, ni?inooaaiiyi aai
ooi/iaiiyi ? ia?aoai?aiiyi ia oai?aio. “Oaoi?/i?” iaoiaee aeine?aeaeaiue
ia/enethaaeueieo iiaeeeainoae ne?i/aiieo ? iaaaceiieo ia?aoai?thaa/?a,
na?aae yeeo i?ia?aeia i?noea iin?aea? oae caaia “oaoi?ea ne?ae?a”,
a?oioothoueny ia ?aeo?neai?e i?e?iae? oeeo ia’?eo?a. Oaeiae ?ioaineaii
aeei?enoiaoaaany iaoiae iiaoaeiae i?eeeaae?a ? eiio?i?eeeaae?a, a
?icae?e? 3 – iaoiae ae?aaiiae?caoe?? Eaioi?a.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. A ?iaio? io?eiaii oae? iia?
?acoeueoaoe:

aeiaaaeaii, ui eean aeeoa?aioe?eiaieo ia aeayeiio ?ioa?aae? ooieoe?e,
ye? caaeathoueny iaaaceiieie R-ia?aoai?thaa/aie, i?noeoueny o eean?
e?i?eieo ia oeueiio ?ioa?aae? ooieoe?e;

?icaeyiooi iaiao?aeio ? aeinoaoith oiiao, ca yei? ao?iia ia?aoai?aiiy c
?aoe?iiaeueieie ia?aiao?aie i?aeiiiaeeie n-aei??iiai i?inoi?o iiaeia
caaeaoe ne?i/aiiei Rnm-ia?aoai?thaa/ai;

aeiaaaeaii ocaaaeueiaiiy oai?aie Iinoianueeiai-Oniainueeiai i?i
ia?aeeaae cia?aaeaiue ae?enieo /enae aeey no?iaeo
R(s,t)-ia?aoai?thaa/?a: aeey ?nioaaiiy no?iaiai R(s,t)-ia?aoai?thaa/a,
eio?ee caaea? oioiaeith ooeoe?th, iaiao?aeii ? aeinoaoiuei, aae iiiaeeia
i?inoeo ae?eueiee?a /enea s i?noeea iiiaeeio i?inoeo ae?eueiee?a /enea
t; oaeiae iieacaii, ui iaiao?aeia ? aeinoaoiy oiiaa iiaeeeaino?
ia?aeeaaeo aeey ne?i/aiieo R(s,t)-ia?aoai?thaa/?a ? a?eueo aei?noeith;

iiaoaeiaaii R21-ia?aoai?thaa/?, eio?? caaeathoue e?eao Iaaii c
eiao?oe??ioii ia?ae?eooy 3 ? nth?’?eoe?th iaeeie/iiai a?ae??ceo ia eeeei
Na?ieinueeiai;

aeiaaaeaii, ui ca aeia?eueiith ???a?o?/iith ?oa?iaaiith
ooieoe?iiaeueiith noaiith (HIFS) iiaeia iiaoaeoaaoe nenoaio
ia?aoai?thaa/?a, ye? caaeathoue oie naiee aaeoi? eiiiaeoieo iiiaeei c
Rn, ui ? caaeaia HIFS; ?icaeyiooi oiiao, ca yei? oe? ia?aoai?thaa/? ?
ne?i/aiieie; iiaoaeiaaii RA[0, 1]-ia?aoai?thaa/, yeee caaea? iacaieiaio
iiiaeeio.

On? iaaaaeai? oaa?aeaeaiiy a?oioothoueny ia ?aeo?neai?e i?e?iae?
ia?aoai?thaa/?a, ia iiaoi?thaaiino? aai “iaeaea iiaoi?thaaiino?” eiiaiae
ia?aoai?thaa/a i?ae /an ia?iaee iane?i/aiiiai cia?aaeaiiy ae?eniiai
/enea. Cie?aia, oaea aeanoea?noue iaoiiaeth? ca’ycie iaeanoae aecia/aiiy
ooieoe?e, ye? caaeathoueny ia?aoai?thaa/aie, oa oe?ieiai eeano
o?aeoaeueieo iiiaeei.

I?aeoe/ia cia/aiiy io?eiaieo ?acoeueoao?a. Cacia/ai? ?acoeueoaoe iiaeia
aeei?enoiaoaoe aeey iiaeaeueoeo aeine?aeaeaiue i?iaeai ia/enethaaiino?
ae?enieo ooieoe?e ? aeine?aeaeaiue o oai??? o?aeoaeueieo iiiaeei. C
i?aeoe/ii? oi/ee ci?o ?ioa?an oyaeythoue aeai?eoie aeia?eueii? oi/iino?
aeey caaeaiiy ao?iieo ia?aoai?aiue aeey caaeaieo iiiaeei P SYMBOL 206 \f
«Symbol» \s 16 I Rn ?aoe?iiaeueieo iao?eoe? M ? aaeoi?o h. Oaeiae aa?oi
ni?iaoaaoe canoinoaaoe aeai?eoi iiaoaeiae e?eai? Iaaii c eiao?oe??ioii
ia?ae?eooy 3 o caaea/ao iaoaiaoe/iiai i?ia?aioaaiiy, ia?iaee
cia?aaeaiue, ia?aaea/? aeaieo oiui, a yeeo aeei?enoiaothoueny i?yi? oa
cai?oi? a?aeia?aaeaiiy a?ae??ceo ia eaaae?ao.

Iniaenoee aianie caeiaoaa/a. Aaoi?ia? ?iaioe iaeaaeeoue oaa?aeaeaiiy i?i
e?i?ei?noue ooieoe?e, ye? caaeathoueny iaaaceiieie ia?aoai?thaa/aie
(aia?oa aiii aoei noi?ioeueiaaii aeey ne?i/aiieo ia?aoai?thaa/?a), iaeei
c aeaio niinia?a eiai aeiaaaeaiiy, oaeiae eiino?oeoe?y e?eai? Iaaii c
eiao?oe??ioii ia?ae?eooy 3 oa oai?aia i?i ca’ycie iiiaeei, ye?
aecia/athoueny nenoaiaie ia?aoai?thaa/?a oa HIFS.

Ai?iaaoe?y ?acoeueoao?a ?iaioe. ?acoeueoaoe ?iaioe aeiiia?aeaeeny ia
??-e I?aeia?iaei?e eiioa?aioe?? “Iaoaiaoe/i? aeai?eoie” (Ieaei?e
Iiaai?iae, /a?aaiue 1995 ?ieo), ia nai?ia?? “O?aeoaeueiee aiae?c ?
noi?aei? ieoaiiy” i?e Iaaeaaia?/iiio oi?aa?neoao? ?i. I.I. Ae?aaiiaiiaa,
ia nai?ia?? i?e a?aeae?e? oai??? oeeo?iaeo aaoiiao?a ?inoeoooo
e?aa?iaoeee IAI Oe?a?ie.

Ioae?eaoe??. Ca oaiith aeena?oaoe?? iioae?eiaaii 4 iaoeiaeo ?iaioe,
nienie yeeo iaaaaeaiee o e?ioe? aaoi?aoa?aoo. ?iaioe [1 – 3] noai?ai? o
ni?aaaoi?noa? c iaoeiaei ea??aieeii. A ?iaioao [1] ? [3] aaoi?ia?
iaeaaeeoue ?aeay aeiaaaeaiiy oai?ai 1 ? 1 a?aeiia?aeii, a yeeo eaeaoueny
i?i ?nioaaiiy ne?i/aiiiai R21-ia?aoai?thaa/a, eio?ee caaea? e?eao Iaaii
c eiao?oe??ioii ia?ae?eooy 3. Oaa?aeaeaiiy oai?aie 5[3] i?i e?i?ei?noue
ooieoe?e, ye? caaeathoueny iaaaceiiei ia?aoai?thaa/ai, ? iaoiae ??
aeiaaaeaiiy iaeaaeaoue aaoi?ia? aeena?oaoe?eii? ?iaioe.

No?oeoo?a oa ianya aeena?oaoe??. Aeaia aeena?oaoe?eia ?iaioa
neeaaea?oueny c ia?ae?eo oiiaieo iicia/aiue, anooio, naie ?icae?e?a oa
?icae?eo, a yeiio aeacaii aeniiaee. ?? ianya noaiiaeoue 151 noi??iee.
?iaioa i?noeoue 8 ?enoie?a ? 5 oaaeeoeue. Nienie aeei?enoaieo aeaea?ae
neeaaea?oueny c? 47 iaeiaioaaiue.

INIIAIEE CI?NO

O anooi? aei ?iaioe iaa?oioiaaii aeooaeuei?noue i?iaeaiaoeee
aeena?oaoe??, noi?ioeueiaaii iaoo ?iaioe, ?icaeyiooi ?? ci?no ca
?icae?eaie c aena?oeaiiyi iaeaaaeeea?oeo ?acoeueoao?a. Oaeiae iaaaaeaii
nienie ae?oeiaaieo i?aoeue aaoi?a, ui a?aeiia?aeathoue oaiaoeoe?
aeine?aeaeaiue. Aeey i?aoeue, noai?aieo o ni?aaaoi?noa?, ?icaeyiooi
iniaenoee aianie aaoi?a aeena?oaoe?eii? ?iaioe.

O ia?oiio ?icae?e? iaeyiooi iniiai? iaoeia? ? i?eeeaaei? i?aoe?, eio??
noinothoueny oaiaoeee aeine?aeaeaiue.

O i?ae?icae?e? 1.1 ?icaeyaeathoueny eiino?oeoeai? i?aeoiaee aei /enea ?
ooieoe??, cai?iiiiiaai? ? aeine?aeaeoaai? ??cieie aaoi?aie. Cie?aia
iaeyiooi aianie a eiino?oeoeaiee aiae?c O’th??iaa, Aaiaoa, Iaco?a,
A?aeaai?/eea. Aeaoaeueii aena?oeaii aecia/aiiy EAe* ? EAeO oa
a?aeiia?aei? aeanoeaino? ca Ia?oeiii-Eueioii.

I?ae?icae?e 1.2 i?enay/aiee iniiaiei ?acoeueoaoai oai???
ia?aoai?thaa/?a. Cie?aia cacia/aii ?acoeueoaoe E?niaeea i?i ?nioaaiiy
R-ia?aoai?thaa/a, eio?ee caaea? iaia?a?aio i?aea ia aeeoa?aioe?eiaio
ooieoe?th, R21-ia?aoai?thaa/a, eio?ee caaea? e?eao Iaaii c eiao?oe??ioii
ia?ae?eooy 4, i?i aeeaaeai?noue eeano ooieoe?e, ye? caaeathoueny
ne?i/aiieie R-ia?aoai?thaa/aie, o eean ooieoe?e, ye? a?aeia?aaeathoue
?aoe?iiaeuei? /enea a ?aoe?iiaeuei?, oa i?i no?iao aeeaaeai?noue eeano
ooieoe?e, ye? caaeathoueny ne?i/aiieie R-ia?aoai?thaa/aie, o eean
ooieoe?e, ye? caaeathoueny iaaaceiieie R-ia?aoai?thaa/aie. Aeanoeaino?
noia?iiceoe?e ooieoe?e, ye? caaeathoueny R-ia?aoai?thaa/aie,
aeine?aeaeoaaee E?niaee ? Ae?iaeoaa. Inoaii?i iiaoaeiaaii ne?i/aiii
2-iaoa?aaeueio ooieoe?th, iiiaeeia oi/ie ?ic?eao yei? eiioeioaeueia, oa
/anoeiao ne?i/aiii 2-iaoa?aaeueio ooieoe?th, iaeanoue aecia/aiiy yei? ?
iiiaeeia aea?eeiai-?aoe?iiaeueieo /enae.

E?niaeeii ? Ae?iaeoaaei aeiaaaeaii aeai?eoi?/io ?ica’ycoaai?noue i?iaeai
aea?aaeaioiino?, iiiioiiiino? ? ??aiino? a eean? ne?i/aiii-iaoa?aaeueieo
ooieoe?e.

O i?ae?icae?e? 1.3 ?icaeyaeathoueny aeai?eoie Iaaii oa A?eueaa?oa
iiaoaeiae iaia?a?aieo nth?’?eoe?e a?ae??cea ia eaaae?ao. Oaeiae
?icaeyiooi i?eioeeie canoinoaaiiy a?aeia?aaeaiue ? ?icai?oie Iaaii a
iaoaiaoe/iiio i?ia?aioaaii? oa ia?aaea/? aeaiieo. Cacia/ai? ?io? aaeoc?
i?eeeaaeii? iaoee, aea canoiniaothoueny oae? a?aeia?aaeaiiy.

O i?ae?icae?e? 1.4 aecia/aia iiiyooy ???a?o?/ii? ?oa?iaaii?
ooieoe?iiaeueii? noaie (HIFS), yea oyaey? niaith nenoaio aeaeyaeo

Xi= SYMBOL 200 \f «Symbol» \s 16 E {fk(Xj) SYMBOL 189 \f «Symbol» \s 16
1/2 (k, j) SYMBOL 206 \f «Symbol» \s 16 I Qi },

aea fk ? noeneaiiy a Rm, Qi SYMBOL 205 \f «Symbol» \s 16 I {1,…d1}
SYMBOL 180 \f «Symbol» \s 16 ? {1,…d2} ca 1 SYMBOL 163 \f «Symbol» \s 16
F k SYMBOL 163 \f «Symbol» \s 16 F d1, 1 SYMBOL 163 \f «Symbol» \s 16 F
i SYMBOL 163 \f «Symbol» \s 16 F d2.

Ca Aaiaeoii oa ?ioeie aaoi?aie a?aeiii, ui oaea nenoaia aecia/a? ?aeeiee
aaeoi? iaii?iaei?o eiiiaeoieo iiiaeei, yeee ? ?? ?ica’yceii. HIFS ?
iaeiei c iniiaieo cania?a caaeaiiy o?aeoaeueieo iiiaeei oa ?ioaineaii
aeei?enoiaothoueny a eiii’thoa?i?e a?ao?oe?.

-ia?aoai?thaa/?a, o caaeaii? o?aeoaeueieo iiiaeei.

Oaeiae o oeueiio ?icae?e? iienaiee iaeei c iniiaieo iaoiae?a
aeine?aeaeaiiiy ia?aoai?thaa/?a c iaiaaeaiiyie ia iai’yo? – “oaoi?ea
ne?ae?a”. ?? aoei ai?iaaaeaeaii iacaeaaeii ??cieie aaoi?aie – Oaii?,
O?aooaia?ioii ? Aa?caeeiai, ?aaeiii. O oe?e ?iaio? aeei?enoaii
oa?i?iieia?th Oaii?.

?icae?ee c 3-ai ii 7-e i?enay/ai? aeine?aeaeaiiyi aaoi?a ?iaioe a oai???
ia?aoai?thaa/?a.

A o?aoueiio ?icae?e? aiae?co?oueny ia/enethaaeueia iiooaei?noue
iaaaceiieo R-ia?aoai?thaa/?a. Iniiaiee ?acoeueoao ?icae?eo, oai?aia 3.1,
aeiaiaeeoueny o i?ae?icae?e? 3.1.

Oai?aia 3.1. Iaoae ooieoe?y f(x) anthaee aeeoa?aioe?eiaia ia aeayeiio
?ioa?aae? (a’, b’), aea a’0, a SYMBOL 185 \f «Symbol» \s 16 ? 1.

O i?ae?icae?e? 3.2 iaaaaeaii i?eeeaae iaia?a?aii? ooieoe?? (eiinoaioe),
eio?a a?aeia?aaea? ?aoe?iiaeuei? /enea a ???aoe?iiaeueia ? caaea?oueny
noaeiaei R -ia?aoai?thaa/ai. Oaeei /eiii aeiaaaeaii, ui eean ooieoe?e,
ye? caaeathoueny iaaaceiieie R -ia?aoai?thaa/aie, no?iai aeeaaeaiee o
eean ooieoe?e, ye? caaeathoueny noaeiaeie R -ia?aoai?thaa/aie.

O /aoaa?oiio ?icae?e? ?icaeyaea?oueny iiaeeea?noue ocaaaeueiaiiy ia
n-aei??i? aaee?aeia? i?inoi?e ?acoeueoaoo i?i i?aaenoaai?noue e?i?eii?
ooieoe?? c ?aoe?iiaeueieie ia?aiao?aie ne?i/aiiei ia?aoai?thaa/ai.

(x)=Mx+h, aea iao?eoey M ?ici??iino? n SYMBOL 180 \f «Symbol» \s 16 ? n
? aaeoi? h=(h1,…hn) ?aoe?iiaeuei?, acaaae? eaaeo/e, ia iiaeia caaeaoe
ne?i/aiiei Rnn-ia?aoai?thaa/ai. I?e/eio oeueiai oaeoo aee?eoi o
i?ae?icae?e? 4.1, aea ?icaeyaeathoueny ooieoe?? aeaeyaeo ax+by,
aecia/ai? ia aeaye?e iiiaeei? P SYMBOL 205 \f «Symbol» \s 16 I R2.

Aoaeaii eacaoe, ui ca’ycia iiiaeeia P SYMBOL 205 \f «Symbol» \s 16 I R2
caaeiaieueiy? oiia? (a), yeui iiiaeeia Pr1(P) iaiaiaaeaia ? iiiaeeia
Pr2(P) iaiaaeaia, aai iaaiaee. Nooue oiiae (a), iieyaa? a oiio, ui ia
ne?i/aii?e iai’yo? ia iiaeia ?aae?coaaoe aeiaeaaaiiy ax+by, aea (x, y)
SYMBOL 206 \f «Symbol» \s 16 I P, eiee ??cieoey ii?yaee?a aeaiaio?a
iiiaeei Pr1(P) ? Pr2(P) ? iaiaiaaeaiith. sseui ??cieoey a ii?yaeeao
aeaio /enae x SYMBOL 206 \f «Symbol» \s 16 I Pr1(P) ? y SYMBOL 206 \f
«Symbol» \s 16 I Pr2(P) ia?aaeuo? e?euee?noue noai?a R21-ia?aoai?thaa/a,
oi R12-ia?aoai?thaa/ “ao?a/a?” ?ioi?iaoe?th i?i ii?yaeie a?eueoiai /enea
aiane?aeie iiaoi?thaaiino? noai?a. Ia oeueiio oaeo? aaco?oueny
aeiaaaeaiiy ianooiiiai oaa?aeaeaiiy.

(x, y)=ax+by.

Aeey caaeaii? iao?eoe? M=(mij)i,j=1n /a?ac Ti(M) aoaeaii iicia/aoe
iiiaeeio ?iaeaen?a {j|1 SYMBOL 163 \f «Symbol» \s 16 F j SYMBOL 163 \f
«Symbol» \s 16 F n SYMBOL 217 \f «Symbol» \s 16 U mij SYMBOL 185 \f
«Symbol» \s 16 ? 0}. Iaoae iao?eoey M ? aaeoi? h=(h1,…hn) ?aoe?iiaeuei?,
a P ? iiiaeeia aeaeyaeo I1 SYMBOL 180 \f «Symbol» \s 16 ? … SYMBOL 180
\f «Symbol» \s 16 ? In, aea Ii SYMBOL 205 \f «Symbol» \s 16 I R ? aai
naaiaio, aai iai?acaieiaiee iaiaiaaeaiee i?ii?iue aai any /eneiaa i?yia,
1 SYMBOL 163 \f «Symbol» \s 16 F i SYMBOL 163 \f «Symbol» \s 16 F n. I?e
oeueiio aaaaea?ii, ui aoaeue-yea c /enae ai=infIi (bi=supIi), a?aei?iia
a?ae SYMBOL 177 \f «Symbol» \s 16 ± SYMBOL 165 \f «Symbol» \s 16 Y , ?
aea?eeiai-?aoe?iiaeueia /enei.

O i?ae?icae?e? 4.2 noi?ioeueiaaii oaeo oiiao (b):

Eacaoeiaii, ui aeey iao?eoe? M ? iiiaeeie P aeeiiaii oiiao (b), yeui
aeey an?o 1 SYMBOL 163 \f «Symbol» \s 16 F i SYMBOL 163 \f «Symbol» \s
16 F n c Card(Ti(M)) SYMBOL 179 \f «Symbol» \s 16 ? 2 aeieeaa?
iaiaaeai?noue on?o iiiaeei Ij, aea j SYMBOL 206 \f «Symbol» \s 16 I
Ti(M).

)=P ?nio? oiae? ? o?eueee oiae?, eiee aeey iao?eoe? M ? iiiaeeie P
aeeiiaii oiiao (b).

Aeiaaaeaiiy aeinoaoiino? ocaaaeueith? eiino?oeoe?th E?niaeea aeey
R-ia?aoai?thaa/a, ui caaea? e?i?eio ooieoe?th c ?aoe?iiaeueieie
ia?aiao?aie. Iaiao?aei?noue nie?a?oueny ia eaio 4.1.

O i’yoiio ?icae?e? ?icaeyaeathoueny aiaeiae oai?aie
Iinoianueeiai-Oniainueeiai aeey no?iaeo R(s, t)-ia?aoai?thaa/?a ?
no?iaeo ne?i/aiieo R(s, t)—ia?aoai?thaa/?a.

Nooue cacia/aii? oai?aie iieyaa? a oiio, ui aeai?eoi ia?aeeaaeo
ia/enethaaiiai s-eiaiai cia?aaeaiiy ae?eniiai /enea a eiai ia/enethaaia
t-eiaa cia?aaeaiiy ?nio? oiae? ? o?eueee oiae?, eiee iiiaeeia i?inoeo
ae?eueiee?a /enea s, yeo iicia/eii /a?ac Ps, i?noeoue iiiaeeio i?inoeo
ae?eueiee?a /enea t, iicia/aio /a?ac Pt.

Aeyaeeiny, ui oeae ?acoeueoao ocaaaeueith?oueny ia an?, ia o?eueee
ia/enethaai?, cia?aaeaiiy ae?enieo /enae.

(x)=x, x SYMBOL 206 \f «Symbol» \s 16 I R, ?nio? oiae? ? o?eueee oiae?,
eiee aeeiiaia aeeth/aiiy Ps SYMBOL 202 \f «Symbol» \s 16 E Pt.

Iaiao?aei?noue aeiaaaeaii iaoiaeii a?ae noi?ioeaiiai. C oiai, ui
iiiaeeia i?inoeo ae?eueiee?a aeey t ia i?noeoueny a iiiaeei? i?inoeo
ae?eueiee?a /enea s, aeieeaa? ?nioaaiiy ?aoe?iiaeueiiai /enea x0, ia s
-eiaiio cia?aaeaii? w eio?iai caaeaiee R(s, t) -ia?aoai?thaa/ A aeaea?
iine?aeiai?noue A(w), yea ia ? t -eiaei cia?aaeaiiyi /enea x0.

Aeey aeiaaaeaiiy aeinoaoiino? iiaoaeiaaii OR(s, t) -ia?aoai?thaa/ A’,
eio?ee caaea? oioiaeith ooieoe?th. O caaeaii? a?aeiia?aeii? nenoaie
i?iaeoeoe?e aeei?enoiaoaaeany ??ai?noue 1/t=d/sb, yea aeeiio?oueny aeey
aeayeeo /enae d, b SYMBOL 206 \f «Symbol» \s 16 I N+ ? aeieeaa? c
aeeth/aiiy iiiaeeie i?inoeo ae?eueiee?a /enea t o iiiaeeio i?inoeo
ae?eueiee?a /enea s. Ia ci?noiaiiio ??ai? iiaoaeiaa t -eiaiai
cia?aaeaiiy caaeaiiai /enea x c a?ae??ceo [0, 1] iieyaa? o caeiaooo?
iecee oe?eeo iaa?ae’?iieo /enae SYMBOL 103 \f «Symbol» \s 16 g 1,
SYMBOL 103 \f «Symbol» \s 16 g 2,…, SYMBOL 103 \f «Symbol» \s 16 g k,…,
aea SYMBOL 103 \f «Symbol» \s 16 g k iicia/a?, “ne?eueee ?ac?a ae??a
1/tk i?noeoueny o caaeaiiio ae?eniiio /ene?” (aea aac o?aooaaiiy noa?oeo
noaiai?a 1/t, oiaoi ae?ia?a aeaeyaeo 1/ti, aea 1 SYMBOL 163 \f «Symbol»
\s 16 F i SYMBOL 163 \f «Symbol» \s 16 F k-1, ine?eueee c “aoiaeaeaiiy j
?ac?a /enea 1/tj oa x” aeieeaa? “aoiaeaeaiiy jt ?ac?a /enea 1/ti oa x”),
aea k SYMBOL 206 \f «Symbol» \s 16 I N+. Oi?iaeueii, SYMBOL 103 \f
«Symbol» \s 16 g k=[x/(1/tk)]-[x/(1/tk-1)] aai SYMBOL 103 \f «Symbol»
\s 16 g k=[x/(1/tk)]-( SYMBOL 103 \f «Symbol» \s 16 g k+…+ SYMBOL 103 \f
«Symbol» \s 16 g 1), aea SYMBOL 103 \f «Symbol» \s 16 g 1=[x/(1/t).
Ca?aene aeeeaa?, ui, iaoi?iaeueiith iiaith, ia?aoai?thaa/ ia? ?aooaaoe
a?aeiia?aei? e?eueeino? ae?ia?a dk/sbk, aea k SYMBOL 206 \f «Symbol» \s
16 I N+. Aeey iiaoaeiae t-eiaiai cia?aaeaiiy oe?ei? /anoeie aeia?eueiiai
ae?eniiai /enea ia?aoai?thaa/ “iaeiie/o? a iai’yo?” ?? s-eiaa
i?aaenoaaeaiiy, ia eiaeiiio e?ioe? ia/eneaiue c/eoothth/e /a?aiaee
ao?aeiee s -eiaee neiaie ? i?/iai ia ae?oeoth/e ia aeoiae? aae aei oiai
iiiaioo, eiee ia cono??ia neiaie SYMBOL 209 \f «Symbol» \s 16 N . I?ney
/eoaiiy SYMBOL 209 \f «Symbol» \s 16 N ia?aoai?thaa/ aeaea? t-eiaa
cia?aaeaiiy oe?eiai, s-eiaee caien yeiai i?noeoueny a iai’yo?, aeaea?
oaeiae i?ney iueiai neiaie SYMBOL 209 \f «Symbol» \s 16 N , “neeaea?”
iai’youe ? ia?aoiaeeoue aei ia?iaee ae?iaiai? /anoeie.

Aeae? a oeueiio ?icae?e? ?icaeyiooi iiaeeea?noue ia?aeeaaeo cia?aaeaiue
ae?enieo /enae c iaei??? nenoaie /eneaiiy a ?ioo ca aeiiiiiaith
ne?i/aiieo R(s, t) -ia?aoai?thaa/?a. O aeiaaeeo ne?i/aiieo R(s, t)
-ia?aoai?thaa/?a iiaiee aiaeia oai?aie Iinoianueeiai-Oniainueeiai i?noey
ia ia?.

(x)=x ? Dom(fB)=Ds, ?nio? oiae? ? o?eueee oiae?, eiee s=tn aeey
aeayeiai /enea n SYMBOL 206 \f «Symbol» \s 16 I N+.

(*a?ac Dom(fB)=Ds iicia/aii iiiaeeio s -eiaeo i?aaenoaaeaiue an?o
ae?enieo /enae.)

I?e aeiaaaeaii? aeei?enoaii iiia?aaeiuei iaa?oioiaaio eaio 5.1.

? any /eneiaa a?nue R, aai iai?acaieiaiee iaiaiaaeaiee i?ii?iue
aeaeyaeo (- SYMBOL 165 \f «Symbol» \s 16 Y , r], aai [r, + SYMBOL 165 \f
«Symbol» \s 16 Y ), aea r SYMBOL 206 \f «Symbol» \s 16 I R, aai naaiaio.
Oiae? ?niothoue oae? /enea m, n SYMBOL 206 \f «Symbol» \s 16 I N+, ui
sm=tn.

Oey eaia oaeoe/ii ? cai?ioiei oaa?aeaeaiiyi aei oaeoo, aeiaaaeaiiai
E?niaeeii, yeee ca oiia ?nioaaiiy /enae m, n SYMBOL 206 \f «Symbol» \s
16 I N+, oaeeo ui sm=tn, aeacaa aeai?eoi iiaoaeiae no?iaiai ne?i/aiiiai
R(s, t)-ia?aoai?thaa/a, ui caaea? oioiaeth ooieoe?th ia a?ae??ceo [0,
1].

Iniiaiei ?acoeueoaoii oinoiai ?icae?eo ? oai?aia 6.1.

: [0,1] SYMBOL 174 \f «Symbol» \s 16 ® [0,1]2 – iaia?a?aia nth?’?eoeaia
ooieoe?y ? aeey aoaeue-yei? oi/ee z c [0,1]2 ?? i?iia?ac f-1(z) i?noeoue
ia a?eueoa o?ueio aeaiaio?a.

?aeay iiaoaeiae oaeiai a?aeia?aaeaiiy iaaaaeo? eiino?oeoe?th A?eueaa?oa.
Iaeeie/iee eaaae?ao ae?eeoueny ia 9 i?aeeaaae?ao?a 1-ai ?aiao (?en. 1).

?en. 1. (O aeena?oaoe?ei?e ?iaio? oeueiio ?enoieo a?aeiia?aea? ?en. 6.1)

A?ae??cie [0, 1] oaeiae ae?eeoueny ia 9 i?aea?ae??ce?a 1-ai ?aiao: [0,
1/16], [1/16, 1/8], [(m-1)/8, m/8], aea 2 SYMBOL 163 \f «Symbol» \s 16 F
m SYMBOL 163 \f «Symbol» \s 16 F 8. I?aeeaaae?aoe ? i?aea?ae??cee
i?eaiaeyoueny o a?aeiia?aei?noue, yea aecia/a?oueny “i?ia?aii” a?ae??cea
[0, 1] o aeiaeaoiiio iai?yieo ? ia?aii i?aeeaaae?ao?a 1-ai ?aiao ca?aeii
c? no??eeaie.

I?ioean ae?eaiiy naaiaio?a n-ai ?aiao ia i?aenaaiaioe (n+1)-ai, n SYMBOL
179 \f «Symbol» \s 16 ? 1, i?iaeiaaeo?oueny oaeei /eiii, aae
a?aeiia?aei? i?aeeaaae?aoe iaoiaeeeeny iini?eue. Oei iaoiaeo
i?aeeaaae?aoo, yeiio a?aeiia?aea? a oi/iino? iaeei noai
R12-ia?aoai?thaa/a, aecia/a?oueny ia?ith aa?oei “ao?ae-aeo?ae”. Ianooiia
i?aaeei caaacia/o? ??ai?noue eiao?oe??ioo ia?ae?eooy o?ueii. Iaoae oi/ea
eaaae?aoo iaeaaeeoue /ioe?ueii i?aeeaaae?aoi iaeii/anii, oiaoi ??
i?iia?ace iaeaaeaoue /ioe?ueii i?aea?ae??ceai c i?iia?aco anueiai
eaaae?aoo. Oiae? aeey yeeoinue aeaio c /ioe?ueio i?aeeaaae?ao?a aiia ia?
aooe oi/eith “aoiaeo-aeoiaeo”, oiaoi iaeei c ?? i?iia?ac? iaeaaeeoue
iaeii/anii aeaii i?aea?ae??ceai (? ?oiueith ni?eueiith a?aieoeath).

[cl, cl’], aea [c0, c0’]=[a, b] oa eiaeiee c a?ae??ce?a aea [cl, cl’]
ca l SYMBOL 206 \f «Symbol» \s 16 I N iaeaaeeoue oeiia? T(l) SYMBOL 206
\f «Symbol» \s 16 I {T1,…, T SYMBOL 76 \f «Symbol» \s 16 L }, ? ca l
SYMBOL 179 \f «Symbol» \s 16 ? 1 ? il -e i?aea?ae??cie a?ae??ceo [cl-1,
cl-1’] i?e ?icaeoo? eiai ca a?aeiia?aeiei oeiii T(l-1).

-ia?aoai?thaa/aie ? ia ? aecia/aieie HIFS.

) ia ? caieiaiith iiiaeeiith.

Ca caaeaiith HIFS iiaeia iiaoaeoaaoe nenoaio ia?aoai?thaa/?a, eio?a
aecia/a? oie naiee aaeoi? iiiaeei.

Oai?aia 7.1. Iaoae caaeaii nenoaio aeaeyaeo

Ci= SYMBOL 200 \f «Symbol» \s 16 E {fk(Cj) SYMBOL 189 \f «Symbol» \s 16
1/2 (k, j) SYMBOL 206 \f «Symbol» \s 16 I Qi }, 1 SYMBOL 163 \f
«Symbol» \s 16 F i SYMBOL 163 \f «Symbol» \s 16 F d2,

aea fk ? noeneoaaeueieie iiae?aiinoyie a R, aea 1 SYMBOL 163 \f «Symbol»
\s 16 F k SYMBOL 163 \f «Symbol» \s 16 F d1 oa Qi SYMBOL 205 \f «Symbol»
\s 16 I {1,…d1} SYMBOL 180 \f «Symbol» \s 16 ? {1,…d2} ca 1 SYMBOL 163
\f «Symbol» \s 16 F i SYMBOL 163 \f «Symbol» \s 16 F d2.

)=Ci, aea 1 SYMBOL 163 \f «Symbol» \s 16 F i SYMBOL 163 \f «Symbol» \s
16 F d2.

)=Ci, aea 1 SYMBOL 163 \f «Symbol» \s 16 F i SYMBOL 163 \f «Symbol» \s
16 F d2.

Oaa?aeaeaiiy, aiaeia?/i? oai?ai? 7.1 ? iane?aeeo 7.1, iathoue i?noea ? a
m-aei??ieo i?inoi?ao, m SYMBOL 179 \f «Symbol» \s 16 ? 1.

AENIIAEE

A ?iaio? iieacaii, ui eean aeeoa?aioe?eiaieo ia caaeaiiio ?ioa?aae?
ooieoe?e, ye? aecia/athoueny R-ia?aoai?thaa/aie c iaaaceiiith iai’yooth,
aeeaaeaiee o eean e?i?eieo ia oeueiio ?ioa?aae? ooieoe?e. Aeiaaaeaii, ui
eean ae?enieo ooieoe?e, ye? caaeathoueny iaaaceiieie R-ia?aoai?thaa/aie,
no?iai aeeaaeaiee o eean ooieoe?e, ye? caaeathoueny R-ia?aoai?thaa/aie
c? noaeiaith iai’yooth.

(x)=Mx+h, aea P SYMBOL 205 \f «Symbol» \s 16 I R2, caaea?oueny
ne?i/aiiei Rnn-ia?aoai?thaa/ai. Oey oiiaa iaeeaaea?oueny ia iiiaeeio P ?
iao?eoeth M.

Aeiaaaeaii oae? oaeoe i?i iiaeeea?noue ia?aeeaaeo /enae c nenoaie
/eneaiiy c iniiaith s a nenoaio /eneaiiy c iniiaith t ca aeiiiiiaith
R(s, t) -ia?aoai?thaa/?a:

(x)=x, ?nio? oiae? ? o?eueee oiae?, eiee iiiaeeia i?inoeo ae?eueiee?a
/enea s aeeth/a? iiiaeeio i?inoeo ae?eueiee?a /enea t (aiaeia oai?aie
Iinoianueeiai-Oniainueeiai);

? Dom(fB)=Ds, ?nio? oiae? ? o?eueee oiae?, eiee s=tn aeey aeayeiai
iaoo?aeueiiai /enea n SYMBOL 206 \f «Symbol» \s 16 I N+.

Iiaoaeiaaii ne?i/aiiee R12-ia?aoai?thaa/, eio?ee caaea? e?eao Iaaii c
eiao?oe??ioii ia?ae?eooy 3, iaeiaioei iiaeeeaei.

)=Ci ca , 1 SYMBOL 163 \f «Symbol» \s 16 F i SYMBOL 163 \f «Symbol» \s
16 F d2.

?icaeyiooi oiiao, ca yei? oaeee ia?aoai?thaa/ ? ne?i/aiiei, ?
iiaoaeiaaii ne?i/aiiee RA[0, 1] -ia?aoai?thaa/, yeee caaea? iacaieiaio
iiiaeeio

Ca ?acoeueoaoaie ?iaioe iiaeia iiaoaeoaaoe oaaeeoeth, ui a?aeia?aaea?
ca’ycie i?ae eeanaie ae?enieo ooieoe?e, iienoaaieo o oa?i?iieia??
eeane/iiai aiae?co, ? eeanaie ooieoe?e, aecia/aieo iaaieie oeiaie
ia?aoai?thaa/?a iaae iaaeneiaaie.

Eeane ae?enieo ooieoe?e

aeeoa?aioe?-eiai? ia ?ioa?aae? oa?aeoa?enoe/i? ooieoe?? o?aeoaeueieo
iiiaeei, aecia/aieo HIFSaie

Eeane ooieoe?e,

aecia/aieo

1 2 3 4 5

Ne?i/aiieie

Rnn ia?aoai?thaa/aie (

n=1

P ? aeaea?o?a aeiaooie a?ae??ce?a, iai?a-caieiaieo i?iiai?a, aai
/eneiaeo inae (i. ? ae. oiiaa aecia/aia o oai?ai? 4.2.) /anoeiiee
aeiaaeie iaaaceiieo Rnn-ia?a-oai?thaa/?a,

n=1 /anoeiiee aeiaaeie

-ia?aoai?thaa/?a

m=n

1 2 3 4 5

iaaaceiieie

Rnn -ia?aoai?thaa/aie caiaeeoueny aei aeiaaeeo ne?i/aiieo Rnn
ia?aoai?thaa-/?a /anoeiiee aeiaaeie ne?i/aiieo Rnn -ia?aoai?thaa/?a ?
e?i?eieie

-ia?aoai?thaa/?a

(o oe?e ?iaio? ia aeine?aeaeoaaeeny)

no?iaeie ne?i/aiieie

R(s,t)mn -ia?aoai?thaa/aie

-ia?aoai?thaa/?a

no?iaeie

-ia?aoai?thaa/aie (?): iinoa? ieoaiiy

i?i aca?ii-aieea ao?aeieo a?ae??ce?a ? aeo?aeii? aea?eeiai? nenoaie
/eneaiiy Ca n=m=1

caiaeeoueny aei aeiaaeeo ne?i/aiieo Rnn -ia?aoai?thaa/?a (?): aiaeia?/ii
ne?i/aiiei Rnn -ia?aoai-?thaa/ai

SYMBOL 202 \f «Symbol» \s 16 E , ae. oiiaa ae-cia/aia

-ia?aoai?thaa/aie (?): iinoa? ieoaiiy

i?i aca?iiaieea ao?aeieo a?ae??ce?a ? ae-o?aeii? aea?eei-ai? nenoaie
/e-neaiiy Ca m=n=1

caiaeeoueny aei ae-iaaeeo ne?i/aiieo — Rnn ia?aoao?thaa/?a m=1
aiaeia?/ii aeniiaeo E?niaeea i?i aecia-/oaai?noue iaia?a?a-ieo ooieoe?e
-ia?aoai-?thaa/aie SYMBOL 202 \f «Symbol» \s 16 E

Oaae. 1. Ciae oai?aoeei-iiiaeeiiiai a?aeiioaiiy o ee?oei? oaaeeoe?
iicia/a? a?aeiioaiiy eeano ooieoe?e, aecia/aiiai a?aeiia?aeiei ?yaeeii,
aei eeano ooieoe?e, aecia/aiiai a?aeiia?aeiei noiai/eeii. (?) iicia/a?
a?aee?eoa ieoaiiy aai a?iioaco.

?acoeueoaoe ?iaioe iiaeia aeei?enoiaoaaoe aeey iiaeaeueoeo
aeine?aeaeaiue ia/enethaaiino? ae?enieo ooieoe?e ca aeiiiiiaith
ia?aoai?thaa/?a, aeey aeine?aeaeaiue o aaeoc? oai??? o?aeoae?a, ?
iaoaiaoe/iiio aiae?c? – aeey iiaoaeiae iaia?a?aieo ooieoe?e c
iane?i/aiieie iiiaeeiaie oi/ie iaaeeoa?aioe?eiaiino?. ?acoeueoaoe, ui
noinothoueny iiaoaeiae aeai?eoi?a ia/eneaiiy ae?enieo ooieoe?e
(iai?eeeaae, e?eai? Iaaii), iiaeia oaeiae canoiniaoaaoe a i?eeeaaeiiio
i?ia?aioaaii?.

NIENIE IIOAE?EIAAIEO I?AOeUe AAOI?A

CA OAIITH AeENA?OAOe?EII? ?IAIOE

L.P. Lisovik, O. Yu. Shkaravskaya. Real functions defined by transducers
// Oacenu Aoi?ie Iaaeaeoia?iaeiie eiioa?aioeee “Iaoaiaoe/aneea
aeai?eoiu”. – 1995. – N. 34-35;

2. Eeniaee E.I., Oea?aaneay I.TH. Ooieoeee, ii?aaeaeyaiua
i?aia?aciaaoaeyie n iaaaceiiie iaiyoueth // Aeiiia?ae? IAI Oe?a?ie. —
1995. – ? 9. – C. 57 — 59;

3. Eeniaee E.I., Oea?aaneay I.TH. I aauanoaaiiuo ooieoeeyo, caaeaaaaiuo
i?aia?aciaaoaeyie // Eeaa?iaoeea e nenoaiiue aiaeec. – 1998. – ? 1. – C.
82 — 93.

4. Oea?aaneay I. TH. Aoeiiua ioia?aaeaiey, caaeaaaaiua eiia/iuie
i?aia?aciaaoaeyie // Eeaa?iaoeea e nenoaiiue aiaeec. – 1998. – ? 5. – C.
178 — 181 .

Oea?aanueea I.TH. Eeane ooieoe?e, aecia/aieo ia?aoai?thaa/aie iaae
iaaeneiaaie. – ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.08 – iaoaiaoe/ia
eia?ea, oai??y aeai?eoi?a ? aeene?aoia iaoaiaoeea. – Ee?anueeee
oi?aa?neoao ?iai? Oa?ana Oaa/aiea, Ee?a, 1999.

O ?iaio? ?icaeyaeathoueny eeane ia?aoai?thaa/?a iaae iane?i/aiieie
neiaieueieie cia?aaeaiiyie ae?enieo /enae. Aeine?aeaeothoueny
ia/enethaaeuei? iiooaeiino? oaeeo ia?aoai?thaa/?a ye cania?a caaeaiiy
iaia?a?aieo ae?enieo ooieoe?e ? o?aeoaeueieo iiiaeei.

-ia?aoai?thaa/aie. ?ic?iaeaii aeai?eoi iiaoaeiae ia?aoai?thaa/a, ui
caaea? ao?iia a?aeia?aaeaiiy i?aeiiiaeeie n-aei??iiai i?inoi?o.
Iiaoaeiaaii ne?i/aiiee R12-ia?aoai?thaa/, yeee caaea? e?eao Iaaii c
iaeiaioei iiaeeeaei eiao?oe??ioii ia?ae?eooy, 3.

-ia?aoai?thaa/.

Oea?aaneay I.TH. Eeannu ooieoeee, ii?aaeaeyaiuo i?aia?aciaaoaeyie iaae
naa?oneiaaie. – ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.08 – iaoaiaoe/aneay
eiaeea, oai?ey aeai?eoiia e aeene?aoiay iaoaiaoeea. – Eeaaneee
oieaa?ceoao eiaie Oa?ana Oaa/aiei, 1999.

A ?aaioa ?anniao?eaathony eeannu i?aia?aciaaoaeae iaae aaneiia/iuie
neiaieueiuie i?aaenoaaeaiyie aauanoaaiiuo /enae. Enneaaeothony
au/eneeoaeueiua aiciiaeiinoe oaeeo i?aia?aciaaoaeae eae n?aaenoa
caaeaiey aeaenoaeoaeueiuo ooieoeee e o?aeoaeueiuo iiiaeanoa.

-i?aia?aciaaoaeyie. Oaeaea ?ac?aaioai aeai?eoi iino?iaiey
i?aia?aciaaoaey, caaeathuaai aooeiiia ioia?aaeaiea iiaeiiiaeanoaa
n-ia?iiai i?ino?ainoaa. Iino?iai eiia/iue R12-i?aia?aciaaoaeue,
caaeathuee e?eaoth Iaaii n iaeiaiueoei aiciiaeiui eiaooeoeeaioii
ia?ae?uoey, 3.

-i?aia?aciaaoaeue.

Shkaravska O. Yu. Classes of the functions defined by transducers over
superwords. – Manuscript.

Thesis nandidate of physics and mathematics by speciality 01.01.08 –
mathematical logics, theory of algorythms and discrete mathematics. –
Kyiv Taras Shevchenko University, Kyiv, 1999.

In this paper, are considered the classes of transducers over infinite
symbolic representations of real numbers. The computational
potentialities of these transducers as means of defining continuous real
functions and fractal sets are studied.

Turing machines, in particular finite automata, can define real
functions. From this point of view the finite automata and generalized
sequential machines were first studied by S.Eilenberg. A finite
automaton sequentially works over the finite binary expansion of x to
define f(x) for the corresponding function f.

-transducer works over the still more general representations of real
numbers. This approach in defining real functions is constructive on the
one hand, but it differs from the traditional theories in constructive
analysis, on the other hand. The constructive real functions are defined
on the constructive real numbers. The latter ones can be roughly
considered as algorithms of approximation of “ordinary” real numbers.
Note that we don’t restrict ourselves by constructive expansions dealing
with transducers.

-transducer. Thus we have a new criterion of classification of these
functions, based on power and structure of the memory of the
corresponding transducers. From the practical point of view the
transducers are algorithms that calculate the values the functions with
an arbitrary precision. Also, the transducers can be considered as means
for defining the recursive fractal sets and functions on these sets.

In this thesis, the following new results are presented:

-transducer is included in the class of the functions linear on the
interval;

-transducer has been considered;

-transducer defining the identity map exists iff the set of the prime
divisors of s is included in the set of the prime divisors of t;

-transducer for Peano curve with the best overlapping coefficient 3 and
the function mapping the unit segment onto Serpinsky carpet hve been
built;

-transducer defining the set that is not closed has been built.

All the statements listed above are based on recursive nature of
transducers, i.e on iteration or “almost iteration” of the commands of
the given transducer while it is working over the infinite expansion of
a real number. Also the “trace technique” has been used.

-transducer.

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