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Методи дослідження рельєфу земної поверхні: Автореф. дис… д-ра техн. наук / Р.М. Рудий, Держ. ун-т ‘Львів. політехніка’. — Л., 1999. — 32 с. — укp.

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I?I?NOA?NOAI INA?OE OE?A?IEAeA?AEAAIEE OI?AA?NEOAO “EUeA?ANUeEA
IIE?OAOI?EA”

Ia i?aaao ?oeiieno

?OAeEE

?iiai Ieoaeeiae/

OAeE 528.7+528.94:551.5(043.3)

IAOIAeE AeINE?AeAEAIIss ?AEUe?OO CAIII? IIAA?OI?

Niaoe?aeuei?noue 05.24.02 – oioia?aiiao??y oa ea?oia?ao?y

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

aeieoi?a oaoi?/ieo iaoe

Euea?a – 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia ia eaoaae?? ?iaeaia?ii? aaiaeac??
?iaeaia?ii-aeieia?/iiai oaeoeueoaoo ?aaii-O?aie?anueeiai aea?aeaaiiai
oaoi?/iiai oi?aa?neoaoo iaooe ? aaco.

Io?oe?ei? iiiiaioe:

Aeieoi? oaoi?/ieo iaoe, i?ioani? *a?aaiueia ?ai? A?eai?iae/,
Oa?e?anueeee aea?aeaaiee oi?aa?neoao, caa?aeoaa/ eaoaae?e
nioe?aeueiiaeiiii?/ii? aaia?ao??, i?e?iaeiioi?iiiiai iaiaaeaeiaioo oa
aai?ioi?iaoeee, i.Oa?e?a;

aeieoi? oaoi?/ieo iaoe, i?ioani? Iaeueiee Aieiaeeie? Ieeieaeiae/,
Aieeinueeee aea?aeaaiee oi?aa?neoao ?i. Ean? Oe?a?iee, caa?aeoaa/
eaoaae?e aaiaeieia??, aeaeai aaia?ao?/iiai oaeoeueoaoo, i.Eooeuee;

aeieoi? oaoi?/ieo iaoe, i?ioani? Noaaio?i ?aai ?ai?iae/, Ee?anueeee
iaoe?iiaeueiee oi?aa?neoao aoae?aieoeoaa ? a?o?oaeoo?e, eaoaae?a
?iaeaia?ii? aaiaeac??, i.Ee?a.

I?ia?aeia onoaiiaa: Aeiiaoeueeee aea?aeaaiee oaoi?/iee oi?aa?neoao,
eaoaae?a aaiaeac??, I?i?noa?noai ina?oe Oe?a?ie, i. Aeiiaoeuee.

Caoeno a?aeaoaeaoueny..12 eenoiiaaea 1999 ?ieo i 14 aiae.

ia can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 35.052.13 Aea?aeaaiiai
oi?aa?neoaoo “Euea?anueea iie?oaoi?ea” ca aae?anith 290646, i.
Euea?a-13, aoe. N.Aaiaea?e, 12.

C aeena?oaoe??th iiaeia iciaeiieoenue a iaoeia?e a?ae?ioaoe?
Aea?aeaaiiai oi?aa?neoaoo “Euea?anueea iie?oaoi?ea”, aoe. I?ioani?nueea
1.

Aaoi?aoa?ao ?ic?neaiee 28 aa?aniy 1999 ?.

A/aiee nae?aoa? niaoe?ae?ciaaii? a/aii? ?aaee

eaiaeeaeao oaoi?/ieo iaoe N.A.Naa/oe

CAAAEUeIA OA?AEOA?ENOEEA I?IAEAIE

Aeooaeuei?noue ? nooi?iue aeine?aeaeaiino? oaiaoeee aeena?oaoe??.

A ca’yceo c oei, ui ai?iaaaeaeaiiy ?ioi?iaoe?eieo oaoiieia?e a
aaiaeace/ia ae?iaieoeoai aeiaaa? oni?oiiai ?ica’yceo caaea/, iia’ycaieo
c aiae?cii oa eeaneo?eaoe??th ?aeue?oo caiii? iiaa?oi?, a eiiieaen?
oiiia?aoi-aaiaeace/ieo ?ia?o ana a?eueoa /anoea i?eiaaea? ia
eaia?aeuei? ?iaioe, ia ?ic?iaeo aeai?eoi?a oa a?aeiia?aeieo i?ia?aiieo
cania?a. Noai?aiiy ?ioi?iaoe?eieo nenoai ni?yiiaaieo ia ai?iaaaeaeaiiy
i?ia?aiiiai, oaoi?/iiai oa oaoiieia?/iiai caaacia/aiiy aoaeoeaiiai
aeei?enoaiiy oeeo?iaeo ea?o, caaacia/aiiy i?eaaoecaoe?? caiaeueieo
ae?eyiie, aaeineiiaeaiiy aai?ioi?iaoe?eieo nenoai (A?N), caiaeueieo
?ioi?iaoe?eieo nenoai (C?N) oa caiaeueieo ?a?no?aoe?eieo nenoai (C?N) ?
aaeueie aeooaeueiei a ca’yceo c i?iaaaeaiiyi a e?a?i? ?ia?o c
i?eaaoecaoe?? caiaeueieo ae?eyiie. C 1997 ?. ii 2007 ?. ae?? I?ia?aia
noai?aiiy aaiiaoeciaaii? nenoaie aaaeaiiy aea?aeaaiiai caiaeueiiai
eaaeano?o, ui caoaa?aeaeaia iinoaiiaith Eaa?iaoo I?i?no??a Oe?a?ie a?ae
2 a?oaeiy 1997 ?. ? 1355, a ye?e aeaco?oueny, ui aaaeaiiy aea?aeaaiiai
caiaeueiiai eaaeano?o caaacia/o?oueny aeeiiaiiyi oiiia?aoi-aaiaeace/ieo,
ea?oia?ao?/ieo, a?oioiaeo oa ?ioeo ianoaaeaiue ? ?ica?aeoaaiue. A
inoaii? ?iee i?e ea?oia?aooaaii? a?oio?a ii/aa oe?iei aeei?enoiaoaaoenue
iaoiae ieanoeee ?aeue?oo. A?oioiciaaoe? anoaiiaeee oe?eaa? oiiieia?/i?
caeiiii??iino? ?icoaooaaiiy a?oioiaiai iie?eooy a caeaaeiino? a?ae
?aeue?oo.

Eiiini?th III I?aeia?iaeiiai oiaa?enoaa oioia?aiiao??? oa
aeenoaioe?eiiai ciiaeoaaiiy (IOOAeC) aenoiooa caaaeaiiy oe?ieiai
ai?iaaaeaeaiiy oai?i? ?iciiciaaaiiy ia?acia, ui eeth/a? a naaa iioeeo,
oai?ith ia?iaee iioi?iaoei? oa iioi i?eeeaaeii aeenoeeieiie. I?i
aaaeeea?noue aeacaieo caaaeaiue na?ae/eoue oi?ioethaaiiy ?o aeey
o?aouei? eii?n?? IOOAeC: Object reconstraction from digital imagery
(aeae?eaiiy ia’?eo?a c oeeo?iaiai cia?aaeaiiy) oa image understanding
(aiae?c cia?aaeaiiy).

Aeey iioeiaeueiiai nei?i/aiiy oeeo?iai? ?ioi?iaoe??, yeith iiaea?oueny
OeI?, iaiao?aeii ca aeaieie, yeeo ? a iaaeeeoeo, aecia/eoe iieiaeaiiy
no?oeoo?ieo e?i?e ?aeue?oo, aeae?eeoe oa?aeoa?i? oi?ie iiaa?oi?. A
e?oa?aoo?? a?aecia/a?oueny, ui ?aeue?o ? i?ia?aeiei aeaiaioii i?e
aeine?aeaeaii? aaieia?/ieo, aeieia?/ieo, eaiaeoaoieo ia’?eo?a ieaiaoe.
Ine?eueee ?aeue?o ? iniiaiei oa iaeoa?aeoa?i?oei eiiiiiaioii
eaiaeoaooo, oi a?i ni?ey? aeeoa?aioe?aoe?? i?e?iee?iaoo. Oiio
aaoiiaoecaoe?y i?ioeano anoaiiaeaiiy aneiao??? eaiaeoaoieo eiiiiiaio?a
aeicaiey? oaeaeei, /anoi a nenoai? ?aaeueiiai ianooaao /ano, io?eiaoe
e?euee?ni? e?eoa???, a cia/eoue ? ia’?eoeai? a?ae?iiaoai?ieia?/i?
aeai?. ?aeue?o caiii? iiaa?oi? /anoi aoaa? aecia/aeueiei oaeoi?ii i?e
aea/aii? iaaacia/ieo a?ae?iiaoai?ieia?/ieo yaeu a a??nueeeo ?aeiiao
iaoi? e?a?ie. I?i aeooaeuei?noue aea/aiiy oe??? i?iaeaie na?ae/aoue
caaaeaiiy, ui iinoaaeai? ia?aae a?aeiia?aeieie eii?n?yie IOOAeC oa
I?aeia?iaeiiai aaia?ao?/iiai oiaa?enoaa.

E??i oeueiai, ia?aae oiiia?aoi-aaiaeace/ieie neoaeaaie e?a?ie aaea
aeaaii iinoaaeaia caaaeaiiy eeaneo?eaoe?? iaaieo oei?a ?aeue?oo
oa?eoi???. Eiaeai oei ?aeue?oo iiaea a?aeiia?aeiei /eiii
a?aeia?aaeaoenue ia niaoe?aeueieo ea?oao ? io?eioaaoe iaaio eaaeano?iao
ioe?ieo. Oaea ?aeiioaaiiy oa?eoi??? e?a?ie, ye ? aaiii?oieia?/ia
?aeiioaaiiy, a oaeiae noai?aiiy a?aeiia?aeieo ea?o ? aeineoue neeaaeiei
oa ia?niaeoeaiei caaaeaiiyi, yea ua a?ey aeaaaeoeyoe ?ie?a oiio aenoioa
a?aeiiee aaiaeaceno i?io. Eaa/oe A.I. Ia’?eoeaiee ?ica’ycie caaea/? c
ioe?iee neeaaeiino? ?aeue?oo noaiiaeoue caoe?eaaeaiiy a ia?oo /a?ao a
oiiia?ao??. Oea aea?? iaae?eieo /eneiaeo e?eoa???a, ye?
oa?aeoa?ecoaaoeiooue nooi?iue ia?an?/aiino?, ai?cio?ii?th oa ?io?
ia?aiao?e ieanoeee ?aeue?oo.

?in?enueeee aaiii?oieia I.I.Eanoi/e?i a?aecia/aa, ui iaei??th c
iniiaieo i?e/ei a?aenooiino? aaaiieo oni?o?a o?ce/ii? aaia?ao?? a
iaeano? aeieia?? ? iaaeinoaoiy e?euee?noue nooo?aeo aeine?aeaeaiue a
ea?oia?aooaaii? ii?oieia?? ?aeue?oo. Naia ?aeue?o ua c /an?a
A.?.Aa?ia?aenueeiai iiaeaaaany iaai?ii ??ciiiai?oieo i?noeaiieiaeaiue
?ineeiieo oa oaa?eiieo eiiieaen?a ni?eueiio. Oiio noaaeyoueny ieoaiiy
i?i /?oea aeae?eaiiy /e iaiaaeaiiy, aecia/aiiy ? nenoaiaoecaoe?th
i?e?iaeiuei-oa?eoi??aeueieo eiiieaen?a, ui ? ia o?eueee eaiaeoaooieie, a
e aeieia?/ieie ia’?eoaie.

Noai?aiiy nenoai aaoiiaoe/iiai i?iaeooaaiiy ?iaeaia?ieo nii?oae
aeeeeea? iaiao?aei?noue aaoiiaoe/iiai ?ici?ciaaaiiy ie?aieo oi?i
?aeue?oo aeey oiai, uia ?ici?uaiiy ie?aieo nii?oae cae?enithaaeinue ia
o?eueee o a?aeiia?aeiino? aei aeaiaio?a neooaoe?? oa neeaaeiino?
?aeue?oo, a e i?enoiniaoth/enue aei iaaieo oi?i ?aeue?oo.

Aeei?enoiaoth/e iniiai? aeaiaioe oai??? ?ici?ciaaaiiy ia?ac?a, iiaeia
?ica’ycaoe aeacai? caaaeaiiy.

Iaoa ? caaea/? aeine?aeaeaiiy. 1) noai?aiiy oai?aoe/ieo iniia
aeae?eaiiy oa?aeoa?ieo e?i?e oa oi?i ?aeue?oo ca aeaieie oeeo?iai?
iiaeae?; 2) aeine?aeaeaiiy iaoiaeeee aeei?enoaiiy
ei?aeyoe?eii-niaeo?aeueiiai aiae?co aeey ioe?iee nooiaiy ia?an?/aiino?
aneiao??? oa ?ioeo ia?aiao??a ieanoeee ?aeue?oo.

Aeey aeinyaiaiiy oe??? iaoe a aeena?oaoe?ei?e ?iaio? ?ica’ycai?
ianooii? caaea/?:

– cai?iiiiiaaii ? oai?aoe/ii iaa?oioiaaii iaoiae aeine?aeaeaiiy
?aeue?oo iiaa?oi? c aeei?enoaiiyi ne?i/aiieo ??cieoeue;

– ?ic?iaeaii iaoiaeeeo aecia/aiiy iieiaeaiiy no?oeoo?ieo e?i?e ?aeue?oo
oa iaaieo oi?i ?aeue?oo;

– noi?ioeueiaai? i?eioeeie ioe?iee nooiaiy ia?an?/aiino? oa aneiao???
?aeue?oo ca iaeiiaei??ieie ? aeaiaei??ieie ei?aeyoe?eieie oa
niaeo?aeueieie oa?aeoa?enoeeaie;

– ?ic?iaeai? oai?aoe/i? aeaiaioe aiae?oe/iiai iaoiaeo caeaaeaeoaaiiy
?aeue?oo;

– ?ic?iaeaii iiaee oai?aoe/iee iai?yiie eeaneo?eaoe?? ae?eyiie
?aeue?oo ca oi?iith iiaa?oi?;

– iieacai? iai?yiee iiaeaeueoeo aeine?aeaeaiue o aaeoc? eeanoa?ecaoe??
oi?i ?aeue?oo;

– ?ic?iaeaii niin?a iiaeaethaaiiy ?aeue?oo c aeiiiiiaith ci?iii?
iioe/ii? u?eueiino?;

– cai?iiiiiaaii aiaeiaiaee i?eeaae aeey iiaeaiiy ?aeue?oo ci?iiith
iioe/iith u?euei?noth;

– cai?iiiiiaaii i?eeaae aeey aecia/aiiy iieiaeaiiy no?oeoo?ieo e?i?e
?aeue?oo;

– cai?iiiiiaaii niin?a aecia/aiiy iieiaeaiiy ii?nueeiai noaeia /e
e?oath/iai aia?aoo ca ?aeue?oii caiii? iiaa?oi?.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. A aeena?oaoe?ei?e ?iaio?
ii?aoeueiaaii oai??th ? ?ic?iaeai? i?eioeeiiai iia? iaoiaee aecia/aiiy
iieiaeaiiy oa?aeoa?ieo e?i?e oa oi?i ?aeue?oo ca oeeo?iaith iiaeaeeth c
aeei?enoaiiyi ne?i/aiieo ??cieoeue, cai?iiiiiaaii niin?a iiaeaethaaiiy
?aeue?oo, yeee aeicaiey? io?eioaaoe eiai niaeo?aeueia cia?aaeaiiy
aiaeiaiaei iaoiaeii, noai?aii aoaeoeaiee iaoiae aeae?eaiiy ie?aieo oi?i
?aeue?oo ca aeaieie OeI? c aeei?enoaiiyi oai??? ?ici?ciaaaiiy ia?ac?a.
Ia ?ica’ycie ia?a?aoiaaieo aeua caaea/ ? ia cai?iiiiiaai? niiniae oa
i?eeaaee aaoi?ii io?eiai? aaoi?nuee? na?aeioeoaa. Noeoii?noue io?eiaieo
iaoeiaeo ?acoeueoao?a, ia?ae?e yeeo i?eaaaeaii a caeeth/i?e /anoei?
?iaioe, iiaeia eaae?o?eoaaoe ye iiaee iaoeiaee iai?yiie a aeine?aeaeaii?
?aeue?oo caiii? iiaa?oi?. Oi?ioaaiiy oeueiai iai?yieo a?aeaoaaeinue i?ae
aieeaii ?aeae A.I.Eaa/oea oa ?.A.*a?aaiueiaa. Iniiaith aeey ?icaeoeo
oeueiai iai?yieo iineoaeeei aeei?enoaiiy iniiaieo aniaeo?a oai???
ia?iaee cia?aaeaiue.

Iniiai? iieiaeaiiy, ui aeiinyoueny ia caoeno.

1. Oai?aoe/i? canaaee aiae?co ?aeue?oo caiii? iiaa?oi? oeaaino? ca
oeeo?iaith iiaeaeeth oa ?o i?aeoe/ia ?aae?caoe?y aeey noai?aiiy
no?oeoo?iiai ?iaa??aioo:

– ?ic?iaeai? iia? oai?aoe/i? aeaiaioe aiae?oe/iiai aeae?eaiiy
no?oeoo?ieo e?i?e ?aeue?oo;

– ?ic?iaeai? iia? oai?aoe/i? aeaiaioe aiae?oe/iiai aeae?eaiiy oi?i
caiii? iiaa?oi?.

2.Oai?aoe/ia oa aenia?eiaioaeueia ?aae?caoe?y aaoiiaoeciaaiiai
aeyaeaiiy caeiiii??iinoae ieanoeee ?aeue?oo ia iniia? iaeii- oa
aeaiaei??ieo niaeo??a, aaoiei?aeyoe?eieo ooieoe?e oa e?eaeo
ia?an?/aiino?. 3. Iiaeaethaaiiy ?aeue?oo cia?aaeaiiyi c? ci?iiith
iioe/iith u?eueiinoth, io?eiaia ca ?acoeueoaoaie aaoiiaoeciaaiiai
aiae?co ieanoeee iiaa?oi? ia’?eoa.

4. Oai?aoeei-iaoiaeieia?/i? iniiae ?ica’yceo i?iaeaie eeaneo?eaoe?? oa
eeanoa?ecaoe?? ?aeue?oo caiii? iiaa?oi? ia oai??? ?ici?ciaaaiiy
ia?ac?a:

5. Eiioeaiooaeuei? iiaeae? i?eeaae?a ? niinia?a aiae?co oa a?aeoai?aiiy
?aeue?oo: i?oioioii?iaeoi?; aeeoa?aioe?aeueiee oioio?ainoi?iaoi?; niin?a
iiaeaethaaiiy ?aeue?oo; niin?a aecia/aiiy iieiaeaiiy e?oath/eo aia?ao?a
oa ii?nueeeo noaeai ca ?aeue?oii i?noeaaino?.

Oai?aoe/ia ? i?aeoe/ia oe?ii?noue aeine?aeaeaiue iieyaa? a ianooiiiio:

– ?ic?iaeai? iaoiaee ?ici?ciaaaiiy oa aeae?eaiiy iaaieo oi?i ?aeue?oo
iiaeooue aooe canoiniaai? ia o?eueee a oiiia?ao?? i?e a?aeia?aaeaii?
?aeue?oo c aeiiiiiaith ai?eciioaeae, oiaoi aeey iie?auaiiy ii?oieia?/ieo
oa ii?oiiao?e/ieo oa?aeoa?enoee, a e i?e aaoiiaoe/iiio i?iaeooaaii?
?iaeaia?ieo nii?oae, iniaeeai i?e aeai?? i?noeue iaooiaeo naa?aeeiaei,
ia yeeo iiaeeea? aaa??ei? neooaoe?? c aeeeaeii eiiiiiaio?a-caa?oaeiee?a.
E??i oeueiai, aeai? iaoiaee iiaeooue aeei?enoiaoaaoenue i?e
?ici?ciaaaii? ie?aieo ae?eyiie iiaa?oiiue ?ioeo ia’?eo?a, iai?eeeaae,
i?e?ineii?/ieo ia’?eo?a, a?ieia?/ieo ia’?eo?a, i?e aeine?aeaeaii? aeia
ii??a, o a?enueeia?e ni?aa?, iniaeeai a oiio aeiaaeeo, eiee iaiao?aeii
aecia/aoe /e eeaneo?eoaaoe ia’?eo a nenoai? ?aaeueiiai ianooaao /ano
“on line”. A inoaiiueiio ?icae?e? ?iaioe iieacaii aeei?enoaiiy aeaieo
iaoiae?a i?e i?iaiicoaaii? iaaacia/ieo a?ae?iiaoai?ieia?/ieo yaeu a
a??nueeeo ?aeiiao.

– cai?iiiiiaai? aaoi?ii aeanyoue ?ie?a oiio aiaeiaia? oioia?aiiao?e/i?
i?eeaaee a aeaiee /an canoa??ee, iaeiae niin?a iiaeaethaaiiy ?aeue?oo c
aeiiiiiaith iioe/ii? u?eueiino? oni?oii aeei?enoiao?oueny i?e
a?aeia?aaeaii? ?aeue?oo a no/anieo eiii’thoa?ieo oaoiieia?yo.

Iniiai? ?acoeueoaoe aeena?oaoe?eii? ?iaioe ciaeoee nai? canoinoaaiiy a
?aaii-O?aie?anueeiio aea?aeaaiiio oaoi?/iiio oi?aa?neoao? iaooe ? aaco,
Ea?iaonuee?e naeanoieia?e noaioe?? (i. ss?ai/a ?aaii-O?aie?anueei?
iae.), ?aaii-o?aie?anueeiio oeaio?? aaiaeac??, eaaeano?o ? A?N i?e
aeeiiaii? noi?niiai oe?a?inueei-eaiaaenueeiai i?iaeoo caiaeueii?
?a?no?aoe?? a 1995-1997 ??, a Iaoeiai-aeine?aeiiio ?inoeooo? aeieia?/ii?
aaciaee ? i?e?iaeieo ?ano?n?a, i. ?aaii-O?aie?anuee.

Ai?iaaoe?y ?iaioe. Iniiai? oai?aoe/i? ? i?aeoe/i? ?acoeueoaoe
aeena?oaoe?eii? ?iaioe iaaiai?thaaeenue ia eiioa?aioe?yo a
?anioae?eainueeiio aeii? aeiiii?/ii? ? iaoeiai-oaoi?/ii? i?iiaaaiaee
oiaa?enoaa “Ciaiiy” O?N?, Ee?anueeiio i?aeaia O?oaeiaiai /a?aiiiai
i?aii?a ?iaeaia?ii-aoae?aaeueiiio ?inoeooo? oa Iaoeiai-aeine?aeiiio
?inoeooo? aaoiiaoecaoe?? nenoai ieaioaaiiy ? oi?aae?iiy a aoae?aieoeoa?
Aea?aeaoaeo O?N? a 1989-1990 ?.?. (i.Ee?a), I?aaeaiii-cao?aeiiio
aa?iaaiaeace/iiio i?aei?e?inoa? AOAE i?e ?aae? I?i?no??a N?N?,
Oe?a?inueeiio ?anioae?eainueeiio i?aae?ii? ananithciiai oiaa?enoaa
aaiaeac?? aa?ieini?/ieo ceiiie ? ea?oia?ao?? a 1990 ?. (i.Ee?a), ia
iaoeiai-oaoi?/ieo eiioa?aioe?yo i?ioani?nueei-aeeeaaeaoeueeiai neeaaeo
?OAeOOIA (i.?aaii-O?aie?anuee, (1978-1998 ?.?.), ia eiioa?aioe??
Cao?aeii? ?aa?iiaeueii? i?aai?caoe?? Oe?a?inueeiai oiaa?enoaa aaiaeac??
aa?ieini?/ieo ceiiie ? ea?oia?ao??, Aea?aeaaiiai oi?aa?neoaoo
“Euea?anueea iie?oaoi?ea” ? aeniaaeeoe?? ?7 Oe?a?inueeiai
aa?iaaiaeace/iiai i?aei?e?inoaa (i.Euea?a, 1997 ?.) “No/ani? aeinyaiaiiy
aaiaeace/ii? iaoee ? ae?iaieoeoaa a Oe?a?i?”, ia ia?o?e I?aeia?iaei?e
iaoeiai-i?aeoe/i?e eiioa?aioe?? “Eaaeano?, oioia?aiiao??y,
aai?ioi?iaoeea – no/ani? oaoiieia?? ? ia?niaeoeae ?icaeoeo”, ia
iaoeiai-i?aeoe/i?e eiioa?aioe?? “Aaiaeac?y, eaaeano? oa aai?ioi?iaoeea”
(i.?aaii-O?aie?anuee, 1997 ?.), ia iaoeiai-i?aeoe/i?e eiioa?aioe??
“Aaiaeace/iee iii?oi?eia, aaiaeeiai?ea ?ao?aeoiiao??y ia iaae? OO?
noie?ooy” Aea?aeaaiiai oi?aa?neoaoo “Euea?anueea iie?oaoi?ea” ?
Euea?anueeiai ano?iiiii-aaiaeace/iiai oiaa?enoaa (i.Euea?a, 1998).

Iniaenoee aianie caeiaoaa/a a ?iaioo. Oai?aoe/i? iieiaeaiiy ?ica’yceo
caaea/ aeae?eaiiy no?oeoo?ieo e?i?e oa ie?aieo oi?i ?aeue?oo ca
oeeo?iaith iiaeaeeth c aeei?enoaiiyi ne?i/aiieo ??cieoeue ?ic?iaeai?
aaoi?ii naiino?eii. Iaoiaeeea ?ici?ciaaaiiy, eeaneo?eaoe?? oa
eeanoa?ecaoe?? ie?aieo oi?i ?aeue?oo, ye ? iaoiaeeea aaoiiaoe/ii?
aaia?ae?caoe?? ?aeue?oo oaae ?ic?iaeai? aaoi?ii naiino?eii.
Aenia?eiaioaeuei? aeine?aeaeaiiy ?ic?iaeaieo iaoiaeee i?iaiaeeeenue c
o/anoth i?aoe?aiee?a ?aaii-O?aie?anueeiai Oeaio?o aaiaeac??, eaaeano?o
? A?N. Cai?iiiiiaai? niiniae oa i?eeaaee aeey a?aeia?aaeaiiy ?aeue?oo
?ic?iaeai? aaoi?ii naiino?eii, i?i ui na?ae/aoue a?aeiia?aei? aaoi?nuee?
na?aeioeoaa.

Ioae?eaoe??. Iniiaiee ci?no aeena?oaoe?? aena?oeaii a 34 ?iaioao, a
oiio /ene? a 3 aaoi?nueeeo na?aeioeoaao ? iaeiiio iaoaio? Oe?a?ie. 17
noaoae iioae?eiaaii a oaoiaeo aeo?iaeao ? ca??ieeao iaoeiaeo i?aoeue
Oe?a?ie oa ?in??. Iaeia ?c noaoae ia?aiioae?eiaaia a oaoiaiio aeo?iae? a
NOA.

No?oeoo?a ? ia’?i aeena?oaoe??. Aeena?oaoe?eia ?iaioa neeaaea?oueny c?
anooio, /ioe?ueio ?icae?e?a, i?aenoieiaeo aeniiae?a, aeiaeaoe?a oa
a?ae?ia?ao??. Caaaeueiee ianya ?iaioe neeaaea? 251 noi?., 50 iae., 75
oaae. Nienie e?oa?aoo?e neeaaea?oueny c 143 iaeiaioaaiue.

Aaoi? aeneiaeth? aaey/i?noue ca iino?eio i?aeo?eieo i?e ?iaio? iaae
aeena?oaoe??th ni?a?ia?oieeai eaoaae?e ?iaeaia?ii? aaiaeac?? ?OAeOOIA,
a oaeiae ni?a?ia?oieeai ?aaii-O?aie?anueeiai Oeaio?o aaiaeac??,
ea?oia?ao?? oa aai?ioi?iaoe?eieo nenoai, a naia eiai ea??aieeia?
Aiaaeaiia? Iaethc?, ?iaeaia?ai A?oae??a? Ea?ea?o, Aiae???a? Iao?uoeo oa
Iaeneiia? ?oaeiio ca ni?eyiiy oa aeiiiiiao i?e aeeiiaii?
aenia?eiaioaeueieo aeine?aeaeaiue.

Noaiaoe/ii iniiai? oeyoe aeeiiaiiy iai?/aieo aeine?aeaeaiue iieacaii ia
?en.1, yeee iaaiei /eiii oa?aeoa?eco? ? ci?no aeaii? ?iaioe.

Ci?no aeena?oaoe??.

O anooi? i?eaaaeaii iaa?oiooaaiiy aeooaeueiino? iinoaaeaieo caaea/ ?
iiaeaii ei?ioeee ci?no aeena?oaoe?eii? ?iaioe. Aeaii oa?aeoa?enoeee
iniiaieo iaoeiaeo iai?yie?a, ui ?icaeaathoueny aaoi?ii aeey ?ica’ycaiiy
iinoaaeaieo iaoeiaeo i?iaeai.

A ia?oiio ?icae?e? iiaeaii iaeyae e?oa?aoo?ieo aeaea?ae, aiae?c
?nioth/eo iaoiae?a aeine?aeaeaiiy ?aeue?oo caiii? iiaa?oi?, iaa?oioiaai?
aea?ai? iai?yiee ?ica’yceo caaea/. Ei?ioei ?icaeyioo? ?noi?e/i? aniaeoe
aeacaieo i?iaeai. I?aee?aneth?oueny aaaeeea?noue ?ia?o, aeeiiaieo
?.A.*a?aaiueiaei oa ?ioeie aeine?aeieeaie i?ae eiai ea??aieoeoaii, oiaoi
oa?e?anueei? oeiee, ? I.I.Eanoi/e?iei, oiaoi Naieo-Iaoa?ao?cueei? oeiee.
A oeueiio iai?yieo c?iaeaii aeoaea aaaaoi ye a?o/eciyieie, oae ?
ca?oa?aeieie a/aieie. I?e oeueiio aeine?aeaeaiiy a iaeano? ?aeue?oo,
oi/iino? aecia/aiiy ia’?i?a ??ciiiai?oieo ia’?eo?a, yea a ia?oo /a?ao
aecia/a?oueny ?aeue?oii iiaa?oi?, i?iaiaeeeenue ? aaia?aoaie, ?
oiiia?aoaie, ? oioia?aiiao?enoaie, ? aaiii?oieiaaie. Iiaeia iacaaoe
I.N.Aaniooa, I.I.Aa?eyioa, I.A.Aieea, I.E.Aei?iaeeinueeiai,
I.I.AEa?iianueeiai, ?.A.AEo?e?ia, E.ss.Eeiiioiaa, I.I.Eiaaiiaa,
A.E.Iaeyanueeiai, A.I.Iaeueieea, N.A.Iiaeeueiiai oa ?ioeo.

A?aecia/ai? ooiaeaiaioaeuei? aeine?aeaeaiiy a iaeano? ieanoeee
?aeue?oo, ui aeeiiai?, ?.I.Aaeioniaei, ?.A.Aonaea?aei, A.E.A?aenooii,
I.N.Aeaaaea??ai?, A.A.Eini?iei, I.I.Eiceiaei, I.I.I?ioiaeueyeiiiaei,
I.?.O?ii/aiei, A.I.Oono, A.I.O?einioiaei, I.Ae.ssiiieuenueeei, oa
aaaaoueia ?ioeie.

Ii?yae c oeei i?aee?aneth?oueny, ui ia i?ioyc? inoaii?o
aeanyoe-i’yoiaaeoeyoe ?ie?a ieoaiiy ieanoeee ?aeue?oo iino?eii
?icaeyaeathoueny ia i?aeia?iaeieo eiia?anao. ?c ca?oa?aeieo aaoi??a
a?aecia/athoueny Ammannati F., Betti B., Dereney E., Frederiksen P.,
Jacobi O., Kubik K. oa ?io?.

Cae?i/o?oueny oeae ?icae?e, ye ? ?io? ?icae?ee, aeniiaeaie, iaeiae ooo
aiie c?iaeai? ye ?acthia, aea iiaeai? iniiai? iai?yiee iaiao?aeieo
aeine?aeaeaiue a iaeano? ieanoeee ?aeue?oo.

A aeoaiio ?icae?e? iiaeaii oai?aoe/i? oa i?aeoe/i? aeine?aeaeaiiy c
aeei?enoaiiy iniiaieo iieiaeaiue aeeoa?aioe?aeueii? aaiiao??? aeey
aeae?eaiiy iaaieo ii?oieia?/ieo oa?aeoa?enoee iiaa?oi?. Aeey oeueiai
/anoeii? iio?aei? iiaa?oi? aecia/athoueny /a?ac ce?i/aii? ??cieoe?
a?aeiia?aeieo ii?yaee?a. Iaoiae ne?i/aiieo ??cieoeue aeey ?ica’yceo
aeeoa?aioe?aeueieo ??aiyiue a /anoeiieo iio?aeieo iieyaa? a
aeene?aoecaoe?? aeeoa?aioe?aeueiiai iia?aoi?a. Ae?eyieo ?aeue?oo caiii?
iiaa?oi? ?icaeyaeathoue i?e oeueiio ye aeene?aoiee neaiae, yeee a
iaoaiaoeoe? aaaaea?oueny ianeaii aai iine?aeiai?noth. Oey
iine?aeiai?noue O aecia/a?oueny noeoii?noth aii?yaeeiaaieo ia? /enae

O =x (n1, n2) –

? ??y???????????? ???????????yyy????.????1????????????? @???&??yyyy?????AyTHy?th???&[email protected]?????? ?? c iaaiith oi/iinoth iiaeathoue iioa?iieyoeieiei iieiiiiii aeey aeaio ciiiieo wmetafile8? ??®??????????? ???????????yyy????.????1????????????? ?`1???&??yyyss?????Ay?y 1????&? [email protected]???? ??wmetafile8? ??P????????????? ???????????yyy????.????1????????????? `@???&??yyyy?????Ay3/4y? ???&??MathType??`????&??yyyy???????????????????? `@???&??yy?yyy?yyyyyyyyyyyyyyyyyyyyyyyyyyywmetafile8? ??A??????????? ???????????yyy????.????1?????? ???????`? ???&??yyyy?????Ay3/4yA ???&??MathType???????u????????????-?????L wmetafile8? ?????????????? ???????????yyy????.????1?????? [email protected] ???&??yyyss?????Ay?y? ????&??MathType???????u—y???? wmetafile8? ?????????????? ???????????yyy????.????1????????????? ?a???&??yyyy?????Ay?y 2???&? ?MathType???????u????????????-?????l(???li???l? [email protected]?????? wmetafile8? ??"??????????? ???????????yyy????.????1????????????? A????&[email protected]???&??MathType??`[email protected]?????? C oeueiai iioa?iieyoeieiiai iieiiiia ciaeaeai? iioiaeii oa ioe?iaia ?o oi/i?noue ca caeeoeiaei /eaiii R wmetafile8? ??Q??????????? [email protected] `.???&??yyyy?????Aycy .a???&? [email protected]????h ???h/???hu???hA???hAE???+wmetafile8? ??Q??????????? ???????????yyy????.????1?????? ???????  ?.???&[email protected] ???&? ?MathType??0???u????????????-?????[@????[ ???[/???[???[TH???[a???w wmetafile8? ??O??????????? ???????????yyy????.????1????????????? aA!???&??yyyy?????Ay¦y?!†???&? ?MathType??a????u????????????-?????¬@????¬o???¬???? ¬????¬N???¬[email protected]??????wmetafile8? ??O??????????? ???????????yyy????.????1????????????? @`!???&??yyyy?????Ay¦y !ae???&? ?MathType?????u????????????-?????¬@????¬o???¬???? ¬E???¬d???¬[email protected]??????wmetafile8? ??u??????????? ???????????yyy????.????1????????????? `?(???&[email protected](???&??MathType?? ???u????????????-?????¬@????¬¤???¬ wmetafile8? ??(??????????? ???????????yyy????.????1????????????? a????&??yyyss?????Ay?yA’???&? [email protected]????l"???l??? l???lwmetafile8? ??(??????????? ???????????yyy????.????1????????????? a????&??yyyy?????Ay?yA’???&? [email protected]????l"???l??? l???lwmetafile8? ????????????? ???????????yyy????.????1????????????? `????&??yyyy?????Ay3/[email protected] ???&??MathType?? ???u????????????-?????¬@????¬J???¬wmetafile8? ?? ??????????? ???????????yyy????.????1????????????? `????&??yyyy?????Ay3/[email protected] ???&??MathType?? ???u????????????-?????¬@????¬J???¬wmetafile8? ?? Aeine?aeaeaiiy iieacaee, ui aaniethoi? cia/aiiy ne?i/aiieo ??cieoeue iiaeooue c?inoaoe c? ca?eueoaiiyi ?o ii?yaeeo. Ioaea aiaeia?/ii ca?eueooaaoeiooueny ? aaniethoi? cia/aiiy ioe?iie /anoeiieo iio?aeieo, ? ?oi? iioeaee. A ca`yceo c oeei eiinoaoo?oueny ianooiia: i?e iaaeeaeaiiio aeeoa?aioe?thaaii? /e ciaoiaeaeiii? /anoeiieo iio?aeieo ca ne?i/aiieie ??cieoeyie ?aeue?oo caiii? iiaa?oi? ia ne?ae, ii-iiaeeeaino?, aeei?enoiaoaaoe ne?i/aii? ??cieoe? aenieeo ii?yaee?a, ii-ae?oaa, iaae?ei?noue /eneiaiai aeeoa?aioe?thaaiiy ne?ae ioe?ithaaoe ca inoaoi/iith ye?noth oiaii?ioeano, aeey yeiai aiie aeei?enoiaothoueny. Oea e aoei c?iaeaii a aeaiiio ?icae?e?. Aeei?enoiaoth/e /anoeii? iio?aei? ia?oiai ii?yaeeo, aoee aeeiiai? aenia?eiaioaeuei? aeine?aeaeaiiy c aeae?eaiiy no?oeoo?ieo e?i?e ?aeue?oo. Aeey oeueiai aoa acyoee o?aaiaio ?aeue?oo c? no?oeoo?ieie e?i?yie, i?iaaaeaieie ca ?acoeueoaoaie a?coaeueieo niinoa?aaeaiue. Ii oeueiio ae ?aeue?oo aoea noai?aia OeI? ? ii i?e aiae?oe/ii, aeei?enoiaoth/e /anoeii? iio?aei? ia?oiai ii?yaeeo, aecia/ai? e?eoe/i? oi/ee, a iio?i no?oeoo?i? e?i??. Ia ?en.2 ? 3 iieacai? no?oeoo?i? e?i??, ui io?eiai? a?aeiia?aeii a?coaeueii ? aiae?oe/ii. Aiae?c ?aeue?oo ca ae?oaith eaaae?aoe/iith oi?iith iiaa?oi? aeicaieea ca OeI? aeae?eeoe oa?aeoa?i? oi?ie ?aeue?oo. C aeei?enoaiiyi (3) io?eiaii ??aiyiiy ae?oai? eaaae?aoe/ii? oi?ie iiaa?oi? wmetafile8? ??[??????????? ???????????yyy????.????1?????????????  ?!???&??yyyss?????Ay?yA J???&[email protected]?????? Aiaeia?/ii aoee aeeiiai? aeine?aeaeaiiy oeyoii aiae?co iao?eoe?, neeaaeaii? c /anoeiieo iio?aeieo ae?oaiai ii?yaeeo. Aeacaia iao?eoey, ui iaceaa?oueny aanniaiii, ia? aeaeyae wmetafile8? ??$??????????? ???????????yyy????.????1????????????? ?????&??yyyss?????Ay¤[email protected]$???&? ?MathType??`???u????????????-?????e???

???/4???/ ????/h ???/Ue???A4???A????Ah ???AUe [email protected]?????? Aieiai? e?eaecie, aeey yeeo ii?iaeueia e?eaecia aeinyaa? aeno?aiaeueiiai cia/aiiy, aoaeooue aeanieie /eneaie iao?eoei (6). ?o ciaoiaeyoue ?ica’ycoth/e oa?aeoa?enoe/ia ?iaiyiiy wmetafile8? ??ae??????????? ???????????yyy????.????1?????????????   ???&??yyyy?????Ay?y`J???&? ?MathType??0???u????????????-?????V?H????BH????V????? [email protected]??????aea aeani? /enea aeacaii? iao?eoe? Aanna. ?iciiciaaaiiy oi?i caiii? iiaa?oii aeeiiaii oaeiae ?icaeyaeii ciaeia e?eaecie Aaonna, yea ia? iiceoeaiee ciae a aeiioe/ieo oi/eao i iaaaoeaiee ciae a aiia?aiei/ieo oi/eao, a a ia?aaiei/ieo oi/eao i oi/eao nieiuaiiy aeacaia e?eaecia aei?iaith? ioeth. Na?aaeiy e?eaecia iiaea aooe ?ic?aoiaaia ye neiae iao?eoei wmetafile8? ??e??????????? ???????????yyy????.????1?????????????  ???&??yyyy?????Ay?y`E???&? ?MathType??????u????????????-?????¬???¬‰ ???¬v ???¬[email protected]?????? Oe? ae ?acoeueoaoe aoee io?eiai? i?e aiae?c? aeaeyaeo aeioe/iiai ia?aaiei?aea. Ii?iaeueia e?eaecia aeioe/iiai ia?aaiei?aea aei?iaith? ii?iaeueiie e?eaecii iiaa?oii, ui aea/a?oueny a aeaiiio iai?yieo. ?iaiyiiy aeioe/iiai ia?aaiei?aea o aea?ai?e nenoaii eii?aeeiao caieno?oueny wmetafile8? ?? Iiaenoaaeaoe a iueiai iaaeeaeai? cia/aiiy /anoeiieo iioiaeieo, ui io?eiaii ca neii/aiieie ??cieoeyie io?eiaii wmetafile8? ????????????? ???????????yyy????.????1????????????? `a???&??yyyss?????Ay3/4y  ???&? [email protected]?????? Aenia?eiaioaeuei? aeine?aeaeaiiy ia c?aceo ?aeue?oo, acyoiio c iaoeiai? e?oa?aoo?e oa ia ae?iaie/iio iaoa??ae? i?aeoaa?aeeee ?ic?iaeai? oai?aoe/i? iniiae iaoiae?a aeae?eaiiy no?oeoo?ieo e?i?e oa oi?i ?aeue?oo ca iaoa??aeaie OeI?. A?eueo iaae?eii aeae?eythoueny no?oeoo?i? e?i?? ?aeue?oo iiaa?oi?. Ui ae noino?oueny aeae?eaiiy oi?i ?aeue?oo c aeei?enoaiiyi ae?oai? eaaae?aoe/ii? oi?ie oa aann?aia, oi aenia?eiaioaeuei? aeine?aeaeaiiy ia aeine?aeiiio c?aceo ?aeue?oo aeaee aaniethoii oi/i? iiceoeai? ?acoeueoaoe a 77 a?eii, a ia ae?iaie/iio iaoa??ae? - oae? ae ?acoeueoaoe io?eiai? o?eueee a 52 a?eii. Iiaeia a?aei?oeoe, ui iaoiaeeea io?eiaiiy no?oeoo?ieo e?i?e aoea ia?aa??aia ca?oa?aeiei aaoi?ii, i?i ui aoei iia?aeiieaii ia O??? Eiia?an? IOOAeC, a a?aeiia?aeia noaooy aaoi?a aeaii? aeana?oaoe?? aoea ia?aiioae?eiaaia a oaoiaiio aeo?iae? a NOA. A o?aoueiio ?icae?e? ?iaioe i?eaaaeai? ?acoeueoaoe aeine?aeaeaiue c aeei?enoaiiy ei?aeyoe?eii-niaeo?aeueiiai aiae?co aeey eeaneo?eaoe?? ?aeue?oo i?noeaaino? a caeaaeiino? a?ae nooiaith ia?an?/aiino?. Aeacaia caaea/a eeaneo?eaoe?? a i?aeoe/i?e ae?iaie/?e ae?yeueiino? aeeiio?oueny a cia/i?e i??? noa’?eoeaieie i??eoaaiiyie. Aeey i?aeaeuaiiy ia’?eoeaiino? ?ica’yceo aeine?aeieeaie i?iiiioaaeinue aeei?enoiaoaaoe niaoe?aeueio e?eao ia?an?/aiino?, aaoiei?aeyoe?eio ooieoe?th oa niaeo?aeuei? oa?aeoa?enoeee. I?e oeueiio iiaeooue aeei?enoiaoaaoenue iaeiiaei??i? oa aeaiaei??i? ooieoe??. Cia/i? ia?aaaae iiaea iaoe aeei?enoaiiy aeaiaei??ieo niaeo?aeueieo oa?aeoa?enoee, ine?eueee aiie iiaeooue aooe io?eiai? ia o?eueee oeeo?iaeie iaoiaeaie, a e aiaeiaiaeie c aeei?enoaiiyi eiaa?aioii? iioeee. Aenia?eiaioaeuei? aeine?aeaeaiiy aeeiioaaeenue ia ae?iaie/iio iaoa??ae?, ia c?aceao ?aeue?oo, acyoeo c ii?i ae?ia?oeo aeey oiiia?ao?a ?en.4, oa ia o?aaiaio? ?aeue?oo oeeo?iai? ea?oe. ?acoeueoaoe ia/eneaiiy e?eaeo ia?an?/aiino?, ui io?eiai? ca oi?ioeith wmetafile8? ??v??????????? ???????????yyy????.????1?????????????  ???&??yyyy?????Ay1/2y`Y???&? ?MathType?????u????????????-??????e????O???‚ae ???–ae???‚g???–[email protected]??????aeicaieeee iiaoaeoaaoe a?ao?ee (?en.5). Oai aea i?eaaaeai? ?acoeueoaoe ai?ieneiaoe?? aeacaieo e?eaeo ooieoe??th iaoo?aeueiiai eiaa?eoia c ioe?ieith oi/iino?. Ca ?aceueoaoaie ia/eneaieo ii?iiaaieo aeaiaei??ieo niaeo?aeueieo oa?aeoa?enoee c?ace?a ?aeue?oo iiaoaeiaai? cia?aaeaiiy, ye? i?eaaaeai? ia ?en.6. Ia/eneaiiy niaeo??a aaeinue ca oi?ioeith wmetafile8? ????????????? ???????????yyy????.????1????????????? @$???&??yyyss?????Ay?y?$I???&? ?MathType??????u????????????-?????«c???«“ ???«v???«o???«a???«"[email protected]?????? N1=N2=N, ?ici??e ae?eyiee a /ania?e iaeano? ? a?aeiia?aeiee ii?ioth/ee iiiaeiee. fx,fy -i?inoi?iai /anoioe. Iaeiiaei??i? niaeo?aeuei? oa?aeoa?enoeee, ui ?ic?aoiaai? ca ianooiiith oi?ioeith wmetafile8? ??s??????????? ???????????yyy????.????1????????????? ????&??yyyy?????Ay1/2yAY???&[email protected]?????? i?eaaaeai? a oaaeeoe? 1. Aeei?enoiaoth/e oai?ith aeai?ee, ?aeue?o iinoeaainoi iiaeia aaciiieeeiai aiaeia?aceoe a oiio aeiaaeeo, yeui iieaoii oi/ee aoaeooue ?iciiuoaaoenue /a?ac iioa?aaee wmetafile8? ??I??????????? ???????????yyy????.????1????????????? a`???&??yyyy?????Ay3/4y ????&? ?MathType?????u????????????-?????LI???LI???La??? L][email protected]?????? aea x0, y0 -ooeaii iioeiaeueii iioa?aaee acaeiaae inae o i o, fx0, fy0 - iaeaeui /anoioe, ui aeanoeai niaeo?o aeaii? iiaa?oii. A ?iaio? i?iiiio?oueny iaoiaeeea aecia/aiiy iioeiaeueiiai ?ioa?aaea ca oi?ioeith, i?eaaaeaiith i?ae oaaeeoeath 1. Ciaeoiaoe a?aie/ia cia/aiiy niaeo?aeueii? u?eueiino? ca oe??th oi?ioeith, iiaeia aecia/eoe ia o?eueee iioeiaeueiee ?ioa?aae i?ae i?eaoieie oi/eaie, a e ai?cio?ii?th ?aeue?oo. Ai?cio?ii?y ?aeue?oo iiaa?oi? aoaea i?iyaeyoenue ??cieie iioeiaeueieie ?ioa?aaeaie, io?eiaieie aeey ??cieo iai?yie?a. A?aie/ia cia/aiiy niaeo?a S0, io?eiaia ca i?eaaaeaiith oi?ioeith, aeicaiey? aecia/eoe a?aie/io /anoioo f0 ? ca oeei cia/aiiyi ciaoiaeyoue iioeiaeueiee ?ioa?aae. Ana neacaia iiaea aiaeiineoenue ia oieueee aei ?aeue?oo caiii? iiaa?oii, a e aei ?aeue?oo aoaeue-yeeo ia’?eoia a oiio /enei i iie?ineiii/ieo. A oe?e aea oaaeeoe? i?eaaaeai? cia/aiiy iioeiaeueieo ?ioa?aae?a aeey ae?iaie/eo ia’?eo?a *i?iiai?a oa Ooi?iie. Iiaeia a?aecia/eoe, ui aeey c?acea nueiii? eaoaai??? neeaaeiino? acaeiaae aa?oeeaeueii? in? cia/aiiy niaeo?aeueii? u?eueiino? ia aeinyaathoue aeiionoeiiai S0 a ?ioa?aae? /anoio a?ae 0 aei . A oeueiio aeiaaeeo ?aeiiaiaeo?oueny caouaoe i?eaoi? oi/ee, oiaoi ciaiooaaoe ?ici??e ee?oeie ?aaoey?ii? n?oee. C?acie ?aeue?oo o?aouei? eaoaai??? neeaaeiino?, ia yeiio cia?aaeaii caoyaeiee noee, ia aaaaea?oueny neeaaeiei aeey ?enoaaiiy ?aeue?oo. Iaeiae niaeo? oaeiai c?acea, aaoiei?aeyoe?eia ooieoe?y, oa e?eaa ia?an?/aiino? na?ae/aoue i?i cia/io ?ic/eaiiaai?noue ?aeue?oo, yei? a ae?eniino? iaia?. Aeey oiai uia iicaooenue aeacaiiai iino?eiiai o?aiaeo, aeeiioaaany iaoee c?acea. Oeae i?ioean cae?enithaaany ca a?aeiieie c oioia?aiiao??? oi?ioeaie. A?i ? ??aiicia/iei ia?aoiaeo aei ?ioi? nenoaie eii?aeeiao, iaoeeaii? ia eoo , ine?eueee iaoee cae?enithaaany acaeiaae in? aanoeen. Oiio a aenia?eiaioaeueieo aeine?aeaeaiiyo aeei?enoiaoaaeenue i’youe c?ace?a, c?acie o?aouei? eaoaai??? neeaaeiino? oa oeae aea c?acie c onoiooei o?aiaeii, oiaoi iaoeeaiee. Ia?ao?ae aei nenoaie IX’Y’Z’ a?ae nenoaie OXYZ cae?enithaaany iiai?ioii wmetafile8? ??I??????????? ???????????yyy????.????1?????????????  ???&??yyyy?????Ay?ya J???&[email protected]?????? aea eiiiiiaioaie iao?eoe? A ? iai?yii? eineione, ui caeaaeaoue a?ae eoo?a iaoeeo c?acea ?aeue?oo a?aeiinii aea?aii? nenoaie eii?aeeiao. Aeei?enoiaoth/e iniiai? aeanoeaino? niaeo?aeueieo ia?aoai?aiue, cai?iiiiiaaii cae?enieoe caeaaeaeoaaiiy, aai aaia?ae?caoe?th ?aeue?oo aiae?oe/iei iaoiaeii. Aiii aeeiio?oueny /eneiaeie iaoiaeaie i?inoi?iaith oieueo?aoei?th /anoio a niaeo?i ca?aeii oi?ioee cai?oee wmetafile8? ??W??????????? ???????????yyy????.????1?????????????   ???&??yyyy?????Ay?y` T???&[email protected]?????? aea y - aeaiaeii?ia aeiaaeeiaa ooieoeiy ia aeoiaei; x - aiaeiai/ia ooieoeiy ia aoiaei; h - iiioeueniee aiaeaoe eiiieii? nenoaie. Aeacaio iia?aoeith iiaea aeeiiaoe c a?aie/iith i?inoioith eiaa?aioia iioe/ia nenoaia, oiaoi aeaiaeii?ia ia?aoai?aiiy Oo?’? iiaea aeeiiaoe cai?ia eiica. I?e oeueiio inaioeaiiy iiaeiii aooe iiiio?iiaoe/iei, a ?iciiaeie aiieiooaee naioea ?icaeyaea?oueny a caaeiie oieaeueiie ieiueii eiice. Aeey oeueiai ?aeue?o iiaa?oi? iiaeaethaaany o aeaeyae? cia?aaeaiiy c? ci?iith iioe/ii? u?eueiino?. Cai?iiiiiaaiee niinia iiaeaethaaiiy ?aeue?oo aiaeiai/iee "aiaeieaoei", aea aieueo oi/iee oa i?iaeoeoeaiee. Ia ?en.9 iieacaii cia?aaeaiiy ?aeue?oo o?aaoa *i?iiai?a, io?eiaia ca aeaiith iaoiaeeeith. Aeei?enoaiiy ei?aeyoe?eii-niaeo?aeueiiai aiae?co aeicaiey? ioe?ieoe ? iiaeaoe e?euee?nio oa?aea?enoeeo /anoioiiai neeaaeo ?aeue?oo iiaa?oi?. Noa? iiaeeeaei e?euee?nii ioe?ieoe ai?cio?ii?th ?aeue?oo aoaeue-yei? iiaa?oi?, iiaeaii? a oeeo?iaiio aeaeyae?. E?euee?ni? oa?aeoa?enoeee, io?eiai? a ?acoeueoao? ai?ieneioaiiy ia/eneaieo ei?aeyoe?eii-niae?aeueieo ooieoe?e, aeicaieythoue ciaeoe iaae?ei? e?eoa??? aeey ?aeiioaaiiy caiii? iiaa?oi? a caeaaeiino? a?ae nooiaith ia?an?/aiino? ?aeue?oo. *aoaa?oee ?icae?e i?ena’y/aiee aeei?enoaiith oai??? ?ici?ciaaaiiy ia?ac?a aeey aeae?eaiiy, eeaneo?eaoe?? oa eeanoa?ecaoe?? oi?i ?aeue?oo. Nenoaia aaoiiaoe/iiai ?iciiciaaaiiy iieacaia ia ?en.7. Ni?iuaii aiia aoaea neeaaeaoenue c aeaio iniiaieo iiaenenoai: aeao/eea i eeaneoieaoi?a. Aeao/ee ? iiaenenoaiith, ui ia?aoai?th? oice/ii oa?aeoa?enoeee ia’?eoa, yeee iiaeeyaa? ?iciiciaaaiith, a iaai? iciae o aeaeyaei aaeoi?a x=(x1,x2,...xn)’. (17) Oo?eo ooo icia/a? o?ainiiioaaiiy. A ?iaio? aeieeaaeii ?icaeyiooa ieoaiiy aeai?o iciae aeey ae?eyiie ?aeue?oo iiaa?oi?. Eeaneoieaoi? ? i?eno?i?i /e i?ia?aiith, yea aiaeiineoue eiaeai ia?ac /e iaai? iciae, ye? iinooiathoue ia eiai aoiae, aei iaeiiai c eeania. No?oeoo?a aeaieo, ui aeei?enoiaothoueny i?e ?ica’yceo caaea/?, ia? ianooiiee aeaeyae. wmetafile8? ????????????? ???????????yyy????.????1?????????????   %???&??yyyy?????Ay«y`%K???&[email protected]?????? wmetafile8? ??e??????????? ???????????yyy????.????1????????????? A????&??yyyy?????Ay?yAj???&??MathType??`[email protected]??????N- oa’?eoe, a?eia, eeane. wmetafile8? ??“??????????? ???????????yyy????.????1?????????????  `???&??yyyy?????Ay3/4y ^???&[email protected]?????? Yj- aaeoi? iaoee?a c oeaio?a ae?eyiee-a?eia a iaaiiio, iai?eeeaae, a i?aaeaiiiio iai?yieo. wmetafile8? ??a??????????? ???????????yyy????.????1????????????? [email protected]???&??yyyy?????Ay?y?????&[email protected]?????? - ie?aiee ia?ac, aaeoi? iciae. Ia?ace, ui a?aeiinyoueny aei aeaii? caaea/?, oiaoi ?aeue?o iaaieo ae?eyiie, aacothoueny ia aeei?enoai? a?iioace i?i iiaeeea?noue iiaeaiiy ia?aca o aeaeyae? aaeoi?a iciae. Iciaeaie a?aeenue iaoeee c oeaio?a ia?aca ia e?aei? oi/ee (?en.8). Aiaenoaiue iiae ia’?eoaie iiaea aooe iiaeaia ?icieie niiniaaie. Ia?oa aecia/aiiy aeaii? aiaenoaii, ia iao iiaeyae, iaei?inoioa i iaeiioe?aiioa, oea aaeeiaeiaa aiaenoaiue, ui aecia/a?oueny ca oi?ioeith wmetafile8? ??h??????????? ???????????yyy????.????1????????????? ???&??yyyy?????Ay1/2yaY???&? ?MathType?????u????????????-?????‚B???–B???u???????? ????-??????????vH???-?????”? ???„ [email protected]??????Xi,Xj - aaeoi?e, iiae yeeie ioeiith?oueny aiaenoaiue; xik- neeaaeiaa aaeoi?a Xi. Ae?oaa aecia/aiiy aeaii? aiaenoaii, aeey yei? ?iiae? aaeeaa?oueny oa?iii "iao?eea iinueeeo eaa?oaeia", caieno?oueny oi?ioeith wmetafile8? ??!??????????? ???????????yyy????.????1????????????? @???&??yyyy?????Ay1/2y?Y???&? ?MathType?????u????????????-?????‚o???–o???/?O?aoy /aaeoaanueea aiaenoaiue ?ic?aoiao?oueny ca oi?ioeith wmetafile8? ??U??????????? ???????????yyy????.????1????????????? ``???&??yyyy?????Ay·y ???&? ?MathType??°????u????????????-?????I?i???Ii???I?y??? [email protected]??????Aiaeiia oaeiae aiaenoaiue Eaiaa??a, ui aecia/a?oueny ca oi?ioeith wmetafile8? ??m??????????? ???????????yyy????.????1????????????? ? ???&??yyyy?????Ay1/[email protected] Y???&? ?MathType?????u????????????-?????‚o???–o???Iw??? Iw???I ???I ???) A aeaiee /an ia ?nio? ?aeeii? oai???, yea a aeicaieeea aecia/eoe, ye? iaoiaee ?ici?ciaaaiiy ne?ae aeei?enoiaoaaoe aei oeo /e ?ioeo ia?ac?a. Aaaaea?oueny, ui nenoaia ?iciiciaaaiiy R1 e?aua iiae R2, yeui iiiai?iinoue iiieeeoenue aeey nenoaie R1 iaioa iiae aeey nenoaie R2. sseui caaeaii M eeania ia?acia w1,w2,...wm . n - aeao/eeia aeaeathoue iioi?iaoeith i?i eiaeai ia’?eo o aeaeyaei aaeoi?ia x=(x1,x2,...,xn), I?e oeueiio eiaeai aeao/ee aeaea? iaaio oa?aeoa?enoeeo. Iioeiaeueiith aoaea oaea d0(o), yea aea? iaeiaioo iiiai?iinoue iiieeee aeey anio o. Aiaeic cia/aiue ooieoeie ?ica’yceo, i?eaaaeaieo a aeiaeaoeo aeena?oaoe??, iieaco? aenieo ?icaeieueio caeaoiinoue acyoiai aaeoi?a iciae i iaaeieiinoue ia/enethaaeueii? i?ioeaaeo?e. Ia oieueee ani i?aaenoaaeaii c?acee eeaneoieoaaeenue i?aaeeueii, aea e cia/aiiy ooieoei? ?ica’yceo aeey ia’?eoia, ui eaeooue aaciina?aaeiuei ca iaeneiaeueiei, oaae oa?aeoa?ecothoue aicoaeueio iiaeiaiinoue ia’?eoia, ui ?iciiciathoueny. Iiiiiaeueia cia/aiiy ooieoeie ?ica’yceo, oiaoi iaeneiaeueia cia/aiiy a?aenoai? oa?aeoa?eco? i?ioeeaaeiinoue ia?aco, ui ?iciicia?oueny, aei aiaeiiaiaeiiai aoaeiio. Iai?eeeaae, aeey iaai?aa i?ioeeaaeiith aoaea yia, aeey o?aaoa - oeiaiaeia i o. ae. Aeeiiaii aenia?eiaioaeueii aeineiaeaeaiiy ii ?iciiciaaaiith oa eeaneoieaoei? aeieyiie ?aeue?oo iinoeaainoi iiaeoaa?aeeee ye i?aaeeueiinoue nenoaie aeao/eea, oiaoi aea?aiiai aaeoi?a iciae, oae i nenoaie eeaneoieaoi?a, oiaoi aea?aii? ooieoei? ?ica’yceo oa iaoiaeeee ia/eneaiue. Iaeiae i?e ?o aeeiiaiii aeieeee aeiaeaoeiai caaea/i, iai?eeeaae, caaea/a aeai?o ii/aoeiai? oe?eie aieia. Ia iao iiaeyae ?? iiaeia ?ica’ycaoe iaoiaeii ei?aeyoeieiiai aai niaeo?aeueiiai aiaeico. Aea?aaoe ii/aoeiao oe?eio aieia o aiaeiiaiaeiinoi c iaeaeueie /anoioaie, yei aeanoeai aeaiie iiaa?oii, iioii, i?e iaiaoiaeiinoi, /anoioa ciaioo?oueny, ui aiaeiiaiaea? caieueoaiith ?icii?ia aieia. ?icii?iinoue aaeoi?a iciae oaae ? neeaaeiith caaea/ath, oiio ui, c iaeii?? noi?iie, yeui aieueoa iciae aeei?enoiao?oueny, oi iaaeieiioa iiaeiii a aanoenue ?iciiciaaaiiy, c iioiai aieo aaeeea eieueeinoue iciae i?ecaiaeeoue aei iaiaa?oioiaaieo aeiiiii/ieo cao?ao, a a aeayeeo aeiaaeeao i aei ciaioaiiy iaaeieiinoi ?iciiciaaaiiy. Oeth caaea/o, iaaooue, neiae ?ica’ycoaaoe ioa?aoeaiei iaoiaeii, caieueooth/e iinooiiai eieueeinoue iciae i eiio?iethth/e i?e oeueiio yeinoue ?iciiciaaaiiy. Iaeei ic aa?iaioia ?ica’yceo aeacaii? caaea/i i?eaiaeeoueny a aeena?oaoe?ei?e ?iaio?. A oeueiio ae ?icae?e? i?eaiaeeoueny oaeiae iaoiaeeea eeanoa?ecaoe?? oi?i ?aeue?oo iiaa?oi? ii noai? eeaneo?eaoe?? "iaa/aiiy aac a/eoaey", oiaoi o aeiaaeeo, eiee eeane aei ii/aoeo eeaneo?eaoe?? iaa?aeii?. ?ioeie neiaaie, a oeueiio aeiaaeeo i?iaiaeeoueny iaeiio?ieueiaaia ?ici?ciaaaiiy. Aianeiaeie oiai, ui oi?ie ?aeue?oo iiaeooue aooe ii-?iciiio i?i?ioiaaii iaeia aiaeiinii iaeii? i, ioaea, aiaeiinii inae aaiaeace/ieo eii?aeeiaao, neiae aeey eiaeiiai ii?iaiyiiy ia/enethaaoe aaeeiaeiai aiaenoaii iiae aiaeiiaiaeieie ia?acaie noieueee ?ac, neieueee aeaiaioia o aaeoi?i iciae, ui oa?aeoa?eco? aeaii ia?ace. I?e oeueiio aaeoi? iciae aoaeiia caeeoa?oueny ia?ooiiei, a aaeoi? iciae ia?aco, yeee ?iciicia?oueny, iiaa?oa?oueny iiaeaiaioii ia aaee/eio aiaenoaii iiae noniaeiiie aeaiaioaie aaeoi?a o, oiaoi iciaeaie. I?aeoe/ii oeae i?ioean iieyaa? a ianooiiiio. Ii oeeo?iaie iiaeaei, i?aaenoaaeaiie aenioaie a ia?ao?anoyo nioee eaaae?aoia /e i?yiieooieeia, noai?ththoue aaeoi?e iciae x wmetafile8? ??P????????????? ???????????yyy????.????1????????????? `@???&??yyyy?????Ay3/4y? ???&??MathType??`????&??yyyy???????????????????? `@???&??yy?????yyyyyyyyyyyyyyyyyyyyyyyyyyywmetafile8? ??5??????????? ???????????yyy????.????1?????? ???????  ???&??yyyy?????Ay«yaK???&? [email protected]?????? aea m - iciaee; n - ia’?eoe, oiaoi cia?aaeaiiy ?aeue?oo iinoeaainoi, ui ?iciicia?oueny a iaaiiio aieii. A iaoie caaea/i ia’?eoe, ui eeaneoieothoueny, iiaeai? iao?eoeath B(i,j). Oiio caiinoue iao?eoei x, noai?th?oueny o?ueioaeii?iee ianea P(i,j,k). A iueiio i,j - ieinei eii?aeeiaoe oeaio?a aieia, a yeiio ?iciiuaiee ia?ac, ui ?iciicia?oueny, k - aaeoi? iciae noai?thaaiiai ia?aco. ?acoeueoaoe aeine?aeaeaiue i?aeoaa?aeeee aenieo ?icae?eueio caeaoi?noue ia?oeo o?ueio ooieoe?e ?ica’yceo. Ii??aiyii a?eueoa a?aenoaiue i?ae i?ioeeaaeieie ia?acaie i?e aeei?enoaii? ooieoe?? d1. Iaea??oa ?icae?eythoueny ia’?eoe c aeiiiiiaith ooieoe?? d4. Iaeiae c aeiiiiiaith aeaii? ooieoe?? iaee?aua ?ici?ciathoueny ia?ace, ye? iathoue oi?io i?ioeeaaeio aei aoaeiia. A oe?eiio aea?? o??? /e ?ioi? ooieoe?? aeey ?ica’yceo caaea/? ?ici?ciaaaiiy caeaaeeoue a?ae iniaeeainoae ia?ac?a oa a?aeiia?aeieo aoaeii?a. I?iiiio?oueny oaeiae ia iaiaaeoaaoenue aeei?enoaiiyi iaei??? ooieoe?? ?ica’yceo, a aeei?enoiaoaaoe iaeii/anii aeae?eueea c ieo. Aeei?enoiaoth/e iniiaii iieiaeaiiy /aoaa?oiai ?icae?eo, aoea cai?iiiiiaaia iaoiaeeea aecia/aiiy iinoeaiieiaeaiiy ii?nueeiai noaeia /e aaciieioiiai eioath/iai aia?aoo. Iaoiaeeea iiaea aeei?enoiaoaaoenue a oiio aeiaaeeo, eiee neeaaeii aeei?enoaoe nenoaio GPS. Nooue aeaii? iaoiaeeee iieyaa? a ianooiiiio. Aecia/eaoe aeeaeio iiae ii?nueeei noaeiii /e aenioo iieueioo aeey iiaio?yiiai aia?aoo, aecia/athoue iaeii/anii aeaeieueea aiaenoaiae aiae ei?aaey aei iiaa?oii Caiei. Oei aiaenoaii iiaeiii aooe iaoeeaii aei i?yiiaenii? eiii? iiae oieniaaieie, iaia?aae caaeaieie eooaie . Io?eiaii aiaenoaii Di aeei?enoiaothoueny aeey noai?aiiy eiiiiiaio aaeoi?a iciae ia?aco aeieyiee ?aeue?oo, iaae yeith ciaoiaeeoueny oa /e iioa noaeii. Oaei ae ia?ace aeey anio iiaeeeaeo cii iinoeaainoi, iaae yeeie iiaea ciaoiaeeoenue noaeii, iiaeiii aooe a iai’yoi i?ioeani?a. Ioaea ?iciiciaaaiiy iinoey ciaoiaeaeaiiy eioath/iai aia?aoo caeienith?oueny oeyoii ii?iaiyiiy ia?aco, ?ic?aoiaaiiai ca iciaeaie io?eiaieie a ?acoeueoaoi i?iaaaeaieo aeii?ia, c ia?acaie, ui ciaoiaeyoueny a iai’yoi i?ioeani?a. A oeueiio ae ?icae?e? i?eaaaeai? ?acoeueoaoe canoinoaaiiy iaoiae?a aiae?co ?aeue?oo caiii? iiaa?oi? i?e aea/aii? iaaacia/ieo a?ae?iiaoai?ieia?/ieo yaeu a Oe?a?inueeeo Ea?iaoao. ?aeue?o caiii? iiaa?oi? /anoi ia? ae??oaeueia cia/aiiy i?e aea/aii? aaaaoueio i?e?iaeieo yaeu oa i?ioean?a. Oiio, aea/ath/e iaaacia/i? a?ae?iiaoai?ieia?/i? yaeua ia oa?eoi??? iaoi? aea?aeaae, ne?ae a ia?oo /a?ao coieieoenue ia i?ia?ao?/iiio aieea? Ea?iao. A oeueiio ?icae?e? i?eaaaeaia oa?aeoa?enoeea aieeao a?? ia aeieeiaiiy iaaacia/ieo ceea, nae?a, ni?aiaeo eaaei, ?aii?o in?ii?o oa i?cieo aaniyieo caii?ice?a ? o. ?i. ? cai?iiiiiaaia iaoiaeeea aecia/aiiy a??nueeeo noee?a c iaaiith aeniiceoe??th aeey i?iaiico ni?aiaeo eaaei. ?ici?ciaaaiiy oa eeaneo?eaoe?y aeeiioaaeenue ia iniia? A?N- oaoiieia?e. Ca oeeo?iaith ea?oith ianooaao 1:100000 c ?aeue?oii, caaeaiei oeeo?iaith iiaeaeeth, noai?thaaeanue ?aaoey?ia n?oea. Oiaoi ?aeue?o aoa iiaeaiee aenioaie a ia?ao?anoyo aeaii? n?oee. Aoaeiiiei ia?acii aoei aea?aii caoyaeiee noee, i???ioiaaiee a i?aaeaiii-no?aeiiio iai?yieo. Ii aeacai?e iaoiaeeoe? ia oa?eoi??? a??nueeiai ianeao *i?iiai?a, yeee aaaaea?oueny iaeiei c iaea?eueo eaaeiiiaaacia/ieo ?aeii?a Oe?a?inueeeo Ea?iao, aoee ?ici?ciai?, oiaoi eeaneo?eiaai? ae?eyiee, iiae?ai? aei acyoiai aoaeiio a caeaaeiino? a?ae acyoi? aaee/ie ooieoe?? ?ica’yceo. Ia ?en.10 iieacaii cia?aaeaiiy ?aeue?oo c iicia/aieie i?aaeaiii-no?aeieie noeeaie e?ooeciith 300 aeey a?eii ?ici??ii 400o400 i. sseui ia aeai? noeee iaianoe c A?N oa? e?n?a, oi i?e a?aenooiino? e?no, aeacai? ae?eyiee aoaeooue eaaeiiiaaacia/ieie. Aeey i?aeaeuaiiy aeieia?/ii? aaciaee ia aeacaieo ae?eyieao iiaeiai aooe ianaaeaeaiee e?n oaa?aeeo ii??ae. AENIIAEE A aeena?oaoe?? ia iniia? iioae?eiaaieo ?ia?o oa aaoi?nueeeo na?aeioeoa ia aeiaoiaee cai?iiiiiaaii ? ?icaeiooi iiaee iaoeiaee iai?yi a aeine?aeaeaii? ?aeue?oo caiii? iiaa?oi?, ui aaco?oueny ia aeei?enoaii? iniiaieo iieiaeaiue oai??? ia?iaee cia?aaeaiue. I?aeoe/iee ?icaeoie oeeo aeine?aeaeaiue iai?aaeaiee ia ?ica’ycaiiy aaaeeeai? ia?iaeii-ainiiaea?nueei? i?iaeaie, iia’ycaii? c? noai?aiiyi A?N ? ia ?? iniia? ae??oaiiy ?yaeo i?eeeaaeieo caaea/ a oiiia?ao??, aaiii?oieia??, a?ae?iiaoai?ieia??. Ca ?acoeueoaoaie aeeiiaieo oai?aoe/ieo oa aenia?eiaioaeueieo aeine?aeaeaiue iiaeia noi?ioethaaoe ianooii? aeniiaee. 1. ?ic?iaeai? oai?aoe/i? iniiae aeine?aeaeaiiy ?aeue?oo i?noeaaino? ca oeeo?iaith iiaeaeeth c aeei?enoaiiyi ne?i/aiieo ??cieoeue oa ?ioeo cania?a iaoaiaoe/iiai aia?aoo aeeoa?aioe?aeueii? aaiiao??? aeicaieeee ae??oeoe i?iaeaio aaoiiaoe/ii? iiaoaeiae no?oeoo?iiai ?iaa??aioo caiii? iiaa?oi? oa aeae?eaiiy ie?aieo oi?i ?aeue?oo: - c aeei?enoaiiyi ne?i/aiieo ??cieoeue ia?oiai ii?yaeeo ?ic?iaeaii iaoiaeeeo aeae?eaiiy no?oeoo?ieo e?i?e ?aeue?oo; - c aeei?enoaiiyi ce?i/aiieo ??cieoeue ae?oaiai ? aeueo ii?yaee?a ?ic?iaeaii iaoiaeeeo aeae?eaiiy oi?i ?aeue?oo caiii? iiaa?oi?; - ca ?acoeueoaoaie oai?aoe/ieo oa aenia?eiaioaeueieo aeine?aeaeaiue c?iaeai? aeniiaee uiaei ioe?iee oi/iino? io?eiaieo /anoeiieo iio?aeieo; - c?iaeaii ?aeiiaiaeaoe?? uiaei aeei?enoaiiy ne?i/aiieo ??cieoeue iaaiiai ii?yaeeo aeey ioe?iee /anoeiieo iio?aeieo ?aeue?oo caiii? iiaa?oi?. 2. Aenia?eiaioaeuei? aeine?aeaeaiiy ?aeue?oo c aeei?enoaiiyi ??cieo ei?aeyoe?eii-niaeo?aeueieo ooieoe?e aeicaieeee ioe?ieoe ?o i?eaeaoi?noue aeey ioe?iee nooiaiy eiai ?ic/eaiiaaiino? /e ia?an?/aiino? oa ai?cio?ii??, a oaeiae ?ic?iaeoe iaoiaeeeo iiaeaiiy /e iiaeaethaaiiy ?aeue?oo oa ?ic?iaeoe niin?a eiai caeaaeaeoaaiiy: - c?iaeai? ?aeiiaiaeaoe?? aeey aeei?enoaiiy iaaieo ooieoe?e aeey ioe?iee /anoioiiai neeaaeo /e ?ic/eaiiaaiino? iiaa?oi?, ai?cio?ii??, ia?aiaaeo aenio; - a ?acoeueoao? aeine?aeaeaiiy ?aeue?oo ei?aeyoe?eii-niaeo?aeueieie iaoiaeaie cai?iiiiiaaii niin?a iiaeaethaaiiy ?aeue?oo ci?iith iioe/ii? u?eueiino?, ui aeicaiey? aeeiioaaoe o?ainoi?iaoe?th Oo?’? ?aeue?oo iaaii? ae?eyiee c aeiiiiiaith e?ice, oiaoi aiaeiaiaei iaoiaeii; - ?ic?iaeaiee niin?a iiaeaethaaiiy ?aeue?oo ci?iith iioe/ii? u?eueiino? iineoaeea iniiaith aeey noai?aiiy aeaio i?eeaae?a, ia ye?, ye ? ia aeacaiee niin?a, aoee io?eiai? aaoi?nuee? na?aeioeoaa; - ?ic?iaeaia iaoiaeeea aeeo/aiiy iino?eiiai o?aiaeo i?e aeine?aeaeaii? nooiaiy ia?an?/aiino? ?aeue?oo iiaa?oi?; - aeeiiai? aenia?eiaioe c aaia?ae?caoe?? ?aeue?oo o?eueo?aoe??th iaoiaeii cai?oee ia aeaee iaae?eieo ?acoeueoao?a ? iio?aaothoue iiaeaeueoiai aea/aiiy. 3. C aeei?enoaiiyi iniiaieo iieiaeaiue oai??? ?ici?ciaaaiiy ia?ac?a ?ic?iaeaii iaoiaeeeo eeaneo?eaoe?? oa eeanoa?ecaoe?? ae?eyiie caiii? iiaa?oi? caeaaeii a?ae oi?ie ?o ?aeue?oo: - aeeiiai? aenia?eiaioaeuei? aeine?aeaeaiiy c aeei?enoaiiy /ioe?ueio ??cieo ooieoe?e ?ica’yceo aeicaieeee aecia/eoe iniaeeaino? ?o canoinoaaiiy i?e ?ici?ciaaaii? ae?eyiie caiii? iiaa?oi?; - i?iaaaeai? aenia?eiaioaeuei? aeine?aeaeaiiy aeey aecia/aiiy iioeiaeueii? ?ici??iino? aaeoi?a iciae oa c?iaeai? a?aeiia?aei? ?aeiiaiaeaoe??; - ?ic?iaeaii niin?a iaa?oaiiy ia?aca iaaeiei aoaeiia aei iaee?auiai ?o ni?aiaaeaiiy i?e ?ici?ciaaaii?, eeanoa?ecaoe?? /e eeaneo?eaoe??; - ia iniia? ?ic?iaeaii? iaoiaeeee cai?iiiiiaaii niin?a aecia/aiiy i?noeaiieiaeaiiy e?oath/iai aia?aoa /e ii?nueeiai noaeia ca ?aeue?oii caiii? iiaa?oi? /e ii?nueeiai aeia, a?aeiia?aeii; - cai?iiiiiaaii niin?a aecia/aiiy ae?eyiie a??nueeeo noee?a c iaaiith aeniiceoe??th, ia yeeo i?e iaaieo ianoaaeiao iiaeeeaa noiaeaeaiiy ni?aiaeo eaaei; - c aeei?enoaiiyi ?ic?iaeaii? iaoiaeeee cae?eniaii ?ici?ciaaaiiy oa aeae?eaiiy a?aeiia?aeii i???ioiaaieo noee?a ia a??nueeiio ianea? *i?iiai?a; - ia cai?iiiiiaaiee niin?a aecia/aiiy iieiaeaiiy e?oath/iai aia?aoa /e ii?nueeiai noaeia ca ?aeue?oii caiii? iiaa?oi? io?eiaii Iaoaio Oe?a?ie. Iniiaiee ci?no aeena?oaoe?eii? ?iaioe aena?oeaii a ianooiieo ioae?eaoe?yo 1. Ao?oouineay O.A., ?oaeue ?.I. E aiaeeco iia?aoiinoae ia?o?ooiie i?ino?ainoaaiiie oioio?eaiaoeyoeee. // Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1981.- ?33.- N.111-114. 2. E?aaaoe ss.N., Ieeeithe ?.A., ?oaeue ?.I. Aaiaeace/aneia iaania/aiea ecuneaiee e no?ieoaeuenoaa iaaeno?aeueiuo aacii?iaiaeia a ai?iuo oneiaeyo. yeni?an eioi?iaoeey. Aaciaay i?iiuoeaiiinoue. // Na?ey: O?ainii?o ia?a?aaioea e eniieueciaaiea aaca a ia?iaeiii oicyenoaa. - 1985. - N. 19-20. 3. ?oaeue ?.I. I ?an/aoa iaiaoiaeeiiai eiee/anoaa oi/ae eniuoaoaeueiiai iieeaiia aeey eaeea?iaee ay?ioioiaiia?aoa. // Eca. aocia. Aaiaeacey e ay?ioioinuaiea. - 1980. - ?3 - N. 107 - 111. 4. ?oaeue ?.I. *anoioiue aiaeec iia?aoiinoae ay?inieiea. Aeai. a AEIEOE 30 ethiy 1980 a. ? 2758-80. 5. ?oaeue ?.I., Oea/aiei TH.O., Aoiaiiay I.I. E aiaeeco ei??aeyoeeiiiie iao?eoeu i?ino?ainoaaiiuo eii?aeeiao oi/ae noa?aiiiaeaee. Aeai. a AEIEOE 30 ethiy 1980 a. ? 2757-80. 6. ?oaeue ?.I. Auai? eioa?aaea iaaeaeo eiio?ieueiuie oi/eaie i?e enneaaeiaaiee noa?aieiiia?aoi?a. // Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1980. - ?31. - N. 148-151. 7. ?oaeue ?.I. Auaeaeaiea no?oeoo?iuo eeiee ?aeueaoa aiaeeoe/aneei iaoiaeii. // Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1984. - ?39. - N. 135-140. 8. ?oaeue ?.I. E aiaeeco ?aeueaoa ianoiinoe. Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1984. - ?40. - N. 93-98. 9. ?oaeue ?.I. I aai?a eiee/anoaaiiuo e?eoa?eaa aeey eeanneoeeaoeee ?aeueaoa. Aeai. Oe?AEIEOE 10.07.84. ?12489-84. 10. ?oaeue ?.I. I eeaneoeeaoeee ?aeueaoa. // Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. 1985. - ?41. - N. 104-108. 11. ?oaeue ?.I. ?aniiciaaaiea ia?acia eieaeueiuo o/anoeia caiiie iiaa?oiinoe. Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1987. - ?46. - N. 127-131. 12. ?oaeue ?.I. Auai? i?eciaeia aeey eeanneoeeaoeee oi?i ?aeueaoa. // Eca.aocia. Aaiaeacey e ay?ioioinuaiea. - 1987.- ?1.-N. 35-36. 13. ?oaeue ?.I. Eeanoa?ecaoeey oi?i ?aeueaoa caiiie iiaa?oiinoe. // Aaiaeacey, ea?oia?aoey e ay?ioioinyuaiea. - 1988. - ?47.- N. 119-123. 14. ?oaeue ?.I. E iino?iaieth ia?acia o/anoeia ?aeueaoa. // Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1988. - ?48.- N. 27-29. 15. ?oaeue ?.I. I aaia?aeecaoeee ?aeueaoa ianoiinoe iaoiaeii naa?oee. // Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1989. - ?49. - N. 120-126. 16. ?oaeue ?.I. Aiaeec oi?i ?aeueaoa ii aoi?ie eaaae?aoe/iie oi?ia iiaa?oiinoe. // Aaiaeacey, ea?oia?aoey e ay?ioioinuaiea. - 1989. - ?50. - N. 92- 96. 17. ?oaeue ?.I. Enneaaeiaaiea ?aeueaoa caiiie iiaa?oiinoe n eniieueciaaieai aanneaia. // Eca. aocia Aaiaeacey e ay?ioioinuaiea. - 1989. - ?3. - N. 49-54. 18. ?oaeue ?.I. Auaeaeaiea oi?i ?aeueaoa ii aeaeo nii?eeanathuaainy ia?aaieieaea. // Eiaeaia?iay aaiaeacey : ?ani. iaaeaaae. iao/i.-oaoi. na., 1989. - aui.32. - N. 79-83. 19. ?oaeee ?.I. I?eeeaaeia oioia?aiiao??y. Iaa/aeueiee iin?aiee. - Ee?a // IIE AI, 1991. 168 n. 20. ?oaeee ?.I. Aeei?enoaiiy oai??? ?ici?ciaaaiiy ia?ac?a i?e noai?aii? aai?ioi?iaoe?eieo nenoai. Oace iaoeiai-oaoi?/ii? eiioa?aioe?? i?ioani?nueei-aeeeaaeaoeueeiai neeaaeo ?inoeoooo iaooe ? aaco. - ?aaii-O?aie?anuee, 1994. - N. 44-45. 21. ?oaeee ?.I. Aeei?enoaiiy oai??? ?ici?ciaaaiiy ia?ac?a i?e aeine?aeaeaii? oiiia?ao?? iiaa?oiiue. Oace iaoeiai-oaoi?/ii? eiioa?aioe?? i?ioani?nueei- aeeeaaeaoeueeiai neeaaeo oi?aa?neoaoo. - ?aaii- O?aie?anuee, 1995. - N. 115. 22. ?oaeee ?.I., Iaey?/oe A.I. Iiooe iaooiaaciaeo ?iaeiaeu c aeiiiiiaith ii?oiiao?e/iiai aiae?co ?aeue?oo. // Oace iaoeiai- iaoiaee/ii? eiioa?aioe??. - ?aaii-O?aie?anuee, 1996. - N. 85. 23. ?oaeee ?.I. Aeae?eaiiy ia’?eo?a c oeeo?iai? iiaeae? ?aeue?oo c aeei?enoaiiyi oai??? ?ici?ciaaaiiy ia?ac?a. // Aaiaeac?y, ea?oia?ao?y ? aa?ioioici?iaiiy. - 1997. - ?58. - N. 207-212. 24. ?oaeee ?.I. Aaoiiaoe/iee aiae?c aeanoeainoae ?aeue?oo i?noeaaino? aeey niaoe?aeueiiai ?aeiioaaiiy c aeei?enoaiiyi oai??? ?ici?ciaaaiiy ia?ac?a. // No/ani? aeinyaiaiiy aaiaeace/ii? iaoee ? ae?iaieoeoaa a Oe?a?i?. Ca??iee iaoeiaeo i?aoeue eiioa?aioe??. - Euea?a, 1997. - N. 142-147. 25. ?oaeee ?.I., Iao?uoe A.A., ?oaeee I.?. Aeine?aeaeaiiy oi/iino? i?inoi?iai? aiae?oe/ii? oioio??aiaoeyoe?? SPACE-M. // No/ani? aeinyaiaiiy aaiaeace/ii? iaoee ? ae?iaieoeoaa a Oe?a?i?. Ca??iee iaoeiaeo i?aoeue eiioa?aioe??. - Euea?a, 1997. - N. 150-154. 26. ?oaeee ?.I. Canoinoaaiiy iaoiae?a aeine?aeaeaiiy ?aeue?oo i?e aea/aii? iaaacia/ieo a?ae?iiaoai?ieia?/ieo yaeu a Oe?a?inueeeo Ea?iaoao. // Ca??iee iaoeiaeo i?aoeue i?aeia?iaeii? iaoeiai-i?aeoe/ii? eiioa?aioe??. Aaiaeace/iee iii?oi??ia, aaiaeeiai?ea ? ?ao?aeoiiao??y ia iaae? OOI noie?ooy. - Euea?a, 1998. - N. 86-88. 27. ?oaeee ?.I., Ea?ea? A.A., ?oaeee I.?. I?iaiic ni?aiaeo eaaei a caeaaeiino? a?ae ?aeue?oo a??nueeeo noee?a. // Ca??iee iaoeiaeo i?aoeue i?aeia?iaeii? iaoeiai-i?aeoe/ii? eiioa?aioe??. Aaiaeace/iee iii?oi??ia, aaiaeeiai?ea ? ?ao?aeoiiao??y ia iaae? OOI noie?ooy. - Euea?a, 1998. - N.136-139. 28. ?oaeue ?.I. I?oioioii?iaeoi?. A.n. ?122309. Athe.?13, 07.04.86. 29. ?oaeue ?.I. Niinia ioia?aaeaiey ?aeueaoa ianoiinoe. A.n. 1278578. Athe. ?47, 23.12.86. 30. ?oaeue ?.I. Aeeooa?aioeeaeueiue oioio?ainoi?iaoi?. A.n. ?1451543. Athe.?2, 15.01.89 31. ?oaeee ?.I. Iaoaio Oe?a?ie ia aeiao?ae. Niin?a aecia/aiiy i?noeaiieiaeaiiy e?oath/iai aia?aoa aai ii?nueeiai noaeia. ?24487 a?ae 21.07.98. 32. Naeaeaeianeee E.ss., ?oaeue ?.I., I?eaiei ?.E. E enneaaeiaaieth iia?aoiinoae a aeaeaieyo eeiaa iioe/aneiai oaiaeieeoa. // Eiaeaia?iay aaiaeacey. - 1981. - ?24. - N. 54-56. 33. Naeaeaeianeee E.ss., ?oaeue ?.I. Eniieueciaaiea niaeo?aeueiuo oa?aeoa?enoee aeey eiee/anoaaiiie ioeaiee ?aeueaoa ianoiinoe. // Eiaeaia?iay aaiaeacey. - 1984. - ?27. - N. 74-77.` 34. Rudyy, R. M., An Analytical Metod for Identifying Structure Lines in Topography. Mapping Science and Remote Sensing. Vol.22, ?1,1987, pp 82-87. ?oaeee ?.I. Iaoiaee aeine?aeaeaiiy ?aeue?oo caiii? iiaa?oi?. - ?oeiien. Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy aeieoi?a oaoi?/ieo iaoe ca niaoe?aeuei?noth 05.24.02. - oioia?aiiao??y oa ea?oia?ao?y.- Aea?aeaaiee oi?aa?neoao "Euea?anueea iie?oaoi?ea". Euea?a, 1999. Aeena?oaoe?y i?enay/aia ?ica’yceo i?iaeai aeae?eaiiy ia’aeo?a c oeeo?iaiai cia?aaeaiiy, eeaneo?eaoe?? ae?eyiie caiii? iiaa?oi? ca ??aiai neeaaeiino? ?aeue?oo oa ca iayai?n?th iaaieo oi?i. A ?iaio? cai?iiiiiaaii iiaee iaoiae aeine?aeaeaiiy ?aeue?oo, ui aaco?oueny ia aeei?enoaii? iniiaieo iieiaeaiue oai??? ia?iaee cia?aaeaiue. ?ic?iaeaii iaoaiaoe/iee i?aeo?ae aeey aeae?eaiiy no?oeoo?ieo e?i?e iiaa?oi? ca aeaieie oeeo?iai? iiaeae? aaoiiaoe/iei iaoiaeii. I?iaaaeai? aenia?eiaioaeuei? aeine?aeaeaiiy neeaaeiino? ?aeue?oo ca aeiiiiiaith ??cieo ei?aeyoe?eieo oa niaeo?aeueieo ooieoe?e. Ia iniia? aeeiiaieo aeine?aeaeaiue cai?iiiiiaaii iaoiae iiaeaethaaiiy ?aeue?oo cia?aaeaiiyi c? ci?iiith iioe/iith u?euei?noth. Aaoi? aeei?enoaa iniiai? iieiaeaiiy oai??? ?ici?ciaaaiiy ia?ac?a aeey aeae?eaiiy iaaieo oi?i ?aeue?oo, ui aeaei iiaeeea?noue noai?eoe iaoiaeeeo aeae?eaiiy noee?a c iaaiith aeniiceoe??th oa ?ioeo oi?i ?aeue?oo ca oeeo?iaith iiaeaeeth a nenoai? ?aaeueiiai aai aeecueeiai aei ?aaeueiiai ianooaao /ano. Io?eiai? i?aeoe/i? ?acoeueoaoe oa ?ic?iaeai? iaoiaeeee ciaoiaeyoue canoinoaaiiy a i?aei?e?inoaao oa onoaiiaao, ui aeei?enoiaothoue A?N- oaoiieia??. Eeth/ia? neiaa: ?aeue?o, noee, oi?ia, iiaa?oiy, oeeo?iaa cia?aaeaiiy, iiaeaethaaiiy, no?oeoo?ia e?i?y, eeaneo?eaoe?y, ia?aoai?aiiy Oo?’?, ?ici?ciaaaiiy ia?ac?a, iciaea. R.M. Rudyy. Methods of Land Surface Relief Investigation. - Manuscript. Thesis for a doctor’s degree by speciality 05.24.02 - Photogrammetry and Cartography. - State University ”Lviv Polytechnics”, Lviv, 1999. The dissertation is dedicated to solution problems of object reconstruction from digital image, image understanding, recognising and classification of land surface areas according to levels of compounds and existing definite forms. Different methods for terrain relief classification and description are presented. In this work a new branch in theory of land surface investigation was created. On the basis of this approach main principles digital images treatment were used. The development of automated mapping methods is connected with creation of structural model, which used the structure lines. They are topographic features that depict the boundaries of existing actual changes in topographic surface, and play an important role in the presentation of morphometric and morphologic characteristics of relief. In this thesis a mathematical approach for determining topographic structure lines by means of digital elevation model (DEM) is introduced. Then methods of differential geometry were used. This methodology suggests to define structure lines of relief in order to use data from DEM. A network of ridges is extracted from DEM data for the first time. An automatic identifying these lines is performed by means of DEM in the form of grid elevation data used for image registration. The first task was to identify all points in the grid elevation model that are on ridges and achieve this in an automated way. The dissertation presents prescribe methods of definite levels of compounds of land surface using correlative and spectral functions, one- and twodimentions, M.M.Protodiakonov’s special curves. Classification methods are based on Fourier-transformation and correlative analysis, classifying the terrain types according to their frequencies of undulations. Frequency distribution allows to classificate terrain according to roughness and genesis. A method for the registration of digital images and digital terrain models (DTM) is presented. It resulted in solving the problem of modelling land surface or surface of other objects. On the basis of making relief analyses suggested a new method of relief modelling any type of surface according to change optical density. At the moment this method is used in geodetic production. The author used main principals of the theory of pattern recognition. In this way was elaborated methodology remarking certain types of relief. This approach gives a possibility in real or close to real scale of time to define slopes with appointed exposition. In experimental results obtained are of interest not only geodetic and cartography enterprises. Interest was shown by another organisations that use GIS technologies. Main results of this work get practical application in different fields. For instance, places of discharge of pollutant's determination near oil pipe-lines or oil wells; prediction of snow avalanches in mountain areas; in agriculture evaluation of areas. In Research Institute of Ecological Safety and Natural Resources tasks determination places of exhaust and polluted near oil wells in Carpathian mountain have been solved. The author takes part in the project initiated by the Government of Canada Department of External Affairs and from the State Ministry of Geodesy, Cartography and Cadastre of Ukraine. Surveying, technical work, and management has been co-ordinated by UMA Engineering and supported by the Ivano-Frankivsk Centre of Geodesy, Cartography and Cadastre in 1995-1997 years. One of the main goals of the project was to apply new technology to the creation and storage of information about land and its owners. Then exact co-ordinates of the land parcels along with their areas, perimeters and relief were obtained. In the dissertation relief of the village of Utoropy in the Kosiv Ryon has being used for investigation. Key words: relief, slope, form, surface, digital image, modelling, structural line, classification, Fourier-transformation, pattern recognition, feature. ?oaeue ?.I. Iaoiaeu enneaaeiaaiey ?aeueaoa caiiie iiaa?oiinoe. - ?oeiienue. Aeenna?oaoeey ia nieneaiea o/aiie noaiaie aeieoi?a oaoie/aneeo iaoe ii niaoeeaeueiinoe 05.24.02. - oioia?aiiao?ey e ea?oia?aoey.- Ainoaea?noaaiiue oieaa?neoao "Eueaeanueea iieeoaoieea". Eueaia, 1999. Aeenna?oaoeey iinayuaia ?aoaieth i?iaeai auaeaeaiey iauaeoia ec oeeo?iaiai ecia?aaeaiey, eeanneoeeaoeee o/anoeia caiiie iiaa?oiinoe ii o?iaith neiaeiinoe ?aeueaoa e ii iaee/eth ii?aaeaeaiiuo oi?i. A ?iaioa i?aaeeiaeaii iiaue iaoiae enneaaeiaaiey ?aeueaoa, eioi?ue aace?oaony ia eniieueciaaiee iniiaiuo iieiaeaiee oai?ee ia?aaioee ecia?aaeaiee. ?ac?aaioai iaoaiaoe/aneee iiaeoiae aeey auaeaeaiiey no?oeoo?iuo eeiee iiaa?oiinoe ii aeaiiui oeeo?iaie iiaeaee aaoiiaoe/aneei iaoiaeii. I?iaaaeaiu aenia?eiaioaeueiua enneaaeiaaiey neiaeiinoe ?aeueaoa n iiiiuueth ?acee/iuo ei??aeyoeeiiiuo e niaeo?aeueiuo ooieoeee. Ia iniiaa auiieiaiuo enneaaeiaaiee i?aaeeiaeai iaoiae iiaeaee?iaaiey ?aeue?oa ecia?aaeaieai n ia?aiaiiie iioe/aneie ieioiinoueth. Aaoi? eniieueciaae iniiaiua iieiaeaiey oai?ee ?aniiciaaaiey ia?acia aeey auaeaeaiey ii?aaeaeaiiuo oi?i ?aeueaoa, /oi aeaei aiciiaeiinoue nicaeaoue iaoiaeeeo aeaeaeaiiey neeiiia n ii?aaeeaiiie yniiceoeeae e ae?oaeo oi?i ?aeueaoa ii oeeo?iaie iiaeaee a nenoaia ?aaeueiiai eee aeeceiai e ?aaeueiiio ianooaaa a?aiaie. Iieo/aiiua i?aeoe/aneea ?acoeueoaou e ?ac?aaioaiiua iaoiaeeee iaoiaeyo eniieueciaaiea a i?aaei?eyoeyo e i?aaiecaoeeyo, eniieuecothueo AEN-oaoiieiaee. Eeth/aaua neiaa: ?aeueao, neeii, oi?ia, iiaa?oiinoue, oeeo?iaia ecia?aaeaiiea, iiaeaee?iaaiea, no?oeoo?iay eeiey, eeanneoeeaoeey, i?aia?aciaaiea Oo?uea, ?aniiciaaaiea ia?acia, i?eciae.

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