IAOe?IIAEUeIA AEAAeAI?ss IAOE OE?A?IE

?INOEOOO IAOAIAOEEE

IAIEIA Ia?e Ieaenaiae?iae/

OAeE 517.164.152+517.6

???AAOEss?I? I?AeIIIAEEIE

IIIAIAEAe?A A?ANIAIA

OA ?O CANOINOAAIIss

A OAI??? A?AeIA?AAEAIUe

01.01.01 — iaoaiaoe/iee aiae?c

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

EE?A 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a ?inoeooo? iaoaiaoeee IAI Oe?a?ie

Iaoeiaee ea??aiee:

aeieoi? o?ceei-iaoaiaoe/ieo iaoe i?ioani?

OA?EI Aieiaeeie? Aaneeueiae/,

i?ia?aeiee iaoeiaee ni?a?ia?oiee

?inoeoooo iaoaiaoeee IAI Oe?a?ie

Io?oe?ei? iiiiaioe:

aeieoi? o?ceei-iaoaiaoe/ieo iaoe

CAE?INUeEEE TH??e Ai?eniae/,

i?ia?aeiee iaoeiaee ni?a?ia?oiee

?inoeoooo iaoaiaoeee IAI Oe?a?ie

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe

EIINOAIOEIIA Ieaen?e TH??eiae/,

anenoaio eaoaae?e iaoaiaoe/iiai aiae?co

Ee?anueeiai oi?aa?neoaoo ?i. Oa?ana Oaa/aiei

I?ia?aeia onoaiiaa:

Euea?anueeee aea?aeaaiee oi?aa?neoao ?i. ?aaia O?ai-

ea, eaoaae?a aeaaa?e ? oiiieia??.

caoeno a?aeaoaeaoueny 23.03.99 i 15 aiae. ia can?aeaii? niaoe?ae?ciaaii?
a/aii? ?aaee Ae 26.206.01 i?e ?inoeooo? iaoaiaoeee IAI Oe?a?ie ca
aae?anith: 252601, i. Ee?a, aoe Oa?auaie?anueea, 3.

C aeena?oaoe??th iiaeia iciaeiieoenue a a?ae?ioaoe? ?inoeoooo
iaoaiaoeee.

Aaoi?aoa?ao ?ic?neaii 19.02.99

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee
IA?AAAA?C?A N. A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Ia ii/aoeo 30-o ?ie?a Iia?e?iaii aoei aeiaaaeaii,
ui aeey eiaeii? k-aei??ii? eiiiaeoii? i?aeiiiaeeie O ( Rn ciaeaeaoueny
k-aei??ia ieiueia l oaea, ui i?oiaiiaeueia i?iaeoe?y iiiaeeie O ia
ieiueio l ia? iaii?iaeith aioo??oi?noue, oiaoi oiiieia?/ia ?ici??i?noue
aeaii? i?iaeoe?? aei??aith? k.

Iaoae N(O, Y) — oiiieia?/iee i?ino?? on?o iaia?a?aieo a?aeia?aaeaiue
aeayeiai oiiieia?/iiai i?inoi?o O a oiiieia?/iee Y. A?aeia?aaeaiiy f (
N(O, Y) iaceaathoue no?eeei, yeui aiii ia? no?eea cia/aiiy o ( Y; oiaoi
aeey eiaeiiai, aeinoaoiuei aeecueeiai aei f, a?aeia?aaeaiiy g ( N(O, Y)
iiiaeeia g(O) i?noeoue oi/eo o. A?aeiii, ui aeey eiaeii? iiiaeeie O (
Rn ia??ai?noue dim O ( k ia? i?noea oiae? ? eeoa oiae?, eiee ?nio?
no?eea a?aeia?aaeaiiy f ( N(O, Rn). Aeieea? i?e?iaeia ieoaiiy: /e ia
?nio? o k-aei??ii? i?aeiiiaeeie O a Rn aeinoaoiuei i?inoeo
a?aeia?aaeaiue c i?inoi?o N(O, Y), iai?eeeaae, i?iaeoe?e ia aeayeo
k-aei??io ieiueio?

Aeaia i?iaeaia iia’ycaia c aeia?a a?aeiiith a?iioacith *iaioa?e?,
ca?aeii c yeith eiaeia k-aei??ia eiiiaeoia iiiaeeia O ( Rn iiaeiia iaoe
no?eeee ia?aoei c aeayeith (n — k)-aei??iith ieiueiith; oiaoi
ciaeaeaoueny ( ( ( oaea, ui aeey eiaeiiai iaia?a?aiiai (-cao?aiiy

f : X ( Rn ,(((x( ( x(( ( (x ( O

iiiaeeia f(X) ia?aoeia? aeaio ieiueio. Ne?ae cacia/eoe, ui cai?ioi?
oaa?aeaeaiiy ia? i?noea aeey eiaeii? k-aei??ii? (ia iaia’yceiai
eiiiaeoii?) i?aeiiiaeeie Rn.

Aeyaey?oueny, ui a?iioaca *iaioa?e? aoaea ni?aaaaeeeaith, yeui aeey
eiaeii? k-aei??ii? eiiiaeoii? i?aeiiiaeeie Rn ciaeaeaoueny no?eea
a?aeia?aaeaiiy a Rk, yea ? i?oiaiiaeueiith i?iaeoe??th iiiaeeie X ia
aeayeo k-aei??io ieiueio (cai?ioi?, acaaae? eaaeo/e, ia a??ii, c
oaa?aeaeaiiy *iaioa?e? ia aeieeaa? ?nioaaiiy o eiaeii? k-aei??ii?
i?aeiiiaeeie Rn i?oiaiiaeueii? i?iaeoe?? ia aeayeo k-aei??io ieiueio,
yea aoea a no?eeei a?aeia?aaeaiiyi oe??? iiiaeeie).

A?aeiii, ui ia?ac eiaeiiai no?eeiai a?aeia?aaeaiiy ia? iaii?iaeith
aioo??oi?noue. Oiio caaaeaia aeua oai?aia Iia?e?iaa o?eaaeee /an aoea
a?aoiaioii ia ei?enoue a?iioace *iaioa?e?. Oey a?iioaca eeoaeanue
a?aee?eoith aae aei inoaiiueiai /ano. A 1995-io ?ioe? I. I. Ae?ai?oi?eia
iiaoaeoaaa eiio?i?eeeaae, ui ni?inoiao? oeth a?iioaco. I?eeeaae
Ae?ai?oi?eiaa iieaco? oaeiae, ui a?iioaca i?i no?ee?noue i?iaeoe?e
k-aei??ieo i?aeiiiaeei ia k-aei??i? ieiueie ia ni?aaaeaeo?oueny.

Oai?aia A. Ao?aae/a noaa?aeaeo?, ui aeey eiaeii? k-aei??ii? eiiiaeoii?
iiiaeeie X a Rn iiiaeeia D(X) on?o iaia?a?aieo a?aeia?aaeaiue f ( C(X,
Rk) oaeeo, ui dim f(X) = k, i?noeoue ianeaio i?aeiiiaeeio. O ca’yceo c
oeei aeieeea ianooiia caaea/a: iieacaoe, ui iiae?aia ia? i?noea aeey
iaia?a?aieo a?aeia?aaeaiue X a Rk, ye? ? i?oiaiiaeueieie i?iaeoe?yie
iiiaeeie X ia k-aei??i? ieiueie. I?e oeueiio ie ia aoaeaii
iaiaaeoaaoenue eeoa eiiiaeoiei aeiaaeeii. I?e ?ica’ycoaaii? oe???
caaea/? ie ia iiaeaii ei?enooaaoeny caaaeaiith aeua oai?aiith A.
Ao?aae/a; ine?eueee on? i?iaeoe?? noai?ththoue o C(X, Rk) i?aea ia
u?eueio i?aeiiiaeeio P(X) ? iiiaeeia D(X) ( P(X) ia ciaia’ycaia aooe
anthaee u?eueiith o P(X).

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. ?iaioa
i?iaiaeeeanue ca?aeii c caaaeueiei ieaiii aeine?aeaeaiue a?aeae?eo
oai??? iaaeeaeaiue.

Iaoa aeine?aeaeaiiy. Iieacaoe, ui aeey eiaeii? k-aei??ii? i?aeiiiaeeie X
( Rn ciaeaeaoueny aeineoue aaeeea iiiaeeia k-aei??ieo ieiuei oaeeo, ui
i?oiaiiaeuei? i?iaeoe?? iiiaeeie X ia oe? ieiueie iathoue ?ici??i?noue
k.

Iaoeiaa iiaecia io?eiaieo ?acoeueoao?a. Aoee aaaaeai?, oae caai?,
???aaoey?i? i?aeiiiaeeie iiiaiaeae?a A?aniaia. ?o aeanoeaino?
aeei?enoiaoaaeenue aeey aeine?aeaeaiiy

— i?iaeoe?e k-aei??ieo i?aeiiiaeei Rn ia k-aei??i? ieiueie,

— ieoaiue, iia’ycaieo c i?iaeaiith Ao?aae/a-Aieiaia,

— No?oeoo?e oeiiai? iiiaeeie ??aiy a?aeia?aaeaiue c Rn a Rm.

I?aeoe/ia cia/aiiy io?eiaieo ?acoeueoao?a. Aeena?oaoe?y ia? oai?aoe/iee
oa?aeoa?. ?acoeueoaoe ?iaioe iiaeooue aooe aeei?enoai? aeey iiaeaeueoeo
aeine?aeaeaiue a oai??? a?aeia?aaeaiue ? oai??? ?ici??iino?.

Iniaenoee aianie caeiaoaa/a. ?acoeueoaoe aeena?oaoe?? ?iaioe io?eiai?
aaoi?ii naiino?eii.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe aeena?oaoe??
aeiiia?aeaeenue ia nai?ia?ao a?aeae?eo iaaeeaeaiue ?inoeoooo iaoaiaoeee
IAI Oe?a?ie, o?ao?e i?aeia?iaei?e iaoeia?e eiioa?aioe??. Nienie
ioae?eaoe?e iaaaaeaii a e?ioe? aaoi?aoa?aoo.

No?oeoo?a ? ia”?i ?iaioe. Aeena?oaoe?y neeaaea?oueny c? anooio,
/ioe?ueio ?icae?e?a, ?icaeoeo ia i?ae?icae?ee, aeniiae?a ? nieneo
aeei?enoaieo aeaea?ae. Ianya aeena?oaoe?? — 110 noi??iie.

CI?NO ?IAIOE

O ia?oiio ?icae?e? ((( 1.1 — 1.4) c?a?ai? aeaye? iaiao?aei? a?aeiiino?
c oai??? ?ici??iino? oa ?ioeo aaeocae iaoaiaoeee, a a (1.5 iaaiaeeyoueny
iniiae oai??? e?eoe/iio oi/ie aeia?eueiiai a?aeia?aaeaiiy Rn a Rm.

Ae?oaee ?icae?e i?enay/aiee aeine?aeaeaiith i?iaeoe?e k-aei??ieo
i?aeiiiaeei Rn ia k-aei??i? ieiueie. Nii/aoeo iaaaaeaii aeae?eueea
icia/aiue.

Iaoae ( ( Rn, l ( Gnk oa s ( Gnn-k . I?iaeoe?y iiiaeeie X ia ieiueio l
acaeiaae ieiueie s aoaea iaceaaoeny c-?aaoey?iith, yeui aiia ?
iiiaeeiith ae?oai? eaoaai??? a l. O i?ioeeaaeiiio aeiaaeeo ie eaaeaii,
ui iaoa i?iaeoe?y c-???aaoey?ia. Iicia/eii /a?ac CRnk(X) iiiaeeio on?o
oeo ieiuei l ( Gnk, aeey yeeo i?oiaiiaeueia i?iaeoe?y iiiaeeie X ia
ieiueio l ? c-???aaoey?iith. I?iaeoe?y iiiaeeie X ia ieiueio l acaeiaae
ieiueie s aoaea iaceaaoeny m-?aaoey?iith, yeui aiia ia? a?aei?iio a?ae
ioey ciai?oith i??o Eaaaaa a l (iaaaaea?ii, ui ciai?oiy i??a aecia/aia
aeey aoaeue-yei? i?aeiiiaeeie Rn). sseui i?iaeoe?y ia? ioeueiao i??o
Eaaaaa, oi ie aoaeaii eacaoe, ui aiia m-???aaoey?ia. A?aeiia?aeii,
iicia/eii /a?ac MRnk(X) iiiaeeio on?o oeo ieiuei l ( Gnk, aeey yeeo
i?oiaiiaeueia i?iaeoe?y iiiaeeie X ia l ? m-???aaoey?iith.

Iaoae DRnk(X) — iiiaeeia, yea neeaaea?oueny c on?o oeo ieiuei l ( Gnk,
aeey yeeo i?oiaiiaeueia i?iaeoe?y iiiaeeie X ia l ia? ?ici??i?noue k.
Oiae? ie ia?ii

CMnk (X) ( Dnk(X) ( CRnk(X) .

Iiiaae oa, yeui X ? F( -i?aeiiiaeeiith Rn, oiae? ia? i?noea ianooiia
??ai?noue:

Dnk (X) = CRnk (X) .

O ae?oaiio ?icae?e? aeiaiaeeoueny ianooiia oaa?aeaeaiy, yea ? iniiaiei
?acoeueoaoii ?iaioe.

Oai?aia 1 (1.2.1) sseui dim X ( k, oiae? iiiaeeie CRnk (X) oa MRnk (X)
anthaee u?euei? a Gnk; e??i oiai, o aeiaaeeao k = 1, n ( 1 aeiiiaiaiiy
aei iiiaeei CRnk (X) oa MRnk (X) aoaeooue i?aea ia u?eueieie a Gnk.

Aeey F(-i?aeiiiaeei ie ia?ii a?eueo neeueia oaa?aeaeaiiy, yea ?
iane?aeeii iiia?aaeiueiai (iane?aeie 2.1.1): yeui X ? k-aei??iith
F(-i?aeiiiaeeiith Rn, oiae? iiiaeeia Dnk(X) anthaee u?eueia a Gnk; e??i
oiai, o aeiaaeeao k = 1, n — 1 ?? aeiiiaiaiiy aoaea ia u?eueiei a Gnk.

Oe? oaa?aeaeaiiy oi?ioeththoueny a ( 2.1. Oaeiae oai iiaeia ciaeoe
aeae?eueea ?o iane?aee?a.

Aeiaaaeaiith oai?aie 1 i?enay/aiee aanue ae?oaee ?icae?e. Naia
aeiaaaeaiiy nie?a?oueny ia aeaye? aeanoeaino? ???aaoey?ieo i?aeiiiaeei
Gnk, ye? aaiaeyoueny oa aeine?aeaeothoueny o ( 2.2.

Iaoae

n!

cnk = (((( .

k! (n-k)!

Eiaeia nenoaia eii?aeeiao a Rn i?noeoue ??aii cnk ??cieo k-aei??ieo
eii?aeeiaoieo ieiuei. Iiiaeeia V ( Gnk aoaea iaceaaoeny ???aaoey?iith,
yeui aiia ia i?noeoue cnk ieiuei, ye? ? ??cieie eii?aeeiaoieie ieiueiaie
aeayei? nenoaie eii?aeeiao a Rn; oiaoi iiiaeeia V ? ???aaoey?iith, yeui
aiia ia i?noeoue on?o k-aei??ieo eii?aeeiaoieo ieiuei eiaeii? nenoaie
eii?aeeiao a Rn.

Ia?oei aoaiii aeiaaaeaiiy oai?aie 1 ? ianooiia oaa?aeaeaiiy.

Oai?aia 2 (2.3.1) sseui dim X ( k, oiae? iiiaeeie

CInk (X) = Gnk \ CRnk (X)

oa

MInk (X) = Gnk \ MRnk (X)

? ???aaoey?ieie.

C oai?aie 2 aeieeaa?, ui o aeiaaeeo eiee X ? k-aei??iith
F(-i?aeiiiaeeiith Rn, iiiaeeia Gnk \ Dnk (X) ???aaoey?ia (iane?aeie
2.3.1). Oi?ioethaaiith oeeo oaa?aeaeaiue oa ?o iane?aee?a i?enay/aiee (
2.3.

A ca’yceo c oai?aiith 2 aeieea? ieoaiiy i?i u?euei?noue ???aaoey?ieo
i?aeiiiaeei. O ( 2.4 aoee io?eiai? ianooii? ?acoeueoaoe.

Oai?aia 3 (2.4.1) O aeiaaeeao k = 1, n — 1 eiaeia ???aaoey?ia
i?aeiiiaeeia i?aea ia u?eueia a Gnk.

Oai?aia 4 (2.4.1) Aeiiiaiaiiy aei eiaeii? ???aaoey?ii? i?aeiiiaeeie
anthaee u?eueia a Gnk.

Oai?aia 1 ? iane?aeeii oai?ai 2,3 ? 4. Aeiaaaeaiiy oai?aie 2 nie?a?oueny
ia aeanoeaino? oae caaieo Xnk-iiiaeei, ye? aaiaeyoueny o ( 2.5. O ( 2.5
— 2.7 anoaiiaeth?oueny, ui ?ici??i?noue eiaeii? Xnk-iiiaeeie aei??aith?
k. I?ney oeueiai aeiaaaeaii, ui o aeiaaeeo, eiee iaeia c iiiaeei CInk
(X), MInk (X) ia ? ???aaoey?iith, iiiaeeia X iiaeiia i?noeoeny o aeaye?e
Xnk-iiiaeei?. Iane?aeeii oeueiai ? ia??ai?noue dim X ( k ? oai?aia 2
aoaea aeiaaaeaia.

A ( 2.8 aoaeo?oueny (n — 1)-aei??ia i?aeiiiaeeia X ( Rn oaea, ui
iiiaeeia Gnk \ Dnk(X) ia ? ???aaoey?iith i?e oiia?

O o?aoueiio oa /aoaa?oiio ?icae?eao iaoiae Xnk-iiiaeei canoiniao?oueny
aeey aeine?aeaeaiiy aeayeeo aeooaeueieo ieoaiue oai??? ?ici??iino? oa
oai??? a?aeia?aaeaiue — i?iaeaie Ao?aae/a-Aieiaia oa no?oeoo?e oeiiai?
iiiaeeie ??aiy a?aeia?aaeaiue Rn a Rm

A o?aoueiio ?icae?e? aoaea ?icaeyiooa ianooiia i?iaeaia: /e ?nio?
k-aei??ia i?aeiiiaeeia X ( Rn oaea, ui eiaeia i?aeiiiaeeia a Rn,
?ici??i?noue yei? ia a?eueo i?ae k, iiaea aooe aeeaaeaia a X
aiiaiii?o?ciii i?inoi?o Rn ia naaa?

O eean? on?o Xnk-iiiaeei ie aa?aii niaoe?aeueieo i?aeeean, aeaiaioe
yeiai aoaeooue iaceaaoeny SXnk-iiiaeeiaie. Icia/aiiy SXnk-iiiaeei
iiaea?oueny a ( 3.1. A ( 3.2-3.4 aeiaiaeyoueny ianooii? aeanoeaino?
Xnk-iiiaeei ? SXnk-iiiaeei:

1) ?ici??i?noue eiaeii? Xnk-iiiaeeie aei??aith? k;

2) o aeiaaeeo n ( 2k + 1 eiaeia Xnk-iiiaeeia ? k-aei??iei
oi?aa?naeueiei

i?inoi?ii;

3) eiaeio SXnk-iiiaeeio iiaeia ia?aaanoe o aoaeue-yeo ?ioo
SXnk-iiiaeeio aiiaiii?o?ciii i?inoi?o Rn ia naaa;

4) eiaeia k-aei??ia neaaio?aeoaeueia i?aeiiiaeeia a Rn aeiionea?
aeeaaeaiiy o aoaeue-yeo SXnk-iiiaeeio aiiaiii?o?ciii i?inoi?o Rn ia
naaa (i?ae neaai- o?aeoaeueiith iiiaeeiith ?icoi??oueny oaea
iiiaeeia X, ui ia? i?noea ia??a- i?noue ((X) — dim X ( 1, aea ((X) —
?ici??i?noue Oaonaei?oa-Aac?eiae/a;

5) yeui k-aei??ia F(-i?aeiiiaeeia X a Rn aeeaaea?oueny a aeayeo
Xnk-iiiaeeio,

oi oey iiiaeeia i?eionea? aeeaaeaiiy a aoaeue-yeo
SXnk-iiiaeeio; a?eueo

oiai, yeui aeaia aeeaaeaiiy a Xnk-iiiaeeio iiaeia
cae?enieoe ca aeiiiiiaith aiiaiii?o?cio i?inoi?o Rn ia naaa, oi
iiiaeeia X iiaea aooe aeeaaeaia a aoaeue-yeo SXnk-iiiaeeio
aiiaiii?o?ciii i?inoi?o Rn ia naaa.

E??i oiai, iiaeia aaaaeaoe, ui o oaa?aeaeaiiyo 3), 4) aiiaiii?o?ci ia
ca?eueoo? a?aenoai? i?ae oi/eaie.

O /aoaa?oiio ?icae?e?, ei?enooth/enue aeayeeie aeanoeainoyie
Xnk-iiiaeei, ie aoaeaii aeine?aeaeoaaoe no?oeoo?o oeiiai? iiiaeeie ??aiy
a?aeia?aaeaiiy Rn a Rm; i?e oeueiio oiiae aeaaeeino? aai iaia?a?aiino?
ia a?aeia?aaeaiiy iaeeaaeaoenue ia aoaeooue.

I?ae oeiiaith iiiaeeiith ??aiy aeayeiai a?aeia?aaeaiiy i?eeiyoi
?icoi?oe ianooiia. Aiai?youe, ui oeiia?e iiiaeei? ??aiy a?aeia?aaeaiy f
: Rn ( Rm i?eoaiaiia aeanoea?noue (, yeui

mes{y( f -1 (y) ia caaeiaieueiy? (} = 0 ,

aea mes — ciai?oiy i??a Eaaaaa.

O ( 4.1 aea?oueny icia/aiiy oa iaaiaeyoueny i?eeeaaee oae caaieo
a?aeia?aaeaiue Oaonaei?oa. A?aeia?aaeaiiy Rn a Rm iaceaa?oueny
oaonaei?oiaei, yeui aiii a?aeia?aaea? eiaeio iiiaeeio ioeueiai? m-i??e
Oaonaei?oa o iiiaeeio ioeueiai? i??e Eaaaaa. O ( 4.2 aoaea aeiaaaeaia

Oai?aia 5 (4.2.1) ?ici??i?noue oeiiai? iiiaeeie ??aiy a?aeia?aaeaiiy
Oaonaei?oa Rn a Rm ia ia?aaeuo? n — m.

Nie?ath/enue ia ?aeath, ui eaaeeoue a iniia? aeiaaaeaiiy oai?aie 5,
iiaeia iiaeaoe, a?aei?iia a?ae i?ea?iaeueiiai, aeiaaaeaiiy ianooiii?
oai?aie A. ss. Aeoaiaeoeueeiai i?i ?ici??i?noue ia?aoeio oeii?i?
iiiaeeie aeaaeeiai a?aeia?aaeaiiy, ui ia caaeiaieueiy? oiiaai oai?aie
Na?aea, c iiiaeeiith e?eoe/ieo oi/ie.

Oai?aia 6 (A. ss. Aeoaiaeoeueeee) Aeey eiaeiiai

f ( Ci (Rn ; Rm) , i = n — m — k + 1

ia? i?noea ianooiia ??ai?noue:

mes {y( dim (f -1 (y) ( ( (f)) ( k} = 0 ,

oiaoi ia?aoei oeiiai? iiiaeeie ??aiy a?aeia?aaeaiiy f c iiiaeeiith
e?eoe/ieo oi/ie ( (f) ia? ?ici??i?noue ia a?eueo, i?ae k — 1.

E??i oiai iaoiaee, ui a?ei?enoiaothoueny o aeiaaaeaii? oai?aie 5 ?
oai?aie A. ss. Aeoaiaeoeueeiai, aeicaieythoue ia?aianoe oeae ?acoeueoao
ia aeiaaeie aeia?eueiiai a?aeia?aaeaiiy.

Oi/ea x ( Rn iaceaa?oueny neaaie?eoe/iith oi/eith
a?aeia?aaeaiiy f : Rn ( Rm, yeui ciaeaeaoueny nenoaia eoeue {Bri
(x)}(i=1 c oeaio?aie o oi/oe? x, ?aae?one yeeo i?yiothoue aei ioey i?e
i, ui i?yio? aei iane?i/aiiino?, ?

Ia? i?noea ianooiia oaa?aeaeaiiy.

Oai?aia 7(4.3.2) Aeey aeia?eueiiai a?aeia?aaeaiiy f : Rn ( Rm
aeeiio?oueny ianooiia ??ai?noue:

mes {y(dim (f -1 (y) ( G( (f)) ( n — m} = 0 ,

oiaoi ia?aoei oeiiai? iiiaeeie ??aiy a?aeia?aaeaiiy f c iiiaeeiith
neaaie?eoe/ieo oi/ie G((f) ia? ?ici??i?noue, ui ia ia?aaeuo? n — m — 1.

Aeiaaaeaiiy oai?ai 6 ? 7 i?noeoueny a ( 4.3.

AENIIAEE

A aeena?oaoe?? io?eiaii ianooii? ?acoeueoaoe:

— Aeiaaaeaii, ui aeey eiaeii? k-aei??ii? i?aeiiiaeeie X ( Rn iiiaeeie
CRnk(X) ? MRnk(X) anthaee u?euei? a Gnk, a o aeiaaeeo, eiee k = 1, n —
1, aeiiiaiaiiy aei oeeo iiiaeei ? i?aea ia u?eueieie.

— Aeey eiaeii? k-aei??ii? F(-i?aeiiiaeeie X ( Rn iiiaeeia Dnk (X)
anthaee u?eueia a Gnk; e??i oiai, o aeiaaeeao, eiee k = 1, n — 1,
aeiiiaiaiiy aei oe??? iiiaeeie ? i?aea ia u?eueiei.

— Aoei aeiaaaeaii, ui i?e k = 1, n — 1 eiaeia ???aaoey?ia i?aeiiiaeeia
Gnk i?aea ia u?eueia. O caaaeueiiio aeiaaeeo aeiiiaiaiiy aei eiaeii?
???aaoey?ii? iiiaeeie anthaee u?eueia a Gnk.

— Aoea iiaoaeiaaia k-aei??ia i?aeiiiaeeia Rn, o yeo aeeaaea?oueny
aeineoue aaeeeee eean k-aei??ieo i?aeiiiaeei Rn, i?e oeueiio aeeaaeaiiy
cae?enith?oueny aiiaiii?o?ciii Rn ia naaa.

— Io?eiaia aa?oiy ioe?iea ?ici??iino? oeiiai? iiiaeeie ??aiy
a?aeia?aaeaiue

Oaonaei?oa.

— Oaa?aeaeaiiy a?aeiii? oai?aie A. ss. Aeoaiaeoeueeiai aoei ia?aianaia
ia aeiaaeie

aeia?eueiiai a?aeia?aaeaiiy.

NIENIE IIOAE?EIAAIEO ?IA?O AAOI?A

CA OAIITH AeENA?OAOe??

1. Iaieia I. A., Iieoeyo A. A. E?eoe/aneea oi/ee i?iecaieuei?o
ioia?aaeaiee ec

Rn a Rm // Ooieoeeii. aiaeec e aai i?ee. — 1997. — N 3. — N.82-85.

2. Pankov M. A. Hausdorff maps. // Methods of Functional Analysis and
Topology. —

1998 — N 3. — P. 58-60.

3. Pankov M. A. Projections of k-dimensional subsets of Rn onto
k-dimentional planes.//

Matematicheskaya fizika, analiz, geometriya. — 1998. — N 1/2. — P.
114-124.

4. Pankov M. A. On one class of universal k-dimentional spase. // Oacenu

iaaeaeoia?iaeiie eiioa?aioeee ii oiiieiaee e aa i?eeiaeaieyi. Eeaa:
Ei-o

iaoaiaoeee IAI Oe?aeiu, Eeaaneee iaoeeiiaeueiue oieaa?neoao ei. O.

Oaa/aiei.-1995.-N.28. Iaieia I. I. ???aaoey?i? i?aeiiiaeeie
iiiaiaeae?a A?aniaia oa ?o canoinoaaiiy o oai??? a?aeia?aaeaiue. —
?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.01 — iaoaiaoe/iee
aiae?c.

-?inoeooo iaoaiaoeee IAI Oe?a?ie, Ee?a, 1998.

A aeena?oaoe?? aaiaeyoueny oa aeine?aeaeothoueny oae caai? ???aaoey?i?
i?aeiiiaeeie iiiaiaeae?a A?aniaia. ?o aeanoeaino? a?ei?enoiaothoueny
aeey aeine?aeaeaiiy i?iaeoe?e k-aei??ieo i?aeiiiaeei Rn ia k-aei??i?
ieiueie. Oey caaea/a iia’ycaia c a?aeiiith a?iioacith *iaioa?e?. E??i
oiai, ?icaeyaea?oueny i?iaeaia Ao?aae/a-Aieiaia oa aeine?aeaeo?oueny
no?oeoo?a oeiiai? iiiaeeie ??aiy a?aeia?aaeaiue Rn a Rm.

Eeth/ia? neiaa: iiiaiaeae A?aniaia, ???aaoey?i? i?aeiiiaeeie
iiiaiaeae?a A?aniaia, k-aei??i? i?aeiiiaeeie, a?aeia?aaeaiiy Oaonaei?oa.

Iaieia I. A. E??aaoey?iua iiaeiiiaeanoaa iiiaiia?acee A?anniaia e eo
i?eiaiaiea a oai?ee ioia?aaeaiee. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.01 — iaoaiaoe/aneee
aiaeec. -Einoeooo iaoaiaoeee IAI Oe?aeiu, Eeaa, 1998.

A ?aaioa aaiaeyony e enneaaeothony e??aaoey?iua iiaeiiiaeanoaa
A?anniaiiauo iiiaiia?acee. Eo naienoaa eniieuecothony aeey eco/aiey
i?iaeoeee k-ia?iuo iiaeiiiaeanoa Rn ia k-ia?iua ieineinoe. Aeaiiay
caaea/a oanii naycaia n ecaanoiie aeiioacie *iaioaeee i onoie/eaii
ia?ana/aiee k-ia?iiai iiaeiiiaeanoaa Rn n (n — k)-ia?iie ieineinoueth.
E?iia oiai, ?anniao?eaaaony i?iaeaia Ao?aae/a-Aieiaia i k-ia?iuo
oieaa?naeueiuo iiaeiiiaeanoaao Rn e enneaaeoaony no?oeoo?a oeie/iiai
iiiaeanoaa o?iaiy ioia?aaeaiee Rn a Rm.

Iaeia ec iniiaiuo oeaeae iieacaoue, /oi aeey eaaeaeiai k-ia?iiai
iiaeiiiaeanoaa X a Rn iiiaeanoai Dnk (X) anao k-ia?iuo ieineinoae,
i?oiaiiaeueiua i?iaeoeee iiiaeanoaa X ia eioi?ua eiatho ?acia?iinoue k
(eee ii e?aeiae ia?a yaeythony iiiaeanoaaie aoi?ie eaoaai?ee)
aeinoaoi/ii aaeeei (o.a. anthaeo ieioii eee ianneaii). A aeaaa 2 aeaiiay
caaea/a naiaeeony e i?iaeaia ieioiinoe e??aaoey?iuo iiaeiiiaeanoa
A?anniaiiauo iiiaiia?acee; o.a. aeieacaii, /oi aeey eaaeaeiai k-ia?iiai
F( iiaeiiiaeanoaa X a Rn aeiiieiaiea e iiiaeanoao Dnk (X) e??aaoey?ii,
aeey i?iecaieueiiai k-ia?iiai (ia iaycaoaeueii F(-iiaeiiiaeanoaa)
aeieacaii iaeioi?ia aieaa neaaia ooaa?aeaeaiea. A nayce n yoei auei
onoaiiaeaii, /oi aeiiieiaiea o e??aaoey?iiio iiaeiiiaeanoao A?anniaiiaa
iiiaiia?acey Gnk anthaeo ieioii, a i?e k = 1 , n — 1 eaaeaeia
e??aaoey?iia iiaeiiiaeanoai Gnk ieaaea ia ieioii.

Oaoieea, ?ac?aaioaiiay aeey aeieacaoaeuenoaa oiiiyioouo auoa
?acoeueoaoia, iicaieyao aeieacaoue iaeii ooaa?aeaeaiea, naycaiiia n
i?iaeaiie Ao?aae/a-Aieiaia. A aeaaa 3 no?ieony k-ia?iia iiaeiiiaeanoai X
a Rn oaeia, /oi aeinoaoi/ii oe?ieee eeann k-ia?iuo iiaeiiiaeanoa Rn
aeiioneaao aeiaeaiea a X aiiaiii?oeciii Rn ia naay.

Aeaaa 4 iinayuaia eco/aieth no?oeoo?u oeie/iiai iiaiaeanoaa o?iaiy
ioia?aaeaiee Rn a Rn. Aeaaony aa?aeiyy ioeaiea ?acia?iinoe oeie/iiai
iiiaeanoaa o?iaiy ioia?aaeaiee Oaonaei?oa, iaiauathuay ecaanoioth
ioeaieo A. ss. Aeoaiaeoeeiai aeey iai?a?uaiiaeeooa?aioee?iaaiuo
ioia?aaeaiee. E?iia oiai, i?eaiaeeony i?inoia aeieacaoaeuenoai oai?aiu
A. ss. Aeoaiaeoeeiai i ia?ana/aiee oeie/iiai iiiaeanoaa o?iaiy
iai?a?uaii aeeooa?aioee?oaiiai ioia?aaeaiey n iiiaeanoaii e?eoe/aneeo
oi/ae, eioi?ia iicaieyao ia?aianoe aeaiiue ?acoeueoao ia neo/ae
i?iecaieueiiai ioia?aaeaiey (iiiyoea e?eoe/aneie oi/ee i?iecaieueiiai
ioia?aaeaiey iiaeii iaeoe a aeaaa 1, iinayuaiiie i?aaeaa?eoaeueiui
naaaeaieyi ec ?acee/iuo iaeanoae iaoaiaoeee).

Eeth/aaua neiaa: A?anniaiiau iiiaiia?acey, e??aaoey?iua iiaeiiiaeanoaa
A?anniaiiauo iiiaiia?acee, k-ia?iua iiiaeanoaa, ioia?aaeaiey Oaonaei?oa.

Pankov M. A. Irregular subsets of the Grassmannian manifolds and them
applications to the mapping theory. -Manuscript.

Thisis the dissertation for obtaining of the degree of candidate of
sciences in physics and mathematics, specislity 01.01.01 — mathematical
analysis. Institute of mathemetics, NAN Ukraine, Kyiv, 1998.

In the dissertation we introduse and stady so-called irregular subsets
of the Grassmannian manifolds. Them properts will be exploit to
investigate of projections of k-dimentional subsets of Rn onto
k-dimentional planes. There is a closely relation between this problem
and well-known Chogoshvili`s conjecture. Moreover, we consider the
Hurevicz — Wallman problem and study structure of a typical level for
maps of Rn into Rm.

Key words: Grassmannian manifolds, srregular subsets of the
Grassmannian manifold, k-dimentional sets, Hausdorff maps.

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