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Хаос, необратимость времени и брюссельская интерпретация квантовой механики

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Mineianeee oeceei-oaoie/aneee einoeooo

Eaoaae?a oeeinioee

A.?.Iaaeaaaeaa, anie?aio 1-ai aiaea

OAIN, IAIA?AOEIINOUe A?AIAIE E A?THNNAEUeNEAss EIOA?I?AOAOeEss EAAIOIAIE
IAOAIEEE. EIIOeAIOeEss E.I?EAIAEEIA

?aoa?ao e eaiaeeaeaoneiio yecaiaio ii oeeinioee

Niaea?aeaiea

0. AAAAeAIEA

1. OAIN

1.1 Eeanne/aneee aeeiaie/aneee oain: iaonoie/eainoue ii ia/aeueiui
oneiaeyi

1.2 Eeanne/aneee oain: iaeioaa?e?oaiua nenoaiu Ioaiea?a

1.3 Noaoenoe/aneia iienaiea. Aeenneiaoeaiue oain

2. IAIA?AOEIINOUe A?AIAIE

2.1 Ia?aoeiinoue a?aiaie a eeanne/aneie e eaaioiaie iaoaieea

2.2 ?ieue iaia?aoeiinoe a noaoenoe/aneie iaoaieea. Iioiee ei??aeyoeee

2.3 I?iaeaia ianaiaeeiiai iienaiey

3. A?THNNAEUeNEAss EIOA?I?AOAOeEss EAAIOIAIE IAOAIEEE

3.1 Aeueoa?iaoeaiua eioa?i?aoaoeee eaaioiaie iaoaieee

3.2 Iaoieoa?iay yaiethoeey e ianaiaeeiia iienaiea

4. CAEETH*AIEA

NIENIE EEOA?AOO?U

IOOE

1997

0. AAAAeAIEA

Ia/eiay n a?ai?i Aaeeeay e Iuethoiia nia?aiaiiay oeceea i?iaeaeaea
ia?iiiue iooue ii iaeiieaieth, nenoaiaoecaoeee, iienaieth e iniuneaieth
oaeoia ia ie?oaeathuai ie?a. Iienaiea iau/ii aeaeaeinue ia ycuea
iaoaiaoeee, e naia no?oeoo?a yoiai ycuea ca/anooth iicaieyea niaa?oaoue
iiaua ioe?uoey a ?aaeueiii ie?a (/oi naii ii naaa aeinoaoi/ii
oaeeaeoaeueii). Ca ianeieueei noieaoee i?aaeneacaoaeueiay ?ieue oeceee
noaea ianoieueei aieueoie, /oi a ianoiyuaa a?aiy ia?aoaaiuo “n/?oiuo”
caaea/ i?aeoe/anee ia inoaeinue – ii e?aeiae ia?a, n oi/ee c?aiey
i?eioeeieaeueiiai iiieiaiey i?ienoiaeyueo yaeaiee – ie a iaoaieea, ie a
eeanne/aneie yeaeo?iaeeiaieea, ie a eaaioiaie oai?ee.

Oeceea i?iaeieaeaao ?acaeaaoueny, e ca iineaaeiea aeanyoeeaoey aic?in
eioa?an e oaeei a? iiaui iaeanoyi, eae neia?aaoeea, aeeiaie/aneee oain e
naiii?aaiecaoeey. A yoeo aaoayo oeceee ca/anooth eniieuecoaony
i?eaeiaeueiue iaoaiaoe/aneee aiia?ao, a a ni/aoaiee n aic?anoathuae
iiuiinoueth eiiiuethoa?ia e aiciiaeiinoae “/eneaiiiai yenia?eiaioa”
i?aaeneacaoaeueiay neea eo ieacuaaaony aiieia “ia o?iaia”, ia?yaeo n
o?aaeeoeeiiiuie oece/aneeie oai?eyie.

A oi aea a?aiy aicieeee iaeioi?ua i?iaeaiu, eaaeauea nei?aa a iaeanoe ia
iaoaiaoeee, a oeeinioee oeceee. ?acee/iua oece/aneea oai?ee – noa?ua e
iiaua – “ia noueothony” ae?oa n ae?oaii a ioiioaiee ii?aaeae?iiuo
ooiaeaiaioaeueiuo iiiyoee e yaeaiee – a /anoiinoe, aeaoa?ieiecia e
iaia?aoeiinoe a?aiaie.

Ia iae?ineiie/aneii o?iaia iaia?aoeiinoue a?aiaie aoiaeeo ia oieueei a
“iiaoth oeceeo”, ii, iai?eia?, e a ?ac?aaioaiioth a i?ioeii aaea
oa?iiaeeiaieeo. O?oaeiinoe aicieeatho i?e ia?aeeaeuaaiee iinoa n
eeanne/aneeo iaoaie/aneeo iiaeaeae, iniiaaiiuo ia ia?aoeiuo ai a?aiaie
aaieeueoiiiauo o?aaiaieyo, e yaii aeenneiaoeaiiio, iaia?aoeiiio,
iiaaaeaieth ?aaeueiuo oece/aneeo nenoai e oai?eyi, eo iienuaathuei. Yoi
iaeei i?eia?.

Ae?oaie i?eia? oece/aneie i?iaeaiu oeeinioneiai ieaia – aicieeiiaaiea
oaioe/aneiai iiaaaeaiey o i?inouo nenoai, iienuaaaiuo aeaoa?ieienoneeie
o?aaiaieyie aeaeaeaiey. E aiiaue – nouanoaothuea oai?ee oaina aiieia
yooaeoeaii ?aaioatho e iienuaatho oaeea nenoaiu, ii “iinoa” e
eeanne/aneie /anoe oeceee iao. Ioeoaea aa??ony oain a aeaoa?ieie?iaaiiuo
nenoaiao?

Aeaiiay ?aaioa iinayuaia acaeyaeai ia yoe aii?inu, ?acaeaaaiui oae
iacuaaaiie “a?thnnaeueneie oeieie”, eaeaeiui ?oeiaiaeeoaeai eioi?ie
yaeyaony ecaanoiue aeioecee, neia?aaoee, eao?aao Iiaaeaaneie i?aiee ii
oeiee ca 1977 a. Eeuey I?eaiaeei.

Iniiaiay iniaaiiinoue iao/iie eiioeaioeee, ?acaeaaaiie E.I?eaiaeeiui –
iaia?aoeiinoue a?aiaie ia iee?ineiie/aneii o?iaia. Ia io?eoeay ie
caeiiia, ie ?acoeueoaoia o?aaeeoeeiiiie oeceee, I?eaiaeei i?aaeeaaaao
iiaoth eioa?i?aoaoeeth yoeo ?acoeueoaoia. Oaoie/anee yoi au?aaeaaony eae
iiene ?aoaiee an? oao aea o?aaiaiee (o?aaiaiee Aaieeueoiia, Eeoaeeey,
O??aeeiaa?a e o.ae.) – ii a iiaii eeanna ooieoeee, a iiaii
ooieoeeiiaeueiii i?ino?ainoaa.

A ?acaeaea 1 ianoiyuae ?aaiou ?anniao?eaathony i?eia?u eeanne/aneiai
aeeiaie/aneiai oaina a i?inoaeoeo iaoaiaoe/aneeo iiaeaeyo naeaeaa
Aa?ioeee e i?aia?aciaaiey iaea?y (iaonoie/eainoue ii ia/aeueiui
oneiaeyi), a oaeaea ooiaeaiaioaeueiia naienoai iaeioaa?e?oaiinoe iiiaeo
aeeiaie/aneeo nenoai (oai?aia Ioaiea?a), oaeaea i?eaiaeyuaa e
oaioe/aneiio iiaaaeaieth.

?acaeae 2 iinayu?i i?iaeaiai naiaeeiinoe “iae?ineiie/aneiai” oaina e
“iee?ineiie/aneiio” e i?iaeaia ia?aoeiinoe a?aiaie. Nouanoaaiii, /oi e a
eeanne/aneie iaoaieea, e a eiiaiaaaaineie eioa?i?aoaoeee eaaioiaie
iaoaieee iienaiea iaia?aoeiiai iiaaaeaiey iae?ineiie/aneeo nenoai
enoiaey ec ia?aoeiuo iee?ineiie/aneeo caeiiia iaoaeeeaaaony ia
nouanoaaiiua o?oaeiinoe.

A ?acaeaea 3 ae?aooea iienaiu iniiaiua eioa?i?aoaoeee eaaioiaie
iaoaieee: eiiaiaaaaineay, noaoenoe/aneay, iiiaiie?iaay eioa?i?aoaoeey
Yaa?aooa. Iniiaiia aea aieiaiea oaeaeyaony a?thnnaeueneie eioa?i?aoaoeee
eaaioiaie iaoaieee, ?acaeaaaiie E.I?eaiaeeiui. Iniaaiiinoe a?
iaoaiaoe/aneiai aiia?aoa iiyniythony ia i?inouo i?eia?ao aeeiaie/aneeo
nenoai, oaea ?anniio?aiiuo a i?aaeuaeoueo ?acaeaeao. Iauay eiioeaioeey
iaoieoa?iie yaiethoeee i?eaiaeeo e oiio, /oi aaeeinoaaiii aaeaeaaoiui
noaiiaeony noaoenoe/aneia iienaiea nenoai – eae eeanne/aneeo, oae e
eaaioiauo. Aeey neo/ay iineaaeieo i?iyniythony iaeioi?ua ecaanoiua
ia?aaeienu ecaanoiuo eioa?i?aoaoeee eaaioiaie iaoaieee, naycaiiua n
?ieueth aiaoiaai iaaethaeaoaey.

E niaeaeaieth, eaeae E.I?eaiaeeia o?aaotho aeey naiaai eceiaeaiey
(aeaaea a iiioey?iii aeaea) nouanoaaiiiai eniieueciaaiey iaoaiaoe/aneiai
aiia?aoa, /oi i?eaaei e iaeioi?ie ia?aa?oaeaiiinoe oaenoa oi?ioeaie.
Aaoi?, iaeiaei, iaaeaaony, /oi “ean” ca “aea?aaueyie” ia ne?ueny, e
iniiaiua iieiaeaiey oece/aneie eiioeaioeee A?thnnaeueneie oeieu iaoee
io?aaeaiea a ianoiyuae ?aaioa.

1. OAIN

Eo eeaa aeeciue e iaiaeath oain…

E.A?iaeneee, “Aeaa /ana a ?aca?aoa?a”

1.1 Eeanne/aneee aeeiaie/aneee oain: iaonoie/eainoue ii ia/aeueiui
oneiaeyi

Oaioe/aneia iiaaaeaiea iiaeao aicieeaoue aeaaea a i/aiue i?inouo
nenoaiao, iai?eia?, ec oece/aneeo iiaeaeae – a eieaaaieyo noa?e/aneiai
iayoieea n aeaoiy noaiaiyie naiaiaeu. Iu aeey ia/aea ?anniio?ei aeaaea
au? aieaa i?inoua iaoaiaoe/aneea iiaeaee n aeene?aoiui a?aiaiai – naeaea
Aa?ioeee e i?aia?aciaaiea iaea?y.

Naeaea Aa?ioeee i?aaenoaaeyao niaie ioia?aaeaiea a iaeiiia?iii
i?ino?ainoaa ia eioa?aaea (0,1) ii caeiio

xn+1=2xn(mod1).

Yoi o?aaiaiea aeaeaeaiey aeaoa?ieienoe/ii: ii caaeaiiiio xn iaeiicia/ii
au/eneyaony xn+1. I?e yoii, iaeiaei, naeaea Aa?ioeee ia yaeyaony
ia?aoeiui ioia?aaeaieai. Neiiao?ey ai a?aiaie ia?ooaia au? ia o?iaia
o?aaiaiey aeaeaeaiey. Yoei naeaea Aa?ioeee ioee/aaony io aeeiaie/aneeo
nenoai n ia?aoeiuie o?aaiaieyie aeaeaeaiey.

Naeaea Aa?ioeee i?aaenoaaeyao niaie i?eia? aeaoa?ieienoe/aneiai oaina.
Iiaeii i?aaenoaaeoue i?eia?u iineaaeiaaoaeueiinoae, ia/eiathueony n
eaeiai-ieaoaeue i?iecaieueiiai /enea, iai?eia?:

{0.13; 0.26; 0.52; 0.04; 0.08; 0.16; 0.32; 0.64; 0.28… }

e

{0.14; 0.28; 0.56; 0.12; 0.24; 0.48; 0.96; 0.92; 0.84… } –

eae aeaeei, iacia/eoaeueiia ioee/ea a ia/aeueiuo oneiaeyo oaea ia 4-i
oaaa ii?iaeaeaao nouanoaaiiia ?acee/ea o?aaeoi?ee, a a aeaeueiaeoai eo
iiaaaeaiea niaa?oaiii ?acee/ii.

Eaaei iieacaoue, /oi ni a?aiaiai ?acieaeoony o?aaeoi?ee ethauo aeaoo
neieue oaiaeii aeeceeo oi/ae. Caieoai /enei x a aeaea aeaie/iie ae?iae:

x=0.u–1u–2u–3…u–k…=u–1/2 + u–2/22 + u–3/23 + … + u–k/2k + …

Iienaiiia auoa ioia?aaeaiea niioaaonoaoao naeaeao u–k’=u–(k+1) , ioeoaea
noaiiaeony iiiyoiui iacaaiea “naeaea Aa?ioeee”. Aeaeii, /oi ioeaaie
?ac?yae /enea i?e yoii oa?yaony, /oi niioaaonoaoao
ia-acaeiiiaeiicia/iinoe ioia?aaeaiey.

Iienaiea yaiethoeee aeeiaie/aneie nenoaiu oeia naeaeaa Aa?ioeee a
oa?ieiao o?aaeoi?ee iaaaeaeaaoii, oae eae aeey aaeaeaaoiinoe o?aaeoi?ey
aeieaeia inoaaaoueny “ii/oe iaeiie e oie aea” i?e iacia/eoaeueiii
eciaiaiee ia/aeueiuo oneiaee.

A aeaiiii aea neo/aa eiaao niune ia?aoeoueny e noaoenoe/aneiio
iienaieth, aaaaey ieioiinoue aa?iyoiinoe ((x) i?aauaaiey nenoaiu a
eaaeaeie oi/ea x eioa?aaea (0,1). Ioia?aaeaiea i?aaenoaaeyao niaie
iia?aoi? U, aeaenoaothuee ia yoo ooieoeeth:

(n+1=U(n(x)= ( (n(x/2)+(n((x+1)/2) ) / 2.

Ieacuaaaony, /oi i?e iiiaie?aoiii i?eiaiaiee iia?aoi?a ioia?aaeaiey e
i?iecaieueiiio ?ani?aaeaeaieth ieioiinoe aa?iyoiinoe iii no?aieony e
eiinoaioa:

(n=Un(0(x)((((x)=const.

A aeaeueiaeoai iu au? aa?iainy e ioia?aaeaieth Aa?ioeee e naienoaai aai
iia?aoi?a, a iiea ?anniio?ei ae?oaoth i?inooth aeeiaie/aneoth nenoaio,
oaia?ue oaea aeaoia?ioth, iacuaaaioth i?aia?aciaaieai iaea?y:

I?aaeei, ii?aaeaeythuaa i?aia?aciaaiea iaea?y, i/aiue i?inoi. Nia/aea
eaaae?ao ni noi?iiie, ?aaiie 1, niethueaaaony a i?yiioaieueiee aeeeiie 2
e aunioie 1/2, caoai i?aaay iieiaeia iieo/aiiiai i?yiioaieueieea
iaeeaaeuaaaony ia eaaoth, ia?acoy iiaue eaaae?ao. I?ioeann a /?i-oi
aiaeiae/ai ?aciaoeaaieth oanoa, ionthaea e iacaaiea.

A ioee/ea io naeaeaa Aa?ioeee i?aia?aciaaiea iaea?y ia?aoeii ai a?aiaie.
Iaeiaei iii oi/ii oae aea ii?iaeaeaao oaioe/aneia aeaeaeaiea, naycaiiia
n iaonoie/eainoueth ii ia/aeueiui oneiaeyi.

I?aia?aciaaiea iaea?y naiaeeony e naeaeao a aeaonoi?iiiae aeaie/iie
iineaaeiaaoaeueiinoe:

x0y = ….u–k…u–3u-2u–1u0u1u2…uk….,

uk’ = u–(k+1).

Aeaeii, /oi i?e yoii ieeaeea aeaie/iua ?ac?yaeu ia oa?ythony, /oi e
niioaaonoaoao ia?aoeiinoe i?aia?aciaaiey iaea?y ai a?aiaie.

Aiaeiae/ii naeaeao Aa?ioeee, i?aia?aciaaiea iaea?y ii?iaeaeaao
aeeiaie/aneee oain, e iienaiea aeaeaeaiey oi/ee a oa?ieiao o?aaeoi?ee
oaeaea iaaaeaeaaoii.

A neo/aa i?aia?aciaaiey iaea?y iienaiea yaiethoeee nenoaiu a
noaoenoe/aneeo oa?ieiao aeaaea aieaa “oece/anee iniuneaiii”, /ai aeey
naeaeaa Aa?ioeee. Aeaei a oii, /oi oaia?ue, a aeaoia?iii neo/aa, iiaeii
?anniao?eaaoue eii?aeeiaoioth ieineinoue eae oaciaia i?ino?ainoai
iaeioi?ie aeeiaie/aneie nenoaiu n iaeiie noaiaiueth naiaiaeu: inue x
niioaaonoaoao eii?aeeiaoa, a inue y – eiioeueno. Aiaeiaey n
“oece/aneeie” aeeiaie/aneeie nenoaiaie oneeeaaaony au? e oai, /oi
auiieiyaony oai?aia Eeoaeeey: nio?aiyaony iau?i a oaciaii i?ino?ainoaa.
Ae?oaeie neiaaie, acya ainaiaeue oi/ae aioo?e iaeioi?ie iaeanoe e
i?iaeaeaa i?iecaieueiia eiee/anoai i?aia?aciaaiee iaea?y, iu iaia?oaeei
oiaea naiia eiee/anoai oi/ae aioo?e iaeioi?ie ae?oaie iaeanoe (oi?ia a?
i?e yoii i/aiue neeueii eciaieony e noaiao e?aeia caiuneiaaoie). Iau?i
yoie iaeanoe (a iaoai aeaoia?iii neo/aa aio niioaaonoaoao ieiuaaeue)
inoaiaony iaeciaiiui.

Ianiio?y ia ia?aoeiinoue i?aia?aciaaiey iaea?y ai a?aiaie, yaiethoeey
i?e t ( +( e i?e t ( –( ieacuaaaony ?acee/iie [1,c.114].

E?iia iienaiiuo auoa, nouanoaoao au? iiiai n?aaieoaeueii i?inouo
iiaeaeae aeeiaie/aneiai oaina. Iaeiaei iu aicaea?aeeiny io eo
iiae?iaiiai ?anniio?aiey, e ia?aeae?i oaia?ue e i?e/eiai, eaaeauei a
iniiaa iai?aaeneacoaiiai iiaaaeaiey oece/aneeo nenoai.

1.2 Eeanne/aneee oain: iaeioaa?e?oaiua nenoaiu Ioaiea?a

*ai i?inoia ioee/aaony io neiaeiiai? O?aaeeoeeiiiue ioaao niaea?aeeo
nnueeo ia ea?a?oeth. Ia iaeiii eiioea oeaeu iu iaoiaeei oaeea iauaeou,
eae iayoiee, iiae/eiythueeny i?inoui aeaoa?ieienoneei caeiiai. Ia
ae?oaii eiioea oeaeu iaoiaeyony ethaee e eo niiauanoaa. Iaaeaeo yoeie
iiethnaie iiaeii iuneaiii aienaoue oeaeoth ea?a?oeth “eiiieaeneoeeaoeee”
– aicieeiiaaiey neiaeiiai ec i?inoiai. A aeaenoaeoaeueiinoe aea aeaei
ianoieo aeaaea aieaa oiiei: i?inoia e neiaeiia iiaoo ninouanoaiaaoue
aianoa, ia aoaeo/e naycaiu iaaeaeo niaie ea?a?oe/anee.

*oi eanaaony /aeiaa/aneeo niiauanoa, oai?ey eo iiaaaeaiey e?aeia o?oaeii
iiaeaea?ony oioue eaeie-ieaoaeue iaoaiaoecaoeee e caneoaeeaaao
ioaeaeueiiai ?anniio?aiey, aia ?aiie ianoiyuae ?aaiou. I?eia? aea
oaioe/aneiai iiaaaeaiey i?inoaeoeo oece/aneeo nenoai oeia iayoieea
aoaeao ?anniio?ai ieaea.

I?e enneaaeiaaiee oiai, eae i?inoia ioiineony e neiaeiiio, iau/ii oe?iei
eniieuecoaony iiiyoea aoo?aeoi?a, oi anoue eiia/iiai ninoiyiey eee oiaea
yaiethoeee aeenneiaoeaiie nenoaiu. Niune yoiai iiiyoey aue aeoaiei
i?aia?aciaai nia?aiaiiie oeceeie e iaoaiaoeeie. A i?ioeii n/eoaeinue,
/oi ana nenoaiu, yaiethoeey eioi?uo naycaia n nouanoaiaaieai aoo?aeoi?a,
iaeeiaeiau. Iuia iiiyoea aoo?aeoi?a naycuaatho n ?aciiia?aceai
aeenneiaoeaiuo nenoai.

Eaeaaeueiue iayoiee aac o?aiey ia eiaao aoo?aeoi?a e eieaaeaony
aaneiia/ii. N ae?oaie noi?iiu, aeaeaeaiea ?aaeueiiai iayoieea –
aeenneiaoeaiie nenoaiu, aeaeaeaiea eioi?ie aeeth/aao o?aiea, –
iinoaiaiii inoaiaaeeaaaony a iieiaeaiee ?aaiiaaney. Yoi iieiaeaiea
yaeyaony aoo?aeoi?ii. Aiaeiae/iui ia?acii, aoo?aeoi?ii yaeyaony e
ninoiyiea oa?iiaeeiaie/aneiai ?aaiiaaney: ainaiaeue ec ieeeea?aeia e
ieeeea?aeia /anoeoe, ia?acothueo eciee?iaaiioth nenoaio,
yaiethoeeiie?oao e ninoiyieth ?aaiiaaney, iienaiea eioi?iai caaeneo
eeoue io iaiiiaeo ia?aiao?ia, oaeeo eae oaiia?aoo?a e aeaaeaiea.

Eaeaaeueiue iayoiee neoaeeo i?eia?ii oae iacuaaaiie no?oeoo?iie
iaonoie/eainoe: a ionoonoaea o?aiey aoo?aeoi? ia nouanoaoao, ii
aaaaeaiea aeaaea naiiai iacia/eoaeueiiai o?aiey eciaiyao aeaeaeaiea
iayoieea e aaiaeeo aoo?aeoi?.

*oiau i?aaenoaaeoue aoo?aeoi? aaiiao?e/anee, iau/ii aaiaeyo
i?ino?ainoai, ?acia?iinoue eioi?iai niaiaaeaao n /eneii ia?aiaiiuo,
iaiaoiaeeiuo aeey iienaiey nenoaiu. Yoi iiaoo auoue eii?aeeiaou,
eiioeuenu, ?acee/iua oa?iiaeeiaie/aneea ia?aiaiiua. Ai aaaae?iiii
i?ino?ainoaa ?aaiiaaniia ninoiyiea aeenneiaoeaiuo nenoai niioaaonoaoao
oi/a/iiio aoo?aeoi?o. Oi aea ioiineony e e noaoeeiia?iui ninoiyieyi
nenoai, aeeceeo e oa?iiaeeiaie/aneiio ?aaiiaaneth e oaeiaeaoai?ythuei
oai?aia i ieieiaeueiii i?iecaiaenoaa yio?iiee. Ai anao neo/ayo, eaeiai
au ie auei ia?aiia/aeueiia i?eaioiaeaiea nenoaiu, a? yaiethoeey iiaeao
auoue iienaia o?aaeoi?eae, aaaeouae ec oi/ee, eioi?ay i?aaenoaaeyao
ia/aeueiia ninoiyiea, e aoo?aeoi?o. Oaeei ia?acii, eiia/iay oi/ea –
aoo?aeoi? – i?aaenoaaeyao niaie oeiaeueiia ninoiyiea anao o?aaeoi?ee.

Ia ana aeenneiaoeaiua nenoaiu i?eaiaeyo e iaeiie-aaeeinoaaiiie eiia/iie
oi/ea. Iai?eia?, neeueii ia?aaiiaaniay aeenneiaoeaiay no?oeoo?a,
ecaanoiay iiae iacaaieai “oeie/aneea /anu”, yaiethoeeiie?oao ia e
eaeiio-ieaoaeue ninoiyieth, a e onoie/eaiio ia?eiaee/aneiio ?aaeeio.
Oaeay neooaoeey i?eaiaeeo e iaiaoiaeeiinoe iaiauaiey eaeae aoo?aeoi?a:
aoo?aeoi? aieaa ia oi/ea, a eeiey, iienuaathuay ia?eiaee/aneia ai
a?aiaie eciaiaiea eiioeaio?aoeee oeie/aneeo aauanoa. I?eia?u iiaeiaiuo
aoo?aeoi?ia eaaei iaeoe, iai?eia?, e a ?aaeeioeceea – eie yaeythony
i?aaeaeueiua oeeeeu aaoiaaia?aoi?ia, – e ai iiiaeo ae?oaeo ?acaeaeao
anoanoaiciaiey.

Nenoaia n i?aaeaeueiui oeeeeii inoa?ony i?aaeneacoaiie e iioiio
aeiioneaao i?inoia iienaiea. Ii ca yoie i?inoioie e?ithony iaiaeeaeaiiua
naienoaa. Iao?oaeii i?aaenoaaeoue naaa oeie/aneia ?aaiiaanea –
iiiaeanoai oeie/aneeo i?ioeannia, eiiiaine?othueo ae?oa ae?oaa iiaeiaii
oiio, eae a ninoiyiee aeaiia?aoe/aneiai ?aaiiaaney ?iaeaeaaiinoue
eiiiaine?oao nia?oiinoue. Ii aiia?aaeaiea aanneeueii i?aaenoaaeoue naaa,
eae ia?iiiua eiee/anoaa iieaeoe, acaeiiaeaenoaothueo oieueei /a?ac
noieeiiaaiey, ia/eiatho aae?oa aeaenoaiaaoue “ae?oaeii” – oae, /oi
n?aaea ia?eiaee/anee eciaiyao naie oeaao.

A ae?oaeo neo/ayo, iuoaynue iino?ieoue ecia?aaeaiea aoo?aeoi?a, iu
iieo/ei ia oi/eo eee caieioooth eeieth, a iiaa?oiinoue eee iau?i.
Iiai?ioiui aea niauoeai noaei ioe?uoea aoo?aeoi?ia, ia ioiinyueony e
noieue i?inoui aaiiao?e/aneei iauaeoai – oae iacuaaaiui no?aiiuo
aoo?aeoi?ia. A ioee/ea io eeiee eee iiaa?oiinoe, no?aiiua aoo?aeoi?u
i?aaenoaaeytho niaie o?aeoaeueiua iauaeou, oa?aeoa?ecothueany ae?iaiie
?acia?iinoueth.

No?aiiua aoo?aeoi?u auee iaia?oaeaiu a iiaaaeaiee iiiaeo aeeiaie/aneeo
nenoai, iienuaaaiuo aeaoa?ieienoe/aneeie o?aaiaieyie aeaeaeaiey.
Iai?eia?, iie aicieeatho aeey oae iacuaaaiiai noa?e/aneiai iayoieea –
iaueiiaaiiiai a?oceea ia ieoea, eioi?ue niaa?oaao eieaaaiey ia a
ieineinoe, a ii iiaa?oiinoe iieonoa?u. I?e aianaiee aiciouaiee a aeaea
eieaaaiee oi/ee iiaeaana a iaeioi?ue e?eoe/aneee iiiaio (caaenyuee io
/anoiou aiciouaiey) aeaeaeaiea iayoieea noaiiaeony oaioe/aneei, a aai
o?aaeoi?ey iienuaaaony no?aiiui aoo?aeoi?ii [1, n.83].

Ei??aeyoeeiiiue aiaeec a?aiaiiu’o iineaaeiaaoaeueiinoae,
oa?aeoa?ecothueo ?aaioo /aeiaa/aneiai iicaa, eciaiaiey eeeiaoa ia
ieaiaoa ca ieeeeiiu eao e eo?na aeoeee ia ae?aea oaeaea i?eaiaeeo e
iaia?oaeaieth no?aiiuo aoo?aeoi?ia. Ai?i/ai, i?e iaee/ee ia?iiiiai
eiee/anoaa aiaoieo i?e/ei, aeeythueo ia iiaaaeaiea anao yoeo nenoai,
neo/aeiinoue eo iiaaaeaiey a?iaea au oaeeaeaiey ia aucuaaao, iiyoiio
iiea ia?aoei aieiaiea ia aieaa caaaaei/iia yaeaiea. Ioeoaea aicieeaao
oaioe/aneia iiaaaeaiea a neo/aa noa?e/aneiai iayoieea?

Eae auei iieacaii auoa, oaioe/aneia iiaaaeaiea ioia?aaeaiee oeia naeaeaa
Aa?ioeee naycaii n iaonoie/eainoueth ii ia/aeueiui oneiaeyi, a
iaia?aoeiinoue eo ai a?aiaie – n iioa?ae eioi?iaoeee i?e naeaeaa
aeaie/iie caiene /enea. Iiaeii, iaeiaei, aic?aceoue, /oi i?eaaae?iiua
i?eia?u ioia?aaeaiee ianeieueei eneonnoaaiiu, oae eae a i?e?iaea ia
ano?a/aaony iiaeiaiuo aeene?aoiuo i?ioeannia, aea e “au/eneeoaeueiie
iiuiinoe” i?e?iaeu ia oaaoeo ia auiieiaiea noieue ioae??iie iia?aoeee,
eae iiaeoeueiay a?eoiaoeea.

Ieacuaaaony, iaeiaei, /oi e ia o?iaia ?aoaiey iau/iuo o?aaiaiee
aeaeaeaiee (auoaeathueo ec caeiiia Iuethoiia) aeey oiai aea iayoieea
aiciiaeii iieo/aiea iaonoie/eauo ?aoaiee, naycaiiuo n oae iacuaaaiie
iaeioaa?e?oaiinoueth nenoaiu ii Ioaiea?a.

Iniiaiay i?iaeaia eeanne/aneie iaoaieee ninoieo a ?an/?oa aeaeaeaiey
acaeiiaeaenoaothueo oae ia iniiaa eo o?aaiaiee aeaeaeaiey (a /anoiii
neo/aa, iai?eia?, yoi iiaeao auoue caeii Iuethoiia F=ma). Iaiauaiea
iuethoiiianeie iaoaieee ia aieaa neiaeiua nenoaiu iieacaei, /oi aieaa
oaeiaiie oi?iie iienaiey yaeyaony ia caaeneiinoue io a?aiaie
i?ino?ainoaaiiie o?aaeoi?ee nenoaiu (a iaoai i?eia?a – eii?aeeiaou), a
aeaeaeaiea oi/ee, ecia?aaeathuae nenoaio, a i?ino?ainoaa aaeaia aieueoae
?acia?iinoe, /ai iau/iia “oece/aneia”. A iauai neo/aa ninoiyiea
aeeiaie/aneie nenoaiu iienuaaaony eii?aeeiaoaie q1, …, qs, eioi?ua
yaeythony iacaaeneiuie ia?aiaiiuie, e niioaaonoaothueie ei eiioeuenaie
p1, …, ps. I?aeiouanoaii oaeiai iiaeoiaea yaeyaony nouanoaaiiia
oi?iuaiea o?aaiaiee aeaeaeaiey.

), e /oi eii?aeeiaou, caaeathuea iieiaeaiea /anoeoe, – eeiaeiua
ooieoeee a?aiaie. Yoio o?eaeaeueiue neo/ae ea?aao, oai ia iaiaa, aanueia
aaaeioth ?ieue a iauae i?iaeaia eioaa?e?iaaiey aaieeueoiiiauo o?aaiaiee
aeaeaeaiey.

, aaea k – ae?noeinoue i?oaeeiu, q – niauaiea a?oca io iieiaeaiey
?aaiiaaney. *oiau oi?inoeoue o?aaiaiey aeaeaeaiey, aaaae?i iiaua
ia?aiaiiua ( e J aianoi noa?uo q e p:

,

,

, iia iaiyaony eeiaeii ii a?aiaie.

Ia?aoiae io ia?aiaiiuo p, q e ia?aiaiiui J, ( iacuaaaony eaiiie/aneei
i?aia?aciaaieai. A aeaiiii neo/aa iii iicaieeei eneeth/eoue ec
aaieeueoiieaia /eai, ioaaonoaaiiue ca iioaioeeaeueioth yia?aeth.
Aiaeiae/iia i?aia?aciaaiea iiaeii eiiaaea i?iaeaeaoue e a neo/aa nenoaiu
ni iiiaeie noaiaiyie naiaiaeu, eneeth/ea ec aaieeueoiieaia iaae/anoe/iia
acaeiiaeaenoaea, e au?aceoue aeaeaeaiea a oeeeee/aneeo ia?aiaiiuo. Eo
iacaaiea ioiineony e ia?eiaee/aneiio oa?aeoa?o aeaeaeaiey, eioi?ue
aeaeaaony yaiui a oaeeo ia?aiaiiuo.

Iniaoth aaaeioth ?ieue ea?atho /anoiou nenoaiu (1, (2, …, (n. Eiaiii
/a?ac yoe /anoiou iu i?eoiaeei e iiiyoeth ?aciiaina, eiathuaai ?aoathuaa
cia/aiea aeey oai?aiu Ioaiea?a.

Aeaeaeaiea eioaa?e?oaiie nenoaiu n aeaoiy noaiaiyie naiaiaeu iiaeii
i?aaenoaaeoue ia oi?a. Aiciiaeiu aeaa neooaoeee. Anee aeey iaeioi?uo
oeaeuo n1 e n2 auiieiyaony oneiaea n1(1+ n2(2=0, oi anoue /anoiou
niecia?eiu, iu eiaai ?aciiain, e aeaeaeaiea ia oi?a ia?eiaee/aneia –
o?aaeoi?ey caieiooay. Anee aea yoa noiia ie i?e eaeeo eiiaeiaoeeyo n1 e
n2 ia ?aaia ioeth, oi o?aaeoi?ey iaaeaaaony ia iiaa?oiinoue oi?a e
ieeiaaea ia caiueaaony. A eiioea eiioeia, eae iieacaii Ioaiea?a, oaeay
o?aaeoi?ey i?ioiaeeo neieue oaiaeii aeecei e i?iecaieueiie oi/ea ia
iiaa?oiinoe oi?a. O?aaeoi?ey i?e yoii iacuaaaony anthaeo ieioiie, a
aeaeaeaiea – eaaceia?eiaee/aneei. Eaaceia?eiaee/aneia aeaeaeaiea i/aiue
neiaeii auaeyaeeo, ii ia naiii aeaea yaeyaony aiieia aeaoa?ieie?iaaiiui.

Aei Ioaiea?a iieaaaeinue, /oi ana aeeiaie/aneea nenoaiu yaeythony
eioaa?e?oaiuie. Iaeiaei a 1889 a. Ioaiea?a iieacae, /oi a iauai neo/aa
iaaiciiaeii iieo/eo eaiiie/aneia i?aia?aciaaiea, nio?aiythuaa aeae
aaieeueoiiiauo o?aaiaiee, eioi?ia i?eaiaeeei au e oeeeee/aneei
ia?aiaiiui. Iai?eia?, nenoaia aeaoo oae (Caiey – Nieioea) eioaa?e?oaia,
a aio nenoaia o??o oae (Caiey – Nieioea – THieoa?) iaeioaa?e?oaia.
Ei?i/a aiai?y, iiaeaaeythuaa aieueoeinoai aeeiaie/aneeo nenoai
iaeioaa?e?oaiu.

Aeaiiay ?aaioa ia iinayuaia aiaeeco iaoaiaoe/aneeo iaoiaeia, eioi?uie
Ioaiea?a aeieacuaae naith oai?aio. Ioiaoei oieueei, /oi ii noi?ioee?iaae
naie aii?in a oa?ieiao oai?ee aiciouaiee, oi anoue iuoaeny aeey
aaieeueoiieaia aeaea

H(J,() = H0(J)+(V(J,()

ii?aaeaeeoue iiaua ia?aiaiiua aeaenoaey J’ aeaea J’ = J + (J1 + (2J2+
…, aiaeeoe/anee ia?aoiaeyuea a enoiaeiua i?e no?aieaiee eiinoaiou
nayce ( (ia?aiao?a, ii?aaeaeythuaai eioaineaiinoue acaeiiaeaenoaey) e
ioeth. Anee oaeay caiaia aiciiaeia, oi iu iiaeai eneeth/eoue
iioaioeeaeueioth yia?aeth aiciou?iiie nenoaiu e aaanoe iiaue
aaieeueoiieai, caaenyuee oieueei io J’. Eioaa?e?iaaiea aiciou?iiie
nenoaiu auei au a yoii neo/aa noieue ae i?inoui, oae eae iiaua
ia?aiaiiua aeaenoaey auee au iinoiyiiuie aeaeaeaiey. Iaeiaei Ioaiea?a
iieacae, /oi oaeay caiaia aiciiaeia aeaeaei ia anaaaea.

I?aaeiieiaeei, /oi Ioaiea?a oaeaeinue au aeieacaoue eioaa?e?oaiinoue
anao aeeiaie/aneeo nenoai. Yoi icia/aei au, /oi ana aeeiaie/aneea
aeaeaeaiey eciii?oiu aeaeaeaieth naiaiaeiuo (ia acaeiiaeaenoaothueo)
/anoeoe. ?acoiaaony, oaeay iiaeaeue ia inoaaeyao ieeaeiai ianoa aeey
aiciiaeiinoe iae?ii?ioeannia, eioi?ua iu iaaethaeaai aaeaieiooii. A
eioaa?e?oaiii ie?a ia iaoeinue au ianoa ie aeey naiii?aaiecaoeee, ie
aeey eiaa?aioiinoe (a neo/aa, iai?eia?, aeenneiaoeaiiai oaina).

Ioaiea?a ia oieueei aeieacae iaeioaa?e?oaiinoue, ii e oeacae ia a?
i?e/eio, a eiaiii – ia nouanoaiaaiea ?aciiainia iaaeaeo noaiaiyie
naiaiaeu nenoaiu. Eiaiii ?aciiainu neeueii naycuaatho noaiaie naiaiaeu e
ia aeatho aiciiaeiinoue eneeth/eoue acaeiiaeaenoaea. A ea/anoaa i?eia?a
?anniio?ei nenoaio n aeaoiy noaiaiyie naiaiaeu, aaieeueoiieai eioi?ie
eiaao aeae

H = H0(J1,J2)+(V(J1,J2,(1,(2),

i?aaenoaaeiue a aeaea noiiu iaaiciou?iiiai eioaa?e?oaiiai aaieeueoiieaia
e iaeiai aiciouaiey (V. Eae iieacae Ioaiea?a, oai?ey aiciouaiee
iaecaaaeii i?eaiaeeo e iiyaeaieth /eaiia n “ieaniuie” ciaiaiaoaeyie
aeaea 1/(n1(1+n2(2). Anee /anoiou niecia?eiu e nouanoaotho ?aciiainu, oi
/eaiu ?yaea oai?ee aiciouaiee ?anoiaeyony, e ei i?eoiaeeony i?eienuaaoue
cia/aiea, ?aaiia aaneiia/iinoe. Ii yoi icia/aao, /oi a oeceea iienaiey
/oi-oi “ia oae”!

I?iaeaia iaeuo ciaiaiaoaeae auea ecaanoia au? ano?iiiiai a XIX a.
Oai?aia Ioaiea?a iieacaea, /oi iniiaiay o?oaeiinoue – iiyaeaiea
?anoiaeeiinoae a ?aoaiee caaea/ aeeiaieee – ia iiaeao auoue ono?aiaia e
aeaeaao iaaiciiaeiui aaaaeaiea oeeeee/aneeo ia?aiaiiuo aeey aieueoeinoaa
aeeiaie/aneeo i?iaeai, ia/eiay n i?iaeaiu o??o oae.

Ioe?uoea iaeioaa?e?oaiinoe aucaaei ii?aaeae?iiue ianneieci e iaaeioiaiea
a ?yaeao iiiaeo oeceeia. Iaen Ai?i, iai?eia?, caiaoee: “Auei au aanueia
no?aiii, anee au I?e?iaea oe?ueanue io aeaeueiaeoaai i?ia?anna iiciaiey
ca aiaeeoe/aneeie o?oaeiinoyie i?iaeaiu iiiaeo oae”. Oieueei n
iiyaeaieai ?aaio Eieiiai?iaa, i?iaeieaeaiiuo A?iieueaeii e Iica?ii (oae
iacuaaaiie oai?ee EAI), i?iaeaio iaeioaa?e?oaiinoe ia?anoaee ioeaieaaoue
eae nii?ioeaeaiea I?e?iaeu i?ia?anno ciaiey, a ia/aee ?anniao?eaaoue eae
iiaue ioi?aaiie ioieo aeaeueiaeoaai ?acaeoey aeeiaieee.

Oai?ey EAI ?anniao?eaaao aeeyiea ?aciiainia ia o?aaeoi?ee. I?inoie
neo/ae aa?iiie/aneiai inoeeeeyoi?a n iinoiyiiie /anoioie, ia caaenyuae
io ia?aiaiiuo aeaenoaey J, yaeyaony eneeth/aieai: /anoiou, aiiaua
aiai?y, caaenyo io cia/aiee, i?eieiaaiuo ia?aiaiiuie aeaenoaey. A iinaio
a iaeieo oi/eao oaciaiai i?ino?ainoaa aeeiaie/aneie nenoaiu ?aciiain
iiaeao nouanoaiaaoue, a a ae?oaeo – iao. ?aciiainu niioaaonoaotho
?aoeeiiaeueiui niioiioaieyi iaaeaeo /anoioaie, eeanne/aneee aea
?acoeueoao oai?ee /enae aiai?eo, /oi ia?a ?aoeeiiaeueiuo /enae ii
n?aaiaieth n ia?ie e??aoeeiiaeueiuo ?aaia ioeth. Yoi icia/aao, /oi
?aciiainu ano?a/athony e?aeia ?aaeei. E?iia oiai, a ionoonoaea
aiciouaiee, eae auei neacaii auoa, ?aciiainu i?eaiaeyo e ia?eiaee/aneiio
aeaeaeaieth, a a iauai neo/aa iu eiaai eaaceia?eiaee/aneia aeaeaeaiea
(ia?aciiainiua oi?u). ?acthie?oy, iiaeii neacaoue, /oi ia?eiaee/aneea
aeaeaeaiey – ia i?aaeei, a eneeth/aiea.

(Eioa?anii auei au i?aaeiieiaeeoue, eaeeie iooyie ?acaeaaeanue au
yaiethoeey aeecie ia Caiea, anee au aeaeaeaiea Caiee aie?oa Nieioea ia
iineei ia?eiaee/aneiai oa?aeoa?a. Aiciiaeia ee, iai?eia?, aeeciue a
oneiaeyo ieaiaoiie nenoaiu aeaieiie caacaeu? Aaoi? ?aoa?aoa iieaaaao,
/oi anee “e?aeiea” oneiaey, a eioi?ua iiiaaeaea au oaeay ieaiaoa, ia
auee neeoeii oae ae?noeeie, oi aeeciue iaoea au aiciiaeiinoue
i?eniiniaeoueny e yaiethoeey auea au an?-oaee aiciiaeia. Iaeiaei ana yoe
?annoaeaeaiey iniiaaiu eeoue ia iioeiecia aaoi?a e aai aa?a a aeoaieoth
i?eniiniaeyaiinoue anaai aeeaiai e aiaoiei oneiaeyi, e eiatho e?aeia
iaei ioiioaiey e iauyaeaiiie a caaeaaee oaia ?aaiou).

I?e aaaaeaiee aiciouaiee oa?aeoa? aeaeaeaiey ia ?aciiainiuo oi?ao ?acei
eciaiyaony (ii oai?aia Ioaiea?a), a oi a?aiy eae eaaceia?eiaee/aneia
aeaeaeaiea eciaiyaony iacia/eoaeueii, ii e?aeiae ia?a, i?e iaeii
ia?aiao?a aiciouaiey (. Iniiaiie ?acoeueoao oai?ee EAI ninoieo a oii,
/oi oaia?ue iu eiaai aeaa niaa?oaiii ?acee/iuo oeia o?aaeoi?ee: neaaea
eciaieaoeany eaaceia?eiaee/aneea o?aaeoi?ee e noioanoe/aneea o?aaeoi?ee,
aicieeoea i?e ?ac?ooaiee ?aciiainiuo oi?ia. Iiyaeaiea noioanoe/aneeo
o?aaeoi?ee iiaeoaa?aeaeaaony /eneaiiuie yenia?eiaioaie [1, c.127].

Oai?ey EAI ia i?eaiaeeo e aeeiaie/aneie oai?ee oaina. A? aeaaiue aeeaae
a ae?oaii: iia iieacaea, /oi i?e iaeuo cia/aieyo ia?aiao?a ( iu eiaai
i?iiaaeooi/iue ?aaeei, a eioi?ii ninouanoaotho o?aaeoi?ee aeaoo oeiia –
?aaoey?iua e noioanoe/aneea. A aeaeueiaeoai ian aoaeao a iniiaiii
eioa?aniaaoue oi, /oi i?ienoiaeeo a i?aaeaeueiii neo/aa, eiaaea niiaa
inoaiaony oieueei iaeei oei o?aaeoi?ee. Yoa neooaoeey niioaaonoaoao oae
iacuaaaiui aieueoei nenoaiai Ioaiea?a (ANI), e ?anniio?aieth eioi?uo iu
e ia?aoiaeei.

I?e ?anniio?aiee i?aaeeiaeaiiie Ioaiea?a eeanneoeeaoeee aeeiaie/aneeo
nenoai ia eioaa?e?oaiua e iaeioaa?e?oaiu iu ioiaoeee, /oi ?aciiainu
ano?a/athony ?aaeei. I?e ia?aoiaea e ANI neooaoeey ?aaeeeaeueii
eciaiyaony: a ANI ?aciiainu ea?atho aeaaioth ?ieue.

?anniio?ei a ea/anoaa i?eia?a acaeiiaeaenoaea iaaeaeo eaeie-ieaoaeue
/anoeoeae e iieai. Iiea iiaeii ?anniao?eaaoue eae noia?iiceoeeth
inoeeeeyoi?ia n eiioeiooiii /anoio. A ioee/ea io iiey, /anoeoea
niaa?oaao eieaaaiey n iaeiie oeene?iaaiiie /anoioie (1. Ia?aae iaie –
i?eia? iaeioaa?e?oaiie nenoaiu Ioaiea?a. ?aciiainu aoaeoo aicieeaoue
anyeee ?ac, eiaaea (1=(k. Enioneaiea eceo/aiey iaoneiaeaii eiaiii oaeeie
?aciiainiuie acaeiiaeaenoaeyie iaaeaeo ca?yaeaiiie /anoeoeae e iieai.
Enioneaiea eceo/aiey i?aaenoaaeyao niaie iaia?aoeiue i?ioeann, naycaiiue
n ?aciiainaie Ioaiea?a.

Iiaay iniaaiiinoue ninoieo a oii, /oi /anoioa (k anoue iai?a?uaiay
ooieoeey eiaeaena k, niioaaonoaothuay aeeeiai aiei inoeeeeyoi?a iiey.
Oaeiaa niaoeeoe/aneay iniaaiiinoue aieueoeo nenoai Ioaiea?a, oi anoue
oaioe/aneeo nenoai, o eioi?uo iao ?aaoey?iuo o?aaeoi?ee, ninouanoaothueo
n oaioe/aneeie o?aaeoi?eyie. ANI niioaaonoaotho a aeaenoaeoaeueiinoe
aieueoeinoao oece/aneeo neooaoeee, n eioi?uie iu noaeeeaaainy a
i?e?iaea. Ii ANI iicaieytho oaeaea eneeth/eoue ?anoiaeeiinoe Ioaiea?a,
oi anoue ono?aieoue iniiaiia i?aiyonoaea ia iooe e eioaa?e?iaaieth
o?aaiaiee aeaeaeaiey. Yoio ?acoeueoao, caiaoii i?eoiiiaeathuee iiuue
aeeiaie/aneiai iienaiey, ?ac?ooaao ioiaeaeanoaeaiea iuethoiiianeie eee
aaieeueoiiiaie iaoaieee e ia?aoeiiai ii a?aiaie aeaoa?ieiecia a aeooa
Eaieana. O?aaiaiey aeey aieueoeo nenoai Ioaiea?a a iauai neo/aa
i?eaiaeyo e i?eioeeieaeueii aa?iyoiinoiie yaiethoeee n ia?ooaiiie
neiiao?eae ai a?aiaie. Aieaa iiae?iaii aii?inu iaia?aoeiinoe a?aiaie
?anniio?ei a neaaeothuai ?acaeaea.

1.3 Noaoenoe/aneia iienaiea. Aeenneiaoeaiue oain

Iiaeii iienuaaoue ie? a oa?ieiao o?aaeoi?ee (a eeanne/aneie oeceea) eee
aieiiauo ooieoeee (a eaaioiaie iaoaieea). Ii/oe noi eao iacaae Aeaan e
Yeiooaei aaaee au? iaeei oei iienaiey – noaoenoe/aneia iienaiea a
oa?ieiao ainaiaeae. Iienaiea ioaeaeueiie aeeiaie/aneie nenoaiu
caiaiyaony iienaieai ainaiaey nenoai, eioi?ua ana niioaaonoaotho iaeiiio
e oiio aea aaieeueoiieaio e ?acee/athony oieueei ia/aeueiuie oneiaeyie
yaiethoeee. Aeey aaaaeaiey ainaiaeaaie oi/ee c?aiey auee aeaa iniiaiua
i?e/eiu. Ai-ia?auo, iienaiea a oa?ieiao ainaiaey iicaieeei oaeiaii
au/eneyoue n?aaeiea cia/aiey. Ai-aoi?uo, iiiyoea ainaiaey noaei
iaiaoiaeeiui aeey iienaiey nenoaiu, aeinoeaoae oa?iiaeeiaie/aneiai
?aaiiaaney. Ieacaeinue, /oi oa?iiaeeiaie/aneea naienoaa iiaeii iiiyoue
oieueei a oa?ieiao ainaiaeae, ii ioithaeue ia a oa?ieiao ioaeaeueiuo
o?aaeoi?ee eee aieiiauo ooieoeee. Ainaiaeaaue iiaeoiae i?eiaiei ei anai
aeeiaie/aneei nenoaiai, eioaa?e?oaiui e iaeioaa?e?oaiui, onoie/eaui e
iaonoie/eaui.

Iniiaiie aaee/eiie a ainaiaeaaii iiaeoiaea noaiiaeony ?ani?aaeaeaiea
aa?iyoiinoae. Iaeiaei ie/oi ia iaoaao aa?iooueny eae e i?aaeaeueiiio
neo/ath. Iiaeoiae Aeaana–Yeiooaeia – aeueoa?iaoeaiue, ii yeaeaaeaioiue
niinia i?aaenoaaeaiey caeiiia oeceee, ii yaeyaony naiaeeiui
noaoenoe/aneei iienaieai.

Eiioeaioeeth ianaiaeeiuo noaoenoe/aneeo iienaiee, ?acaeaaaioth oeieie
E.I?eaiaeeia, iu iiae?iaiaa ?anniio?ei a o?aoueai ?acaeaea. Iiea /oi
ae?aooea ia?aoeiny e eeanne/aneiio aeenneiaoeaiiio oaino, aeey eioi?iai
noaoenoe/aneia iienaiea yaeyaony aaeeinoaaiii aiciiaeiui iiaeoiaeii.
Aaaae?i oaeaea iaeioi?ua iiiyoey, iaiaoiaeeiua aeey aeaeueiaeoeo
?annoaeaeaiee i noaoenoe/aneii iienaiee. (Iiae?iaiaa – ni. [4]).

Eae e i?aaeaea, eaaeaeiio ninoiyieth nenoaiu niioaaonoaoao oi/ea a
oaciaii i?ino?ainoaa. Ii a oai?ee ainaiaeae Aeaana nenoaia eae oeaeia
i?aaenoaaeia eeoue “iaeaeii” oi/ae a oaciaii i?ino?ainoaa. Yoi “iaeaei”
iienuaaaony iai?a?uaiui ?ani?aaeaeaieai ieioiinoe aa?iyoiinoe
((q1,…,qs,p1,…,ps) a oaciaii i?ino?ainoaa. Eaaeaeay oi/ea oaciaiai
i?ino?ainoaa aeaeaeaony ai a?aiaie ii naiae aeeiaie/aneie o?aaeoi?ee,
eioi?ua ieeiaaea ia ia?anaeathony. Aeaa ia?aiia/aeueii ?acee/iua oi/ee
iaanaaaea inoathony ?acee/iuie. Yoi ooiaeaiaioaeueiia naienoai i?eaiaeeo
e oai?aia Eeoaeeey, eioi?ay oaea oiiieiaeanue i?e iienaiee
i?aia?aciaaiey iaea?y. Yoa oai?aia ooaa?aeaeaao, /oi ieioiinoue ( aaae?o
naay eae ianaeeiaaiay aeeaeeinoue: aeey ethaie aeeiaie/aneie nenoaiu
iau?i iaeanoe, caiyoie i?aaenoaaeythueie oi/eaie a oaciaii i?ino?ainoaa,
nio?aiyaony a oiaea yaiethoeee. Iaeiaei oai?aia Eeoaeeey ioithaeue ia
eneeth/aao eciaiaiey oi?iu iaeanoe, caiyoie i?aaenoaaeythueie oi/eaie.

Aa?i?iny e oaino. I?eia?u oaioe/anee aaaeoueo naay aeeiaie/aneeo nenoai,
iienaiiua auoa, ioiineoaeueii iiau e, eae oaea oiiieiaeinue, ia anaaaea
“oece/iu”. Oa?iiaeeiaieea aea e noaoenoe/aneay oeceea i?eia?ii ia noi
eao ?aiueoa noieeioeenue n i?iaeaiie oaioe/aneiai iiaaaeaiey nenoai.

Ca i?eia?aie aeaeaei oiaeeoue ia neaaeoao – ie?oaeathuay ian aoiinoa?a
aaae?o naay aiieia oaioe/anee, i?aaeneacaiea i?iaiica iiaiaeu ia
neieueei-ieaoaeue aieueoie n?ie – caaea/a ia?iiiie neiaeiinoe (oioy a
i?eioeeia e iaaaciaae?aeiay).

Iaeiaei aeaaea a aoiinoa?a ano?a/athony ioiineoaeueii onoie/eaua
ia?aciaaiey e ia iaeioi?ii o?iaia iienaiey iiaaaeaiea aoiinoa?u ia
nianai oaioe/ii. Ae?oaei i?eia?ii oiai, /oi (oa?iiaeeiaie/aneee) oain e
aanii?yaeie – a oeceea ia neiiieiu, yaeythony oe?iei ecaanoiua y/aeee
Aaia?a (ianoieueei ecaanoiua, /oi aaoi? ii/aio-oi niaa?oaiii ia aeaeaao
a i/a?aaeiie ?ac aeaaaoue iienaiea yoiai yaeaiey – ni., iai?eia?, [1,
n.68]). E y/aeee Aaia?a, e aoiinoa?iua aeo?e, e iiiaea ae?oaea iiaeiaiua
yaeaiey ioiinyony e oae iacuaaaiui aeenneiaoeaiui no?oeoo?ai –
no?oeoo?ai, nouanoaiaaiea eioi?uo iai?yioth iaoneiaeaii iaee/eai a
nenoaia i?ioeannia aeenneiaoeee yia?aee e i?iecaiaenoaa yio?iiee.

Oaeei ia?acii, i?inoia e neiaeiia, aeaoa?ieie?iaaiiia e oaioe/aneia
iiaaaeaiea ninouanoaotho a nia?aiaiiie oeceea ?yaeii. Caeii/ei yoio
i/aiue e?aoeee iaci? neiaaie E.I?eaiaeeia [1, n.59]: “…oioaeinue au
iiae/a?eiooue caia/aoaeueiue aeoaeeci, eioi?ue iu iaia?oaeeaaai a
i?e?iaea, – ninouanoaiaaiea ?aaiiaaniuo neooaoeee oeia eceo/aiey
aaniethoii /??iiai oaea e aunieii?aaieciaaiiuo iauaeoia, iaeiei ec
iaeaieaa caia/aoaeueiuo n?aaee eioi?uo, ii-aeaeeiiio, yaeyaony
/aeiaa/aneee iica n aai 1011 naycaiiuo iaaeaeo niaie iae?iiaie. Ii?yaeie
e aanii?yaeie ia iiaoo auoue iiiyou a oa?ieiao Aieueoeiaia: ii?yaeie eae
iaiaa aa?iyoiia ninoiyiea, aanii?yaeie eae aieaa aa?iyoiia ninoiyiea. E
ii?yaeie, e aanii?yaeie yaeythony iaiouaieaiuie ninoaaiuie /anoyie e
i?iaeoeoaie ei??aee?iaaiiuo yaiethoeeiiiuo i?ioeannia”.

2. IAIA?AOEIINOUe A?AIAIE

Iaae oai aio iaeaoe oaie, ii ia ioeuena. A aio /anu,

Iie eaeoo, e aeiaieueii auno?i – y i?iaa?ye…

I.Ua?aaeia, “Oa?iaoeaao”

2.1 Ia?aoeiinoue a?aiaie a eeanne/aneie e eaaioiaie iaoaieea

Oeaio?aeueiay oaia ?aciuoeaiee E.I?eaiaeeia e iai?aaeaiea ?aciuoeaiee
“a?thnnaeueneie oeieu” ninoieo a ?aoaiee aeeeaiiu: io?eoeaiea –
iaio?eoeaiea no?aeu a?aiaie. Au?aaeaiea “no?aea a?aiaie” auei aaaaeaii a
1928 a. Yaeaeeiaoiiii a aai eieaa “I?e?iaea oece/aneiai ie?a”. A yoie
eieaa Yaeaeeiaoii i?aaeneacuaae eiiaoe ainiiaenoaa a oeceea “ia?ae/iuo”
(aeaoa?ieienoe/aneeo) caeiiia e ianooieaiea y?u “aoi?e/iuo”
(noaoenoe/aneeo) caeiiia, iienuaathueo iaia?aoeiua i?ioeannu.

A oii aeaea, a eaeii a?aiy aoiaeeo a ooiaeaiaioaeueiua caeiiu oeceee io
eeanne/aneie aeeiaieee aei oai?ee ioiineoaeueiinoe e eaaioiaie oeceee,
a?aiy ia niaea?aeeo a naaa ?acee/ey iaaeaeo i?ioeui e aoaeouei. Aeey
iiiaeo oeceeia yoi oaea ii/oe aii?in aa?u: aei oao ii? e iineieueeo
?a/ue eae?o i ooiaeaiaioaeueiii o?iaia iienaiey, “no?aeu a?aiaie” ia
nouanoaoao.

Ii ia iae?io?iaia, a ie?a iauaeoia, n eioi?uie iu eiaai aeaei
aaeaaeiaaii, ia o?iaia aeeauo i?aaieciia iaia?aoeiinoue a?aiaie
niiiaiee ie o eiai ia aucuaaao. I?ioeannu noa?aiey, ?aniaaea, ?annayiey
yia?aee iaecaaaeiu. Eae neacaii a ia?iaeee ia ecaanoioth ianith, “oa?o
iaaiciiaeii i?iaa?iooue iacaae”. No?aea a?aiaie ia naiii aeaea
i?enoonoaoao e ai anao oece/aneeo oai?eyo, iienuaathueo ?aaeueiue ie?.
Ii i?enoonoaoao iia oai ia a aeaea /eaiia a o?aaiaieyo, a a aeaea
i?eia/aiee e eiiiaioa?eaa e yoei o?aaiaieyi, i?aaenoaaeyy niaie
auneacuaaiey oeia: “…Ec yoeo aeaoo ?aoaiee iu aeieaeiu aua?aoue
ia?aia, iineieueeo iii niioaaonoaoao i?yiiio iai?aaeaieth oiaea a?aiaie”
eee “…A oi?ioea (…) ia?aue /eai ioaa/aao ca i?yiia, a aoi?ie – ca
ia?aoiia ?annayiea, a ?aaeueiinoe ia iaaethaeathuaany, iiyoiio iu aoaeai
?anniao?eaaoue oieueei ?aoaiey aeaea (…)”.

A aieaa yaiii aeaea no?aea a?aiaie iiyaeyaony a oa?iiaeeiaieea, a
?acee/iuo oi?ioee?iaeao a? aoi?iai ia/aea e a H-oai?aia Aieueoeiaia.
Oaeeaeoaeueiui ieacuaaaony oi, /oi i?e iiiuoea aiaeece?iaaoue oaeea
i?ioeannu, eae aeeooocey eee ayceinoue – aiieia iae?ineiie/anee
iaia?aoeiua – oeceea oniaoii eo iienuaaao n iiiiuueth ia?aoeiuo ai
a?aiaie iee?ii?ioeannia.

A iniiaa eeanne/aneie iaoaieee (enoi?e/anee, aeaaea anee e ia eiae/anee)
eaaeeo caeii Iuethoiia. Ii ia?aoei ai a?aiaie e aeaoa?ieienoe/ai. Caeii
Iuethoiia iiaeii ?anniao?eaaoue eae i?ioioei iaeiaai Oieaa?naeueiiai
Caeiia I?e?iaeu.

Iiiyoea caeiia i?e?iaeu caneoaeeaaao iaeioi?iai ionooieaiey. Iu
ianoieueei i?eaueee e iaio, /oi iii aini?eieiaaony eae ia/oi naii niaie
?acoiathuaany. Iaeiaei a ae?oaeo acaeyaeao ia ie? (ia anaaaea aiieia
iao/iuo – n iuiaoiae oi/ee c?aiey) oaeay eiioeaioeey “caeiia i?e?iaeu”
ionoonoaoao. Ii A?enoioaeth, aeeaua nouanoaa ia iiae/eiythony ieeaeei
caeiiai, eo aeayoaeueiinoue iaoneiaeaia eo nianoaaiiuie aioo?aiieie
i?e/eiaie. Eaaeaeia nouanoai no?aieony e aeinoeaeaieth naiae nianoaaiiie
enoeiu. A Eeoaa ainiiaenoaiaaee acaeyaeu i niiioaiiie aa?iiiee einiina,
naiaai ?iaea noaoenoe/aneii ?aaiiaanee, naycuaathuai aiaaeeii i?e?iaeo,
iauanoai e iaaana. I?eia?u iiaeii iiiaeeoue e iiiaeeoue…

Eaeay i oii, /oi a ie?a iiaoo aeaenoaiaaoue caeiiu, auc?aea a iaae?ao
caiaaeiie iunee. Io/anoe iia ainoiaeeo e noieeai, ianiio?y ia oo ?ieue,
eioi?oth iie ioaiaeeee ?ieo. Iaiaeiaaaeioth ?ieue, aa?iyoii, nua?aee e
eoaeaio?enoeaineea i?aaenoaaeaiey i Aiaa eae aaniethoiii Anaaea?aeeoaea,
onoaiaaeeaathuai caeiiu aeey anaai nouaai. Oae eee eia/a, ioe?uoea
iaeciaiythueony aeaoa?ieienoe/aneeo caeiiia eae au naeeaeaei
/aeiaa/aneia ciaiea n aiaeanoaaiiie, aiaa?aiaiiie oi/eie c?aiey.

Iaia/aiiay i?ia?aiia ieacaeanue iaiau/aeii oniaoiie. Iaeiaei ia
i?ioyaeaiee anae enoi?ee caiaaeiie iunee iaiaeiie?aoii aicieeae iaeei e
oio aea aii?in: eae neaaeoao iiieiaoue iiaia, ea?athuaa oeaio?aeueioth
?ieue, a ie?a, oi?aaeyaiii aeaoa?ieienoe/aneeie caeiiaie?

Aia?aua yoio aii?in aiciee caaeieai aei ?iaeaeaiey nia?aiaiiie iaoee.
Au? Ieaoii naycuaae ?acoi e enoeio n aeinooiii e “auoeth”, iaeciaiiie
?aaeueiinoueth, noiyuae ca “noaiiaeaieai”. Noaiiaeaiea, iioie
aini?eieiaaiuo iaie yaeaiee, ioiineony e noa?a “/enoiai iiaiey”. Iaeiaei
Ieaoii niciaaae ia?aaeienaeueiinoue oaeie iiceoeee, iineieueeo iia
i?eieaeaea aeeciue e iuneue, eioi?ua i?aaenoaaaee eae iaioaeaeeiua io
i?ioeanna noaiiaeaiey. A “Nioenoa” Ieaoii i?eoiaeeo e caeeth/aieth, /oi
iai iaiaoiaeeiu e auoea, e noaiiaeaiea.

N oie aea o?oaeiinoueth noieeioeenue e aoiienou. *oiau aeiionoeoue
aicieeiiaaiea iiaiai, Eoe?aoeeth i?eoeinue aaanoe “eeeiaiai”,
aiciouathuee aeaoa?ieienoe/aneia iaaeaiea aoiiia a ionoioa. Ia?auaiea e
eeeiaiaio /anoi iiaeaa?aaeinue e?eoeea eae aaaaeaiea /oaea?iaeiiai
yeaiaioa a noaio aoiienoe/aneiai iienaiey. Ii e /a?ac aeaa ouny/aeaoey
iu ano?a/aai aiaeiae/iia ooaa?aeaeaiea a ?aaioa Yeiooaeia, iinayu?iiie
naiii?iecaieueiiio enioneaieth naaoa aicaoaeae?iiui aoiiii, aaea
aiai?eony, /oi “a?aiy e iai?aaeaiea yeaiaioa?iuo i?ioeannia ii?aaeaeaiu
neo/aeiui ia?acii” [6, n.386]

E eeeiaiai, e niiioaiiia enioneaiea naaoa ioiinyony e niauoeyi,
niioaaonoaothuei aa?iyoiinoiiio iienaieth. Niauoey e aa?iyoiinoe
o?aaothony e aeey yaiethoeeiiiiai iienaiey, aoaeue oi aea?aeiianeay
oai?ey yaiethoeee eee enoi?ey /aeiaa/anoaa. Anoa?o aii?in: iiaeii ee
iieoe aeaeueoa, /ai Eoe?aoeee e Yeiooaei, “aeiaaaeaoea” niauoey e
aeaoa?ieienoe/aneei caeiiai? Iiaeii ee aeaeieciaieoue naii iiiyoea
oece/aneeo caeiiia oae, /oiau aeeth/eoue a ooiaeaiaioaeueiia iienaiea
i?e?iaeu iaia?aoeiinoue, niauoey e no?aeo a?aiaie?

Aeey ioaaoa ia yoio aii?in ia?aoeiny nia/aea e oie iaeanoe oeceee,
eioi?ay eiaao aeaei n “iaeaieaa iaia?aoeiuie” ec ano?a/athueony a
iianaaeiaaiie aeecie nenoaiaie – a eiaiii, e oa?iiaeeiaieea e
noaoenoe/aneie oeceea.

2.2 ?ieue iaia?aoeiinoe a noaoenoe/aneie iaoaieea. Iioiee ei??aeyoeee

Oai?ey ainaiaeae Aeaana e Yeiooaeia i?aaeiacia/aeanue aeaaiui ia?acii
aeey aeinoeaeaiey eo/oaai iiieiaiey ?aaiiaaniie oa?iiaeeiaieee a
oa?ieiao ?aaiiaaniuo ainaiaeae. Eieue nei?i ?aaiiaaniia ?ani?aaeaeaiea
caaeaii, iu iiaeai au/eneeoue ana oa?iiaeeiaie/aneea naienoaa:
aeaaeaiea, oaeaeueioth oaiei?ieinoue e o.ae. Iu iiaeai aeaaea aueoe ca
?aiee iee?ineiie/aneie oa?iiaeeiaieee, iineieueeo ie/oi ia iaoaao iai
au/eneyoue oeoeooaoeee ?aaiiaaniuo aaee/ei. Ii iauaio iiaieth, a
iaoe?iie iaeanoe ?aaiiaaniie “noaoenoe/aneie” oa?iiaeeiaieee ia
inoaeinue eaeeo-eeai eiioeaiooaeueiuo o?oaeiinoae, au/eneeoaeueiua aea
eaaei nieiathony /eneaiiui iiaeaee?iaaieai. Oaeei ia?acii, i?eiaiaiea
oai?ee ainaiaeae e ?aaiiaaniui ?ani?aaeaeaieyi ieacaeinue aanueia
oniaoiui.

Ii oa?iiaeeiaie/aneea aaee/eiu, “niioaaonoaothuea” iaia?aoeiiio
oa?aeoa?o a?aiaie – oaeea, eae yio?iiey – iaeaaeatho ooiaeaiaioaeueii
aaaeiuie naienoaaie e aia ?aaiiaaney. Anoa?o aii?in: eae iiaeii iiiyoue
a oa?ieiao oai?ee ainaiaeae i?eaeeaeaiea e ?aaiiaaneth?

I?e iienaiee ?aaiiaaniiai ninoiyiey iniiaiie aaee/eiie yaeyaony
?ani?aaeaeaiea nei?inoae f(v,t). Iee?ineiie/aneei aiaeiaii yio?iiee
Aieueoeiai iauyaee ciaiaieooth H-ooieoeeth:

Aieueoeiai iieacae, /oi aeey ?ac?aae?iiuo aacia ?ani?aaeaeaiea nei?inoae
yaiethoeeiie?oao aei oao ii?, iiea ia aeinoeaaao ?aaiiaaniiai
?ani?aaeaeaiey nei?inoae Iaenaaeea-Aieueoeiaia, i?e yoii H(t) iiiioiiii
oauaaao.

Eiiiuethoa?iia iiaeaee?iaaiea e /eneaiiua yenia?eiaiou iiaeoaa?aeaeatho
ooaa?aeaeaiea Aieueoeiaia [1, n.167], oi anoue iaee/ea iaia?aoeiuo
i?ioeannia ia iee?ineiie/aneii o?iaia. Iaeiaei oaeay i?iaa?ea ia iiaeao
ian iieiinoueth oaeiaeaoai?eoue: anaaaea iiaeii nienaoue iiyaeythuothny
iaia?aoeiinoue ia n/?o iaoi/iinoe au/eneaiee (aiaeiae/ii iioa?a
eioi?iaoeee i?e naeaeaa Aa?ioeee, ?anniio?aiiii auoa).

Oai?aia Aieueoeiaia iiaeaa?aaeanue e?eoeea (a /anoiinoe, ni noi?iiu
Eioieaeoa) ia oii iniiaaiee, /oi iia i?ioeai?a/eo ia?aoeiui ai a?aiaie
caeiiai aeeiaieee. Eioieaeo auaeaeioe aic?aaeaiea, iniiaaiiia ia oii,
/oi ia?auaiea anao nei?inoae icia/aei au, /oi aeey eaaeaeie
“aieueoeiaiianeie” yaiethoeee e ?aaiiaaneth nouanoaiaaea au ae?oaay
yaiethoeey, oiaiueoathuay yio?iieth.

Aa?iyoii, Eioieaeo aue i?aa. Ia oi anoue na?ue?ciua iniiaaiey, eaaeauea
a iniiaa oie naiie aaieeueoiiiaie iaoaieee, ia aaca eioi?ie no?ieeanue
eeanne/aneay noaoenoe/aneay iaoaieea. Aeaei a oii, /oi eioaa?e?oaiua
nenoaiu ia iiaoo i?eaeeaeaoueny e ?aaiiaaneth, iineieueeo aeey oaeeo
nenoai ana ia?aiaiiua aeaenoaey J1, …, Js yaeythony eiaa?eaioaie
aeaeaeaiey: anee ia?aiia/aeueii ( anoue ooieoeey oieueei ia?aiaiiuo
aeaenoaey, oi yoa ooieoeey inoa?ony iinoiyiiie ai a?aiaie e ia iiaeao
yaiethoeeiie?iaaoue a ooieoeeth oieueei yia?aee, eae aeieaeii auoue aeey
?aaiiaaniiai ninoiyiey.

Iuoaynue oaycaoue aeaoa?ieieci iiaaaeaiey aeeiaie/aneeo nenoai n
iaia?aoeiinoueth nenoai noaoenoe/aneeo, Iaenaaee e Aieueoeiai aaaee
iiiyoea y?aiaee/iinoe – oi anoue naienoaa nenoaiu n oa/aieai a?aiaie
neieue oaiaeii aeecei iiaeoiaeeoue e ethaie oi/ea ia yia?aaoe/aneie
iiaa?oiinoe. I?e yoii a i?aaeaea, i?e aieueoeo a?aiaiao, n?aaeiea io
aeeiaie/aneeo naienoa ii a?aiaie niaiaaeatho ni n?aaeieie ii ainaiaeth.
Y?aiaee/aneay oai?ey e ?acee/iua a? iaiauaiey iicaieytho aeaeaoue
caeeth/aiey i iiaaaeaiee aeeiaie/aneeo nenoai i?e aieueoeo a?aiaiao (i?e
yoii aac?acee/ii, t ( (( eee t ( –( ), ii ia aeatho ieeaeie eioi?iaoeee
ioiineoaeueii iiaaaeaiey nenoaiu i?e eiia/iuo a?aiaiao. E?iia oiai,
eioaa?e?oaiua nenoaiu, aiiaua aiai?y, iay?aiaee/iu.

Iaaeaeo oai, eiaiii iiaaaeaiea nenoai ia eiia/iuo a?aiaiao yaeyaony
oeaio?aeueiie iaoaiaoe/aneie i?iaeaiie iaia?aoeiinoe. Ioaeia iaiau?iiay
niaeo?aeueiay oai?ey, aeeth/athuay a niaeo? oaeea aeenneiaoeaiua
naienoaa, eae a?aiaia aeecie, a?aiaia ?aeaenaoeee e o.ae.
(A?thnnaeueneay oeiea eae ?ac e i?aaeeaaaao oaeia eiiieaeniia
niaeo?aeueiia i?aaenoaaeaiea aeey iaonoie/eauo aeeiaie/aneeo nenoai – ia
yoii neacaii a neaaeothueo ?acaeaeao aeaiiie ?aaiou).

Iinea aic?aaeaiee Eioieaeoa aeey iienaiey ?acee/ey iaaeaeo
“aieueoeiaiianeeie” e “aioeaieueoeiaiianeeie” ia/aeueiuie ninoiyieyie
auea i?aaei?eiyoa iiiuoea ainiieueciaaoueny ei??aeyoeeyie a nei?inoyo
/anoeoe, aicieeathueie a ?acoeueoaoa iaae/anoe/iuo noieeiiaaiee.
Iineaaeiaaoaeueiua noieeiiaaiey ii?iaeaeatho ia?iua, o?ieiua,…,
n–a?iua ei??aeyoeee iaaeaeo /anoeoeaie. Ia?auaiea nei?inoe i?eaaei au e
noieeiiaaieyi, ?ac?ooathuei ei??aeyoeee.

A oa?ieiao ooieoeee ?ani?aaeaeaiey yoi iiaeii au?aceoue oae:
i?ieioaa?e?oai ii eii?aeeiaoai ooieoeeth ((q1, …, qn, …, p1, …,
pn,, t). Iieo/ei a ?acoeueoaoa ooieoeeth (0(p1, …, pn,, t), caaenyuoth
oieueei io eiioeuenia. A iae ia niaea?aeeony ieeaeie eioi?iaoeee i
iieiaeaiee /anoeoe a i?ino?ainoaa, iiyoiio a? iiaeii iacaaoue aaeooiii
ei??aeyoeee. Iiaeii oaeaea ii?aaeaeeoue ooieoeeth, niaea?aeauoth
eioi?iaoeeth i iieiaeaiee iaeiie i-e /anoeoeu, ooieoeeth (2(qi.,qj,, p1,
…, pn,, t), iienuaathuoth aeaa /anoeoeu e o.ae. Ooieoeey (2 niaea?aeeo
oaea eioi?iaoeeth i ia?iuo noieeiiaaieyo, (3 – i o?ieiuo, … A
?acoeueoaoa, iu iiaeai ?aceiaeeoue ( ia aaeooi ei??aeyoeee (0 e ia
ninoiyiey ei??aeyoeee. Ioee/ea a eaaioiaie iaoaieea, eae iau/ii, naycaii
n /eneii iacaaeneiuo ia?aiaiiuo. Iao?eoea ieioiinoe niioaaonoaoao
iao?e/iia i?aaenoaaeaiea – iai?eia?, a oa?ieiao eiioeuenia –
((p1,…,pn,p1′,…,pn’). Iu eiaai aeeaaiiaeueiua yeaiaiou n p1=p1′,
p2=p2′,… e iaaeeaaiiaeueiua, o eioi?uo ii e?aeiae ia?a iaeii ec yoeo
niioiioaiee ia?ooaii. A eaaioiaie iaoaieea aaeooio ei??aeyoeee (0
niioaaonoaoao aeeaaiiaeueiui yeaiaioai iao?eoeu (, a (( –
iaaeeaaiiaeueiui yeaiaioai, a eioi?uo ( ia?aiaiiuo p1, p2, …, p( ia
?aaiu niioaaonoaaiii p1′, p2′, …, p(‘. A ?acoeueoaoa acaeiiaeaenoaee
?acee/iua ninoiyiey ei??aeyoeee ia?aoiaeyo ae?oa a ae?oaa. (N oi/ee
c?aiey iia?aoi?iiai oi?iaeecia ia iao?eoeu pi aeaenoaoao noia?iia?aoi?
Eeoaeeey – ni. ieaea). Eiaaea /anoeoea, oaea ei??aee?iaaiiay n ae?oaie
/anoeoeae, noaeeeaaaony n o?aoueae, aicieeaao o?ieiay ei??aeyoeey, e
o.ae.

Oaia?ue iao?oaeii onoaiiaeoue naycue iaaeaeo iioieii ei??aeyoeee e
oai?aiie Ioaiea?a. Eioaa?e?oaiua nenoaiu – yoi nenoaiu, a eioi?uo iu
iiaeai eneeth/eoue acaeiiaeaenoaea, iiyoiio eneeth/aaony e iioie
ei??aeyoeee. Neaaeiaaoaeueii, anee yaiethoeey eioaa?e?oaiie nenoaiu
ia/eiaaony n aaeooia ei??aeyoeee, a oiaea yaiethoeee ieeiaaea ia
aicieeioo aeaieiua, o?ieiua e o.ae. ei??aeyoeee. Iioiea ei??aeyoeee a
eioaa?e?oaiuo nenoaiao ia nouanoaoao.

A ioee/ea io eioaa?e?oaiuo nenoai, a iaeioaa?e?oaiuo nenoaiao Ioaiea?a
nouanoaoao iai?a?uaiue i?ioeann ?iaeaeaiey ei??aeyoeee.
Iaeioaa?e?oaiinoue icia/aao, /oi iu ia iiaeai eneeth/eoue iioie
ei??aeyoeee n iiiiuueth ethaiai (eaiiie/aneiai) i?aia?aciaaiey. Iioie
ei??aeyoeee, eae e ana iaia?aoeiua i?ioeannu, iineo aioo?aiiee oa?aeoa?.

E?iia oiai, a iaeioaa?e?oaiuo nenoaiao aaeooi ei??aeyoeee noaiiaeony
caaenyuei io a?aiaie. Oaeei ia?acii, aeaeaaony caeeth/aiea, /oi
eeiaoe/aneea o?aaiaiey oeia o?aaiaiee Aieueoeiaia iiaoo auiieiyoueny
oieueei aeey “iaeioaa?e?oaiuo” nenoai, eae eeanne/aneeo, oae e
eaaioiauo.

2.3 I?iaeaia ianaiaeeiiai iienaiey

Yaiethoeey ai a?aiaie ieioiinoe ?ani?aaeaeaiey aa?iyoiinoe ii?aaeaeyaony
o?aaiaieai Eeoaeeey, eioi?ia neaaeoao ec eeanne/aneie aaieeueoiiiaie
aeeiaieee. A iia?aoi?iie caiene iii eiaao aeae

i?e yoii yaiue aeae iia?aoi?a Eeoaeeey L iiaeao auoue auaaaeai ec
aaieeueoiieaia. Neaaeoao ioiaoeoue, /oi eae e iia?aoi?u eaaioiaie
iaoaieee, iia?aoi? Eeoaeeey y?ieoia.

. Oae eae iiaue iia?aoi? Eeoaeeey aeaenoaoao ia ia aieiiaua ooieoeee, a
ia iao?eoeo ieioiinoe, eioi?ay naia ii naaa iia?aoi?, L iau/ii iacuaatho
noia?iia?aoi?ii. Iia?aoi? L – y?ieoia, a i?ino?ainoai iao?eoe ieioiinoe
– aeeueaa?oiai. [5]

Eniieueciaaiea iia?aoi?iiai oi?iaeecia iicaieyao a noaoenoe/aneie
iaoaieea i?eiaiyoue e eeanne/aneei nenoaiai iaoiaeu, ?ac?aaioaiiua aeey
eaaioiauo nenoai: ii?aaeaeaiea nianoaaiiuo ooieoeee e nianoaaiiuo
cia/aiee aeey iia?aoi?a Eeoaeeey.

Eae e a eaaioiaie iaoaieea, iu iiaeai ?anniio?aoue caaea/o ia
nianoaaiiua cia/aiey:

I?e yoii, iineieueeo L – y?ieoia iia?aoi?, aai nianoaaiiua cia/aiey ln
aeaenoaeoaeueiu. E?iia oiai, ec ooieoeee ((n > iiaeii ninoaaeoue iieioth
i?oiii?ie?iaaiioth nenoaio, ii eioi?ie ?aneeaaeuaaaony ethaay ooieoeey
?ani?aaeaeaiey:

.

Yaiethoeey aea ?ani?aaeaeaiey ai a?aiaie ii?aaeaeyaony niioiioaieai

((t)=U(t)((0)=e–iLt((0).

Eae e a eaaioiaie iaoaieea, U(t) – oieoa?iue iia?aoi?, e iiyoiio

.

Oaeei ia?acii, ?ani?aaeaeaiea aa?iyoiinoe ?aceaaaaony a noiio iacaaeneii
?acaeaathueony ai a?aiaie iiae, eaaeaeay ec eioi?uo aoiaeeo n aanii cn,
iinoiyiiui ai a?aiaie. Iineieueeo nianoaaiiua cia/aiey aauanoaaiiu,
eaaeaeay iiaea “a?auaaony” a oaciaii i?ino?ainoaa. Aaeeinoaaiiia ioee/ea
io eaaioiaie iaoaieee ninoieo a oii, /oi a aeaiiii neo/aa eaaeaeay iiaea
aiineo naie aeeaae iaiin?aaenoaaiii a aa?iyoiinoue (, a ia a aiieeooaeo
aa?iyoiinoe (, eae a eaaioiaie iaoaieea.

I?iaeaia ninoieo a oii, /oi ?aoaiea o?aaiaiey Eeoaeeey aeey iao?eoeu
ieioiinoe a aeeueaa?oiaii i?ino?ainoaa ia iienuaaao i?eaeeaeaiey e
?aaiiaaneth [1, n.166].

Iu noaeeeaaainy caeanue n iniiaiie o?oaeiinoueth oai?ee iaia?aoeiuo
i?ioeannia. A?auaiea ii oaca nio?aiyao neiiao?eth ai a?aiaie. *oiau
iieo/eoue ia?ooaiea neiiao?ee ai a?aiaie, auei au iaiaoiaeeii eiaoue
eiiieaeniua nianoaaiiua cia/aiey ln = ln’ + iln”, oiaaea
exp(–ilnt)=exp(–iln’t)exp(–ln”t), e aoi?ie iiiaeeoaeue ii?iaeaeaao
yeniiiaioeeaeueiia caoooaiea. Ii yoi iaaiciiaeii, iineieueeo iu eiaai
aeaei n y?ieoiaui iia?aoi?ii e eniieuecoai oi?iaeeci aeeueaa?oiaa
i?ino?ainoaa.

Iaeia ec aiciiaeiinoae, e i?eiyoeth eioi?ie neeiiythony iiiaea aaoi?u,
ninoieo a ooaa?aeaeaiee, /oi iineieueeo o?aaiaiea Eeoaeeey ia?aoeii ai
a?aiaie, iaia?aoeiinoue aicieeaao a ?acoeueoaoa a?oaie ca?ienoinoe, oi
anoue i?eaeeae?iiiai iienaiey. Ii ia iee?ineiie/aneii o?iaia iu niiaa
aica?auaainy e ia?aaeieno a?aiaie. ?aoeoue aai iiaeii oieueei aeaoiy
niiniaaie: aua?aoue a ea/anoaa enoiaeiuo iiaua o?aaiaiey aeaeaeaiey, n
naiiai ia/aea niaea?aeauea iaia?aoeiinoue, eee ioeacaoueny io
aeeueaa?oiaa i?ino?ainoaa. Eiioeaioeey I?eaiaeeia ?aaeecoao aoi?oth
aiciiaeiinoue.

Aeey eioaa?e?oaiuo eeanne/aneeo nenoai ?aoaiea caaea/e ia nianoaaiiua
cia/aiey iia?aoi?a L i?eaiaeeo e o?aaeoi?eyi. A eaaioiaie oai?ee
ainaiaeae neooaoeey aiaeiae/ia. Anee caaea/a ia nianoaaiiua cia/aiey
aeey aaieeueoiieaia H ?aoaia, oi iu iiaeai ?aoeoue a? e aeey L e
i?aaenoaaeoue ?aoaiea a oa?ieiao aieiiauo ooieoeee. Aeey eaaioiauo
nenoai n aeene?aoiui niaeo?ii ieeaeeo o?oaeiinoae i?e yoii ia aicieeaao,
ii i?e ia?aoiaea e aieueoei nenoaiai Ioaiea?a (n iai?a?uaiui niaeo?ii e
iai?a?uaiuie iiiaeanoaaie ?aciiainia) ia nouanoaoao oaea eiino?oeoeaiiai
iaoiaea ?aoaiey caaea/e ie aeey H, ie aeey L [1, n.164].

Ioee/ea noaoenoe/aneiai iienaiey, aeaaaaiiai oeieie I?eaiaeeia, io
eeanne/aneiai yeiooaeiianei-aeaanianeiai eiaiii a oii, /oi iii
ianaiaeeii. Iii iai?eiaieii e ioaeaeueiie o?aaeoi?ee. Yoi ooaa?aeaeaiea
i?aaenoaaeyao niaie no?iaee iaoaiaoe/aneee ?acoeueoao, iieo/aiiue a
?acoeueoaoa i?eiaiaiey e aiaeeco oaina iaoiaeia nia?aiaiiiai
ooieoeeiiaeueiiai aiaeeca. E?iia oiai, a oaeii iaia?aoeiii aa?iyoiinoiii
iienaiee i?ioeia e aoaeouaa ea?atho ?acee/iua ?iee. Oain i?eaiaeeo e
aeeth/aieth no?aeu a?aiaie a ooiaeaiaioaeueiia aeeiaie/aneia iienaiea.

Eaaei iieacaoue, /oi oain, ii?aaeaeyaiue eae iau/ii, i?eaiaeeo e
ianaiaeeiiio aa?iyoiinoiiio iienaieth. I?eaiaeei ia?auaao yoi
ooaa?aeaeaiea e auaeaeaaao iiaia ii?aaeaeaiea: ana nenoaiu,
aeiioneathuea ianaiaeeiia aa?iyoiinoiia iienaiea, ii ii?aaeaeaieth
n/eoathony oaioe/aneeie [1, n.9].

3. A?THNNAEUeNEAss EIOA?I?AOAOeEss EAAIOIAIE IAOAIEEE

Y.O??aeeiaa?

3.1 Aeueoa?iaoeaiua eioa?i?aoaoeee eaaioiaie iaoaieee

Aa?iyoii, eaaioiaay iaoaieea – iaeia ec iaiiiaeo, anee ia aaeeinoaaiiay
?aaioathuay oece/aneay oai?ey, ii iiaiaeo eioa?i?aoaoeee eioi?ie ia
ooiaeaiaioaeueiii o?iaia aei neo ii? aaaeoony niaea?aeaoaeueiua nii?u.
Aeaiiay ?aaioa iinayuaia e?aoeiio eceiaeaieth iiceoeee e neaaenoaee
oieueei iaeiie ec eioa?i?aoaoeee, iaeiaei aaoi?o eaaeaony iaaiciiaeiui
i?e yoii ia oiiiyiooue naiua ?ani?ino?ai?iiua aeueoa?iaoeaiua
eioa?i?aoaoeee. (Aieaa iiae?iaii – ni.[2]).

Iaeaieaa ecaanoiu neaaeothuea iiaeoiaeu e eaaioiaie iaoaieea:

– eiiaiaaaaineay eioa?i?aoaoeey;

– noaoenoe/aneay eioa?i?aoaoeey;

– “iaieeanne/aneea” eioa?i?aoaoeee ni ne?uouie ia?aiao?aie;

– iiiaiie?iaay eioa?i?aoaoeey;

– a?thnnaeueneay eioa?i?aoaoeey, ?acaeaaaiay oeieie I?eaiaeeia.

Inoaiiaeiny ae?aooea ia eaaeaeie ec yoeo eioa?i?aoaoeee.

a) Eiiaiaaaaineay eioa?i?aoaoeey yaeyaony iaeaieaa ?ani?ino?ai?iiie, ii
a oi aea a?aiy i?aaenoaaeyao (a neeo enoi?e/aneeo i?e/ei) niaie nei?aa
eiiaeiia?ao ?acee/iuo iiaeoiaeia, iaaeaee iiiieeoioth eiioeaioeeth.
Aeaoiy aaaeiaeoeie i?eioeeiaie yaeythony iauaoeeinioneee i?eioeei
aeiiieieoaeueiinoe Ai?a e iinooeao ?aaeoeoeee aieiiaiai iaeaoa.

I?eioeei aeiiieieoaeueiinoe ia?aiia/aeueii aiciee eae enoieeiaaiea
niioiioaiey iaii?aaeae?iiinoae Aaecaiaa?aa. A aeaeueiaeoai Ai? ?acaee
yoio i?eioeei eae iauaiao/iue e i?ecuaae e aai i?eiaiaieth a aeieiaee,
ineoieiaee e aoiaieoa?iuo iaoeao. Niaea?aeaiea aai i?eia?ii oaeiai:
ieeaeay eeanne/anee iai?ioeai?a/eaay nenoaia iiiyoee ia iiaeao iienaoue
?aaeueiinoue, anaaaea nouanoaotho ?acee/iua, acaeiieneeth/athuea e
acaeiiaeiiieiythuea iiaeoiaeu, eaaeaeue ec eioi?uo io?eoeaao ae?oaie.
Oieueei niaianoiia ?anniio?aiea yoeo iienaiee iiaeao aeaoue iai iieioth
ea?oeio i?ienoiaeyueo a ie?a niauoee.

Iinooeao ?aaeoeoeee aieiiaiai iaeaoa iienuaaao i?ioeann iaaethaeaiey
eaaioiaie nenoaiu aiaoiei iaaethaeaoaeai e ooaa?aeaeaao, /oi a oaeii
i?ioeanna i?ienoiaeeo ia?aoiae aieiiaie ooieoeee eaaioiaiai iauaeoa a
iaeii ec nianoaaiiuo ninoiyiee – oi anoue nenoaia ia?aoiaeeo ec
niaoaiiiai ninoiyiey a /enoia, e ia?aoiae yoio iaia?aoei. Nianoaaiii, a
eiiaiaaaaineie eioa?i?aoaoeee yoio iinooeao e yaeyaony oai
“i?eia/aieai”, aiinyuai iaia?aoeiinoue a?aiaie (ni. ?acaeae 2.1) a
oai?eth. N iinooeaoii ?aaeoeoeee aieiiaiai iaeaoa naycaii iiiai
aeeneonnee e ia?aaeienia. Eiiaiaaaaineay eioa?i?aoaoeey eaaioiaie
iaoaieee iaiaeiie?aoii iiaeaa?aaeanue e?eoeea ca iaiaoiaeeiinoue
i?enoonoaey a iae ia?yaeo n eaaioiauie iauaeoaie noaoai eeanne/aneiai
aiaoiaai iaaethaeaoaey.

a) Noaoenoe/aneay eioa?i?aoaoeey, eee eioa?i?aoaoeey noaoenoe/aneeo
ainaiaeae, iniiaaia ia i?aaeiieiaeaiee, /oi aieiiaay ooieoeey eaaioiaie
nenoaiu iienuaaao ia eiaeeaeaeoaeueiue iauaeo, a ainaiaeue iaeeiaeiaui
ia?acii i?eaioiaeaiiuo iauaeoia. I?e yoii i?ecia?ony ooiaeaiaioaeueiue
oa?aeoa? aa?iyoiinoiuo i?aaeneacaiee a eaaioiaie iaoaieea, e a yoii
niunea eaaioiaiiaoaie/aneia iienaiea ?aaeueiinoe n/eoaaony iieiui.
Aa?iyoiinoe oiai eee eiiai ?acoeueoaoa anoanoaaiiui ia?acii aea?ony
ioiineoaeueii-/anoioiia oieeiaaiea. N oi/ee c?aiey noaoenoe/aneie
eioa?i?aoaoeee eaaioiaay iaoaieea aiiaua ia iienuaaao eiaeeaeaeoaeueiua
eaaioiaua iauaeou.

Ioaeii caiaoeoue, /oi a ?aieao noaoenoe/aneie eioa?i?aoaoeee aaiaeeony
iinooeao i oii, /oi a i?ioeanna ecia?aiey iae?ii?eai? auaeaeyao ec
noaoenoe/aneiai ainaiaey iaeioi?ue iiaeainaiaeue, niioaaonoaothuee
aeaiiiio ?acoeueoaoo ecia?aiey. Yoio iinooeao oaeoe/anee caieiaao ianoi
iinooeaoa ?aaeoeoeee a eiiaiaaaaineie eioa?i?aoaoeee.

a)Iaieeanne/aneea eioa?i?aoaoeee eaaioiaie iaoaieee enoiaeyo ec oiai,
/oi eaaioiaiiaoaie/aneia iienaiea a aeaenoaeoaeueiinoe ia yaeyaony
iieiui. Neaaeiaaoaeueii, aeieaeia nouanoaiaaoue aieaa iauay oai?ey,
iaania/eaathuay iaee/ea aeaoa?ieiecia eeanne/aneiai ia?acoea. Ii
ioiioaieth e oaeie oai?ee eaaioiaay iaoaieea auea au iaeioi?ui
noaoenoe/aneei i?eaeeaeaieai. Iaeaieaa ?ani?ino?aiaiu iaieeanne/aneea
oai?ee ni ne?uouie ia?aiao?aie. A ieo i?aaeiieaaaaony, /oi aieiiaay
ooieoeey (( > ia iieiinoueth ii?aaeaeyao ninoiyiea nenoaiu. Ia?yaeo n
iae nouanoaotho ne?uoua ia?aiao?u ( , oaeea, /oi eo oi/iia ciaiea iiaei
au aeaoue aiciiaeiinoue i?aaeneacaiey ?acoeueoaoia ecia?aiey ethaie
oece/aneie aaee/eiu. I?e yoii naie ia?aiao?u yaeythony noaoenoe/anee
?ani?aaeae?iiuie ii iaeioi?iio caeiio, e iu ia iiaeai ia i?aeoeea oi/ii
ii?aaeaeeoue cia/aiea ( . Iiyoiio nio?aiythony ana neaaenoaey eaaioiaie
iaoaieee, a oii /enea iaaiciiaeiinoue iaeiia?aiaiiiai oi/iiai ecia?aiey
iaeiiiooe?othueo aaee/ei. I?eioeeieaeueiui a oaeie iaieeanne/aneie
eioa?i?aoaoeee yaeyaony oaeo, /oi nouanoaoao iienaiea ninoiyiey nenoaiu
((( >, ( ), iicaieythuaa ecaaaeaoue iaaeaoa?ieie?iaaiiinoe a
i?aaeneacaiee ?acoeueoaoia ecia?aiee.

Aii?in ia ia?aoeiinoe a?aiaie a eioa?i?aoaoeee ni ne?uouie ia?aiao?aie
ia yaeyaony eeth/aaui, e inoa?ony noieue aea ioe?uoui, neieue e a
eiiaiaaaaineie eioa?i?aoaoeee (iniaaiii anee ec iineaaeiae “oaeaeinue au
ecuyoue” i?eioeei ?aaeoeoeee aieiiaie ooieoeee).

a) Iiiaiie?iaay eioa?i?aoaoeey eaaioiaie iaoaieee (eiioeaioeey Yaa?aooa)
enoiaeeo ec i?eioeeia ?aaeueiinoe aieiiaie ooieoeee. I?e yoii
iinooee?oaony, /oi nouanoaoao oaeay ooieoeey n?aco aeey anae Anaeaiiie,
e iao iaiaoiaeeiinoe a ienoe/aneii “aiaoiai iaaethaeaoaea”, ioaa/athuai,
iai?eia?, ca eaaioiaua yooaeou a iiiaio a? ?iaeaeaiey. A iiiaiie?iaie
eioa?i?aoaoeee ianoi iinooeaoa ?aaeoeoeee aieiiaiai iaeaoa caieiaao
iiiyoea “aaoaeaiey aieiiaie ooieoeee Anaeaiiie”, eioi?ia iiaeii
oieeiaaoue eeai ia?acii – eae iiyaeaiea “ia?aeeaeueiuo eaaioiauo ie?ia”,
eeai /enoi iaoaiaoe/anee, eae i?ioeaaeo?o aeaoaeoi?ecaoeee aieiiaie
ooieoeee iaaethaeaaiiai iauaeoa [2, n.29]. I?e yoii aicieeatho naie
iaoaiaoe/aneea oiieinoe, naycaiiua n i?aaeii/oeoaeueiui auai?ii aacena
nianoaaiiuo ninoiyiee aeey eaaeaeiai iauaeoa ai Anaeaiiie,
eneeth/athuaai “eeoiea” aaoaeaiey aeey iaiaaethaeathueony a eiie?aoiii
yenia?eiaioa iauaeoia (naiaa?aciia i?eiaiaiea oi?ioi ecaanoiie “a?eoau
Ieeaia”).

Iaeiiaoe, a?thnnaeueneay eioa?i?aoaoeey ia?aie/eaaao i?eiaieiinoue
/enouo ninoiyiee (oi anoue oi/ae a oaciaii i?ino?ainoaa eeanne/aneie
iaoaieee e aieiiauo ooieoeee a eaaioiaie iaoaieea) aaaaeaieai iaeiaai
iiaiai i?eioeeia, eioi?ue iiaeii iacaaoue “iee?ineiie/aneei aoi?ui
ia/aeii oa?iiaeeiaieee”. I?e yoii ioaa?aaaony i?aaenoaaeaiea eae i
?aaeueiinoe aieiiaie ooieoeee a noa?ii niunea yoiai neiaa, oae e i
“eeanne/aneii eaeaaea” – a iieueco iiaie eiioeaioeee, a iniiaa eioi?ie
eaaeeo iaia?aoeiinoue a?aiaie.

3.2 Iaoieoa?iay yaiethoeey e ianaiaeeiia iienaiea

Iaia?aoeiinoue, au?aaeaaiay no?aeie a?aiaie – naienoai noaoenoe/aneia.
Iia ia iiaeao auoue aaaaeaia ia o?iaia ioaeaeueiuo o?aaeoi?ee (eee
aieiiauo ooieoeee) e iiyoiio o?aaoao ?aaeeaeueiiai iooiaea io
iuethoiiianeie iaoaieee eee i?oiaeienaeueiie eaaioiaie iaoaieee, a
iniiaa eioi?uo eaaeao iiiyoey o?aaeoi?ee eee ioaeaeueiie aieiiaie
ooieoeee. Au? Aieueoeiai iiiye, /oi iaiaoiaeei iiaeoiae ia iniiaa
ainaiaeae. Oeiea I?eaiaeeia ?aaeecoao yoo i?ia?aiio n iaiaoiaeeiie
iaoaiaoe/aneie no?iainoueth.

Iaonoie/eainoue e oain auioaeaeatho ioeacaoueny io iienaiey eeanne/aneie
iaoaieee a oa?ieiao o?aaeoi?ee e ia?aeoe e iienaieth a oa?ieiao
?ani?aaeaeaiey aa?iyoiinoe. I?eia?ii iiaeao neoaeeoue ?anniio?aiiia
?aiaa ioia?aaeaiea naeaeaa Aa?ioeee. A ?acaeaea 1.1 aue i?eaaae?i yaiue
aeae iia?aoi?a n aeene?aoiui a?aiaiai, iienuaathuaai yaiethoeeth
ieioiinoe aa?iyoiinoe aeey naeaeaa Aa?ioeee (i?eiaieoaeueii e
ioia?aaeaieyi iiaeiaiue iia?aoi? iacuaaaony iia?aoi?ii
Ia??iia–O?iaaieona). A noaoenoe/aneie iaoaieea iia?aoi? yaiethoeee eiaao
aeae U(t) = e–iLt, a a eaaioiaie iaoaieea U(t) = e–iHt. Aeaa iineaaeieo
iia?aoi?a oieoa?iu, oi anoue nio?aiytho neaey?iia i?iecaaaeaiea, e a
aeeueaa?oiaii i?ino?ainoaa eiatho nianoaaiiua cia/aiey, ii iiaeoeth
?aaiua 1 – oi anoue i?eaiaeyo e ia?eiaee/aneei ooieoeeyi io a?aiaie oeia
exp(–iEnt). A ioee/ea io ieo iia?aoi? yaiethoeee oaioe/aneeo nenoai
aeieaeai iienuaaoue i?eaeeaeaiea e ?aaiiaaneth e, neaaeiaaoaeueii,
niaea?aeaoue a?aiy ?aeaenaoeee. Aeey yoiai o?aaothony eiiieaeniua
niaeo?aeueiua i?aaenoaaeaiey.

Ieacaeinue, /oi aeey naeaeaa Aa?ioeee a aeeueaa?oiaii i?ino?ainoaa
niaeo?aeueiiai ?aceiaeaiey ioia?aaeaiey ia nouanoaoao. Nianoaaiiua
ooieoeee yoiai iia?aoi?a ia oaeiaeaoai?ytho oneiaeth eaaae?aoe/iie
eioaa?e?oaiinoe, iiyoiio aianoi aeeueaa?oiaa i?ino?ainoaa o?aaoaony
ia?aeoe e oae iacuaaaiiio iaiau?iiiio i?ino?ainoao, aeeth/athuaio
ia?yaeo n eaaae?aoe/ii eioaa?e?oaiuie ooieoeeyie, iai?eia?, au? e
(-ooieoeee oeia aee?aeianeie. Nianoaaiiua cia/aiey aeey iino?iaiiuo a
yoii i?ino?ainoaa nianoaaiiuo ooieoeee ieacuaathony iai?yioth naycaiiuie
n a?aiaiai Eyioiiaa a oaioe/aneie nenoaia.

Ia ycuea ?ani?aaeaeaiee aa?iyoiinoe ioaeaeueiay o?aaeoi?ey aeey naeaeaa
Aa?ioeee i?aaenoaaeyaony ooieoeeae (n=((x–xn), naeaea Aa?ioeee
i?aia?acoao a? a (n+1=((x–xn+1)= ((x–2xn) i?e xn<1/2 e a (n+1=((x–xn+1)= ((x+1–2xn) i?e 1/2 = ). Iao?oaeii iieacaoue, /oi
ii eiaao aeae:

Iiaeii oaeaea iieacaoue, /oi iia?aoi? U+ – eciiao?e/aneee, oi anoue
nio?aiyao neaey?iia i?iecaaaeaiea (iaeiaei a ioee/ea io oieoa?iiai
eciiao?e/aneee iia?aoi? ia aeiioneaao ia?aoiiai, ec /aai neaaeoao, /oi
naeaea Aa?ioeee – ia ia?aoeiia ioia?aaeaiea). Caaea/a ia nianoaaiiua
cia/aiey U+f(x)=(f(x) ia eiaao ae?oaeo ?aoaiee a eeanna iai?a?uaiuo
ooieoeee, e?iia iinoiyiiie. Oaeei ia?acii, naeaea Aa?ioeee ia eiaao
niaeo?aeueiiai i?aaenoaaeaiey a aeeueaa?oiaii i?ino?ainoaa. Iaeiaei U+
eiaao nianoaaiiua ooieoeee e nianoaaiiua cia/aiey a iaiau?iiuo
i?ino?ainoaao. Iai?eia?:

U+[((x–1)–((x)]=1/2 [((x–1)–((x)],

neaaeiaaoaeueii, iu eiaai nianoaaiioth ooieoeeth iia?aoi?a U+, eioi?ay
i?eiaaeeaaeeo e eeanno iaiau?iiuo ooieoeee e eiaao oaeia aea
nianoaaiiia cia/aiea, eaeia ia?aue iiiai/eai Aa?ioeee eiaao aeey
iia?aoi?a U. Iaicia/ei iiyoiio iaeaeaiioth ooieoeeth B(1)(x).

Nouanoaoao oeaeia naiaenoai iaiauaiiuo ooieoeee B(n)(x), eioi?ua
yaeythony nianoaaiiuie ooieoeeyie iia?aoi?a U+ e niioaaonoaotho
nianoaaiiui cia/aieyi 1/2n. Yoe ooieoeee ia eiatho eiia/iie ii?iu, /oi
auioaeaeaao e ia?aoiaeo a iaiau?iiia i?ino?ainoai. Eo naiaenoai,
iaeiaei, iaeaaeaao naienoaaie i?oiaiiaeueiinoe e iieiiou.

Oaeei ia?acii, eae e a eaaioiaie iaoaieea, iu iiaeai ?aceiaeeoue
aa?iyoiinoue ((x) ii aei?oiii?ie?iaaiiiio naiaenoao ooieoeee:

.

?ani?ino?aiyy neaey?iia i?iecaaaeaiea ia iaiau?iiua ooieoeee,
iaiaoiaeeii naeaeaoue iaeioi?ua nouanoaaiiua caia/aiey. Iniiaiia
naienoai (-ooieoeee ninoieo a oii, /oi i?e eioaa?e?iaaiee n iau/iie
iai?a?uaiie ooieoeee iia “au?acaao” a? cia/aiea a oi/ea x=x0. Aeey
ei??aeoiinoe neaey?iiai i?iecaaaeaiey , aaea f – iaiau?iiay
ooieoeey, iaiaoiaeeii, /oiau g auea iiaeoiaeyuae ooieoeeae,
iaania/eaathuae noiaeeiinoue neaey?iiai i?iecaaaeaiey. Iia, i/aaeaeii,
ia aeieaeia i?eieiaoue aaneiia/iuo cia/aiee – ai anyeii neo/aa, a oi/ea
x=x0. Iacia?i oaeea ooieoeee i?iaiuie.

Iu iiaeai ii?aaeaeeoue aeaenoaea iia?aoi?a A ia iaiau?iioth ooieoeeth f
n iiiiuueth niioiioaiey = – ii oaeia niioiioaiea aiieia
ii?aaeaeaii oieueei i?e oii oneiaee, /oi A+g inoa?ony i?iaiie ooieoeeae.
Caaea/a ia nianoaaiiua cia/aiey A|f> = (|f> oaeaea eiaao niune oieueei a
oii neo/aa, anee iieueciaaoueny i?iaiuie ooieoeeyie g oaeeie, /oi
= (.

Aica?auaynue e niaeo?aeueiiio i?aaenoaaeaieth yaiethoeee i?e naeaeaa
Aa?ioeee, aeaeaai auaiae: oae eae B(n) – iaiau?iiua ooieoeee, ((x)
aeieaeia auoue i?iaiie ooieoeeae, oae eae a i?ioeaiii neo/aa ae au
niioaaonoaiaaea (-ooieoeey, aeey eioi?ie neaey?iia i?iecaaaeaiea n B(n)
?anoiaeeony.

Niaeo?aeueiua oai?ee I?eaiaeeia i?eiaieiu oieueei aeey ainaiaeae
o?aaeoi?ee – yoi ooiaeaiaioaeueiue ?acoeueoao. Aeey oaioe/aneeo nenoai,
a naeaea Aa?ioeee – i?inoaeoee ec i?eia?ia oaeeo nenoai, aa?iyoiinoiia
iienaiea neaaeoao no?ieoue ia a aeeueaa?oiaii, a a iaiau?iiii
i?ino?ainoaa, e iii ianaiaeeii. A yoii – i?eioeeieaeueiia ioee/ea
a?thnnaeueneiai iiaeoiaea io iiaeoiaea ia iniiaa oai?ee ainaiaeae
Aeaana–Yeiooaeia: eo iienaiea auei naiaeeii, iineieueeo iiaei auoue
?aceiaeaii ia iienaiey ioaeaeueiuo o?aaeoi?ee.

Iu iiaeoiaeei e aaaeiiio aii?ino: /oi icia/aao aeaenoaea iia?aoi?a
yaiethoeee U(t) ia iaiau?iioth ooieoeeth? Yoi niioiioaiea eiaao aiieia
ii?aaeae?iiue niune, anee U+(t)g inoa?ony i?iaiie ooieoeeae. Aeey
oaioe/aneeo nenoai yoi oneiaea, eae i?aaeei, ia auiieiyaony e i?e t>0, e
i?e t<0. I?iaiua ooieoeee aeey i?ioeiai ioee/athony io i?iaiuo ooieoeee aeey aoaeouaai. Yoio oaeo i?eaiaeeo e ia?ooaieth neiiao?ee ai a?aiaie e eaaeeo a iniiaa ?aoaiey ia?aaeiena a?aiaie, i?aaeeaaaaiiai a?thnnaeueneie oeieie.?anniio?aiiia auoa ioia?aaeaiea iaea?y oaeaea aeiioneaao niaeo?aeueiia i?aaenoaaeaiea a aeeueaa?oiaii i?ino?ainoaa, iaeiaei nianoaaiiua cia/aiey aai iia?aoi?a Ia??iia–O?iaaieona ia eiatho i?e yoii ioiioaiey e a?aiaie Eyioiiaa – oaeei ia?acii, oaioe/aneea naienoaa inoathony "ca eaae?ii". Ieacuaaaony an?-oaee, /oi iaeioi?ua oaioe/aneea nenoaiu – e i?aia?aciaaiea iaea?y a /anoiinoe – aeiioneatho aeiiieieoaeueiua niaeo?aeueiua i?aaenoaaeaiey. Iiieii niaeo?aeueiiai i?aaenoaaeaiey iia?aoi?a yaiethoeee a aeeueaa?oiaii i?ino?ainoaa iiaeii iino?ieoue iiaia i?aaenoaaeaiea a iaiau?iiii aeeueaa?oiaii i?ino?ainoaa, eioi?ia naycuaaao yaiethoeeth ai a?aiaie n a?aiaiai Eyioiiaa.Iiaeao aicieeiooue aii?in – oae eaeia aea i?aaenoaaeaiea i?aaeeueiia? N iaoaiaoe/aneie oi/ee c?aiey iie iaa aiieia ei??aeoiu. Iaeiaei eiiieaeniua i?aaenoaaeaiey a iaiau?iiii i?ino?ainoaa iicaieytho i?iaeaeiooueny cia/eoaeueii aeaeueoa, oae eae aeeth/atho a niaeo? iia?aoi?a yaiethoeee a?aiy Eyioiiaa, eioi?ia oa?aeoa?ecoao a?aiaiiie ai?eciio oaioe/aneeo nenoai. Iiaua i?aaenoaaeaiey iicaieytho iienuaaoue i?eaeeaeaiea e ?aaiiaaneth, yaii iienuaatho ia?ooaiea neiiao?ee ai a?aiaie e aeeth/atho iaia?aoeiinoue ia ooiaeaiaioaeueiii o?iaia iienaiey.Aanueia aaaeii, /oi iiaua i?aaenoaaeaiey ianaiaeeiu. Iaiaeiie?aoii ooaa?aeaeaeinue, /oi oain, naycaiiue n /oanoaeoaeueiinoueth e ia/aeueiui oneiaeyi, i?eaiaeeo e "iaau/eneeiui" o?aaeoi?eyi. Eacaeinue, /oi yoi /enoi oaoie/aneay o?oaeiinoue. Eae oaia?ue iiiyoii, i?e/eia ai?acaei aieaa aeoaieay. Nouanoaoao naiaai ?iaea niioiioaiea aeiiieieoaeueiinoe a ai?ianeii niunea iaaeaeo iaia?aoeiinoueth ia o?iaia noaoenoe/aneeo ainaiaeae, n iaeiie noi?iiu, e o?aaeoi?ee – n ae?oaie.Ia i?inoaeoeo oaioe/aneeo i?eia?ao iu i?ieeethno?e?iaaee, eae a eiioeaioeee I?eaiaeeia aicieeaao iaiaoiaeeiinoue ianaiaeeiiai iienaiey e eae a yoii ianaiaeeiii iienaiee i?iyaeyaony no?aea a?aiaie. Ia?aoeiny oaia?ue e auaiaeai, eioi?ua aiaeiae/iue iiaeoiae aea?o a eaaioiaie oai?ee (iau?i ianoiyuae ?aaiou ia iicaieyao iiae?iaii iienaoue iaoaiaoe/aneea iniaaiiinoe i?eiaiaiey yoiai iiaeoiaea). I?eaaae?i oieueei iaeei i?eia?.A iia?aoi?a yaiethoeee U(t)=e–iHt aoaeouaa e i?ioeia ea?atho iaeio e oo aea ?ieue, oae eae iacaaeneii io oiai, eaeea ciaee eiatho t1 e t2 auiieiyaony naienoai U(t1+t2) = U(t1) + U(t2). I?eiyoi aiai?eoue, /oi iia?aoi? yaiethoeee U(t) ia?acoao aeeiaie/aneoth a?oiio. I?iaiua ooieoeee aea i?eiaaeeaaeao aeaoi ?acee/iui eeannai a caaeneiinoe io oiai, eaeoth yaiethoeeth – i?yioth (a aoaeouaa) eee ia?aoioth (a i?ioeia) – iu ?anniao?eaaai. Yoi icia/aao, /oi aeeiaie/aneay a?oiia, ii?iaeaeaaiay iia?aoi?ii yaiethoeee U(t), ?aniaaeaaony ia aeaa iieoa?oiiu – iaeio aeey iia?aoi?a U(+t), ae?oaoth – aeey U(–t).i?e ( ( (. Yoi ono?aiyao ?anoiaeeiinoue – ii oaeay aeiaaaea anoue ia /oi eiia, eae aaaaeaiea o?iiieiae/aneiai oii?yaei/aiey ia iee?ineiie/aneii o?iaia! A ?acoeueoaoa neiiao?e/iia ai a?aiaie o?aaiaiea O??aeeiaa?a ii?iaeaeaao aeaa eeanna ?aoaiee, iaeii ec eioi?uo niioaaonoaoao i?yiiio. a ae?oaia – ia?aoiiio ?annayieth. ?aoaiea o?aaiaiee iaeaaeaao iaiueoae neiiao?eae, /ai o?aaiaiey aeaeaeaiey.Aiaeiae/iue iiaeoiae a eaaioiaie noaoenoe/aneie oai?ee – ?aoaiea caaea/e ia nianoaaiiua cia/aiey noia?iia?aoi?a Eeoaeeey – oaeaea i?eaiaeeo e iaiaoiaeeiinoe iieiie aeiaaaee a ciaiaiaoaeue, e nianoaaiiua ooieoeee noia?iia?aoi?a Eeoaeeey ia?anoatho auoue i?iecaaaeaieyie aieiiauo ooieoeee. Iieo/athueany o?aaiaiey Eeoaeeey–oii Iaeiaia ia iiaoo auoue auaaaeaiu ec o?aaiaiey O??aeeiaa?a. A yoii niunea eiioeaioeey I?eaiaeeia i?eaiaeeo e aeueoa?iaoeaiie eaaioiaie oai?ee.CAEETH*AIEAA eiioeaioeee E.I?eaiaeeia iaia?aoeiinoue i?ioeannia ai a?aiaie aaiaeeony ia iee?ineiie/aneii o?iaia. A eaaioiaie oai?ee yoi aeinoeaaaony ?anniio?aieai i?ino?ainoaa iaiau?iiuo ooieoeee aianoi iau/iiai aeeueaa?oiaa i?ino?ainoaa, i?e yoii iia?aoi? yaiethoeee nenoaiu ia?anoa?o auoue oieoa?iui, a aai nianoaaiiua cia/aiey noaiiayony eiiieaeniuie. Iieiay /anoue yoeo nianoaaiiuo cia/aiee iinea iiaenoaiiaee a o?aaiaiea O??aeeiaa?a ioaa/aao ca caoooaiea, /oi niioaaonoaoao iaia?aoeiinoe a?aiaie.Ae?oaay aaaeiay /a?oa eaaioiaie oai?ee a eiioeaioeee I?eaiaeeia – i?eioeeieaeueiay ianaiaeeiinoue iieo/aaiuo ?aoaiee e aieiiaui ooieoeeyi ioaeaeueiuo /anoeoe. Noaoenoe/aneia iienaiea n eniieueciaaieai iao?eoeu ieioiinoe noaiiaeony iaiaoiaeeiui n naiiai ia/aea, iu aieueoa ia iiaeai ?annoaeaeaoue eia/a, eae a oa?ieiao ainaiaeae.A ioee/ea io eiiaiaaaaineie eioa?i?aoaoeee eaaioiaie iaoaieee, ia o?aaoaony iinooeaoa i ?aaeoeoeee aieiiaiai iaeaoa e nouanoaiaaiey aiaoiaai iaaethaeaoaey n eeanne/aneei i?eai?ii. A yoii anoue iaeioi?ia noiaenoai n iiiaiie?iaie eioa?i?aoaoeeae Yaa?aooa, oae eae iiaeii aaiaeeoue iiiyoea aieiiaie ooieoeee Anaeaiiie. Iaeiaei, iaoaiaoe/aneee aiia?ao oai?ee I?eaiaeeia ia o?aaoao aaaaeaiey i?ioeanna aeaoaeoi?ecaoeee aieiiaie ooieoeee e neiaeiuo i?ioeaaeo? auai?a aacena, naycaiiiai n iauaeoii.Aaaaeaiea aa?iyoiinoae a eiioeaioeee I?eaiaeeia aiieia niaianoeii n oece/aneei ?aaeeciii, e aai ia o?aaoaony iauyniyoue iaiieiioie iaoaai ciaiey. Iaaethaeaoaeue aieaa ia ea?aao aeoeaiie ?iee a yaiethoeee i?e?iaeu – ii e?aeiae ia?a, ea?aao ?ieue ia aieueooth, /ai a eeanne/aneie oeceea. Yoa ?ieue e?aeia aeaeaea io ?iee aeaieo?aa, eioi?ie eiiaiaaaaineay eioa?i?aoaoeey eaaioiaie oeceee iaaeaeyao iaaethaeaoaeae, n/eoay eo ioaaonoaaiiuie ca ia?aoiae io iioaioeeaeueiie aiciiaeiinoe i?e?iaeu e aeooaeueiinoe.Naiui aea, aa?iyoii, aaaeiui, yaeyaony oi, /oi iaeia e oa aea iaoaiaoe/aneay no?oeoo?a, aeeth/athuay a naay oain, iicaieyao ?aoeoue e ia?aaeien a?aiaie, e eaaioiaue ia?aaeien – aeaa i?iaeaiu, eioi?ua ii?a/aee ai?eciiou oeceee ia i?ioyaeaiee iiiaeo-iiiaeo eao.EEOA?AOO?A1. I?eaiaeei E., Noaiaa?n E. A?aiy, oain, eaaio – I.: I?ia?ann, 19942. Aa?aeineee A.I., Eaiaiuee A.TH., Iiiiia??a A.I. Ooiaeaiaioaeueiua i?iaeaiu eioa?i?aoaoeee eaaioiaie iaoaieee. Nia?aiaiiue iiaeoiae – I.: Ecae-ai IAIE, 19883. Eaiaeao E.Ae., Eeooeoe A.I. Oai?aoe/aneay oeceea. O.1, Iaoaieea – I.: Iaoea, 19884. Eaiaeao E.Ae., Eeooeoe A.I. Oai?aoe/aneay oeceea. O.3, Eaaioiaay iaoaieea. Ia?aeyoeaenoneay oai?ey – I.: Iaoea, 19905. Eaiaeao E.Ae., Eeooeoe A.I. Oai?aoe/aneay oeceea. O.5, Noaoenoe/aneay oeceea. *anoue 1 – I.: Iaoea, 19886. Yeiooaei A. Nia?aiea ni/eiaiee a /aou??o oiiao, o.3 – no. Enioneaiea e iiaeiuaiea eceo/aiey ii eaaioiaie oai?ee – I.: Iaoea, 1966PAGEPAGE 2PAGE

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