Y~ 517.946.9
T. M.
HeTecona (HH-T HaTeHaTHKH HAH YKpaHHbl, KtleB)
ITYIIHOBOI~ AHAJIH3 KPAEBbIX
3Aj]Aq MATEMATHtIECKOI~ |
Invariance conditions and conditions of invariant solvability are obtained for boundary-value problems
in mathematical physics.
OTpHHaui yHosu iHBapiaHTXOCTi Ta iHnapiaHTHOi pOan'a3H0CTi Kpaflonax 3a~la,~ MaTeMaTHqH0i
B o6mapao~t cdpepe npaaomemna Teoprlri rpynnosoro aHaJm3a ~aqbqbcpcrmHa2mHHX
ypaBHCHH~I [1] aMeeTca ~enuR pJ~; 3a~aq arrrepecnux H Ba3KHblX KaK
B
o6nacTn
c1)yH]~aMCHTaJIbHhlX HCCJIC~OBaHH~, TaK H a npaKTHqeCKOH nnaHc, HayqeHHC KOTO-
pI~lX r ~aJICK0 /~0 3aBcplIICHHJt. O/][HH H3 TaKHX BOHpOCOB ~ FpyIIIIOBOl~ aHaJIH3
KpaCBhlX 3a~atI MaTCMaTHqCCKOIt dpH3HKH. ~0 CHX Hop BCTpeHalOTC~ J-lHllIb OT-
~e/Ibrlr~e rlprlHepbt Hay,4eHIrUI CBOItCTB HI-mapHaHTHOCT~ KpaeBhlX
3a~a,~ [2, 3].
~aHHaJ~ pa6oTa nocBJtmeHa Hccne/~oBamno
~onpocoB HHBapHaaTaOCTH
KpaeB~x
3a~a~
HaTeHarrrqccKolt
C~DH3HKH.
Hpe~npHmrra nonmaxa ycTaHoB.rleHH$I KpHTepHeB
HX HHBapHaHTH0~ paapcmnHoca-a.
IIycTh
B
HeKOTOpOM IIpOCTpaHCTBe
RN(x,
u), X
=
(Xl, X 2 ..... Xn), U
=
(Ul, U 2 ....
.... Um), N = n + m,
paccHaTpHBaerca KpaeBaa (naqa~bao-~paeBaz) aa/Iaqa ~na
CHCTeMt~I//~I(~C1DCpCHI/~aYI~H/~X ypaBHeHI~:
~ff~: FV(x,u,u ',
.... u (~)) = 0, v
=
1,2 ..... k, (1)
C KpaeBhlMrl (Haqa.rlbHO-KpaeBhlHa) yc.rl0Brl~IHa
~: ~rt(x,u,u',
...) = 0, ~t = 1,2 ..... k, (2)
,,
u(~) _-
F~C
U', U , ..., IIpOH3BO~HhlC IICpeMCHHHX U 1 , U2, ..., /~m IIO Xl, X2, ...,
X n
go
nop~Ka ~: BKJ/IOUj4TC/~HO.
PaccMaTpHBa~ ~'~ H ~ KaK MHOFOO6pa3H~ B HCKOTOpOH Hpo/~oII~KeHHOM
npoc'rpaHcrBe Rg(x, u, u', u", .... u00), M = n + m + (~: - 1),
6yl~cH
Ha3uBaT~,
KpacBym (Ha'~a.m,HO-KpaCBy',o) 3a~a,~ ~ - ~ - (1), (2)
uu~apuanmno~ (cu~em-
pu~noa),
ecJm cymecTsyer rpyrma npeoSpa30BaHma G~ OTHOCHTeJIbHO KOT0p01~
mmapHanTri~ KaZ ypaBHem~a ~]~- (I), TaX H zpaeB~ac (HaaonmHo-Kpaes~e) yCnOBnJi
- (2) 3a/Iaqrl. E/II4~ICTBCHHOe p~IIICHHC U TaKOI~ ~a~a~a, ccJm OHO CylRCCTByeT,
6y~;cM Ha3blBaTb
unoapuaumn~.~t (cu~t~tempul~nbt~t)pelIIeHHeM
3a/~aqH ~- ~, a
aa~a,~y
~ ~puanmno pazpeu~u~tozL
~a.ra,me a/Ipo OCHOBI-II~X FpyIIH
G N
npeo6pa3osaHrd~ HCXO~HOFO ypaBHeHaJt (CH-
CTCHI~ ypaBHeHrn~) 6ygeH o6o3Ha~laTb qepc3
G
.
Oqenmmo, wro aJ~a paccMarprmaeHoll 3a~aqH npeo6pa3oBar!H)~HH HHBapHaHT-
Hocra 6y/lyr T0SmKO TC npeo6pazosarm~ P Ha J~pa G ocnosmax rpynn cHcreHu
ypaaHern~ (I), KOTOpUe XSmnOTCS TazomaMrI ~ana nocramucnmax KpaeBuX
(Ha,~am,-
H0-KpacBhlX) ycn0nHlt
(2)
3atla~H. B
3T01t CBH3H I~IHOBOI~/aHaYlH3 Kpaesblx
~aa
~)q~IP.,CTHO B~ B T~pMP~aX o/moro OT~e,21bHO B3JITOFO IIpeo6pa3oBalta.q P H3 J~tpa
G
OCHOSmax rpynn aCXO/IaOtt cHerema ypa~erma pacc~aTprlaaeMott
aa/Ia~m.
J;[CHO, trro 1t KOpHr
aeeae~tyeHoro
sonpoca
ae~3rr gpltTCpP3t HHBapHaHTHOCTH
~aoroo6pa3rta. Hanomm~, wro paccHarprmaeMoe n
E.
MHOroo6pa3rlC (noBepx-
nocr~) ~2, aazIamaoc ypaaacmtam~
T. M. ~OBA~ 1999
140 I$SN 0041.6053, Yxp. ~m. .,wylm,, 1999, m. 51, IV ~ ]
FPYl'IFIOBO~I AHAJIH3 KPAEBbIX 3A,/IAq MATEMATHqECKO~I ~H3HKH 141
9 ~ ~= 1,2 ..... s, x=x I,x2 .....
Xn,
(3)
Ha3hIBaCTC$1 HHBapHaHTHrJM MHOFOO6pa3HCM
HeKOTOpOI~ FpyIIHKI
G r
npeo6paaoBa-
HI4fl
(n-
pa3MCpHOCTb npocTpaHCTBa, r- napaMcTp rpynn~),
cc~H ~
.mo6oro
npco6paaosaaHa
Tae G~
(Ta:
xi=j~(x,a),
a=al,a 2
.....
ar), x
e
~'d,
cne~yeT
T a x
9
~ff~.
CornacHo KpHTepHIO rlHBaprlaHTH0CTrI MH0roo6pa3nJl/~.rlJt TOrO, Wl'O6bl
3a~aHHoe MHOroo6paaae (3) 6ranO HrmapHarrrHh~, Heo6xomrMo H ~OCTaTO~mO, ~rro-
6m ~na Bcex r0~eK 3Tor0 Mnoroo6pazHa mano0mmmcb paBeHCTBa
Xa~FO(x)
= 0, a = 1,2 ..... r, 6 < 1,2 ..... s, (4)
r~e X a -- HHqbHHrrre3aMara~Hralt oneparop rpynwa G~ n, COOTSeTCTSymttmil npe-
o6pa3oBam o
Ta.
B HatuHx
HCCo-le/~0BaHHJ~X
0FpaHHqI4MC.q paccMoTpeHHeM o/monapaMeTpHuecKHx
rpyrm npeo6pa3oBaHHia (r = 1 ). ~TO o3aaaaeT, aTO ec0m rrpeo6paaoBarm~
T a
H3 r-
napaMeTpauecKo~t rpyrmta G~ npeo6paaosaHH~ COOTBeTCTByeT miqbmmTezrtMa-
0mH~ll onepaTop
X a,
TO upa HatUHX npezmoaoaceHHaX npeo6pazonaHam P Ha
O~HonapaMeTprrqecKott rpynma G 6y~eT COOTSeTCTBOBaTb orIepaTop
i
Xp = ~i(x,u)~x, + ;(x,U)Ouj,
i = 1,2 ..... n, j = 1,2 ..... m. (5)
Tor~a MHOF006pa3HJ/ ~ -
(1) a ~
-
(2)
5y~yT
HHBapHaHTHI~I
OTHOCHTeYlbHO
O~HO-
ro H TOt 0 ~Ke npeo6paaoBanaa P, ecom B IIpo~0JDKeHHOM np0cTpaHCTBe R M Bbl-
nonaaiOTCa paBericTBa
Xt,^r FV(x, u, u', ....
u(~))]~ = 0, (6)
A~
o~g (x,
u, u .... )[S~
Xp "
=
0, (7)
r~e ,, ^" o3HaqaeT npo~to~enae onepaTopa X v ~to nopaz~Ka ~=
PaBeHCTBa (6) H (7), KOTOpBIe nl~gCTaBn.qiOT co6olt ycnosHJt HHBapHaHTHOCTH
ypaBHeHma (1)H (2) OTHOCHTe~HO rrpeo6pa3oBarma P c G, 6y~eM Haa~maT~
paaen-
cmsa~tu (yc.aosu~tu) unsapuanmnocmu
KpaeBoR (HaqaYmHO-KpaeBo~I) 3a~aaH OTHO-
CHTenBHO rlpeo6pa3oBaHHJt
P H3 a~pa G OCHOBHI~IX rpynn npeo6pazoBanHll CHcre-
TpaBHeHHI~I (1).
HOO6XOgHMO OTMOT~T~, '-IYO HOHnT~O mmapHaHTHOCTH KpaoBoro ycnomIa n aaga-
~ax MaTeMaTHqecKofl
CIDH3HKH HMeeT
cnelii4dpI4qecKyIO
OCO~HHOCTb,
KOTOpa.q ilpe/~-
nonaraeT ssmoomeH.e ~On0~HaTen~HOr0 yco~onHa. A HMeHHO, TpeSoBarme arma-
p.aHT~OCTH KpaeBoro ~HorooSpaaHx ~ oTaocrrreomHO npeo6pa3o~aHna P c G,
~onycKae~oro ypaSHeHHJ~H (2), B~mouaeT B ce6a eme H n~osmeHHe a'pe6oBaaHa
onpo~eneHHOlt cor~aco~armocr~ nocras~eHmax KpaeBraX (a Haaasmmax) ycn0~Hi~
paccMaTpH~ae~olt 3a~a.ra. ~ro aono0mrrresmHoe Tpe6onarrae 3a~sno~aeTc~ B TO~,
qTO ssrtayyMermmerm~ ,tocsin Hc3aBHCHM~X ncpestCHmaX B peayamTaTe IIpHMCHeHI~t
~eToao~ rpyrmosoro aHa0maa COOTSeTCTSermo ~O~'KHO y~ermmaT~ca ~ac~o nocTa-
Bnemnax ~ono0mrrrem, max ycaoBma pacc~aapmmeMoit 3a~a~m. ~pyr~ enoBa~m,
/~Ba (H~.rl 6once) ycnosHJl rIOCTaI~CHHOfl xpaesolt (Haqa.llbHO-XpacBOIt) 3a~aqH
~O~KHr~ TpaHcqbopM~posaT~CX B O/toO yc~oBHe B pe~aytmpoBamao~l 3a~a,~c. HanpH-
Mop, B c~y~ae HatlaYlSHO-Xpacsl~X 3a/~aq IUI~ O/IHOMCpHI~X ~BO/IIOI~HOHH~X ypaBHC-
HHa MaTeMaTrraecKolt ~bn~ot (rrpH WrOM HeKOMOe pemeHHe u = u(x, t ) -- qbyHKtma
IIpocTpaI-ICTBeHHOI~ FIeI~MOHHOI~ X H BI~MCI-IH t) CTaB$1T~R/]~Ba Kpacs~tX H O/~HO Ha-
qa3I~HOC yCnOBHC. B pe3ym~TaTe pe;ayral~ npH~eM
K KpaeBl,IM 3a~a~aM ~
06UK-
HOBeHIIOrO /~llqbqbepeHl/~a~soro ypaBHeHH~l, S KOTOplax Ha IICKOMy~o ~)y~
ISSN 0041-6053. Yxp. ~.aun. affptt, 1999, m. 51,1~1
142 T.M. HETECOBA
HaKJIa/I~IRaIoTC~[ TOJISKO ~Ba yC/IOBH$L OI'IHCaHHOr TaKHM
o6pa3oM TpC6OBaHHC
yMe-
HbmCH~ ~Hc~a nocTaB~cm~x ~OHOJIHHTC/IbHI~IX yC~OBHR Kpae~ofl 3aRaqH 6y~CM
Ha3HBaTb c8o~cm~934 w~apuawnnoa pei~ymcuu
KpacB~X (a HaqaJIbHHX) yCIIOBHH 3a-
~aqH, a casm KpacB~e yc~ostta
~
un6apuawnno peDyt~upye~a~va.
rlptme~eHHUe paccyx~aeuHa MOmHO CdpOpMysmpOBar~ B BH~C c~c~3notuero ymc-
p~eHam
Teope~a. ]/n.a
moeo ~mo6~ rpaeoa~ (na~a,u, no-rpaee~) za~a~,a ~tame~tamu-
~ecrotl
~u~Ku (1), (2) 6~za
unoapuaum~ pazpemu:~to~ omRocumemmo neromo-
pozo npeo@azooanu~ P, neoSxoSu~w u Oocmamo~no, wnoS~:
a) npeo@a3o~anue P npunaS/~e~aao ,~Opy ocnoon~x epynn G npeo@a3ooa-
null cucme~4b~ ypaonenu~
(1)--P c
G;
6) o ~,Dot~ mo~re pacc~ampuoae~toz] o6~acmu obmoAn,~UCb yC,~OOUJ~ unoa-
puawnnocmu
(6) u (7);
o) rpaeeb~e u (~a~a~b~b~e) yc,wou,~ (2) saOar 6b~nu unoapuaum~o pec3y~u-
pye~b~U.
3a,~e,~anue.
HOCKOabKy rpynnomott aHa~ma OCHOaaH Ha JIOma/lbHOl~ Teop~H
FpyrlH /IH rlpr TO RCHO, qTO TCXHHKa r MOYKCT 61dTb IIpHMCHHMa
/IHmb K 3a~aqaM, ~oIIycKaIOIIIHM JIOKaJIbHOO paCCMOTpeHHr (HalIpHMop, 3a~aqa
KOUIH, Fypca).
B Kaqccq3e npti~epa paccMoTpHM Haqaylbno-Kpae~yIo aa~aqy, a~n~ottty~c~ Ma-
TOMaTHqCCK0i~ M0]]OJI~IO II~I~CCCOB TOIHI0-MaCCOIICpCH0Ca B cTpaTH~t~HIJ~IpOBaHH0~
BO~Ott cpe~e
[4]:
[f(ux)Ux] x - u t = O, O ~ x < **, t > O;
(8)
u(x,0) = 0, x > 0;
: lf(Ux)Ux
Ix=0 = 0, t > 0; (9)
[ limu(x,t)
= 0, lim
[f(ux)u x] = O, t > O.
i.x "~
x --~ oo
Kaa ycrauosaetmz tmeap.atrr~ott pa3peum~ocam HOCTaBJIOHHOI~ 3a/IaqH npe~e
scero aeo6xo~o onpe~e0mTb ~po ocnoBmax rpyrm G rrpeo6pa3oaa~m~ HCXO/IHO-
ro ypa~HeHHa (8) 3a~awa.
I/IH(IDHHHTO3HMaJIbPJ:d~
onepaTop ~pynn~ G 6yReM HCKaTB B Bl4~e
X = ~x +
not
+ ;0u, (10)
r~e ~, rl . ~
-- qSyHKIItIH OT X, t,'U; ~X) ~t H ~u -- rlpOH3BO]~fla$1 rlO COOTBeTCTBylo-
~ett nepe~enHott.
Bse~aa o6omm.~etma
u x = p. u t = q, uxz = r, uxt = s. u. = l. ~a-mtne~
~TOpOe npo-
aOJL~Kerme ncxo~oro onepaTopa:
r~e npo~onxerm~e K0~l~dpHRaeHTta a a I~ JmJL~OrCa qbyammmda x, t, u, p H q,
a p, ff H X -- qbyHmm~ nepeMeHmax x, t, u, p, q, r, s, I.
Tor~a. r~exo~aa r~a ycJ~omtR mmapHaHTHOerH (6), no.uy~ae~ onpe/zeJmmmee ypae-
HCKI4e]~JLq HCKOMHXKOOp]DIHaT ~, 1] 14 ~
onepaTopa(10)
X2{[f(P)P]x
-q}l~
-
(~
+
2f~,,
+ p2;u u + ;u - 2r~x - 2~x - P~ --
- qrln)tPf'
+f] - 4, - q;u + q~x + qrl, + (~ + p;. - qqt)[rf" + rpf"] = 0.
IIoayqetmoe ypaenetme pemae~ca ~ro~o~ noc~e~o~aTeJmnoro paculermerma
cow.echo c onepanam~ 14cr~moaetm~ 14 ~clxl>epemmpoeauv~, wro n pe~y~sTaxe
rrp~omaT
x CaCTe~e
HeTp~eHa~H~X
~qb~pepeHtma.~Hux ypaBHeHHg, o6mee pe-
metme
KOT0pOg 14 oIIp~e.2IRU'W HCKOMM0 KO~]XI~HI~eHTH
onepaTopa X. A 14biermo,
ISSN 0041-6053, Y~p. ~un. ucytm, 1999, m. 51. N~I
l'~Yl'lHOBOl~ AHAJIH3 KPAEBHX 3A/~H MATEMATHHECKOfl OH3HKH 143
O~a4x
+ (a- 1 )a I , ~l
0~2a4 t + 0~2-1
= = a 2, ~ = aa4u+ (a-1)a 3.
2
CooTaeTcrsymmae aTUM KOaClaqbHmteH'raM HHqbHHHTeaaManbHUe
onepaTop~
n
onpc~e~moT Jt/Ipo
G
OCHOBHIxlX Fpyl'IIl npeo6paaosanHfl ypaBHeHng (8):
PI: ~=x+a, ~=t, ~=u;
P2: .~=x, t'=t+o~ 2, ~'=u;
P3: ~=x, ?=t, ~'= u+a;
P4 : .~=17.x, T=a2t, ~'=au.
BTopoe ycYtoBae HHSapHaHTHOCTH (7)KpaeBrax 3a~aq rIOaBOJIJteT ar.r~e~aa-s rt3
a'roro sz~pa rpyrnay npeo6paaoaaHHlt P4, OTHOCnTem,HO KOTOpOit KpaeBue ycno-
BHJ~ (9) paccMaTpHaaeMofl 3a/iaqri rmBapHaaTHra H, B TO ~Ke BpeMz, aHBaprmnTaO
pe~ytmpye~a,I. ~e~tc'rBHTem,HO, rpynna npeo6paaoBannlt P4 aazse'rcs rpynno~
Macttrra6max npeo6pa3osaHrIfl c nHBapHaHTamf:
X U
rl=~t a V=~. (11)
B~apaa,B petuerme paccMarprmaeMo~ aa~a'~n U =
u(x, t)
,aepea rmBapnanTu (11)
B BHae
U(X,
t) = xff-V(rl), .nerKo Brt~e~, vro nepBoe ri rpe~e aa ycnoBrI~ (9) aa~a-
'au rpancqbopMupy~oTCS B O~HO yc.noane ~m~ qby~KUrIH V(rl), a rIMeUHo,
lim V('q) = 0.
11--)**
Taxr~M o6pa3oM, uenrmeaHaJ~ KpaeBa.a aa~aqa (8), (9) rmaap~arrrno paapetuaMa rt
OThICKaI-IHC ee HHBapHatlTHOFO (B ~aHHOM
cnyqae
aBTOMO,/~eJIbHOFO)
petUenHa CBO-
/XHTCa K pemeHmo cne/w~omefl 3aRaqa ~ O6taKHOaeHHoro/~qbqbepem~azmHoro
ypa~Herma:
[f(V')V'] +
1/2(rlV'- V)= 0, (12)
f(V')Vln=o= Q,
lim V(ri) = 0, lim
[f(V')Vq
= 0. (13)
11-+** 1]-.-***
TeXHHKa rpyrmOBOrO aHa~43a oKa3raBaeTcJ~ yCTlemHO npHMeHHMO~ TaK~Ke npH
paCCMOTpgHHH CrI~T~HaJIbHOFO
K~Tacca 3a~aq,
B KOTOpHX
pemeHHe U nopo~aeTc~
RelllGTBHeM HeKOTOpOFO (MFHOBOHHO HJIH IIOCTO~qHHO
~I~ffTByK)III{eFO) HCTOqHHKa. B
TaKHX 3a~aqax Ha HCKOMOe pemeHHe U HaK3I~aIOTCYl/~OTIOJIHHTCJIbHbIO yCJ~OBHJL
o6ycnosnermhte 3aKOHaMH coxpaHeHHJL PaccMoTprrg MaTeMaTH'4ecKy~O Mo/~e.rn, rrpo-
~ccca
BO3HHKHOBeHHR 3~IeKTpOMarHHTHOFO TIOYIJt
B
dpeppoMar~HTHOfl cpenr (j = (~E,
D = ~E, B = bH ~/n,
n > 1 ) nor ~eltCTBHeM II~OCKOFO HYlH ToqeqHOFO HCTOqHHKa
~YleKTpOMarHHTHOfl
3HepFHH HHTeHCHBHOGTH
Wu(t )
[5]:
1 t,,~H. ~ b~HO-")/nHt
= 0, k = 1, 2, 3; (14)
I'~
e/e- n
n(e,0) = 0, n(o., t) = 0, (Okno)e = 0; (i5)
,~ n+l t ** t
bo, fHTO,_~dO +
O_.k~
~dt
fHgdO-
~Wu(t)dt.
(16)
n+l ~ o
0 0 0 0
,~][po OCHOBH/~X
l"pyllH G npeo6pa~osaHHit ypaBHeH~ (14) npe~eramaaeTCg npeo6-
pa~olmmla~a:
: w ~=at, "g=H;
1'2: "O -" a-b/2o, ~ = t,
HfoJ-/;
P3: w /'=t, ~=H.
ISSN 0041-6053.
Yrp. ~tam. ~yp..,
1999, m. 51,1~1
144 T.M. HETECOBA
Ecm~ cmwravb, qTO
HHTeHCHBHOCTb
HC'rOqHtIKa
Wu(t) --
cTeneHHa~I
qbyHKI~HZ,
T.e. W=(t) =
pwt/'-l,
TO paccMaTptmaeMa~ 3a~a~a (14)-(16) 6y~eT nHBapHanTnO
pa3pemHMott oTuocIcrem,HO rpynnu npeo6pa3osam~tt
c! -be 2
P = ClP 1 + c2P2: 0=iX 2 0, [=o~Ctt, H=t~C2H, (17)
r]]c
c 1 H
c 2-
npOn3SO.rlbH~r
nOCTO~HH~C.
~cttcTstrreJmno, BI~a3HB pcmeHHe H(O, t) qcpr HI-mapHanTu V H Z I
H(0, t) = tmv(TI), 11 =
02/t l, m = c2/c 1, l = 1-bc2/cr
(18)
H
HO]ICTaBHB B HCXO]~IOC
ypaBHCHHr H
yCJIOBH.q
(15), (16),
nony, mM peI1ylJ~IpoBalIHyIo
3a~aqy
/
v(oo) = o, =0, (20)
\ :TI=0
:|_b.fVn 2d + • =w:. (2,)
~Po 03
H3 paseHcrs (20) c~e~yer, ,fro HaqaJmHo-zpaeBa~ 3a~aqa (14)--(16) 6yA~r vraBa-
pHa~rrno pa3pemm~oft OTHOCnTeJZbHO rpynnu Macurra6aux npeo6pa3oBaHHR (17)
npH yc~oamt
p = 2m + 1 + (k + 1 )l / 2, o~y~Ia c 1 = c 2 H
pemCHHe ee CaO~TCZ
K
pcmcemo tpaesott aa~amt (19) - (2I).
BO3MO)F~0CT~ nptIMeUeHH~ TeX~m~ n MeTOAOB rpynnosoro aHa.rIH3a rlpH p~IIIe-
Hm~
zpacmax aajzaq ~o Hac'rosmero Spe~CUH OKOH'4aTCJZI~HO CmC
He
OUCHellbl. Haps-
]~y c ~)aKTOM yrrpomenns HCXO~OI~ 3a~aqH rlyTcM rlOIII4_3KeHH.~ nop~za ypaBHCHH~I
no~mn~eTCS HC06X0]~IMOCTb B ~OHOJ-IHHTeJ'II=HOM HCCJ'IC/~0BaHHH BOHpOCOB Cyl~CCTBO-
BaHH~I H IlOHCKa pr pcAyI~OBOHHO~t 3a~aHH. ~OIIOJIHHTr HCCJIr
HH~I Tpe6ymT TaKH~.r H rlo~lyHaeMI~r HHBapHaHTH~C p~l.I.ICHHSI Ha IlpeKM~I" HX (~H3H-
qe,~KOrO CMI~C./la H yc'roRqHBOCTti.
B o6meM
cJ~yqac rpynnoBott
aHSJIH3 RBJ-L~eTC~I
annapaToM, ]~atouiabl Jn~ub MaTeMaT~qeCKOe pemeHHC 3a~a~ MaTCMaTIIqCCKOI~ QbH-
H ~0Y~r 6blTb rlpHMeHeH npn nOCTpOeHHH d~yH~aMeHTaX~bH0tt biaTe~aT~qeczott
TCOpI414 KpaeshlX 3a]laq
blaTeldaTHqeCKOR
d~)H3HKH.
I. Osc~uroe ./I. 8.
rpynnoaott aHa~n3 n~qbqbepeHuna~bHUX ypaaHeHHlt. -- M.: Hayza, 1978. -
400c.
2. Me~btauros B. M. 0
HenpepUBHO~ conps)ZCH~H ,neapaaH~nux pemeum~//]~Ha~Ka cnaom-
HOlt cpe~u. - 1972. - B~n. I0. - C. 70 - 84.
3, Ilyx~aqee B.B.
HeycTaHOe,ameec~ aaHxetme ~aazolt )Z~KOC~4 co ceo6o/molt rpaHuttetl, orm-
ct~aeM~xe qaffrHqHO-HHBapHaHTHMMH petuenHaMtt ypaBneHti~l HaBbe -- CToKca II TaM ~Ke. --
1972.-B~n. I0.-C. 125- 137.
4. Kepe'~oecruti A. A., Hemecoea T. M. AB'robloAedlbnue pememt~ aa/taqu onpecHeHtL~ t4OpCKI4X BO/I
//Teope'mqecKae H
npaz~a~n~e
aonpocu/~di~pcnmta.n~nux ypalmennlt a a~re6pa.
-
K~eB:
Hayz./13n4Ka, 1978. -C. 41-46.
5. ~epe~oecruii A. A.
JIexu~, no aeJltlneltHldl~l xpacsldM 3a/laqabl MaTeMaTHtleCKOlt diattatIKtt, q. II.
- KHea: HH-T
~aTe~aTmo! AH YCCP, 1974.
-
282 c.
l'lo~yqe.o
18.04.97
I$SN 0041-6053. Yrp,
~m.
~.'vps., 1999 , m. 51, N el