Pairwise products of moduli of families of curves on a Riemannian Möbius strip
We investigate pairwise products of moduli of families of curves on a Riemannian Möbius strip and obtain estimates for these products. As one of the factors, we consider the modulus of a family of arcs from a broad class of families of this sort (for each of these families, we determine the modulus...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4587 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510732908494848 |
|---|---|
| author | Okhrimenko, S. A. Tamrazov, P. M. Охрименко, С. А. Тамразов, П. М. Охрименко, С. А. Тамразов, П. М. |
| author_facet | Okhrimenko, S. A. Tamrazov, P. M. Охрименко, С. А. Тамразов, П. М. Охрименко, С. А. Тамразов, П. М. |
| author_sort | Okhrimenko, S. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We investigate pairwise products of moduli of families of curves on a Riemannian Möbius strip and obtain estimates for these products. As one of the factors, we consider the modulus of a family of arcs from a broad class of families of this sort (for each of these families, we determine the modulus and extremal metric). |
| first_indexed | 2026-03-24T03:01:41Z |
| format | Article |
| fulltext |
Y~K 517.5
IL M. TaMpaaoe, C. A. Oxpr~eHKO (H.-'r blaTeMaTHKH HAH YKpamml, Kaea)
I I A P ~ I E IIPOH3BE~EHHH MO~IY2IEI~ CEMEI~CTB
KPHBbIX HA PHMAHOBOM 2IHCTE MEBHYCA"
Pairwise products of moduli of families of curves on a Riemannian MObius strip are investigated and
estimates for these products are obtained. As one of factors, the modulus of a family of arcs is
considered belonging to a wide class of families of this sort, for which the moduli and the extremal
metrics are also found.
~ocai/DgeHo napHi /IOOyTKn Mo~yaia ciMelt gpanax na piMaaouoMy nt4CTgy Mbo6iyca xa o/iep>gaai
OtdaKrl/llla tll'lX/i06yTgiB. ~[Z 0/it41[ i3 MnO~llaZin po3rlia~ae'rbca Mo~ylib CiM'i ~yr a turlpogoro
gliacy TagrlX ciselt (i ~ a zox~uoi a .ax auafl/Leao Mo/iy.rlb Ta egCTl:~Ma-qblly Me'rpagy).
1. Bae~enae. OReHKH napmax npoHaBe/~eHH~ Mo/xyae~ conpaxeHauX ceMe~aCT8
KpHBhIX HI'paloT Ba~HylO (qacTO petualottly10) po.rlb B MHOrOqI4C2IeHHblX rrp14,rlo:~geHrI-
.qx. B/IamloR paOoTe 80npoc 0 TaKHX oRertKax paccMaTp14aaeTca Z~az HeopHerrr14py-
cMoro MHoroo6pa314a. B KaqecTae 0~1140r0 143 KOMnOHewroB n a p u SmcTynaeT npo14a-
BO./IbH0e FOMOTOH14qeCKOe ceMeI~ICTBO HeCT.qr'HBaeMHX IIeTe3Ib Ha p14MaHOBOM .r114CTe
Me614yca 14~ IlpOH3BOJIbHOe o6~e/g14He1414e TaK14X ceMe~CTH. Mo~yn14 ~TaX ceMeflCTB
na_q~enH paaee [1 - 3]. B KaqecTBe conpax~eHaoro Ko~ln014eHTa n a p u BmcTynaeT
ceMe~CTBO ~ y r 143 mapoKoro K:mcca ceMeItCTB, 14 ~ n a za~r~toro aa H14X n~aqncneH
Mo~yab (anJ~ o ~ n o r o na TaK14X ceMe~cT~ MO~yab 6~aa yKaZaH paHee B [4]).
Ho~yqeHmae HHaMi4 ottenK14 npo143Be~e1414~ Mo/~yne~t ZBJIalOTCa ycnneH14eM 14 0606-
menneM ocao~noro peay~mTaTa 143 [4] o n14~xne~ o u e a z e o~aoia Ha xapa~TepaacTa~
p a ~ a a o ~ a aacTa Me6ayca (c~. naa<e).
I I y c r b 0 < a < +**, 0 < [~ < +~o. B ~ a ~ n e ~ m e ~ ncnon~aye~ cae~tymttme no-
HJtTH~q H o6oaHaqerl14~1143 [1 - 3]: HH0:~eCTBO
Flo.a := { ( x , y ) : - a < x < C t , - [ ~ < y < [ ~ }
B R 2, paManOB :merMe614yca H a , Knacc P ( l ' l a ) MeTpnK p n H a , I~yHKU14OHa2I
A a ( p ) , g014KpeTHym neTmo yl C l ' l a , ceMe~ca~o F l , a neTenb a a I ' l a , nonaTHa
~KCTpeMa21bHO~I MeTp14KH a Mo~yna ceMe~crBa KpHBHX Ha l'Ict.
2. Mo~ayan nonepemIbtX ce~e~ca~ ~yr. IIycr~ 0 < a < + o..
L~)'zo~ S I-[ a yCnOBaMCYt HHa3bma'rl, nenpcpuBnoe oro6pamerme nenyczoro OT-
zp~rroro ~cflcrBrrreJw14oro arrrcpsana s Fla.
a+b
Flycr~ a<b r~ ~: (a,b)-> Hct~1~yraB Ha. Hoaom14M c= 2 r~,~epea 5-
H ~+ yCnOBHMCJ~ 0603HaqaTb cymeH14J~ OTo6pa)Ke1414)~ ~ COOTBeTCTBCHHO Ha nony-
14HTCpl~Im (a, c] I,I [c, b). OTO6pa~KCH14~I ~+ H 5 - yCJIOB14MCa HaahIBaTb no~yOy-
?a~tu, lIp14qebl "re me 0603HaHeHH~q H TepMHH 6y/~eM np14~CHffrb 14 m COOTBeT6TBy~O-
m r ~ o6paza~ ygazaa14~x no~ya14TepsanoB. H a m e Ha ztyr14 8 14ag~a~usae~ HegO-
x o p u e KOM614Hal/HH cae/xymtuax y c a o ~ i .
L / I n s BCJIKOFO KOM14aKTa K CFlc t cyIIiecrByeT KoMnaKT T C (a , b ) TaKoI~,
".fro ~)(t)~ K V t r ( a , b ) \ T .
IL Cytaec '~yeT KOMIIaKT K I C l'[ ct TagO~l, qTO: KaKOB 6I:,I HH 61~IJI KOMIIaKT
T~ C (a,b), ria (a,b)\T~ cymec'mymTTOaKrt a~ 14 b~>ab ~aaagoTopux ~caKaa
R y r a s Ha cHaqa.rloM 5(a~) HKOHROM 6(b~), r o M o T o r m a a s l ' I a ~ y r e ~ : ( a l ,
b~ ) --~ l ' la , I~Me~gT Henyeroe n e p e c e a e a a e c K~.
Bunoanena npu qaeTaqHog no/t31ep~xe Focy/iapcxneanoro qbo./ia qbyH/iaHenX~tbHUX nc-
cneaoeauaa npa Mmmereperne Yxpaanta no aeaa~ nayxa a xexuoaoraa (npoegT 1.4/263) s INTAS
(rpam" 94-1474).
I'L M. TAMPA3OB, C. A. OXPHMEHKO, 1999
110 I$SN 0041-6053. Yrp. ~Jam. ~.'vpu., 1999, m. 51, N ~- 1
I'IAPHbIE rIPOH3BF__~EHH,q MOBYJ'IEI~t CEME~CTB KPHBHX ... 111
III. MHoxcec'rBa ~((a , c]) n 8 ( [ c , b ) ) ne aB:XaIOTCa OTHOcrrreJn,HO KoMnaKT-
HHMH B rIa .
IV. IIyra 8 HMeeT HenycToe nepece,~emm c neT~e~ ,/l.
06oaHaam~ qepea Bct ceMe~CTBO Bcex ~yr n l ' la, ytlon.neTBopalomnx yc~onaJ~
III n IV, a uepea Box ~ ceMeI~CTBO ncex acop~aanoBux ~y r n H a en~a By:
( - a , a ) ---> Ilct, 3aaaBaeMtaX yczoeneM ~y( t ) = (t, y ) E H a n.nn ~ y ( t ) = ( - t , y ) E
E I[a, IRe y ~ napaMeTp, npHrmMa~otuafl .mo6oe IIOCTOJtHHOe aHaqem~e na npoMe-
acyaxa [-13, I~).
06oaHaqaM qepea B~ ceMe~CTBO ecex ~yr B Flct, y~OB.neTBopzIomHx yc.noeaaM
I a II.
Moac.ao y6el~rrrr~ca, qTO B e C B~ C B et.
3aMe'mM, aTO paccMoTpeHHOe n [4] ceMeflCTBO B~ ayr , aMetomnx nH~eKC nepe-
ceqeHH~q --+ 1 C Ka;,K/IOfl KpHBOIt H3 ceMeflCTBa FI,a , y~aOB.neTBOpaeT yc.noBmO B a C
C B~ C B~ (B [4] ]I.nz B~ HCIIOJIb3OBaHO m~oe o6o3Haaeane).
YlycTb Tenepl, F~ - - npowaBO~,Hoe ceMe~CTBO ~oKa.nbHO cnpzM:meM~x ~yr B
I-la, y/~oB~eTBopa~ottIee yC.nOBHaM B a C F~ C Bet.
H c n o ~ , 3 y a KOMn.neKCHyIO nepeMeHHylo w = x + i y, yCTaHOBrl~ c.ne~y~I4fl
peayzbTaT.
Teope~a 1. B H a ztempu~a
p~(w) := 1 V w e ri~
2~
,~cmpe~,aabna On~t ce , teticmoa F~ u M(F~) = -~
(x
~[oxa~ame~bcmoo. CHaqa.na /~OKa~KeM ]IoIIyCTHMOCTb MeTpHKH p~ ]I.rlJt ce-
~eflCTBa B a. qepea I-I 2 o6o3HaaaM ~By.rtaCTHOe HaKp~na~omee MHoroo6paana
Ilct , pea.rlH3OBaHHOe B KOMI'I.rleKCHOfl rlJIOCKOCTH C B BH/Ie KOJIb~a:
rI~ := { z~ c, e -'~<lzl<e~~ r := 2-'~'
c npoeK'mpoBarmeM p : 1-12 ---> H a , p(1 ) = 0 n rpynnofl CKO~U, xCeHWti Z z, o6paay-
1
tomaa KOTOpOfl ~IB.rDteTCJt aHTHKOHC~OpMHbIM FoMeoMoIX19rI3MOM Z I----> - - - .
IlycT~ ~ : (a, b) ---> I'I a ~ ~yra rm B a. ]~e3 orparmqeHrm o6mHOCTr~ cqr~Taer, t,
wro a = - l , b = 1. rlycT~ lt0,,, a
~+. Tor~a ~ , ~ He aB.na~OTCa OT~OCnTe:mHO x o ~ n a K T a ~ n B Yl2a H nepexoaJrr
1
~pyr B ~pyra r~prt OTo6pametmrt Z ~ - - . Awa.nornqHue yTBepa~erma c n p a ~ e ~ a -
Bla I4 no OTHOUleHrUO K no.ny/~yre 5- := ~[ (-1,01 r l ee nO/~HJtTI4JtM ~1, 5~ Ha 1-I 2.
rlpv~ ~TOM ]~yrrl 81 [.J ~5~" r~ 5~ [.J 5~" nepeceKalOT oKpymHOCT~ [Z] = 1, TaK KaK
OHa .,qBJI~IeTC~ IIOJII-IHM IIpoo~pa3oM I'[~rJIH ~fl B OTo6pa~eHHH p : Z I---> W. HM00M
(2131ogz)" ] 213 Idwl = 2131dlogzl, Ip '(z) l
noaTobiy
ISSN 0041-6053. YKp, .~tam. ~.'vpn.. 1999. m. 51, ! r z I
112 rl. M. TAMPA3OB, C. A. OXPHMEHKO
CJIel~OBaTCJIbHO, S Ha MCTpHKa p~(W) = (2a) -! /IoIIycTHMa/IJ]~l ce~e~tc~a B a, a
nO~TOt~ H ~Ia~ F~ C B a. Hp~ ~TO~
Aa(p~) = 4ot[~ = ~~
3HaqIIT,
M(B a) < --. (1)
Cc
Hycz~ Tenep~ p ~ npoH3~OJ~Haz nonycTH~a~ ~ZJ~a Bu ~evpnxa. Tor~a nocJm-
~oaa~eJ~HO H~ee~4
f f pdxdy = pdx dy >- f ldy -- 2J];
( 12 pdxdy <_ Aa(p)Aa(1) = 4a~Aa(p);
~c, J
( 1 ? (213) ~ <_ ~ pd.xdy - 4a13,~(p);
Aa(p) > ~ ~
cr
~HaqHT,
M(B~) >_ - . ~ (2)
IIoczoJr~zy Bc~C F~ C B ~, TO
M(Bc,) < M(F~) < M(Ba). (3)
Conoc'ras.nss (1) - (3), noJvlnacM
M(Ba) = M(F~) = M ( B r = fJ
CneJIonaTe~Ho, ~evp~Ka p ~ ( w ) = ( 2 a ) -1 ~KCZpeMan~Ha ~ln~ Ka~x~oro H3 ce-
MCflCTS Ba, Fa, B a. TcopcMa 1 jIoKa~a~a.
3. IIpomne/IeHea ~o~yJ1eit H HX O~eHKH. B HaCTO~mCM rlyHKTC F0, a, l"~a,
Ts, m 0603HaqatOT BBC~CHHHC B [1'-- 3]ccMcl~c'rsa ne'rcm, (k, s, m ~ HaTypa.rlbH1,Ie).
B [I - 3] Haf~ema ~Kc'rpcMa.rIbHHO MC~rDI4KH H Mo/IyJIH CCMCI~CTB r0. a, I"~a,
Ts.m. ~e cymccvScHHO HcnoJu~yel, i ~ pc3ym,xa'ru.
Bsc~XcM 0603Ha'4cH~C
~p := "2-'~.
H a ~ yc-raHoEmeH~ c~e~ymmHc pe~ym,TaT~ o HapHT~LX I'[pOI,13BC~eHH~IX Mo~y~efl, n
KOTOpHX th o6ozHa~laeT rtglCl)6OJXl~ecKl~ TaHFeHC.
ISSN 0041.6053. Ylcp. :,tam. ~:ypH., !999, m. 51, N e I
FIAPHI:,IE rIPOH3BI~.I~HH.q MO}~YJIEI~I CEMF_~CTB KPHBHX ... 113
TeopeMa 2. B otYosna,~enuax
:= ~ ( a ) : = M(ro,~)M(r~),
c := log(2 + ~ (,, 1,317)
cnpaoeOauota coomnoutenu.~
= 2 og(i- 43) (-o,658)
9 ~ 0 : [ 4 " ~ ( 4 t h c - c ) + 1 (>I)
NO ~ crapozo yS~oatou4a~ dpynKt~u.~ om U
( 1 / 4, 1 ), npu=te~t
1
g[o ,~ l npu ~ "~ 0.
TeopeMa 3. B o5osna,~enuax
:= o~(a):= u(rl*.)u(r;),
cnpaaeOAuau coomnoutenuR:
1
~2m = ~m2 Vm = 1,2 . . . . ;
1 th q) < 1
9~s- I = (2s - l)r 2s - 1 (2s - I) 2
u (2s - 1 )2~ . s_ l ~ c m p o z o yStaeatoula.q ~ynx~u.~ om r
2s - 1
unmepeane (0, 1), npu,~e,*t
( 2 s - I)2 ~ , - I Xa 0 npu
Vcp <c,
Vq~>c
(p co 3na~enu,~t,~tu o unmepaane
r ~ +-0,
2s - 1
k = 1 , 2 . . . . .
V s = 1 , 2 . . . . .
( 2 s - 1)2.q~s_l 21 1 npu q~ xaO.
2s - I
c o 3HazleHuJI3IU 8
B =tacmnocmu, 9 ~ = 1 1 4, a N I - - cmpozo y6uaamu4aa dp)'ntcuua om r co
sna=tenu~u a unmepeane (0, 1 ), npu~e~L
~.a = ~ ,
94a2'1 npu r O,
9~ "~0 npu r +**.
TeopeMa 4. B o6oana*~enuxx "
:= m r , . , ) M ( r : ) .
Is, m : - l o g ~ - 1 + V k 2 a - l j
1 Ylycmb menepb m > s. Ecnu sepno caeOytou4ee. Ecnu m < s, mo ~ . m =~m "
~O/(2s - 1) ~ Is. . , mo
ISSIV 0041.6053. Y~p. ,~tam. ,~.'vp,., 1999, re. $1. IV e I
114 17. M. TAMPA3OB, C. A. OXP14MEHKO
= - 1 th
(2s - 1)q~ 2s- 1
u (2s - 1 ) 2 9~(s 'm - - cmpom y6~oa~ou4aa dpynr~ua om q~
2s - 1
mepoane (0, 1), npuqe~t
2 r 7+~
( 2 s - l ) 9 ~ , m ~ O npu 2 s - 1
U
2 ~P XaO. (2s-l) ~?~,m ,/~ 1 npu 2S--I
Ecau ace 9 > Is m , mo:
2s - 1 "
u zm(,vV~,m - 4m 2 f p - - c m p o e o pacmyu~aa cpyn~
unmepoane (0, +~ ), npu~e,~l
2m(~.~,m 412)q~ XaO npu
2) i < 4m2~(a,m <-( 2m ~2thls, m
k2s - l} Is, m
/d
) 2 s - 1 l 1 2m this m 2 m S , m 1) 0 < 2m ~ ,m 4m2 r = 2 s - I '
( ~ 1 ) 2m
2m , m 2 q~ bynm4ua om 2 s - 1
2m - - X a l ;
2s- 1
CO 31td~leHua31U O Ult-
CO 3 H a L l e n u a 3 t u O
4m29(.~,m ---> 1 npu 2m ~ 1,
2 s - 1
3) 0 < (~.m--4'Im2~2S--l)tP = thlsm-' ~, 2m J(2s-l~21am"
_ 2m u ~(~. m . 1 e~(2s- 1) 9 - - cmpozo pacmyu~aa qbynr~ua om co snaqenua-
4m" ) 2s- 1
~tu 6 unmep~a,~e (0, 1 ), npuqe~t
3 a ~ @ t t t a l t U e . B Tr 2 -- 4 3aBHCHMOCTb BCJIHqHH NO, ~-k, ~.a,m OT KOH-
OpOpMHOR (H MOTpHqOCK01~) ffrpyKTypH 1"[~ cocpcROT0qCHa TO3IbKO B napaMerpc
(BCC 0cTaJIbI-IHC IIapaMCTpbl 3aBHC$1T ~qHl.IIb OT qHCJIOBbIX HH~CKC0B, BbI~paHH~X Ha
Ha ceMe~c'rn KpHBblX).
MeTpHKH p e P(H,,) BI~ROM cmw~no~e O00aHaqCHH~:
L0(a,P) := inff ~pds: ,oeFo,~ },
Lk(a.p):= inf~[pds:u [, k=l,2 .....
L'(ct, p ) : = inff I pds: u er~ }
t< jura n=O, 1,2 . . . . nostowum~
ISSN 0041-6053. Yrp, ~am. a~ylm., 1999, m, 51,1~ 1
HAPHhlE rIPOH3BELIEHHJ:I MO~YJIEI~ CEMEI~CTB KPHBI:tIX ... 115
A~(~) := sup{Ln(o:'p)L*(ot'P): PeP(I"I~)} ' A ~ ( p )
~i~((~) := __L_I
^;,(~)"
JIerKo rlpoBepi4Tl,, qTO
_> Vn = 0 ,1 ,2 . . . . .
a c noMomB~ pcayJIBTaTOB Ha [2, 3] n TCOpCM 1 -- 3 Moryr 6~Tb yCTaHOB~eHU c~eay-
IOmHC yTBCp3K~CHHH.
TeopeMa S. > Vn=0, 1,2 . . . . .
1
TeopeMa6. Mo(a)>~-~0(~)>~ Vcx~ (0,+oo),
inf ffd'o(OO = i n f ~ = 1.
ct 2
B pa6oTC [4] BBe~eHIaI H BhlHHCJIeHM BeJIHHHHI~, aHa~orHqHI~C ffC(;(~) H fft(;(Ct)
(B MeHee o6m~4x npc~nono~KemIHx n HHHX 06OaHaqeHHHX), H noKa:mHO, qTO ncpBa~
Ha HHX > 1 /2 ~nH Bcex ~ e (0, +oo), npHqeM nOCTOHHHaH 1/2 ncynyqmacMa.
3fro OCHOBHOfl peaynbTaT pa6oTH S~aTTepa [4]. TeopeMa 6 HanHCTCH yCH.neHHebi H
O00~I/~CHHeM ~TOF0 pc3yYmTaTa.
~OKaaaTenBCTBO TeopcM 2 -- 40nHpaeTC~ Ha TeOpCMy 1, pcaynbTaTU Ha [2, 3] H
CJICJ~yIOI/.l[yIO HH.-~Ke 2ICMMy, B KOTOpOI~/O~O3HaqCHO
l ( t ) := log(t + t 2 ~ - l ) V t > l .
HH~Ke ch H s h 0603HaqalOT rHnep6o~nqecKHe KOCHHyC H CHHyC COOTBeT-
CI'BeHHO.
JIeM~a. CnpaoeO,~ue~ coomnouaenua
th__tt/z 1
t
th_.~t ",a 0
t
t th l - ! ,~ o
t
t2th l
l
( t h l - t - f / ) , 0 V t , 1 ,
th I - t/-~/~ 1
0 < th__tt < 1 Vt>O,
t
npu
npu
txaO,
t ,~ +~;
> 0 V t > l ,
npu t ~ l ;
> 0 Vt>l,
npu t'~l;
0 < t h l - ~ < l
npu t ,~ +~ .
V t > l ,
15SN 0041-6053. Y~p. ~tam. ~.'vpu.. 1999. m. $1. N ~ 1
116 H.M. TAMPA3OB, C. A. OXPHMEHKO
fl{o~z~ame~bcmeo. HMeeM
ch/ chlog(, + ~ �89 -I- -~+ (t-i- %f~-l)
I' = = (sh/) -~.
HpH t > 0 BCpH~ COOTHOmeHHa (sh t)" = ch t > 1 = t " a s TO~IKe t = 0 C l D ~ t H
sh t paev, u . ]'IO3TOI~y t < sh t V t > 0 H
( ~ 1 ' t(ch t) - 2 - t h t
= t 2
Kpo~e Toro, x~ccM
lira th.~t = 0,
t,--) +** t
3~a,~-r. O< th...~t < 1 Vt>O.
t
]Ia~ee, npa. t > 1 H~eeM
2t - sh 2t
< 0 V t > 0 .
2t 2 ch 2 t
l im th__.~t = l im t h ' t = l ira ( ch t ) - 2
t~O t t~O f t~O
= 1 .
= = 2 / ' ) > 0 Vt> l , -57-
I t h , - ~ 2 ) = [ ' , c h l ) - 2 + 2 , - 3 / - t - 2 / ' = / ' t - 2 + 2 t - 3 I - , - 2 " =
= 2t-31 > 0 V t > l ,
( ,) l i ra t h l - ~ -- 1,
t -
t'~l \
flora/IOKa3aHa.
PosyJIbTaT~ ~aHHO~ p a 6 o T ~ HCpBOHaqaJIbHO ony6JZHKOBa.t~ B BH]IO npenpHH-
Ta [ 5 ] .
1. Ta~tpa3o~ 17. M. M~ro~w HCCJIC~OBaHH~I 3KCTpCMaJIbHIMX MeT~HK H MoRyJIr CeMCIaCTB KpHBIMX B
,,cKpyqeHHOM" pHMaHOBOM MSOrOO6pa~H H MaT. c6. - 1992. - 183, N ~ 3. - C. 55 - 75.
2. Tasq~3oo 17. M. Mo~y~x u 3KffrpeMa,/IbHldr MeTpHKI4 B cKpyqeHHldX pHMaHOBblX MHOrOo6pa-
aHax//MoayJm HcopMeHTHpyeMux H cKpyqeHHeX pHMaHOBblX MHOrOO6pa3Hg. -- KHCB, 1997. -
C. 3-25. - (Ilpenp.wr / HAH Yxpau.u. Hn-T MaTCMa'r.xa; 97.9):
3. Ta~pasoo 17. M. Mo/IyJm H 3K~'DCMaJlbHbI~ MCTDHKH B HeopHewrgpyeMhlx u cKpyqeHHblX
pHMaltoBblX MHoroo6paai4ax//YKp. MaT. )Zylm. - 1998. - 50, N ~ 10. - C. 1388-1398.
4. Blatter C. Zur Riemannschen Geometric im Grossr auf dcm Mobiusband fl Compos. math. -
1960.- 15, N g 1 . -P . 88-107 .
5. Oxpu~uesr.o C. A., Ta~tpazos/7. M. OI~eHKH IIpOH31~eHH~ MO/][yJlr CeMC~IL'rB KpHBINIX Ha puma-
HOnOM aMCTe MeOHyca//Mo~y.nM HeopaewrupyeM~x H cxpyqeHHldlX ptlMatloeblx MUOrOO6pa-
~ I ~ . - Kaee, 1997. - C . 26-40 . - (l'Ipenpmrr /HAI l Yxpatmu. FIH-T MaTeMaTHKH; 97.9).
HOJlyqeHo 07.07.97
ISSbl 0041-6053. Yxp. ~am. ~typ&, 1999. m. 51, bl ~ 1
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| id | umjimathkievua-article-4587 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:01:41Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6f/28d935acc6cf75f41614de9529b4216f.pdf |
| spelling | umjimathkievua-article-45872020-03-18T21:09:14Z Pairwise products of moduli of families of curves on a Riemannian Möbius strip Парные произведения моду лей семейств кривых на римановом листе Мебиуса Okhrimenko, S. A. Tamrazov, P. M. Охрименко, С. А. Тамразов, П. М. Охрименко, С. А. Тамразов, П. М. We investigate pairwise products of moduli of families of curves on a Riemannian Möbius strip and obtain estimates for these products. As one of the factors, we consider the modulus of a family of arcs from a broad class of families of this sort (for each of these families, we determine the modulus and extremal metric). Досліджено парні добутки модулів сімей кривих на рімановому листку Мьобіуса та одержані оцінки для цих добутків. Як один із множників розглядається модуль сім'ї дуг з широкого класу таких сімей (і для кожної з них знайдено модуль та екстремальну метрику). Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4587 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 110–116 Український математичний журнал; Том 51 № 1 (1999); 110–116 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4587/5878 https://umj.imath.kiev.ua/index.php/umj/article/view/4587/5879 Copyright (c) 1999 Okhrimenko S. A.; Tamrazov P. M. |
| spellingShingle | Okhrimenko, S. A. Tamrazov, P. M. Охрименко, С. А. Тамразов, П. М. Охрименко, С. А. Тамразов, П. М. Pairwise products of moduli of families of curves on a Riemannian Möbius strip |
| title | Pairwise products of moduli of families of curves on a Riemannian Möbius strip |
| title_alt | Парные произведения моду лей семейств кривых на римановом листе Мебиуса |
| title_full | Pairwise products of moduli of families of curves on a Riemannian Möbius strip |
| title_fullStr | Pairwise products of moduli of families of curves on a Riemannian Möbius strip |
| title_full_unstemmed | Pairwise products of moduli of families of curves on a Riemannian Möbius strip |
| title_short | Pairwise products of moduli of families of curves on a Riemannian Möbius strip |
| title_sort | pairwise products of moduli of families of curves on a riemannian möbius strip |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4587 |
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