Approximately Inner Two-parameter C0
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1 urli Jourl of ic d pplid Scic, 5(9: 0-6, 0 ISSN pproximly Ir Two-prmr C0 -group of Tor Produc of C -lgr R. zri,. Nikm, M. Hi Dprm of Mmic, Md rc, Ilmic zd Uivriy, P.O.ox 4-975, Md, Ir. rc: I i ppr, w udy ifiiiml gror of wo-prmr or produc C0 - group. I priculr, w provd cry d uffici codiio for wo-prmr C0 -group o pproximly ir. U i fc w prov, wo-prmr or produc of wo pproximly ir o-prmr C0 -group i pproximly ir. Ky word: Two-prmr C 0 -group, pproximly Ir C 0 -migroup, Tor Produc, C- lgr. Iroducio Suppo {,, 0 i rogly coiuou wo-prmr group of -uomorpim o C -lgr wi ui. W y {, i pproximly ir, if r xi quc { d { of rmii lm of uc,, ( 0, for c, wr for fixd, covrgc i uiform for, i compc u of R R. W oci wo o-prmr group u,0 d v 0, o,. T group propri of, impli,, uv. I i ppr, w ow, i pproximly ir if d oly if u d v r pproximly ir (orm.. W provid wo-prmr group of or produc of wo o-prmr C0 -group d udy i ifiiiml gror (orm.. y uig orm (., i ow or produc of wo pproximly ir o-prmr C0 -group i pproximly ir (orm.4. T -prmr C0 -group of opror wic i xio of o-prmr c v udid y E.Hill i 944 d i 946 N. Duford d I. Sgl (946 pplid m o prov orm of Wirr. X -prmr migroup w udid y Hill d Pillip (957. O.. Ivov oid om or rul i -prmr group i Ivov, (966. T migroup of opror v my pplicio i vrl r of pplid mmic, uc ory of rdom fild (Mim, 978; V. Cr, d uful o dcri iicl mcic, pril diffril quio d im voluio of quum fild ory (Nikm, 997; Pzy, 984; Scwrz, 00; Jfd, 004. Prlimiri: I i cio w provid om prlimiri wic r dd i r of ppr. Dfiiio.: L X c pc. wo-prmr fmily {,, 0 wo-prmr migroup of opror, if i I, ( I i idiy opror o X 0,0 of oudd opror i (X i clld Corrpodig uor: Roul. zri, Dprm of Mmic, Md rc, Ilmic zd Uivriy, P.O.ox 4-975, Md, Ir. E-mil: r_zri@mdiu.c.ir 0
2 u. J. ic & ppl. Sci., 5(9: 0-6, 0 ii,,,. Morovr, if {, i rogly coiuou, i., (,, ( x i coiuou for ll x X, i i clld C0 -migroup d if (,, i orm coiuou, i i clld uiformly coiuou. For wo-prmr group {,, w u,0 d v 0,. T migroup propri of {, impli u v, d o c ow i rogly (rp. uiformly coiuou if d oly if, {, { u d { v r rogly (rp. uiformly coiuou. L d ifiiiml gror of { u d { v rpcivly, w do pir (, ifiiiml gror of {,. I x dfiiio w provid dfiiio of pproxim ir of wo-prmr C0 -group. Dfiiio.: C0 -group {, lm of C -lgr uc, i pproximly ir if r r quc d { { of lf-doi, ( 0, uiformly o compc u of R R for c. Mi Rul: Torm.: Suppo {, i rogly coiuou wo-prmr group of -uomorpim o L u,0 d v 0, pproximly ir. Proof. L {, lf-doi lm of, uc : C -lgr., i pproximly ir if d oly if u d {, { { v r pproximly ir, y dfiiio r xi quc d { { of (, 0, uiformly o compc u of R R, for c. If w k 0 or 0, w v: or i i i i u ( 0, v ( 0, uiformly o compc u of R, for c. Hc { u d { v r pproximly ir. For covr, uppo { u d { v r pproximly ir, c r xi quc { d { of lf-doi lm of, uc ; d i i u ( 0,
3 u. J. ic & ppl. Sci., 5(9: 0-6, 0 i i v ( 0, uiformly o compc u of R, for c. W v v ( lim rfor lim lim 0, i i lim, ( { i i ( ( i i i i ic u i pproximly ir. Hc i i u ( v ( u ( i i (, 0, uiformly o compc u of R R, for c. Rmrk.: Suppo d r grl, for rirry C -lgr d C -orm o c xd o -uomorpim o, r -uomorpim o d rpcivly. I, or produc my o uomorpim. u wi pil C -orm. Morovr if d r o uil, o c doi idiy o d d w my um d v idiy lm. I followig y uig mod wic i providd i proof of orm. i [8], w urvy C0 - group propri of wo-prmr or produc of uomorpim d i ifiiiml gror. Torm.: Suppo u d v r rogly coiuou o-prmr group of -uomorpim o { { lgr d wi ifiiiml gror d rpcivly. T, u v coiuou wo-prmr group of -uomorpim o d clour of (, I ifiiiml gror, wic I d I r idiy opror o d rpcivly. Proof. y uig group propri of u d v w v, { i 0,0 u0 v0 I I, ii For <,,, < d ( w v, { C - i rogly I i i, ( ( u ( u ( u v ( ( u ( v ( v ( ( u ( v v ( u ( v (, o,.,,, iii To uify rogly coiuiy of, l X lm of,,
4 u. J. ic & ppl. Sci., 5(9: 0-6, 0 ( X X, ( u [ v ( X X [( u ( [( u ( u ( v ( ( ] v ( ( ( v ( ] v ( ]. Sic { u d { v r rogly coiuou, i follow (,, i rogly coiuou woprmr migroup of -uomorpim of. W c um ( H, H i gror of, d ( rp. i gror of u ( rp.{ v, { ( X X,0 Trfor [( u ( [( u ( ( I ( ( X ] ] ( ( ( u ( [( ( ] u ( (.,0( X X I ( X lim H ( X, 0 ( Similrly 0, ( X X ( X lim H ( X, 0 ( I ( u ( ( I ( X X ( I ( X for X D( D(. Trfor ( I H d ( I H. Sic H d H r clod, o ( I d ( I r clol. Suppo K (rp. K i clour of ( I (rp. ( I, K H d K H d y rolv codiio for gror H d H ( [], [4], [] w v X K ( X X, ( X D( K D( K, 0 R,,.
5 u. J. ic & ppl. Sci., 5(9: 0-6, 0 W do -ulgr of ll lyic lm for (rp. To compl proof, i i uffici o ow { :, N, D (, D (, y D (rp. ( ( D. i d -ulgr of lyic lm for K d K. L D ( d D (, r xi poiiv umr d uc, 0 (! < d 0 (! <. T for 0 Hc K wi > 0, w v, (!, ( ( 0 0 k0 0 k0 0 0 <. 0 k k k k! k! ( ( ( 0 k k!( k! ( (! 0! ( (!! i lyic lm for K. Similrly k i lyic lm for K. Trfor i d -ulgr of lyic lm for K d K, c w coclud K, K i ifiiiml gror (rli, 976. I x orm, pproxim ir of udid. ( {, Torm.4: Suppo Suppo { u d { v r rogly coiuou o-prmr group of -uomorpim o C -lgr d rpcivly. Dfi, u v o. If { u d { v r pproximly ir, i pproximly ir. Proof. L { u d { v r pproximly ir, c r r quc { d { of lfdoi lm of d rpcivly, uc, d i i i i u ( 0, v ( 0, 4
6 u. J. ic & ppl. Sci., 5(9: 0-6, 0 uiformly o compc u of R for c d. S k ( d k (, wic d r ui lm of d rpcivly. Sic d r lf-doi, o k d k r lf-doi lm of. O or d, ic ( u I ( I v, y orm., i i uffici o prov, { u I d I v r pproximly ir. For i, w v followig quliy, ( ( { m0 m0 m0 ( i m ( i ( m! m m ( i ( m! m ( i m!. So, for c ( w v, m ( ( ( i i ( ( i. i Hc ( ( ( i i i ( ( u I i ( u ( i u ( i u ( Sic { u i pproximly ir, rfor, ( ( ( u I ( 0, uiformly o compc u of R for ll ( (. Hc { u I i pproximly ir. Similrly { I v i lo pproximly ir. Trfor y orm., w coclud {, i pproximly ir. REFERENCES rli, O., D.W. Roio, 976. Uoudd drivio of C -lgr, II, Commu. M. Py., 46: -0. Duford, N., I.E. Sgl, 946. Smigroup of opror d Wirr orm, mr. M. Soc. Colloq., 5: Hill, E., R.S. Pillip, 957. Fuciol lyi d Smigroup, ull. mr. M. Soc., : Providc R,I. Ivov, O.., 966. Cri Torm o -prmric migroup of oudd lir opror d ir pplicio o ory of fucio (Rui, Tor. Fukcii Fukciol. l. i Prilozu, Vyp., : 4-4. Jfd, M.,. Nikm, 004. O wo-prmr Dymicl ym d pplicio, Jourl of Scic, Ilmic Rpulic of Ir, 5(: Mim,.G., 978. Hrmoiiliy, V-oudd d iory dlio, Idi. Ui. 5
7 u. J. ic & ppl. Sci., 5(9: 0-6, 0 Nikm,., 997. Ifiiiml gror of C -lgr, Poil lyi, 6: -9. Pzy,., 984. Smigroup of lir opror d pplicio o Pril Diffril Equio, Sprigr vrlg, 984. Scwrz,., 00. Spc d im from rlio ymmry, Jourl of Mmicl Pyic., 5: V. Cr, Gror of rogly coiuou migroup, Pim dvcd Puliig, Rrc No. 6
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