Başlık: On The Embedding of Complements of Some Hyperbolic Planes IIYazar(lar):ANAPA, Pınar;GÜNALTILI, İbrahim;OLGUN, Şükrü Cilt: 57 Sayı: 1 Sayfa: 023-032 DOI: 10.1501/Commua1_0000000183 Yayın Tarihi: 2008 PDF

Tam metin

(1) < : : B ; . 0 & 0 6( ; 6C ; 8 & 2 ?62 @ . )< 9B : 2 " B : / 2 ? $ . 4 2 @

(2) J

(3) & & "

(4)

(5) J .

(6) $"% "$ N %! R" ' " &R%R # (" ; A56@ =.=2? D2 @AB1621 A5.A . 96;2.? @=.02 D5605 6@ A52 0<: =92:2;A <3 96;2.? @=.02 D5<@2 =<6;A@ .?2 ;<A <; . =2;A.4<; 52E.4<; <? . 52=A.4<; 6; . =?<720A6C2 @B/=9.;2 <3 <?12? 6@ 2:/211./92 6; .; B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? ; .116A6<; D2 @5<D21 A5.A A56@ 96;2.? @=.02 6@ A52 0<:=92:2;A <3 02?A.6; ?24B9.? 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- D6A5 ?2@=20A A< . H;6A2 =?<720A6C2 =9.;2. '52 0<:=92:2;A.A6<; =?</92: D6A5 ?2@=20A A< . =?<720A6C2 =9.;2 6@ A52 3<99<D6;4 %2:<C2 . 02?A.6; 0<;H4B?.A6<; <3 =<6;A@ .;1 96;2@ 3?<: A52 =9.;2 12A2?:6;2 A52 =.?.:2A2?@ <3 A52 ?2@B9A6;4 @=.02 '52 =?</92: <3 2:/2116;4 A52 <3 C.?6<B@ 0<;H4B?.A6<; 6; A52 =?<720A6C2 =9.;2@ 5.@ /22; @AB1621 /F C.?6<B@ .BA5<?@ , -, -,

(7) -, -,- ; 6082F @<9C21 A52 =?</92: 3<? A52 0.@2 D52?2 A52 0<;H4B?.A6<; ?2:<C21 D.@ . B;6A.9 ,- ; ! .AA2; 05.?.0A2?6G21 96;2.? @=.02@ D5605 .?2 A52 0<:=92:2;A@ <3 .K;2 <? =?<720A6C2 @B/=9.;2@ <3 H;6A2 =?<720A6C2 =9.;2@ .;1 @5<D21 A5.A A52@2 @=.02@ 0.; /2 2:/211./92 6; .; B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? , - 42;2?.96G21 <3 .AA2;I@ '52<?2: , - D.@ 46C2; /F U;.9AM9M .;1 #94B; ,- ; ,- U;.9AM9M ;.=. .;1 #94B; @5<D21 A5.A . 96;2.? @=.02 D5605 6@ A52 0<:=92:2;A <3 . 96;2.? @=.02 5.C6;4 =<6;A@ .?2 ;<A <; . A?69.A2?.9 <? . >B.1?69.A2?.9 6; . =?<720A6C2 @B/=9.;2 <3 <?12? 6@ 2:/211./92 6; .; B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? ; .116A6<; 6A D.@ 12A2?:6;21 A5.A A56@ 96;2.? @=.02 6@ A52 0<:=92:2;A <3 02?A.6; ?24B9.? 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- D6A5 ?2@=20A A< . H;6A2 =?<720A6C2 =9.;2 ; A56@ @AB1F D2 @5<D21 A5.A . 96;2.? @=.02 D5605 6@ A52 0<:=92:2;A <3 . 96;2.? @=.02 D5<@2 =<6;A@ .?2 ;<A <; . =2;A.4<; 52E.4<; <? . 52=A.4<; 6; . =?<720A6C2 @B/=9.;2 <3 <?12? 6@ 2:/211./92 6; . B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? ; .116A6<; D2 12A2?:6;21 A5.A A56@ 96;2.? @=.02 6@ A52 0<:=92:2;A <3 02?A.6; %2026C21 /F A52 216A<?@ =?69 %2C 2/

(8) 002=A21 !.?05 " $?6:.?F &20<;1.?F ! 6;2.? @=.02 =?<720A6C2 =9.;2 5F=2?/<960 =9.;2 0 ; 8. ?. ( ; 6C2 ?@6AF .

(9).

(10) . $"% "$ N %! R" ' " &R%R # (". ?24B9.? 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- D6A5 ?2@=20A A< . H;6A2 =?<720A6C2 =9.;2 "<D D2 46C2 @<:2 12H;6A6<;@ ?2>B6?21 -&#+#'& " 6@ . =.6?. D52?2 6@ . H;6A2 @2A <3 =<6;A@ .;1 6@ . 3.:69F <3 =?<=2? @B/@2A@ <3 D5605 .?2 0.9921 96;2@ @B05 A5.A .;F AD< =<6;A@ .?2 <; B;6>B2 96;2 2.05 96;2 5.@ .A 92.@A AD< =<6;A@ .;1 A52?2 .?2 .A 92.@A AD< 96;2@ 3 6@ . H;6A2 96;2.? @=.02 A5.A 6@ '52 ;B:/2? <3 96;2@ =.@@6;4 A5?<B45 . =<6;A 6@ 0.9921 A52 124?22 <3 .;1 12;<A21 /F '52 ;B:/2? <3 =<6;A@ <; . 96;2 6@ 0.9921 A52 124?22 <3 .;1 12;<A21 /F .;1 12;<A2 A52 A<A.9 ;B:/2? <3 =<6;A@ .;1 96;2@ <3 ?2@=20A6C29F '52 ;B:/2? .;1 .?2 0.9921 A52 =.?.:2A2?@ <3 '52 =.?.:2A2?@ <3 H;6A2 96;2.? @=.02

(11)

(12) .?2 12H;21 .@ @A.A21 /29<D

(13)

(14) '52 A2?: =<6;A <? 96;2 :.F .9@< B@21 A< ?232? ?2@=20A6C29F A< . =<6;A <? . 96;2 <3 124?22 !<?2<C2? 12;<A2@ A52 A<A.9 ;B:/2? <3 96;2@ 12;<A2@ A52 A<A.9 ;B:/2? <3 =<6;A@ .;1 12;<A2@ A52 A<A.9 ;B:/2? <3 96;2@ =.@@6;4 A5?<B45 . =<6;A '52 6;A242? 12H;21 /F 6@ A52 <3 A52 96;2.? @=.02 -&#+#'& H;6A2 ?24B9.? ! 6; A52 @2;@2 <3 ?.C2@ 6@ . ;<;A?6C6.9 ?24B9.? 96;2.? @=.02 @B05 A5.A '52?2 .?2 3<B? =<6;A@ ;< A5?22 <3 D5605 .?2 0<996;2.? 3 6@ . =<6;A ;<A <; . 96;2 A52; A52?2 2E6@A .A 92.@A AD< 96;2@ ;<A :22A6;4 .;1 A5?<B45

(15) 3 . @B/@2A <3 A52 =<6;A@ <3 0<;A.6;@ A5?22 ;<;0<996;2.? =<6;A@ .;1 0<;A.6;@ .99 =<6;A@ <; A52 96;2@ A5?<B45 =.6?@ <3 16@A6;0A =<6;A@ <3 0<;A.6;@ .99 =<6;A@ <3 )'('*#+#'& B:0?<A , - ;F H;6A2 96;2.? @=.02 @.A6@3F6;4 A52 3<9 9<D6;4 0<;16A6<;

(16)

(17)

(18) 6@ . 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- 96;2.? @=.02 6@ @.61 A< /2 2:/211./92 6; . 96;2.? @=.02 63 0.; /2 </A.6;21 3?<: /F .116A6<; <3 @<:2 =<6;A@ 0.9921 .@ 612.9 =<6;A@ .;1 @<:2 96;2@ 0.9921 .@ 612.9 96;2@.

(19) #" ' ! " # #! $ ! "'& # &#! +$%# $ "& . . #& *,$+* ; A56@ @20A6<; D2 @5<D21 A5.A . 96;2.? @=.02 D5605 6@ A52 0<:=92:2;A <3 96;2.? @=.02 D5<@2 =<6;A@ .?2 ;<A <; . =2;A.4<; 52E.4<; <? . 52=A.4<; 6; . =?<720A6C2 @B/=9.;2 <3 <?12? 6@ 2:/211./92 6; .; B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? ; .116A6<; D2 12A2?:6;21 A5.A A56@ 96;2.? @=.02 6@ A52 0<:=92:2;A <3 02?A.6; ?24B9.? 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- D6A5 ?2@=20A A< . H;6A2 =?<720A6C2 =9.;2 )'('*#+#'& ;F ?24B9.? 96;2.? @=.02 D6A5 96;2 124?22 <? 6@ . 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- )''! 2A /2 .; ?24B9.? 96;2.? @=.02 D6A5 96;2 124?22 . <? A 6@ 092.? A5.A

(20) .;1

(21)

(22)

(23) @6;02

(24) .;1

(25) F A52 $?<=<@6A6<; 6@ . 5F=2?/<960 =9.;2 .;1 6A 6@ 0.9921 . 5F=2?/<960 =9.;2 <3

(26) AF=2 )'('*#+#'& ?2.9 0<:=92:2;A <3 96;2.? @=.02 D5<@2 =<6;A@ .?2 <; . =2;A.4<; 6; . =?<720A6C2 <3 <?12? 6@ . 5F=2?/<960 =9.;2 )''! 2A /2 . =?<720A6C2 =9.;2 <3 <?12? .;1 /2 . ?2.9 0<:=92:2;A <3 . =2;A.4<; 6; . =?<720A6C2 =9.;2 <3 <?12? '52 A<A.9 ;B:/2? <3 =<6;A@ <; . =2;A.4<; 6@

(27) '5B@ 6@ .; ?24B9.? 96;2.? @=.02 D6A5 =<6;A@ 96;2@ .;1 6AI@ 96;2 124?22 6@ <? 9@< 6@ . 5F=2?/<960 =9.;2 <3

(28) AF=2 /F A52 $?<=<@6A6<; )'('*#+#'& ;F ?24B9.? 96;2.? @=.02 D6A5 96;2 124?22 . <?

(29) 6@ . 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- )''! 2A /2 .; ?24B9.? 96;2.? @=.02 D6A5 96;2 124?22 . <?

(30) A 6@ 092.? A5.A

(31) .;1

(32)

(33)

(34) @6;02

(35)

(36) .;1

(37) F A52 $?<=<@6A6<; 6@ . 5F=2?/<960 =9.;2 .;1 6A 6@ 0.9921 . 5F=2?/<960 =9.;2 <3 AF=2 )'('*#+#'& ?2.9 0<:=92:2;A <3 . 96;2.? @=.02 D5<@2 =<6;A@ .?2 <; . 52E.4<; 6; . =?<720A6C2 =9.;2 <3 <?12? 6@ . 5F=2?/<960 =9.;2 )''! 2A /2 . =?<720A6C2 =9.;2 <3 <?12? .;1 /2 . ?2.9 0<:=92:2;A <3 . 52E.4<; 6; '52 A<A.9 ;B:/2? <3 =<6;A@ <; . 52E.4<; 6@ '5B@. 6@ .; ?24B9.? 96;2.? @=.02 D6A5

(38) =<6;A@

(39) 96;2@ .;1 6AI@ 96;2 124?22 6@ <?

(40) 9@< 6@ . 5F=2?/<960 =9.;2 <3 AF=2 /F A52 $?<=<@6A6<;

(41) )'('*#+#'& ;F ?24B9.? 96;2.? @=.02 D6A5 96;2 124?22

(42) <? 6@ . 5F=2?/<960 =9.;2 6; A52 @2;@2 <3 ?.C2@ ,- )''! 2A /2 .; ?24B9.? 96;2.? @=.02 D6A5 96;2 124?22

(43) <? A 6@ 092.? A5.A

(44) .;1

(45)

(46) @6;02

(47) .;1

(48) F A52 $?<=<@6A6<; 6@ . 5F=2?/<960 =9.;2 .;1 6A 6@ 0.9921 . 5F=2?/<960 =9.;2 <3 AF=2.

(49) $"% "$ N %! R" ' " &R%R # (". . )'('*#+#'& ?2.9 0<:=92:2;A <3 . 96;2.? @=.02 D5<@2 =<6;A@ .?2 <; . 52=A.4<; 6; . =?<720A6C2 =9.;2 <3 <?12? 6@ . 5F=2?/<960 =9.;2 )''! 2A /2 . =?<720A6C2 =9.;2 <3 <?12? .;1 /2 . ?2.9 0<:=92:2;A <3 . 52=A.4<; 6; '52 A<A.9 ;B:/2? <3 =<6;A@ <; . 52=A.4<; 6@ '5B@ 6@ .; ?24B9.? 96;2.? @=.02 D6A5

(50) =<6;A@ 96;2@ .;1 6AI@ 96;2 124?22 6@

(51) <? 9@< 6@ . 5F=2?/<960 =9.;2 <3 AF=2 /F A52 $?<=<@6A6<; " ') % 2A /2 .; ?24B9.? 96;2.? @=.02 @B05 A5.A 6

(52) 66

(53)

(54) .;1

(55) 666 2C2?F 96;2 5.@

(56) =<6;A@ '52; 6@ 2:/211./92 6; . B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? .;1 6@ 0<:=92:2;A <3 . 5F=2?/<960 =9.;2

(57) AF=2 )''! 2A /2 A52 @2A <3 =<6;A@ <3 @B05 A5.A A52?2 .?2 96;2@ <3 124?22 96;2@ <3 124?22 96;2@ <3 124?22

(58) 96;2@ <3 124?22 .;1 96;2@ <3 124?22 A5?<B45 2C2?F =<6;A <3 6A '52; ?<: A52 ./<C2 2>B.96A62@ A52 3<99<D6;4 ?2@B9A@ .?2 </A.6;21. . .

(59) . 9@< /F A52 @6:=92 0<B;A6;4 :2A5<1@ . . .

(60) . . . .

(61) . . . . . . .

(62) . . .;1. . . . . . . .

(63) #" ' ! " # #! $ ! "'& # &#! +$%# $ "& . . '52; A52 3<99<D6;4 ?2@B9A@ .?2 </A.6;21 . . .

(64)

(65).

(66)

(67)

(68)

(69) .

(70). . . . &6;02 .;1 A52?2 2E6@A@ .A 92.@A <;2 96;2 <? 2C2?F 96;2 D2 12H;2 6@ . 96;2 16@7<6;A A< &6;02 2.05 =<6;A 5.@ 124?22 2.05 =<6;A <BA@612 962@ <; 2E.0A9F <;2 96;2 D5605 6@ =.?.9929 A< '56@ @5<D@ A5.A 6@ . =.?A6A6<; <3 A52 =<6;A@ <3. 6;A< 16@7<6;A 96;2@ .;1 6;1B02@ .; 2>B6C.92;02 ?29.A6<; .:<;4 A52 96;2@ 6; <3 @6G2 '56@ 2>B6C.92;A ?29.A6<; <; A52 96;2@ <3 @6G2 6@ ?232?21 .@ =.?.99296@: &6;02 :22A@ <A52? 96;2@ 2;02 2.05 96;2 6;1B02@ . =.?A6A6<; <3 A52 =<6;A@ 6;A< 96;2@ ?232?21 .@ A52 =.?.9929 09.@@ .@@<06.A21 D6A5 A5.A 96;2 &B==<@2 A5.A .;1 .?2 AD< 16L2?2;A 96;2@ D5605 :22A '52; :22A@ 96;2@ <3 @<

(71) 2A 2.05 @B05 =.?.9929 09.@@ 0<??2@=<;1@ A< . <;@612? A52 @A?B0 AB?2 D52?2 6@ .9<;4 D6A5 A52 ;2D =<6;A@ .;1 0<;@6@A@ <3 A52 96;2@ <3 /F A5<@2 =.?.9929 09.@@2@ A< D5605 A52F /29<;4 *2 H?@A <3 .99 =?<C2 A5.A 6@ . 96;2.? @=.02 A 6@ 092.? A5.A AD< <91 =<6;A@ =<6;A@ <3 .?2 <; . B;6>B2 96;2 <3 2A .;1 /2 16@A6;0A ;2D =<6;A@ *2 @5<D A5.A A52F 12A2?:6;2 . B;6>B2 96;2 <3 2A .;1 /2 96;2@ D5605 12A2?:6;2 A52 =.?.9929 09.@@2@ 0<??2@=<;16;4 A< .;1 3 .;1 1< ;<A :22A A52; D5605 6@ . 0<;A?.160A6<; &< .;1 :22A &6;02 A52 =<6;A 124?22 <3 6@ 2.05 =<6;A <3 6@ <; . B;6>B2 96;2 <3 A52 =.?.9929 09.@@ 12A2?:6;21 /F '56@ 92.C2@ =?206@29F <;2 96;2 <3 A52 =.?.9929 09.@@ =.?.9929 A< /<A5 .;1 '56@ 6@ A52 ?2>B6?21 96;2 A 3<99<D@ 3?<: <B? :2A5<1 <3 0<;@A?B0A6<; A5.A 2.05 =<6;A <3 6@ <; 96;2@ 6;.99F 6A 6@ @5<D; A5.A .;F AD< 96;2@ <3 .9D.F@ :22A 2A .;1 /2 96;2@ <3. D5605 1< ;<A :22A 6; '52; ;26A52? <? .?2 96;2@ '< =?<C2 A5.A A52F :22A 6; 6A @BK02@ A< H;1 .; 96;2 =.?.9929 A< /<A5 2A .;1 /2 AD< 16@7<6;A 96;2@ D5605 5.C2 @6G2 92@@ A5.; 6; A 6@ 092.? A5.A

(72) .;1

(73) @6;02

(74) 3<? .99 2A /2 A52 ;B:/2? <3 96;2@ :22A6;4 2E09B16;4 6A@293 92A /2 A52 ;B:/2? <3 96;2@ :22A6;4 2E09B16;4 6A@293 .;1 /2 A52 ;B:/2? <3 96;2@ :22A6;4 /<A5.

(75) $"% "$ N %! R" ' " &R%R # (". . .;1 '52 3<99<D6;4 A5?22 2>B.A6<;@ .?2 </A.6;21 /F . @6:=92 0<B;A6;4 :2A5<1

(76)

(77) .;1.

(78)

(79) . '52?23<?2

(80)

(81) 2A .;1 /2 A52 A<A.9 ;B:/2? <3 96;2@ .;1 96;2@ ?2@=20A6C29F :22A6;4 <? 2E09B16;4 .;1 A52:@29C2@ &6;02 A52 3<99<D6;4 ?2@B9A 6@ </A.6;21

(82)

(83)

(84) &6;02 .;F 96;2 <3 @6G2 :22A@ 2C2?F <A52? 96;2 .99 A52 96;2@ <3 @6G2 :22A /<A5 .;1 '52?23<?2 @6;02

(85) .;1

(86) 6A 6@ 092.? A5.A

(87) '5B@ A52?2 6@ .A 92.@A <;2 96;2 =.?.9929 A< /<A5 <;@2>B2;A9F 6@ . =?<720A6C2 =9.;2 <3 <?12? <;@612? A52 0<:=92:2;A <3 6; 6@ .; ?24B9.? 96;2.? @=.02 D5<@2 96;2@ .?2 @2A <3 <? =<6;A@ D5605 .?2 2EA2;@6<;@ <3 96;2@ 96;2@ .;1

(88) 96;2@ <3 ?2@=20A6C29F '52?23<?2 6@ . 5F=2?/<960 =9.;2 <3

(89) AF=2 /F A52 $?<=<@6A6<; " ') % 2A /2 .; ?24B9.? 96;2.? @=.02 @B05 A5.A 6

(90) 66

(91)

(92)

(93) .;1

(94)

(95)

(96) 666 2C2?F 96;2 5.@

(97) =<6;A@ '52; 6@ 2:/211./92 6; . B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? .;1 6@ 0<:=92:2;A <3 . 5F=2?/<960 =9.;2 AF=2 )''! 2A /2 A52 @2A <3 =<6;A@ <3 @B05 A5.A A52?2 .?2 96;2@ <3 124?22 96;2@ <3 124?22 96;2@ <3 124?22

(98) 96;2@ <3 124?22 96;2@ <3 124?22 .;1 96;2@ <3 124?22 A5?<B45 2C2?F =<6;A <3 6A '52;

(99) ?<: A52 ./<C2 2>B.96A62@ A52 3<99<D6;4 ?2@B9A@ .?2 </A.6;21

(100)

(101) . . .

(102)

(103) . 9@< /F A52 @6:=92 0<B;A6;4 :2A5<1@ . . .

(104) .

(105) #" ' ! " # #! $ ! "'& # &#! +$%# $ "& . . .

(106). . . .

(107)

(108) . . .

(109)

(110)

(111) . . .

(112). . .;1. . . . . . . . . . . '52; A52 3<99<D6;4 ?2@B9A@ .?2 </A.6;21 . .

(113) .

(114)

(115).

(116)

(117)

(118)

(119)

(120).

(121). . .

(122)

(123)

(124). &6;02

(125) .;1 A52?2 2E6@A@ .A 92.@A <;2 96;2 <? 2C2?F 96;2 6A 0.; /2 12H;21 =.?.9929 09.@@2@ .;1 0<;@A?B0A21. .@ 6; A52 =?<<3 <3 '52<?2: (@6;4 A52 A205;6>B2 6; A52 '52<?2: 6A 6@ 2.@69F @5<D; .;F AD< =<6;A@ <3 .?2 <; 2E.0A9F <;2 96;2 '5B@ D2 :B@A @5<D A5.A .;F AD< 96;2@ <3 .9D.F@ :22A 2A .;1 /2 96;2@ <3 D5605 1< ;<A :22A 6; '52; ;26A52? <? .?2 96;2@ '< =?<C2 A5.A A52F :22A 6; 6A @BK02@ A< H;1 .; 96;2 =.?.9929 A< /<A5 2A .;1 /2 96;2@ <3 @B05 A5.A .;1 A 6@ 092.? A5.A

(126)

(127) .;1

(128)

(129) @6;02

(130) 3<? .99 4.6; B@6;4 A52 A205;6>B2 6; A52 '52<?2: 6A 6@ 2.@69F 0.90B9.A21 A5.A @6;02

(131) .;1 '52?2 6@ .A 92.@A <;2 96;2 =.?.9929 . A< /<A5 '5B@ 6@ . =?<720A6C2 =9.;2 <3 <?12? <;@612? A52 0<:=92:2;A <3 6; 6@ .; ?24B9.? 96;2.? @=.02 D5<@2 96;2@ .?2 @2A <3 <?

(132) =<6;A@ D5605 .?2 2EA2;@6<;@ <3 96;2@ 96;2@

(133) 96;2@ .;1 96;2@ <3 ?2@=20A6C29F '52?23<?2 6@ . 5F=2?/<960 =9.;2 <3 AF=2 /F A52 $?<=<@6A6<;

(134) " ') % 2A /2 .; ?24B9.? 96;2.? @=.02 @B05 A5.A 6 66

(135)

(136)

(137) .;1

(138)

(139).

(140) $"% "$ N %! R" ' " &R%R # (".

(141) . 666 2C2?F 96;2 5.@

(142) =<6;A@ '52; 6@ 2:/211./92 6; . B;6>B2 D.F 6; . =?<720A6C2 =9.;2 <3 <?12? .;1 6@ 0<:=92:2;A <3 . 5F=2?/<960 =9.;2 AF=2 )''! 2A /2 A52 @2A <3 =<6;A@ <3 @B05 A5.A A52?2 .?2 96;2@ <3 124?22 96;2@ <3 124?22

(143) 96;2@ <3 124?22 96;2@ <3 124?22 96;2@ <3 124?22 .;1 96;2@ <3 124?22 A5?<B45 2C2?F =<6;A <3 6A '52;

(144) ?<: A52 ./<C2 2>B.96A62@ A52 3<99<D6;4 ?2@B9A@ .?2 </A.6;21.

(145)

(146) . .

(147)

(148) . 9@< /F A52 @6:=92 0<B;A6;4 :2A5<1@ . . .

(149) . . . .

(150). . .

(151)

(152)

(153) . . .

(154) . . . . . . .

(155). . . . .;1. . . . . '52; A52 3<99<D6;4 ?2@B9A@ .?2 </A.6;21 . .

(156) .

(157)

(158).

(159)

(160)

(161).

(162). . .

(163) . &6;02

(164) .;1 A52?2 2E6@A@ .A 92.@A <;2 96;2 <? 2C2?F 96;2 6A 0.; /2 12H;21 =.?.9929 09.@@2@ .;1 0<;@A?B0A21. .@ 6; A52 =?<<3 <3 '52<?2: (@6;4 A52 A205;6>B2 6; A52 '52<?2: 6A 6@ 2.@69F @5<D; .;F AD< =<6;A@ <3 .?2 <; 2E.0A9F <;2 96;2.

(165) #" ' ! " # #! $ ! "'& # &#! +$%# $ "& .

(166). '5B@ D2 :B@A @5<D A5.A .;F AD< 96;2@ <3 .9D.F@ :22A 2A .;1 /2 96;2@ <3 D5605 1< ;<A :22A 6; '52; ;26A52? <? .?2 96;2@ '< =?<C2 A5.A A52F :22A 6; 6A @BK02@ A< H;1 .; 96;2 =.?.9929 A< /<A5 2A .;1 /2 96;2@ <3 @B05 A5.A .;1 A 6@ 092.? A5.A

(167) .;1

(168) @6;02

(169) 3<? .99 4.6; B@6;4 A52 A205;6>B2 6; A52 '52<?2: 6A 6@ 2.@69F 0.90B9.A21 A5.A @6;02 .;1 '52?2 6@ .A 92.@A <;2 96;2 =.?.9929 . A< /<A5 '5B@ 6@ . =?<720A6C2 =9.;2 <3 <?12? <;@612? A52 0<:=92:2;A <3 6; 6@ .; ?24B9.? 96;2.? @=.02 D5<@2 96;2@ .?2 @2A <3

(170) <? =<6;A@ D5605 .?2 2EA2;@6<;@ <3 96;2@

(171) 96;2@ 96;2@ .;1 96;2@ <3 ?2@=20A6C29F '52?23<?2 6@ . 5F=2?/<960 =9.;2 <3 AF=2 /F A52 $?<=<@6A6<; . B S.9MO@:.1. :2?A2/212; =?<728A63 .9A1UG92:1286 /2O@ 42; .9AM42; C2 F21642;2 .6A <9:.F.; ;<8A.9.?M ;<8A.9.? 8U:2@6 <9.?.8 8./B9 212; /6? 96;22? BG.FM; AU:92F2;6 <9.; C2 :2?A2/212; =?<728A63 1UG92:2 A28 <9.?.8 4T:U92/692; 96;22? BG.F9.? UG2?6;12 S.9MO@M9:MO@AM? F?M0. /B 96;22? BG.FM; =?<728A63 1UG92:92?2 4T?2 ?.C2@ .;9.:M;1. ?24B9.? 56=2?/<968 1UG92:6; AU:92F2;6 <91BP 4B 4T@A2?69:6O@A6?

(172) , - ;.=. $ .;1 U;.9AM9M #; A52 2:/2116;4 <3 0<:=92:2;A@ <3 @<:2 5F=2?/<960 =9.;2@ ?@ <:/6;.A<?6. , - .AA2; ! :/2116;4 =@2B1<0<:=92:2;A@ 6; H;6A2 =?<720A6C2 =9.;2@ ?@ <:/6;.A<?6.

(173) ,

(174) - .AA2; ! .;1 2BA29@=.052? '52 A52<?F <3 H;6A2 96;2.? @=.02 .:/?6142 (;6C2?@6AF $?2@@ "2D+<?8!29/<B?;2

(175) , - <@2 % .;1 &5?685.;12 && :/2116;4 A52 0<:=92:2;A <3 .; <C.9 6; . =?<720A6C2 =9.;2 <3 2C2; <?12? 6@0?2A2 !.A5

(176)

(177)

(178) ,- B:0?<A % 6;6A2 5F=2?/<960 @=.02@ AA6 <;C24;< 2<: 0<:/ 2 @B2 ==9 $2?B46. .

(179)

(180) ,- 6082F :/2116;4 A52 0<:=92:2;A <3 . B;6A.9 6; . =?<720A6C2 =9.;2 AA6 129 0<;C24;< 16 2<:2A?6. 0<:/6;.A<?6. 2 @B2 ==960.G6<;6 $2?B46.

(181) ,- ?.C2@ ! H;6A2 <9F.6 </.052C@8F =9.;2 :2? !.A5 !<;A59F

(182)

(183) ,- U;.9AM9M N .;1 #94B; & #; A52 2:/2116;4 <3 @<:2 96;2.? @=.02@ 6; H;6A2 =?<720A6C2 =9.;2@ 2<: ,- U;.9AM9M N ;.=. $ .;1 #94B; & #; A52 2:/2116;4 <3 0<:=92:2;A@ <3 @<:2 5F=2?/<960 =9.;2@ ?@ <:/6;.A<?6. , - U;.9AM9M N ;.=. $ .;1 #94B; & #; A52 2:/2116;4 <3 0<:=92:2;A@ <3 @<:2 5F=2?/<960 =9.;2@ B;12? ?6C62D 6; 6@0?2A2 !.A52:.A60@ ,. - U;.9AM9M N $@2B1<@2:60<:=92:2;A@ 6; H;6A2 =?<720A6C2 =9.;2@ .002=A21 6; ?@ <:/6 ;.A<?6. .

(184)

(185) . $"% "$ N %! R" ' " &R%R # (". , - .F. % .;1 #94B; & <;@A?B0A6<; <3 @<:2 5F=2?/<960 =9.;2@ B@6;4 .2? @B/=9.;2@ <:/6;.A<?60I@ )2?9.;4 !216A2??.;2.; $?2@@ . ,

(186) - .F. % .;1 #G0.; #; A52 0<;@A?B0A6<; <3 <9F.6 </.052C@8F =9.;2@ 3?<: =?<720A6C2 =9.;2@ %2;160<;A6 129 &2:6;.?6< !.A52:.A60< 16 ?2@06.

(187) , - %.9@A<; ' #; A52 2:/211./696AF <3 A52 0<:=92:2;A <3 . 0<:=92A2 A?6.;492 6; . H;6A2 =?<720A6C2 =9.;2 ?@ <:/6;.A<?6.. , - #@A?<: ' #C.9@ .;1 H;6A2 <9F.6 </.052C@8F $9.;2@ :2? !.A5 !<;A59F , - QG0.; #94B; & .;1 .F. % #; A52 96;2 09.@@2@ 6; @<:2 H;6A2 5F=2?/<960 =9.;2@ <::B; .0&06(;6C;8&2?62@ )

(188)

(189) , - &.;192? % 6;6A2 5<:<42;2<B@ <9F.6 </.052C@8F =9.;2@ :2? !<;A59F

(190)

(191) , - &24?2 #C.9@ 6; . H;6A2 =?<720A6C2 =9.;2 .;.1. !.A5 @86O@256? #@:.;4.G6 (;6C2?@6AF 2=.?A:2;A <3 !.A52:.A60@ @86O@256? '(%+

(192)

(193)

(194)

(195)

(196)

(197) .

(198) INSTRUCTIONS TO CONTRIBUTORS Mathematics and Statistics of COMMUNICATIONS accepts original research articles written in English in the fields of Mathematics and Statistics. Review articles written by eminent scientists can also be invited by the Editor. Three copies of the manuscripts must be submitted in AMS Article Tex format. Formulas should be numbered consecutively in parentheses ( ) and each rnanuscript should be accompanied by classification numbers from the American Mathematical Society's classification scheme. lt is a fundamental condition that articles submitted to Communications have not been previously published and will not be simultaneously submitted or published elsewhere. After the manuscript has been accepted for publication, i.e., after referree-recommended revisions are complete, the author will not be permitted to rnake any new additions to the manuscript. Before publication the gal1ey proof is always sent to the author for correction. Thus it is solely the author's responsibility for any typographical mistakes which occur in their article as it appears in the Journal. 1. Title Page: The title should not be long but should be informative. Each title page must contain (i) The title of the paper in English, (ii) The complete name(s) of the author(s), (iii) The name and the address of the University. 2. Abstract : The abstract should not exceed 200 words and it should condense the essential features of the articles with the focus on the major advances in the field. 3. References : References must be listed in alphabetical order at the end of the article and then numbered in brackets [1]. Within the manuscript, refer to the references by their given number, again in brackets [1]. They should be styled and punctuated according to the following examples: [1] Kel1ey, J.L., General Topology, 1970, New York: Van Nostrand. [2] Maddox,l.J., Some incusion theorems, Proc. Glasgow Math. Assn. 6 (1964), 161-168. Abstracts, unpublished data and personal communications should not be given in the references but they may be mentioned in the text. 4. Footnotes: Footnotes, except a possible acknowledgement of assistance or financial support on the first page, should be avoided by being incorparated into the text..

(199) 5. Tables and Figures: All tables and figures must be numbered consecutively throughout the paper ( Table 1, Figure 2) and also have a caption or legend. 6. Acknowledgements: Acknowledgements should be given as short as possible at the end of the text. 7. Reprints: 25 free reprints will be provided for each paper. Irrespective of their acceptance, manuscripts will not be returned to the authors. 8. Computer Disk: After the acceptance of manuscripts for publication, we will ask you to submit a revised electronic copy of the manuscript which is written in AMS tex format. For electronic submission please visit our WEB page at http://math.science.ankara.edu.tr/dergi/dergi.htm 9.Charges: Each paper is due to be charged for the amount of which is determined by the administiration each year. 10. Address: Text should be sent to the following address: Prof.Dr. Öner ÇAKAR- Editor-in-Chief, Communications Ankara University, Faculty of Sciences 06100, Tando÷an, ANKARA - TURKEY.

(200)