Intuitionistic Fuzzy G δ -e-locally Continuous and Irresolute Functions
|
|
- Suzanna Johns
- 6 years ago
- Views:
Transcription
1 Intern J Fuzzy Mathematcal rchve Vol 14, No 2, 2017, ISSN (P), (onlne) Publshed on 11 December 2017 wwwresearchmathscorg DOI http//dxdoorg/ /jmav14n2a14 Internatonal Journal o Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons G Saravanakumar 1, S Tamlselvan 2 and Vadvel 3 1 Department o Mathematcs, nnamala Unversty nnamalanagar, Taml Nadu , Inda 2 Mathematcs Secton (FET), nnamala Unversty nnamalanagar, Taml Nadu , Inda 3 Department o Mathematcs, Government rts College (utonomous) Karur, Taml Nadu , Inda Emal avmaths@gmalcom Receved 15 November 2017; accepted 7 December 2017 bstract The purpose o ths paper s to ntroduce the concepts o an ntutonstc uzzy G -e-locally neghborhood, ntutonstc uzzy G -e-locally quas neghborhood, ntutonstc uzzy G -e-locally contnuous and ntutonstc uzzy G -e-locally rresolute unctons n ntutonstc uzzy topologcal spaces lso some nterestng propertes are establshed Keywords ntutonstc uzzy G -e-locally neghborhood, ntutonstc uzzy G -elocally quas neghborhood, ntutonstc uzzy G -e-locally contnuous, ntutonstc uzzy G -e-locally rresolute uncton MS Mathematcs Subject Classcaton (2010) 5440, 5499, 03E72, 03E99 1 Introducton The concept o uzzy sets was ntroduced by Zadeh [10] and later tanassov [1] generalzed the dea to ntutonstc uzzy sets On the other hand, Coker [3] ntroduced the notons o an ntutonstc uzzy topologcal spaces, ntutonstc uzzy contnuty and some other related concepts The concept o an ntutonstc uzzy e -closed set was ntroduced by Sobana et al, [8] Ganster and Relly used locally closed sets n [5] to dene LC-contnuty and LC-rresoluteness alasubramanan [2] ntroduced and studed the concept o uzzy G set n a uzzy topologcal space In ths paper, the concepts o an ntutonstc uzzy G locally quas neghborhood, ntutonstc uzzy G locally contnuous and ntutonstc uzzy G rresolute uncton are ntroduced and studed Some nterestng propertes among contnuous uncton are dscussed 2 Prelmnares Denton 21 [1] Let X be a nonempty xed set and I be the closed nterval [0, 1] n ntutonstc uzzy set (IFS) s an object o the ollowng orm 313
2 G Saravanakumar, S Tamlselvanand Vadvel = { x,, where the mappng µ X I and γ X I denote the degree o membershp (namely µ (x ) ) and the degree o nonmembershp (namely γ (x )) or each element x X to the set, respectvely, and 0 µ ( x) + 1 or each x X Obvously, every uzzy set on a nonempty set X s an IFS o the ollowng orm, = { x, 1 µ ( x) For the sake o smplcty, we shall use the symbol = { x, 314 = x, µ, γ or the ntutonstc uzzy set Denton 22 [1] Let X be a nonempty set and the IFSs and n the orm = { x,, = { x, Then 1 and only µ ( x) ( x) and or all x X ; µ 2 = { x,, µ ( x) x ; 3 = { x, µ ( x) ; 4 = { x, µ ( x) γ ( x) ; Denton 23 [1] The IFS s 0 and 1 are dened by, 0 = { x, 0,1 x and 1 = { x,1, 0 Denton 24 [3] n ntutonstc uzzy topology (IFT) n Coker s sense on a nonempty set X s a amly T o ntutonstc uzzy sets n X satsyng the ollowng axoms 1 0, 1 T ; 2 G 1 G 2 T, or any G 1, G 2 T ; 3 G T or arbtrary amly { G J} T In ths paper by ( X, T ) or smply by X we wll denote the Coker s ntutonstc uzzy topologcal space (IFTS) Each IFS whch belongs to T s called an ntutonstc uzzy open set (IFOS) n X The complement o an IFOS n X s called an ntutonstc uzzy closed set (IFCS) n X Denton 25 [3] Let ( X, T ) be an IFTS and = { x, µ, ν be an IFS n X Then the ntutonstc uzzy closure and ntutonstc uzzy nteror o are dened by 1 IFcl( ) = { C C sanifcsn Xand C } ; 2 IFnt( ) = { D D sanifosn Xand D } ; Proposton 21 [1] For any IFS n ( X, T ) we have 1 cl ( ) = nt( ) 2 nt ( ) = cl( )
3 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons Corollary 21 [3] Let, ( J ) be IFSs n X,, j ( j K) IFSs n Y and X Y a uncton Then 1 1 ( ( )) (I s njectve, then = ( ( ))) 2 ( ( )) (I s surjectve, then ( ( )) = ) 3 ( ) = ( ) 4 j j ( j ) = ( j ) 5 (1 ) = 1 6 (0 ) = 0 7 ( ) = ( ) Denton 26 [4] Let X be a nonempty set and x X a xed element n X I r I 0, s I1 are xed real numbers such that r + s 1, then the IFS xr, s = x, xr,1 x 1 s s called and ntutonstc uzzy pont (IFP) n X, where r denotes the degree o membershp o x, s denotes the degree o non membershp o x and x X the support o x The IFP x s contaned n the IFS ( xr, s ) and only r < µ ( x), s > Denton 27 [6] n IFS U o an IFTS X s called 1 neghborhood o an IFP c ( a, b), there exsts an IFOSG n X such that c( a, b) G U 2 q -neghborhood o an IFP c ( a, b), there exsts an IFOSG n X such that c( a, b) qg U Denton 28 [3] Let X and Y be two nonempty sets and X Y be a uncton 1 I = { y, µ ( y), γ ( y) y Y} s an IFS n Y, then the premage o under (denoted by 1 ( ) ) s dened by ( ) = { x, ( µ )( x), ( γ )( x) x 2 I = { x, λ ( x), ν ( x) s an IFS n X, then the mage o under (denoted by () ) s dened by ( ) = { y, ( λ ( y)), (1 (1ν ))( y) Denton 29 [9] Let be IFS n an IFTS ( X, T ) s called an 1 ntutonstc uzzy regular open set (brely IFROS ) = ntcl ( ) 2 ntutonstc uzzy regular closed set (brely IFRCS ) = clnt( ) Denton 210 [2] Let ( X, T ) be a uzzy topologcal space and λ be a uzzy set n X λ s called uzzy F σ G set = λ = λ where each λ T The complement o uzzy G λ s 1 315
4 G Saravanakumar, S Tamlselvanand Vadvel Denton 211 [5] subset o a space ( X, T ) s called locally closed (brely lc ) = C D, where C s open and D s closed n ( X, T ) Denton 212 [9] Let ( X, T ) be an IFTS and = x, ν ( x) be a IFS n X Then the uzzy closure o are denoted and dened by cl ( ) = { K K s an IFRCS n X and K} and nt ( ) = { G G s an IFROS n X and G } Denton 213 [8] Let be an IFS n an IFTS ( X, T ) s called an ntutonstc uzzy e -open set (IFeOS, or short) n X clnt ( ) ntcl ( ) Denton 214 [3] Let ( X, T ) and ( Y, S) be two IFT s and let X Y be a uncton Then s sad to be ntutonstc uzzy contnuous the premage o each IFS n S s an IFS n T Denton 215 [8] Let ( X, T ) and ( Y, S) be two IFT s and let X Y be a uncton Then s sad to be ntutonstc uzzy e -contnuous the premage o each IFS n S s an IFeOS n T Denton 216 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be ntutonstc uzzy e - locally closed set (n short, IF lcs ) = C D, where C = { x, s an ntutonstc uzzy e -open set and D = { x, x s an ntutonstc uzzy e -closed set n ( X, T ) Denton 217 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, x be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space X Then s sad to be an ntutonstc uzzy D D C C eg - set =, 1 where = { x, x s an ntutonstc uzzy e -open set n an ntutonstc uzzy topologcal space ( X, T ) Denton 218 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, x be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy eg -locally = 316
5 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons closed set (n short,if- eg - lcs ) s an ntutonstc uzzy = C D, where C = { x, eg set and D = { x, x s an ntutonstc uzzy e -closed set n ( X, T ) The complement o an ntutonstc uzzy eg -locally closed set s sad to be an ntutonstc uzzy eg -locally open set (n short, IF eg -los) Denton 219 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, x be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy G locally closed set (n short,if G lcs ) = C, where = { x, x s an ntutonstc uzzy G set and C = { x, s an ntutonstc uzzy e -closed set n ( X, T ) The complement o an ntutonstc uzzy G locally closed set s sad to be an ntutonstc uzzy G locally open set (n short, IFG los) Denton 220 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) The ntutonstc uzzy G locally closure o s denoted and dened by IFG -e -lcl( ) = { = x, µ ( x), γ ( x) X s an ntutonstc uzzy G locally closed set n X and } Denton 221 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space Let = { x, be an ntutonstc uzzy set on an ntutonstc uzzy topologcal space ( X, T ) The ntutonstc uzzy G locally nteror o s denoted and dened by IFG lnt( ) = { = { x, µ ( x), γ ( x) x s an ntutonstc uzzy G -e -locally open set n X and Proposton 22 [7] Let ( X, T ) be an ntutonstc uzzy topologcal space For any two ntutonstc uzzy sets = { x, x and = { x, o an ntutonstc uzzy topologcal space ( X, T ) then the ollowng statements are true 1 IFG lcl (0 ) = 0 2 IFG lcl( ) IFG lcl () C C D D C C 317
6 G Saravanakumar, S Tamlselvanand Vadvel 3 IFG lcl( IFG lcl ( )) = IFG -e - lcl () 4 IFG lcl ( ) = ( IFG lcl ( )) ( IFG lcl ()) Remark 21 [7] 1 IFG lcl ( ) = and only s an ntutonstc uzzy G locally closed set 2 IFG lnt( ) IFG lcl () 3 IFG lnt (1 ) = 1 4 IFG lnt (0 ) = 0 5 IFG lcl (1 ) = 1 3 Intutonstc uzzy G locally contnuous unctons Denton 31 Let ( X, T ) be an ntutonstc uzzy topologcal space Let = x, µ, γ be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy G -e -locally neghbourhood o an ntutonstc uzzy pont x there exsts an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) such that x, It s denoted by IFG lnbd Denton 32 Let ( X, T ) be an ntutonstc uzzy topologcal space Let = x, µ, γ be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Then s sad to be an ntutonstc uzzy G locally quas neghbourhood o an ntutonstc uzzy pont x there exsts an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) such that x s q, It s denoted by IFG lqnbd r, Remark 31 1 The amly o all ntutonstc uzzy G locally neghbourhood o an ntutonstc uzzy pont x, s denoted by N x ) r s IFG el ( r,s 2 The amly o all ntutonstc uzzy G locally quas neghbourhood o an ntutonstc uzzy pont x, s denoted by N x ) r s IFG elq ( r,s Denton 33 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then s sad to be an ntutonstc uzzy G locally contnuous uncton, or each ntutonstc uzzy pont ( ) x, n X and N ), there exsts N x ) such that r s ( x r,s IFG elq ( r,s 318
7 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons Theorem 31 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then the ollowng are equvalent 1 s an ntutonstc uzzy G locally contnuous uncton 2 ( ) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ), or each ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) 3 ( ) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ), or each ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Y, S) 4 IFG lcl( ( )) ( IFcl( )), or each ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S) 5 ( IFnt( )) IFG lnt( 1 ( )), or each ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S) Proo () () Let be an ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) Let x be an ntutonstc uzzy pont n an ntutonstc uzzy topologcal space ( X, T ) such that x 1 s q ( ) Snce s an ntutonstc uzzy r, IFG elq ( r,s G locally contnuous uncton, there exsts N x ) such that ( ) Then xr, s (1) 1 ( ( )) (2) Thus, ( ( )) ( ) Ths mples 1 ( ) whch x IFG elq x shows that ( ) N ( xr, s ) Hence ( ) s an ntutonstc uzzy G -e - locally open set n an ntutonstc uzzy topologcal space ( X, T ) () () Ths can be proved by takng complement o () () (v) Let be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S) Snce IFcl( ), ( ) ( IFcl( )) y (), ( IFcl( )) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Thus, IFG lcl( ( )) ( IFcl( )) (v) (v) Usng (v), IFG -e - lcl( ( )) ( IFcl( )) Then IFG e lcl( ( )) ( IFcl( )), IFG lnt( ( )) ( IFnt( )), IFG e lnt( ( )) ( IFnt( )) mples that ( IFnt( )) IFG e lnt( 1 ( )), puttng =, we have ( IFnt( )) IFG e lnt( ( )) 319
8 G Saravanakumar, S Tamlselvanand Vadvel (v) () Let be an ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) Then IFnt = Usng (v), ( IFnt( )) IFG -e - ( 1 lnt ( )) mples that ( ) IFG lnt( 1 ( )) ut, IFG lnt( ( )) ( ) mples that ( ) = IFG lnt( 1 ( )) that s, ( ) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) Let x be any ntutonstc uzzy pont n 1 ( ) Then x 1 ( ) We have 1 x s q ( ) mples that ( x ) q ( ( )) ut ( ( )) Thus, r, r, s or any ntutonstc uzzy pont x and N ( x r,s ), there exsts IFG elq = r, s ( ) N ( x ) such that ( ( )) Thereore, ( ) Thus, s an ntutonsc uzzy G locally contnuous uncton Theorem 32 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy bjectve uncton Then s an ntutonstc uzzy G locally contnuous uncton and only IFnt ( ( )) ( IFG lnt ()), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Proo ssume that s an ntutonstc uzzy G locally contnuous uncton and be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Hence, ( IFnt( ( ))) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) From Theorem (v) o (01) ( IFnt ( )) IFG lnt( ( ( ))) Snce s an ntutonstc uzzy surjectve uncton, ( IFnt ( )) IFG, lnt () ( ( IFnt ( ))) ( IFG - e - lnt ()) That s, IFnt ( ( )) ( IFG lnt ()) Conversely, assume that IFnt ( ( )) ( IFG lnt ()), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Let be an ntutonstc uzzy open set n an ntutonstc uzzy topologcal space ( Y, S) Then = IFnt( ) Snce s an ntutonstc uzzy surjectve uncton, = IFnt( ) = IFnt( ( ( ))) ( IFG lnt( 1 ( ))) Snce s an ntutonstc uzzy njectve uncton, ( ) ( ( IFG - lnt( 1 ( )))) From the act that, s an ntutonstc uzzy njectve uncton, we have ( ) IFG e lnt( ( )) (3) but - e 320
9 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons IFG e lnt( ( )) ( ) (4) From (3) and (4) mples that ( ) = IFG lnt( 1 ( )) That s, ( ) s an ntutonstc uzzy G locally open set n an ntutonstc uzzy topologcal space ( X, T ) Thus, s an ntutonstc uzzy G locally contnuous uncton Theorem 33 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy bjectve ucton Then s an ntutonstc uzzy G locally contnuous uncton and only ( IFG lcl( )) IFcl( ( )), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Proo ssume that s an ntutonstc uzzy G locally contnuous uncton and be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Hence, ( IFcl( ( ))) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) From Theorem (v) o (01) IFG lcl( ( ( ))) ( IFcl ( )) Snce s an ntutonstc uzzy njectve uncton, IFG lcl( )) ( IFcl ( )) Takng on both sdes, ( IFG -e - lcl( )) ( ( IFcl ( ))) Snce s an ntutonstc uzzy surjectve uncton, ( IFG lcl( )) IFcl( ( )) Conversely, assume that ( IFG lcl( )) IFcl( ( )), or each ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ) Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Y, S) Then = IFcl( ) Snce s an ntutonstc uzzy surjectve uncton, and by assumpton = IFcl( ) = IFcl( ( ( ))) ( IFG lcl ( ( ))), ( ) ( ( IFG lcl( 1 ( )))) Snce s an ntutonstc uzzy njectve uncton, ut ( ) ( ( )) (5) IFG e lcl ( ( )) IFG e cl( ( )) (6) From (5) and (6) mples that s an ntutonstc uzzy ( ) = IFG lcl( 1 ( )) That s, ( ) G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Thus, s an ntutonstc uzzy G locally contnuous uncton Theorem 34 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy bjectve ucton I s an 321
10 G Saravanakumar, S Tamlselvanand Vadvel ntutonstc uzzy G locally contnuous uncton Then uzzy closed set, then ( ) = IFG lcl( 1 ( )) 322 Y I s an ntutonstc Proo Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Y, S) y Theorem(v)o (01) Snce = IFcl( ) ut From (7) and (8) mples that IFG e lcl IFcl ( ( )) ( ( )) = ( ) ( ) ( ( )) IFG e lcl ( ) = IFG lcl( 1 ( )) Proposton 31 Let ( X, T ), ( Y, S) and ( Z, R) be any three ntutonstc uzzy topologcal spaces Let ( X T ) ( Y, S) be an ntutonstc uzzy G locally contnuouus uncton I ( X ) Z Y then g ( X, T ) ( Z, R) where R = S/ Z restrctng the range o s an ntutonstc uzzy G locally contnuous uncton Proo Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( Z, R) Then = S/ Z, or some ntutonstc uzzy closed set o an ntutonstc uzzy topologcal spaces ( Y, S) I ( X ) Z Y, ( ) = g ( ) Snce ( ) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Hence, g ( ) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Thereore, g s an ntutonstc uzzy G locally contnuous uncton Proposton 32 Let ( X, T), ( X1, T1 ) and ( X 2, T2 ) be any three ntutonstc uzzy topologcal spaces and P X X X be an ntutonstc uzzy projecton o 1 2 X 1 X 2 onto X I X X1 X 2 s an ntutonstc uzzy G locally contnuous uncton Then P X X s also an ntutonstc uzzy G locally cotnuous uncton Proo Let be an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal spaces ( X, T ) ( = 1, 2), ( P ) ( ) = ( P ( )) Snce P s an ntutonstc uzzy mappng P ( ) s an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal spaces X 1 X 2 Hence, ( P ( )) s an ntutonstc uzzy G -e - locally closed set n an ntutonstc uzzy topologcal space ( X, T ) Hence, P s an ntutonstc uzzy G -e -locally contnuous uncton Proposton 33 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces I ntutonstc uzzy graph uncton g X X Y s an ntutonstc uzzy (7) (8)
11 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons G locally contnuous uncton Then ( X, T ) ( Y, S) s an ntutonstc uzzy G locally contnuous uncton Proo Let g be an ntutonstc uzzy G locally contnuous uncton and x be any ntutonstc uzzy pont n an ntutonstc uzzy topologcal space ( X, T ) I IFG elq N x ) n an ntutonstc uzzy topologcal space ( r,s ( Y, S), X N g( xr,s ) n an ntutonstc uzzy topologcal space X Y Snce g s an ntutonstc uzzy G - e -locally contnuous uncton, there exsts N ) such that g ( ) X y 323 IFG elq ( x r,s Denton 024, we have ( ) Thereore, s an ntutonstc uzzy G locally contnuous uncton Denton 34 Let ( X, T ) and ( Y, S) be two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then s sad to be an 1 Intutonstc uzzy G -e -locally rresolute uncton, or each ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal 1 ( space ( Y, S), ) s an ntutonstc uzzy ntutonstc uzzy topologcal space ( X, T ) G locally closed set n an 2 Intutonstc uzzy weakly G locally contnuous uncton, or each ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( Y, S), 1 ( ) s an ntutonstc uzzy closed set n an ntutonstc uzzy topologcal space ( X, T ) Example 31 Let X = { a, b, c} = Y, and = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), C = x,(, ),(, ) Now, the amly T = {0,1,,,, } o IFS s n X s an IFT on X and the amly S = {0,1, C} o IFS s n Y s an IFT on Y I we dene the uncton X Y be the dentty uncton Now, s an ntutonstc uzzy G locally rresolute uncton, because C s an IFG locally closed set n Y, ( C ) s also an IFG locally closed set n X
12 Example 32 Let G Saravanakumar, S Tamlselvanand Vadvel X = { a, b, c} = Y, and = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), = x,(, ),(, ), C = x,(, ),(, ) D = x,(, ),(, ) Now, the amly T = {0,1,,,,, C} o IFS s n X s an IFT on X and the amly S = {0,1, D} o IFS s n Y s an IFT on Y I we dene the uncton X Y be the dentty uncton, Now, s an ntutonstc uzzy weakly G -e - locally rresolute uncton, because D s an IFG locally closed set n Y, ( D ) s ntutonstc uzzy closed set n X Theorem 35 Let ( X, T ) and ( Y, S) be any two ntutonstc uzzy topologcal spaces Let ( X, T ) ( Y, S) be an ntutonstc uzzy mappng Then the ollowng statements are equvalent 1 s an ntutonstc uzzy G locally rresolute uncton 2 or every ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( X, T ), ( IFG lcl( )) IFG -e - lcl ( ( )) 3 or every ntutonstc uzzy set o an ntutonstc uzzy topologcal space ( Y, S), IFG lcl ( ( )) ( IFG lcl ()) Proo () () Let be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) Suppose s an ntutonstc uzzy G locally rresolute uncton Now, IFG lcl ( ( )) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( Y, S) y hypothess, (IFG lcl ( ( ))) s an ntutonstc uzzy G locally closed set n an ntutonstc uzzy topologcal space ( X, T ) and hence, ( ( )) ( IFG ( lcl ( ( )))) Now, IFG lcl ( ) ( IFG lcl ( ( ))) That s, ( IFG lcl( )) IFG -e - lcl ( ( )) () () Let be an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( Y, S), then ( ) s an ntutonstc uzzy set n an ntutonstc uzzy topologcal space ( X, T ) y (), ( IFG lcl( ( ))) IFG lcl( ( 1 ( ))) Snce s an ntutonstc uzzy bjectve uncton, IFG lcl ( ( )) ( IFG lcl ()) 324
13 Intutonstc Fuzzy G -e-locally Contnuous and Irresolute Functons () () Suppose s ntutonstc uzzy IFG locally closed set n an ntutonstc uzzy topologcal space ( Y, S) Then IFG lcl ( ) = y hypothess, IFG lcl ( ( )) ( IFG lcl ( )), IFG - e lcl( ( )) ( ) 4 Concluson Intutonstc uzzy topology s an mportant and a major area o mathematcs In ths paper, we ntroduce ntutonstc uzzy G locally neghborhood, ntutonstc uzzy G locally quas neghborhood, ntutonstc uzzy G locally contnuous and ntutonstc uzzy G locally rresolute unctons n ntutonstc uzzy topologcal spaces are studed cknowledgement The authors express ther earnest grattude to the revewers, edtor-nche and managng edtors or ther constructve suggestons and comments whch helped to mprove the present paper REFERENCES 1 KTtanassov, Intutonstc uzzy sets, Fuzzy Sets and Systems, 20 (1986) Galasubramanan, Maxmal uzzy topologes, Kybernetka, 31 (1995) DCoker, n ntroducton to ntutonstc uzzy topologcal spaces, Fuzzy Set and Systems, 88 (1997) DCoker and MDemrc, On ntutonstc uzzy ponts, Notes IFS, 2(1) (1995) MGanster and ILRelly, Locally closed sets and and LC-contnuous unctons, IntrJ Math and Math Sc, 12(3) (1989) IMHanay, Completely contnuous uncton n ntutonstc uzzy topologcal space, Czechoslovak Math J, 53 (2003), GSaravanakumar, STamlselvan and Vadvel, Intutonstc uzzy G locally closed sets, submtted 8 DSobana, VChandrasekar and Vadvel, On uzzy e -open sets, uzzy e - contnuty and uzzy e -compactness n ntutonstc uzzy topologcal spaces, Communcatons n Mathematcal nalyss (n Press) 9 SSThakur and SSngh, On uzzy sem-pre open sets and uzzy sem-pre contnuty, Fuzzy Sets and Systems, (1998) LZadeh, Fuzzy sets, Inormaton and Control, 8 (1965)
LEVEL SET OF INTUITIONTISTIC FUZZY SUBHEMIRINGS OF A HEMIRING
LEVEL SET OF INTUITIONTISTIC FUZZY SUBHEMIRINGS OF HEMIRING N. NITH ssstant Proessor n Mathematcs, Peryar Unversty PG Extn Centre, Dharmapur 636705. Emal : anthaarenu@gmal.com BSTRCT: In ths paper, we
More informationBitopological spaces via Double topological spaces
topologcal spaces va Double topologcal spaces KNDL O TNTWY SEl-Shekh M WFE Mathematcs Department Faculty o scence Helwan Unversty POox 795 aro Egypt Mathematcs Department Faculty o scence Zagazg Unversty
More informationSome Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphs
IOS Journal of Mathematcs (IOS-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 6 Ver. IV (Nov. - Dec. 05), PP 03-07 www.osrournals.org Some Concepts on Constant Interval Valued Intutonstc Fuzzy Graphs
More informationOn wgrα-continuous Functions in Topological Spaces
Vol.3, Issue.2, March-Aprl. 2013 pp-857-863 ISSN: 2249-6645 On wgrα-contnuous Functons n Topologcal Spaces A.Jayalakshm, 1 C.Janak 2 1 Department of Mathematcs, Sree Narayana Guru College, Combatore, TN,
More informationاولت ارص من نوع -c. الخلاصة رنا بهجت اسماعیل مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد 22 (3) 2009
مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد 22 (3) 2009 الت ارص من نوع -C- - جامعة بغداد رنا بهجت اسماعیل قسم الریاضیات - كلیة التربیة- ابن الهیثم الخلاصة قمنا في هذا البحث بتعریف نوع جدید من الت ارص
More informationResearch Article Relative Smooth Topological Spaces
Advances n Fuzzy Systems Volume 2009, Artcle ID 172917, 5 pages do:10.1155/2009/172917 Research Artcle Relatve Smooth Topologcal Spaces B. Ghazanfar Department of Mathematcs, Faculty of Scence, Lorestan
More informationSmooth Neutrosophic Topological Spaces
65 Unversty of New Mexco Smooth Neutrosophc opologcal Spaces M. K. EL Gayyar Physcs and Mathematcal Engneerng Dept., aculty of Engneerng, Port-Sad Unversty, Egypt.- mohamedelgayyar@hotmal.com Abstract.
More informationn-strongly Ding Projective, Injective and Flat Modules
Internatonal Mathematcal Forum, Vol. 7, 2012, no. 42, 2093-2098 n-strongly Dng Projectve, Injectve and Flat Modules Janmn Xng College o Mathematc and Physcs Qngdao Unversty o Scence and Technology Qngdao
More informationINTERVAL-VALUED INTUITIONISTIC FUZZY CLOSED IDEALS OF BG-ALGEBRA AND THEIR PRODUCTS
ITEVL-VLED ITITIOISTIC FZZY CLOSED IDELS OF G-LGE D THEI PODCTS Tapan Senapat #, onoranjan howmk *, adhumangal Pal #3 # Department of ppled athematcs wth Oceanology Computer Programmng, Vdyasagar nversty,
More informationTHE RING AND ALGEBRA OF INTUITIONISTIC SETS
Hacettepe Journal of Mathematcs and Statstcs Volume 401 2011, 21 26 THE RING AND ALGEBRA OF INTUITIONISTIC SETS Alattn Ural Receved 01:08 :2009 : Accepted 19 :03 :2010 Abstract The am of ths study s to
More informationWeakly continuous functions on mixed fuzzy topological spaces
cta Scentarum http://wwwuembr/acta ISSN prnted: 806-563 ISSN on-lne: 807-8664 Do: 0405/actasctechnolv3664 Weakly contnuous functons on mxed fuzzy topologcal spaces Bnod Chandra Trpathy and Gautam Chandra
More informationA Simple Research of Divisor Graphs
The 29th Workshop on Combnatoral Mathematcs and Computaton Theory A Smple Research o Dvsor Graphs Yu-png Tsao General Educaton Center Chna Unversty o Technology Tape Tawan yp-tsao@cuteedutw Tape Tawan
More information= s j Ui U j. i, j, then s F(U) with s Ui F(U) G(U) F(V ) G(V )
1 Lecture 2 Recap Last tme we talked about presheaves and sheaves. Preshea: F on a topologcal space X, wth groups (resp. rngs, sets, etc.) F(U) or each open set U X, wth restrcton homs ρ UV : F(U) F(V
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationAntipodal Interval-Valued Fuzzy Graphs
Internatonal Journal of pplcatons of uzzy ets and rtfcal Intellgence IN 4-40), Vol 3 03), 07-30 ntpodal Interval-Valued uzzy Graphs Hossen Rashmanlou and Madhumangal Pal Department of Mathematcs, Islamc
More informationDouble Layered Fuzzy Planar Graph
Global Journal of Pure and Appled Mathematcs. ISSN 0973-768 Volume 3, Number 0 07), pp. 7365-7376 Research Inda Publcatons http://www.rpublcaton.com Double Layered Fuzzy Planar Graph J. Jon Arockaraj Assstant
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776
More information2 MADALINA ROXANA BUNECI subset G (2) G G (called the set of composable pars), and two maps: h (x y)! xy : G (2)! G (product map) x! x ;1 [: G! G] (nv
An applcaton of Mackey's selecton lemma Madalna Roxana Bunec Abstract. Let G be a locally compact second countable groupod. Let F be a subset of G (0) meetng each orbt exactly once. Let us denote by df
More informationSeparation Axioms of Fuzzy Bitopological Spaces
IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 Separaton Axom of Fuzzy Btopologcal Space Hong Wang College of Scence Southwet Unverty of Scence and Technology Manyang
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationThe L(2, 1)-Labeling on -Product of Graphs
Annals of Pure and Appled Mathematcs Vol 0, No, 05, 9-39 ISSN: 79-087X (P, 79-0888(onlne Publshed on 7 Aprl 05 wwwresearchmathscorg Annals of The L(, -Labelng on -Product of Graphs P Pradhan and Kamesh
More informationOn the Operation A in Analysis Situs. by Kazimierz Kuratowski
v1.3 10/17 On the Operaton A n Analyss Stus by Kazmerz Kuratowsk Author s note. Ths paper s the frst part slghtly modfed of my thess presented May 12, 1920 at the Unversty of Warsaw for the degree of Doctor
More informationInternational Journal of Mathematical Archive-4(12), 2013, Available online through ISSN
Internatonal Journal o Mathematcal Archve-(2, 203, 7-52 Avlable onlne throuh www.jma.no ISSN 2229 506 ON VALUE SHARING OF MEROMORPHIC FUNCTIONS Dbyendu Banerjee* and Bswajt Mandal 2 Department o Mathematcs,
More informationMath 205A Homework #2 Edward Burkard. Assume each composition with a projection is continuous. Let U Y Y be an open set.
Math 205A Homework #2 Edward Burkard Problem - Determne whether the topology T = fx;?; fcg ; fa; bg ; fa; b; cg ; fa; b; c; dgg s Hausdor. Choose the two ponts a; b 2 X. Snce there s no two dsjont open
More informationINTUITIONISTIC FUZZY GRAPH STRUCTURES
Kragujevac Journal of Mathematcs Volume 41(2) (2017), Pages 219 237. INTUITIONISTIC FUZZY GRAPH STRUCTURES MUHAMMAD AKRAM 1 AND RABIA AKMAL 2 Abstract. In ths paper, we ntroduce the concept of an ntutonstc
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More information42. Mon, Dec. 8 Last time, we were discussing CW complexes, and we considered two di erent CW structures on S n. We continue with more examples.
42. Mon, Dec. 8 Last tme, we were dscussng CW complexes, and we consdered two d erent CW structures on S n. We contnue wth more examples. (2) RP n. Let s start wth RP 2. Recall that one model for ths space
More informationGoal Programming Approach to Solve Multi- Objective Intuitionistic Fuzzy Non- Linear Programming Models
Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 Goal Prorammn Approach to Solve Mult- Objectve Intutonstc Fuzzy Non- Lnear Prorammn Models S.Rukman #, R.Sopha Porchelv
More informationA Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"
Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: 10.1007/s11401-005-0025-z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules,
More informationDIFFERENTIAL SCHEMES
DIFFERENTIAL SCHEMES RAYMOND T. HOOBLER Dedcated to the memory o Jerry Kovacc 1. schemes All rngs contan Q and are commutatve. We x a d erental rng A throughout ths secton. 1.1. The topologcal space. Let
More informationComplement of Type-2 Fuzzy Shortest Path Using Possibility Measure
Intern. J. Fuzzy Mathematcal rchve Vol. 5, No., 04, 9-7 ISSN: 30 34 (P, 30 350 (onlne Publshed on 5 November 04 www.researchmathsc.org Internatonal Journal of Complement of Type- Fuzzy Shortest Path Usng
More informationTHERE ARE INFINITELY MANY FIBONACCI COMPOSITES WITH PRIME SUBSCRIPTS
Research and Communcatons n Mathematcs and Mathematcal Scences Vol 10, Issue 2, 2018, Pages 123-140 ISSN 2319-6939 Publshed Onlne on November 19, 2018 2018 Jyot Academc Press http://jyotacademcpressorg
More informationIntuitionistic fuzzy gα**-closed sets
International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 2, Issue 2(February 2013), PP.11-16 Intuitionistic fuzzy gα**-closed sets M. Thirumalaiswamy
More informationMemoirs on Dierential Equations and Mathematical Physics Volume 11, 1997, 67{88 Guram Kharatishvili and Tamaz Tadumadze THE PROBLEM OF OPTIMAL CONTROL
Memors on Derental Equatons and Mathematcal Physcs Volume 11, 1997, 67{88 Guram Kharatshvl and Tamaz Tadumadze THE PROBLEM OF OPTIMAL CONTROL FOR NONLINEAR SYSTEMS WITH VARIABLE STRUCTURE, DELAYS AND PIECEWISE
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More information2 S. S. DRAGOMIR, N. S. BARNETT, AND I. S. GOMM Theorem. Let V :(d d)! R be a twce derentable varogram havng the second dervatve V :(d d)! R whch s bo
J. KSIAM Vol.4, No., -7, 2 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM S. S. DRAGOMIR, N. S. BARNETT, AND I. S. GOMM Abstract. In ths paper we establsh
More informationK-Total Product Cordial Labelling of Graphs
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 7, Issue (December ), pp. 78-76 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) K-Total Product Cordal Labellng o Graphs
More informationCase Study of Markov Chains Ray-Knight Compactification
Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, 753-76 HIKAI Ltd, www.m-har.com http://dx.do.org/.2988/cms.24.46 Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and
More informationGenericity of Critical Types
Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f
More informationDIEGO AVERNA. A point x 2 X is said to be weakly Pareto-optimal for the function f provided
WEAKLY PARETO-OPTIMAL ALTERNATIVES FOR A VECTOR MAXIMIZATION PROBLEM: EXISTENCE AND CONNECTEDNESS DIEGO AVERNA Let X be a non-empty set and f =f 1 ::: f k ):X! R k a functon. A pont x 2 X s sad to be weakly
More informationSemicompactness in Fuzzy Topological Spaces
BULLETIN of the Malaysan Mathematcal Scences Socety http://math.usm.my/bulletn Bull. Malays. Math. Sc. Soc. (2) 28(2) (2005), 205 23 Semcompactness n Fuzzy Topologcal Spaces R.P. Chakraborty, Anjana Bhattacharyya
More informationPROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 7, July 1997, Pages 2119{2125 S (97) THE STRONG OPEN SET CONDITION
PROCDINGS OF TH AMRICAN MATHMATICAL SOCITY Volume 125, Number 7, July 1997, Pages 2119{2125 S 0002-9939(97)03816-1 TH STRONG OPN ST CONDITION IN TH RANDOM CAS NORBRT PATZSCHK (Communcated by Palle. T.
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationSoft Neutrosophic Bi-LA-semigroup and Soft Neutrosophic N-LA-seigroup
Neutrosophc Sets and Systems, Vol. 5, 04 45 Soft Neutrosophc B-LA-semgroup and Soft Mumtaz Al, Florentn Smarandache, Muhammad Shabr 3,3 Department of Mathematcs, Quad--Azam Unversty, Islamabad, 44000,Pakstan.
More informationInfinitely Split Nash Equilibrium Problems in Repeated Games
Infntely Splt ash Equlbrum Problems n Repeated Games Jnlu L Department of Mathematcs Shawnee State Unversty Portsmouth, Oho 4566 USA Abstract In ths paper, we ntroduce the concept of nfntely splt ash equlbrum
More informationOn Intuitionistic Fuzzy Semi-Generalized Closed. Sets and its Applications
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 34, 1677-1688 On Intuitionistic Fuzzy Semi-Generalized Closed Sets and its Applications R. Santhi Department of Mathematics, Nallamuthu Gounder Mahalingam
More informationOn Tiling for Some Types of Manifolds. and their Folding
Appled Mathematcal Scences, Vol. 3, 009, no. 6, 75-84 On Tlng for Some Types of Manfolds and ther Foldng H. Rafat Mathematcs Department, Faculty of Scence Tanta Unversty, Tanta Egypt hshamrafat005@yahoo.com
More informationInternational Journal of Algebra, Vol. 8, 2014, no. 5, HIKARI Ltd,
Internatonal Journal of Algebra, Vol. 8, 2014, no. 5, 229-238 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ja.2014.4212 On P-Duo odules Inaam ohammed Al Had Department of athematcs College of Educaton
More informationEXTENSION DIMENSION OF INVERSE LIMITS. University of Zagreb, Croatia
GLASNIK MATEMATIČKI Vol. 35(55)(2000), 339 354 EXTENSION DIMENSION OF INVERSE LIMITS Sbe Mardešć Unversty of Zagreb, Croata Abstract. Recently L.R. Rubn and P.J. Schapro have consdered nverse sequences
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationComplement of an Extended Fuzzy Set
Internatonal Journal of Computer pplatons (0975 8887) Complement of an Extended Fuzzy Set Trdv Jyot Neog Researh Sholar epartment of Mathemats CMJ Unversty, Shllong, Meghalaya usmanta Kumar Sut ssstant
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationExistence results for a fourth order multipoint boundary value problem at resonance
Avalable onlne at www.scencedrect.com ScenceDrect Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx www.elsever.com/locate/jnnms Exstence results for a fourth order multpont boundary value problem
More informationInterval Valued Neutrosophic Soft Topological Spaces
8 Interval Valued Neutrosoph Soft Topologal njan Mukherjee Mthun Datta Florentn Smarandah Department of Mathemats Trpura Unversty Suryamannagar gartala-7990 Trpura Indamal: anjan00_m@yahooon Department
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationNeutrosophic Bi-LA-Semigroup and Neutrosophic N-LA- Semigroup
Neutrosophc Sets Systems, Vol. 4, 04 9 Neutrosophc B-LA-Semgroup Neutrosophc N-LA- Semgroup Mumtaz Al *, Florentn Smarache, Muhammad Shabr 3 Munazza Naz 4,3 Department of Mathematcs, Quad--Azam Unversty,
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationGeneralized Intuitionistic Fuzzy Ideals Topological Spaces
merican Journal of Mathematics Statistics 013, 3(1: 1-5 DOI: 10593/jajms013030103 Generalized Intuitionistic Fuzzy Ideals Topological Spaces Salama 1,, S lblowi 1 Egypt, Port Said University, Faculty of
More informationA note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 7, Number 2, December 203 Avalable onlne at http://acutm.math.ut.ee A note on almost sure behavor of randomly weghted sums of φ-mxng
More informationSolutions of exercise sheet 3
Topology D-MATH, FS 2013 Damen Calaque Solutons o exercse sheet 3 1. (a) Let U Ă Y be open. Snce s contnuous, 1 puq s open n X. Then p A q 1 puq 1 puq X A s open n the subspace topology on A. (b) I s contnuous,
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationCOUNTABLE-CODIMENSIONAL SUBSPACES OF LOCALLY CONVEX SPACES
COUNTABLE-CODIMENSIONAL SUBSPACES OF LOCALLY CONVEX SPACES by J. H. WEBB (Receved 9th December 1971) A barrel n a locally convex Hausdorff space E[x] s a closed absolutely convex absorbent set. A a-barrel
More informationMatrix-Norm Aggregation Operators
IOSR Journal of Mathematcs (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. PP 8-34 www.osrournals.org Matrx-Norm Aggregaton Operators Shna Vad, Sunl Jacob John Department of Mathematcs, Natonal Insttute of
More informationCHAPTER 4 MAX-MIN AVERAGE COMPOSITION METHOD FOR DECISION MAKING USING INTUITIONISTIC FUZZY SETS
56 CHAPER 4 MAX-MIN AVERAGE COMPOSIION MEHOD FOR DECISION MAKING USING INUIIONISIC FUZZY SES 4.1 INRODUCION Intutonstc fuzz max-mn average composton method s proposed to construct the decson makng for
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationApproximations of Set-Valued Functions Based on the Metric Average
Approxmatons of Set-Valued Functons Based on the Metrc Average Nra Dyn, Alona Mokhov School of Mathematcal Scences Tel-Avv Unversty, Israel Abstract. Ths paper nvestgates the approxmaton of set-valued
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationModulo Magic Labeling in Digraphs
Gen. Math. Notes, Vol. 7, No., August, 03, pp. 5- ISSN 9-784; Copyrght ICSRS Publcaton, 03 www.-csrs.org Avalable free onlne at http://www.geman.n Modulo Magc Labelng n Dgraphs L. Shobana and J. Baskar
More informationDirichlet s Theorem In Arithmetic Progressions
Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,
More informationOn Similarity Measures of Fuzzy Soft Sets
Int J Advance Soft Comput Appl, Vol 3, No, July ISSN 74-853; Copyrght ICSRS Publcaton, www-csrsorg On Smlarty Measures of uzzy Soft Sets PINAKI MAJUMDAR* and SKSAMANTA Department of Mathematcs MUC Women
More informationSystem of implicit nonconvex variationl inequality problems: A projection method approach
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 6 (203), 70 80 Research Artcle System of mplct nonconvex varatonl nequalty problems: A projecton method approach K.R. Kazm a,, N. Ahmad b, S.H. Rzv
More informationRegular product vague graphs and product vague line graphs
APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Regular product vague graphs and product vague lne graphs Ganesh Ghora 1 * and Madhumangal Pal 1 Receved: 26 December 2015 Accepted: 08 July 2016
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More informationSTEINHAUS PROPERTY IN BANACH LATTICES
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS
More informationOn Finite Rank Perturbation of Diagonalizable Operators
Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), 49 53 Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: http://wwwpmfnacrs/faac On Fnte Rank Perturbaton of Dagonalzable
More informationA FINITE TO ONE OPEN MAPPING PRESERVES SPAN ZERO
Volume 13, 1988 Pages 181 188 http://topology.auburn.edu/tp/ A FINITE TO ONE OPEN MAPPING PRESERVES SPAN ZERO by Mara Cuervo and Edwn Duda Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology
More informationwhere a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets
11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationAPPLICATIONS OF SEMIGENERALIZED -CLOSED SETS
Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : 22776982 Volume Issue 4 (Apl 202) http://www.mes.com/ https://stes.google.com/ste/mesounal/ APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM,
More informationSPECTRAL PROPERTIES OF IMAGE MEASURES UNDER THE INFINITE CONFLICT INTERACTION
SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER THE INFINITE CONFLICT INTERACTION SERGIO ALBEVERIO 1,2,3,4, VOLODYMYR KOSHMANENKO 5, MYKOLA PRATSIOVYTYI 6, GRYGORIY TORBIN 7 Abstract. We ntroduce the conflct
More informationEndogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract
Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors Yuanzhu Lu Chna Economcs and Management Academy, Central Unversty o Fnance and Economcs Abstract We nvestgate endogenous
More informationSelf-complementing permutations of k-uniform hypergraphs
Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty
More informationP.P. PROPERTIES OF GROUP RINGS. Libo Zan and Jianlong Chen
Internatonal Electronc Journal of Algebra Volume 3 2008 7-24 P.P. PROPERTIES OF GROUP RINGS Lbo Zan and Janlong Chen Receved: May 2007; Revsed: 24 October 2007 Communcated by John Clark Abstract. A rng
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationLinear programming with Triangular Intuitionistic Fuzzy Number
EUSFLAT-LFA 2011 July 2011 Ax-les-Bans, France Lnear programmng wth Trangular Intutonstc Fuzzy Number Dpt Dubey 1 Aparna Mehra 2 1 Department of Mathematcs, Indan Insttute of Technology, Hauz Khas, New
More informationOn statistical convergence in generalized Lacunary sequence spaces
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 2(87), 208, Pages 7 29 ISSN 024 7696 On statstcal convergence n generalzed Lacunary sequence spaces K.Raj, R.Anand Abstract. In the
More informationOn the Connectedness of the Solution Set for the Weak Vector Variational Inequality 1
Journal of Mathematcal Analyss and Alcatons 260, 15 2001 do:10.1006jmaa.2000.7389, avalable onlne at htt:.dealbrary.com on On the Connectedness of the Soluton Set for the Weak Vector Varatonal Inequalty
More informationCorrespondences and groupoids
Proceedngs of the IX Fall Workshop on Geometry and Physcs, Vlanova la Geltrú, 2000 Publcacones de la RSME, vol. X, pp. 1 6. Correspondences and groupods 1 Marta Macho-Stadler and 2 Moto O uch 1 Departamento
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationTHE FUNDAMENTAL THEOREM OF CALCULUS FOR MULTIDIMENSIONAL BANACH SPACE-VALUED HENSTOCK VECTOR INTEGRALS
Real Analyss Exchange Vol.,, pp. 469 480 Márca Federson, Insttuto de Matemátca e Estatístca, Unversdade de São Paulo, R. do Matão 1010, SP, Brazl, 05315-970. e-mal: marca@me.usp.br THE FUNDAMENTAL THEOREM
More informationEXPANSIVE MAPPINGS. by W. R. Utz
Volume 3, 978 Pages 6 http://topology.auburn.edu/tp/ EXPANSIVE MAPPINGS by W. R. Utz Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology Proceedngs Department of Mathematcs & Statstcs
More information