FuzzyStacked Set and Fuzzy Stacked Semigroup

Transcription

1 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December 204 ISSN FuzzySacked Se and Fuzzy Sacked Semgroup Aymen AAhmed Imam*, Mohammed AlBshr**, Emad Eldeen A A Rahm** * PhD sudena he Unversy ofsudanof Scence and Technology ** Dean of hefaculy of Graduae Sudeshe Unversy ofneelan *** Head ofdeparmen of Mahemacs, Faculy of Scence, Unversy of Sudan Absrac- To develophe concep of sacked se and fuzzy sacked se n he prevous wopapers [] and [2], also generalzehe dea ofsacked sysems o becomemore comprehensven addressngsmlarssues, and he addon ofsomealgebracrelaonsandalgebracoperaons, whle mananng hebascshape of hesacked sysems Index Terms- sacked se, fuzzy sacked se, fuzzy se, fuzzy sysem, sacked T I INTRODUCTION hs sysemha we have developedn prevouspapers, observanandpresence neveryhng n hsunverse Our worlds full ofoverlappngrelaonshpsbeween heelemens, ands dffcul odeermne he specfcarrangemenof he mporance ofhese elemenso each oher, andso musgeneralze hssysemo ncludemany ssuesn scence I s noed ha sacked syle buldng close o he dea of fuzzy se,, so you should urn hs se relaed o he ordnal se sackng no obscury or fuzzy se And snce he general dea for hs sudy based on he denfcaon of ways o arrange an elemen whn he se, hey need o follow hs order, o he syle of fuzzy se Thereforewe needo fndoheroperaonsandrelaonshpsmake a more posvedea 2 Defnon [] II PRELIMINARIES le T α be a fne se, where T α be a sacked se f and only f a α T α, α N/0, α s he number of mehods sackng elemens n he se, and s called pahs ( P,P 2,, P α ) 22 Defnon [] The sysem ( T α, ) called sacked- sysem f and only f a γ b = mn 0 ( a γ, b ), and he sysem lookng for ( zero convergence), and The sysem ( T α, Ʈ ) called sacked- sysem f and only f a γ Ʈ b = max 0 (a γ, b ), and he sysem lookng for ( zero spacng ) a γ and b T α 2 Defnon [] The sysem ( T α, ) called sacked- sysem f and only f a γ b = mn ( a γ, b ), and he sysem lookng for (convergence of ), and The sysem ( T α, Ʈ ) called sacked- sysem f and only f a γ Ʈ b = max (a γ, b ), and he sysem lookng for (spacng of) a γ and b T α, R 24 Defnon [] The order elemen on sacked sysem T α,where he sysem lookng for zero convergence or zero spacng, s amoun conrbues o hs elemen n he sysem, and hs esmae s calculaed relaonshp of hs elemen n every pah ha conans hs elemen, hen he elemen order of a γ, ( O 0 (a γ ) ) : O 0 (a γ ) = a γ 0 = [ a 25 Defnon [] a + a 2 a a [ ] / a + + a a ]/α = The order elemen on sacked sysem T α,where he sysem lookng for convergence of ( or spacng of ), s amoun conrbues o hs elemen n he sysem, and hs esmae s calculaed relaonshp of hs elemen n every pah ha conans hs elemen, hen he elemen order of a γ,(o (a γ )) : [ a a 26 Defnon [] + O (a γ ) = a γ = a 2 a + + a [ ] / a a a ]/α = The order sacked se T α n zero convergence sysem s se O 0 (T α) = {x,x 2,,x n } f x 0 x 2 0 x n 0 And

2 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN he order sacked se T α n zero spacng sysem s se O 0 [T α] = {x n,x n-,,x } f x 0 > x 2 0 > > x n 0 The order sacked se T α, where he sysem lookng for convergence of ( or spacng of ), s se O 0 (T α) = {x,x 2,,x n } f x x 2 x n And he order sacked se T α n zero spacng sysem s se O 0 [T α] = {x n,x n-,,x } f x > x 2 > > x n 27 Defnon [] - If ( T α, ) or (T α, Ʈ ) s sacked sysem hen a, b T α : Max (a, b ) = a Ʈ b, and Mn ( a, b ) = a b - If ( T α, ) or (T α, Ʈ ) s sacked sysem hen a, b T α : a max ( a, b) b a mn ( a, b) b : f b a : f b a : f b a : f b a - If a α =,( n a ) hen we suppose ha a α =, and where a, α {, } hen, we compensae a α = 0 - If a = b ( one order elemen n wo dfferen places)so we have many ype of hs sysem, and f a b he sysem s ype- 28 Defnon [] A sacked-semgroup s a sacked-sysem T α, wh assocave bnary operaon 29 Theorem [] () If he sysems ( T α, ) s a sacked-sysem ( ype ), hen ( T α, ) s a semgroup and called a sackedsemgroup () If he sysems ( T α, Ʈ )s a sacked-sysem ( ype ), hen ( T α, Ʈ ) s a semgroup and called a sackedsemgroup Prove hs heorem earler n paper [] 20 Defnon [2] Le T α ={ x, x 2,, x n, x 2,, x 2n,, x nn } be called a sacked se f heelemens n T αaresackedn erms ofheplace(horzonally and vercally) 2 Defnon [2] If T α s a sacked sysem of elemen denoed genercally by x hen a fuzzy sacked sysem T μ n T α s a sysem of ordered pars: T μ = {( x, μ T (x)) x T α } μ T (x) s called he membershp funcon or grade of membershp ( also degree of compably or degree of ruh ) of x n T μ whch maps T α o he membershp space M (When M conans only he wo pons 0 and, T μ s nonfuzzy and μ T (x) s dencal o he characersc funcon of a nonfuzzy sacked se) The range of he membershp funcon s a subse of he nonnegave real numbers whose supremum s fne Elemens wh a zero degree of membershp are normally no lsed T μ = {( x, μ T (x)) x T } 22 Theorem [2] If T α s a sacked sysem, x, a T α : μ T (x)= ( + a x ) - μ T 2 (x)= ( + ( a x ) 2 ) - μ T n (x)= ( + ( a x ) n ) - are ypes of a funcon such ha μ T (x) [0, ], {, 2,, n } proof :le T α s a sacked sysem, x, a T α : a x 0 (+ a x ) 0 ( + a x ) - μ T (x)= ( + a x ) - [0, ] ( a x ) 2 0 (+( a x ) 2 ) 0 ( + ( a x ) 2 ) - μ T (x)= ( + ( a x ) 2 ) - [0, ], so μ T n (x)= ( + ( a x ) n ) - ) [0, ], {, 2,, n } 2 Theorem [2] If T α s a sacked sysem, x T α, μ T (a) μ T (x) hen : μ T (x)= ( + ( a x ) n ) -, n N/0,s convex funcon Proof : Le x < a < x 2,and x T α, μ T (a) μ T (x) hen : μ T (a) μ T (x ), and μ T (a) μ T (x 2 ) So :[μ T (a) μ T (x )] / [a x ]< 0 < [μ T (x 2 ) μ T (a)] / [x 2 a] From defnon 26 :[μ T (a) μ T (x )] / [a x ] [μ T (x 2 ) μ T (a)] / [x 2 a],hen μ T (x)= ( + ( a x ) n ) - s convex funcon 24 Defnon [2] If T α s a sacked sysem, hen he fuzzy sacked sysem T μ n T α s a sysem of ordered pars: T μ = {( x, μ T (x)) x T } Such ha : μ T (x) = ( + ( x a ) n ) -, n N/0 and x Tα

3 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December 204 ISSN Defnon [] 25 Defnon [2] For a fne fuzzy sacked se T μ he column(row) cardnaly \T μ ( C or R ) \ s defned as : 26 Defnon [2] \T μ ( C or R ) \ = x ( CorR ) μ T( C or R ) (x) Le (x) T μ, γ s a row and β s column hen he order fuzzy sacked of μ(x γβ) s : O(x) = μ(x γβ) =[ μ(x γβ)/ \T μ (γ ) \ + μ(x γβ)/ \T μ (β ) \ ]/2 27 Theorem [2] If T s a sacked sysem, x, a T : O(x) = μ(x γβ) =[ μ(x γβ)/ \T μ (γ ) \ + μ(x γβ)/ \T μ (β ) \ ]/2 s a ype of a funcon such ha O(x) [0, ], proof :from defnons (5), (7), (8) 0 μ(x γβ) \T μ (γ ) \ 0 / \T μ (γ ) \ μ(x γβ) / \T μ (γ ) \ \T μ (γ ) \ / \T μ (γ ) \ 0 μ(x γβ)/ \T μ (γ ) \ (A) And so 0 μ(x γβ) \T μ (β ) \ 0 / \T μ (β ) \ μ(x γβ) / \T μ (β ) \ \T μ (β ) \ / \T μ (β ) \ 0 μ(x γβ)/ \T μ (β ) \ (B) Then (A) +(B) 0+0 = 0 μ(x γβ)/ \T μ (γ ) \ + μ(x γβ)/ \T μ (β ) \ + =2 0 [μ(x γβ)/ \T μ (γ ) \ + μ(x γβ)/ \T μ (β ) \]/ 2 O(x) ) [0, ] 28 Defnon [2] If T α s a sacked sysem, hen he fuzzy level sacked sysem l(t μ) n T α s a sysem : l(t μ) = {( x, μ T (x), O(x)) x T α } If à s a collecon of objecs denoed genercally by x hen a fuzzy se A n à s a se of ordered pars: à = {( x, μ à (x)) x T } μ à (x)s called he membershp funcon or grade of membershp (also degree of compably or degree of ruh) of x n A whch maps X o he membershp space M (When M conans only he wo pons 0 and, à s nonfuzzy and μ à (x) s dencal o he characersc funcon of a nonfuzzy se) The range of he membershp funcon s a subse of he nonnegave real numbers whose supremum s fne Elemens wh a zero degree of membershp are normally no lsed 22 Defnon [] The suppor of a fuzzy se Ã, S(Ã), s he crsp se of all x X such ha : μ à (x) > Defnon [] The (crsp) se of elemens ha belong o he fuzzy se A a leas o he degree a s called he α - level se : A α = {x X μ à (x) } Ã= {x X μ à (x) >α } s called "srong α - level se" or "srong α -cu" 22 Defnon [] A fuzzy se à s convex f : μ à ( λx + ( λ )x 2 ) mn(μ à (x ), μ à (x 2 )), x, x 2 X, λ [ 0, ] 29 Defnon [2] Le l(t μ) n T α s a fuzzy level sacked sysem [l(t μ)={(x, μ T (x), O(x))x T α } ], hen : max [O(T)] = { x, x 2,, x n } Tha's where : Rx, C max[o(t)] = x, max[o(t)/{ x }] = x2,, max[o(t)/{ R, C, R, C,, R, C }]= x n, x x x2 x2 xn xn C mean column and R mean row,and f x β max (O(T)), hen R β max (O(T))= {x β}, and C β max (O(T))= {x β},c = R = Max (O(T)) = n 224 Defnon [4] Fgures Convex fuzzy se By groupod ( S, * ) we shall mean a non-empy se S on whch a bnary operaon * s defned Tha s o say, we have a mappng *: S S S We shall say ha ( S, * ) s a semgroup f * s assocave, e f ( x, y, z S), ((x, y)*, z)* = (x, (y, z)*)* 225 Defnon [4] S s a fne semgroup f has only a fnely many elemens

4 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN Defnon [4] A commuave semgroups a semgroup S wh propery : ( x, y S) ( xy = yx ) 227 Defnon [4] If here exss an elemen of S such ha ( x S) x = x = x We say ha s an deny (elemen) of S and ha S s a semgroup wh deny 228 Defnon [5] Le S be a se and σ : S S S a bnary operaon ha maps each ordered par (x, y) of S o an elemen σ(x, y) of S The par (S, σ) (or jus S, f here s no fear of confuson) s called a groupod 229 Defnon [6] We denoe by L[a] (R[a], J[a], I[a], Q[a], B[a]) he prncpal lef (rgh, wo-sded, neror, quas-, b-) deal of a semgroup S generaed by he elemen a S, ha s, L[a] = {a} Sa, R[a] = {a} as, J[a] = {a} Sa as SaS, I[a] = {a} {a 2 } SaS, Q[a] = {a} ( as Sa ), B[a] = {a} {a 2 } asa [0] 20 Defnon [7] A real funcon f defned on a real nerval I s convex on I ff: x,x 2,x I:x <x 2 <x :[f(x 2 ) f(x )] / [x 2 x ] [f(x ) f(x 2 )] / [x x 2 ] or: x,x 2,x I:x <x 2 <x :[f(x 2 ) f(x )] / [x 2 x ] [f(x ) f(x )] / [x x ] The funcon f s srcly convex on I f, n he above nequales, equaly canno hold 2 Defnon [8] For a se A,we defne a membershp funcon μ A such as ff x A A ( x ) 0 ff x A ( ff s shor for f and only f ) X be a classcal se of objecs, called he unverse, whose generc elemens are denoed x Membershp n a classcal subse A of X s ofen vewed as a characersc funcon, μ A from X o {0,} such ha : μ A : X { 0, } 22 Axoms for Unon Funcon [8] In general sense, unon of A and B s specfed by a funcon of he form μ A B (x)= [ 0,] [0,] [0,] Ths unon funcon calculaes he membershp degree of unon A B from hose of A and B μ A B (x) = [μ A (x), μ B (x)] hs unon funcon should obey nex axoms (Axom U) (0,0) = 0, (0,) =, (,0) =, (,) = so hs unon funcon follows properes of unon operaon of crsp ses (boundary condon) (Axom U2) (a, b) = (b, a) Commuavy holds (Axom U) If a a and b b, (a, b) (a, b ) Funcon s a monoonc funcon (Axom U4) ( (a, b), c) = (a, (b, c)) Assocavy holds The above four saemens are called as axomac skeleon I s ofen o resrc he class of fuzzy unons by addng he followng axoms (Axom U5) Funcon s connuous (Axom U6) (a, a) = a (dempoency) 25 Some Algebrac Operaons [8] () Probablsc sum A B (Algebrac sum) Fuzzy unon A B s defned as, x X, μ A B (x) = μ A (x) + μ B (x) - μ A (x) μ B (x) I follows commuavy, assocavy, deny, and De Morgan s law Ths operaor holds also he followng : A X = X (2) Bounded sum A B (Bold unon) x X, μ A B (x) = Mn[, μ A (x) + μ B (x)] Ths operaor s dencal o Yager funcon a w = Commuavy, assocavy, deny, and De Morgan s Law are perfeced, and has relaons, A X = X A A = X bu does no dempoency and dsrbuvy a absorpon () Drasc sum A B Drasc sum s defned as follows: A( x ), when B( x ) 0 x X, ( x ) B( x ), when A( x ) 0 A B, for ohers (4) Hamacher s sum A B

5 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN A ( x ) B ( x ) (2 ) A ( x ) B ( x ) x X, AB( x ), 0 ( ) ( x) ( x) (5) Algebrac produc A B (probablsc produc) x X, μ A B (x) = μ A (x) μ B (x) Operaor s obeden o rules of commuavy, assocavy, deny, and De Morgan s law (6) Bounded produc A B (Bold nersecon) Ths operaor s defned as, x X, μ A B (x) = Max[ 0, μ A (x) + μ B (x) ] and s dencal o Yager nersecon funcon wh w =, I (a, b) = Mn[, 2 - a b] commuavy, assocavy, deny, and De Morgan s law hold n hs operaor The followng relaons A = A Ᾱ = are also sasfed, bu no dempoency, dsrbuably, and absorpon 26 Defnon [9] The mahemacal sysems s a se of neracng or nerdependen componens formng an negraed whole or a se of elemens (ofen called 'componens' ) and relaonshps whch are dfferen from relaonshps of he se or s elemens o oher elemens or ses Example: ( R, + ) s a sysem, R s a se of all he reel numbers, and ( + ) he relaon beween he elemens 27 Defnon [7] A bnary operaon on a se S s a mappng of he Caresan produc S S no S 28 Defnon [0] Le X be a space of pons (objecs),wh a generc elemen of X denoed by x Thus, X = {x} A fuzzy se (class) A n X s characerzed by a membershp (characersc) funcon f A (x) whch assocaes wh each pon n X a real number n he nerval [0, ], wh he value of f A (x) a x represenng he grade of membershp of : x n A Thus, he nearer he value of f A (x) o uny, he hgher he grade of membershp of 0 n A When A s a se n he ordnary sense of he erm, s membershp funcon can ake on only wo values 0 and, wh f A (x) = or 0 accordng as x does or does no belong o A Thus, n hs case f A (x) reduces o he famlar characersc funcon of a se A (When here s a need o dfferenae beween such ses and fuzzy ses, he ses wh wo-valued characersc funcons wll be referred o as ordnary ses or smply ses) A B III BASICCONCEPTSIN THE STACKED FUZZY SET Theorem If T α be a sacked se hen he se of ordered pars : T μ() = {( x γ, μ (x γ)) x γ T α, μ (x γ) = x γ } s a fuzzy sacked se Proof : We only need o prove haμ (x γ) [ 0, ],and ha's enough o prove T μ() s a fuzzy sacked se From he defnon of he sacked se, x γ T, 0 x x 2 Example 0 x γ 0 xγ x + 2 x x x + + ( o α mes )= α x x 0 x [ ] / x 0 x γ Then x γ = μ (x γ) [ 0, ] If here are hree dsrbuon ceners,consumer producsofheype(a,b, C), whereheyareransporedosales ceners(x, Y, Z)aacos, asnhefollowngable: Table X Y Z A 2 B 4 5 C 7 ++ So heresaprocess of ransferrngbeween(a, B, C) and (X, Y, Z) Transporaon beweenaandxcos, so cos ( A,X) = cos ( A,Y) = 2 cos ( A,Z) = 2 2

6 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN cos ( B,X) = cos ( B,Y) = cos ( B,Z) = 2 cos ( C,X) = 7 7 cos ( C,Y) = 2 cos ( C,Z) = And T 2, ( Non-Order )= {, 2, 2, 4 2, 5 22, 2, 7, 2, } In hs example, f requredheransferofconsumer producs ahelowescos, so ha heransferofa sngle producfrom each sngle dsrbuoncenero he sngle cener ofhesale,we godreclyo : Mn 0 {cos[ (A,B,C ),(X,Y,Z)]} = {, 2, 2 } Bu whenheaverageransfer requesarelmedoaspecfcvalue, we denfyhrough : T () = {( x, μ (x)) x T α } Now supposeha he averageransporaonnendedfordsrbuon unsof producs, fromeachdsrbuon ceneroeach cenersale μ (x γ) = a γ = x [ ] / x If x a =, hen we suppose ha x a =, and where x, a {, } hen, we compensae x a = 0 - T,2 = Table4 when O ( T,2 ) = { 2, 2, 4 2, 2,, 2,, 7, 5 22 } so T μ() = { ( 2,0 0 ), ( 2, ), (4 2, 07), (2, 67 ), (, 067 ), ( 2, 0400 ), (, 0476 ), ( 7, 069 ), ( 5 22, 0700 ) } Then f we have denfedaverageransporaonexen uns, mos suableconducfor ransporare: Mn {cos[ (A,B,C ),(X,Y,Z)]} = {( 2,0 0 ), (4 2, 07), ( 2, 67 )} Example In example 2, f requredheransferofconsumer producs ahelowescos, so ha heransferofa sngle producfrom each sngle dsrbuoncenero he sngle cener ofhesale,we godreclyo : Mn 0 {cos[ (A,B,C ),(X,Y,Z)]} = {, 2, 2 } Bu whenheaverageransfer requesslmedoaspecfcvalue, we denfyhrough: T μ = {(x, μ T (x)) x T} Now supposeha he averageransporaonnendedfordsrbuon unsof producs, fromeachdsrbuon ceneroeach cenersale And ha does no exceed6 uns forransporaonproducs Table2 2- T,2 [ x γ ] = Table - T,2 [ μ T (x γ)] = T,2 [ a γ ] = x T,2 [ [ ] / x ] = So: μ T (x) = Then μ T ( ) = ( + ( ) 2 ) - =, μ T ( 2 )=,, μ T (7 )= 0, when T,2 = {, 2, 2, 4 2, 5 22, 2, 7, 2, 2 } (non-order ) So: T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), ( 2, ) } Or T μ = 0 : x > 6 ( + ( x ) 2 ) - : 0 x Table5

7 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN Then f we have denfedaverageransporaonexen uns,mos suableconducfor ransporare: Max {cos[ (A,B,C ),(X,Y,Z)]} = {( 2, ),( 4 2, 05 ), ( 2, 05 )} I has already been menoned ha he membershp funcon s no lmed o values beween 0 and lf sup x μ T (x) = he fuzzy sacked se T μ s called normal A nonempy fuzzy sacked se T μ can always be normalzed by dvdng μ T (x) by sup x μ T (x): As a maer of convenence we wll generally assume ha fuzzy sacked ses are normalzed For he represenaon of fuzzy sacked ses we wll use he noaon llusraed n example above respecvely A fuzzy sacked se s obvously a generalzaon of a classcal se and he membershp funcon a generalzaon of he characersc funcon Snce we are generally referrng o a unversal (crsp) se T some elemens of a fuzzy se may have he degree of membershp zero Ofen s approprae o consder ha elemen of he unverse whch has a nonzero degree of membershp n fuzzy sacked se 4 Defnon The suppor of a fuzzy sacked se T μ (), S (T μ () ), s he crsp se of all x T such ha μ (x) > 0 5 Example In example above S(T μ) = {, 2, 2, 4 2, 5 22, 2, 2, 2 } The elemen {7 } s no par of he suppor of T μ 6 Defnon The (crsp) se of elemens ha belong o he fuzzy sacked se T μ a leas o he degree α level sacked se: T α = {x T μ T (x) α} T α = {x T μ T (x) >α} s called (srong α level sacked se or srong α cu) 7 Example: The ls possble α level sacked se n he above example: T = {, 2, 2, 4 2, 5 22, 2, 2, 2 }, T 05 = { 2, 2, 4 2, 2, 2 }, T 0 = { 2, 2 } The srong α level sacked se for α = 05 s T 05 = { 2, 2 } Convexy also plays a role n fuzzy se heory By conras o classcal se heory, however, convexy condons are defned wh reference o hen membershp funcon raher han he suppor of a fuzzy se 8 Example Supposeha he averageransporaonnendedfordsrbuon unsof producs, fromeachdsrbuon ceneroeach cenersale And ha does no exceed6 uns forransporaonproducs So: μ T (x) = when T,2 = {, 2, 2, 4 2, 5 22, 2, 7, 2, 2 } So: T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), ( 2, ) } The cardnaly s: T μ = μ T (x) = xx + = 8 Is relave cardnaly s: T μ = 8 9 = 0422 IV THEORETIC OPERATIONS FOR FUZZY STACKED SETS The membershp funcon s obvously he crucal componen of a fuzzy se I s herefore no surprsng ha operaons wh fuzzy ses are defned va her membershp funcons 4 Defnon The membershp funcon μ CT (x) of he nersecon C T = A T B T s pon wse defned by: μ CT (x) = mn {μ AT (x), μ BT (x) } Such ha: f a μ AT (x) and b μ BT (x) a : f a < b b : f b < a If a = b ( one order elemen n wo dfferen places)so we have many ype of hs sysem, and f a b he sysem s ype- 42 Example Supposeha he averageransporaonnendedfordsrbuon unsof producs, fromeachdsrbuon ceneroeach cenersale And ha does no exceed6 uns forransporaonproducs: μ T (x) = Mn (a, b )= 6 0 : x > 6 ( + ( x ) 2 ) - : 0 x 6 0 : x > 6 ( + ( x ) 2 ) - : 0 x when T,2 = {, 2, 2, 4 2, 5 22, 2, 7, 2, 2 } so T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), ( 2, ) } Or:

8 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN T μ = Table6 In he same se T,2 = {, 2, 2, 4 2, 5 22, 2, 7, 2, 2 } Assume ha hedsrbuonofproducs fromalldsrbuon ceners,opons of saleahe largescosand akngnoaccounha he bggescosfor dsrbuons7 uns So: μ T2 (x) = ( + (x 7) 2 ) - : 0 x 7 T μ2 = { (,0027 ), ( 2, ), ( 2, ), ( 4 2, 0 ), ( 5 22, ), ( 2, 0027 ), ( 7, ), ( 2, ), ( 2, 0027 ) } Or: T μ2 = Table7 he order elemen on sacked sysem T μ,where he sysem lookng for zero convergence or zero spacng, s amoun conrbues o hs elemen n he sysem, and hs esmae s calculaed relaonshp of hs elemen n every pah ha conans hs elemen, hen he elemen order of a γ, ( O(a γ ) ) : a aγ 0 = [ ] / a Then: (0027 ) 0 = [(0027 / ( )) + (0027 / ( ))] = / 2 And so: Tγ2 [ aγ 0 ]= Table8 And Tγ [ aγ 0 ] = Table μ T(x) μ T2(x) = mn { μ T (x), μ T2(x) } = { (, ), ( 2, 9 ), ( 2, 0627 ), ( 4 2, 097 ), ( 5 22, 0566 ), ( 2, 0082 ), ( 7,0 ), ( 2, 0966 ), ( 2, 058 ) } 4 Defnon The membershp funcon DT(x) of he unon D T = A T B T s pon wse defned by μ DT(x) = max { μ AT(x), μ BT(x) } Such ha: f a μ AT(x) and b μ BT(x) a : f a > b Max b : f (a b, b > ) = a f a = b ( one order elemen n wo dfferen places)so we have many ype of hs sysem, and f a b he sysem s ype- 44 Example From example above Tμ (,2) [ aγ 0 ] = Table0 Tμ 2(,2) [ aγ 0 ] = Table μ T(x) μ T2(x) = max { μ T (x), μ T2(x) } = { (,0 ), ( 2, 052 ), ( 2, 0425 ), ( 4 2, 0649 ), ( 5 22, 062 ), ( 2, 22 ), ( 7,0904 ), ( 2, 0644 ), ( 2, 094 ) } 45 Defnon The membershp funcon of he complemen of a fuzzy sacked se Tμ, μ c T(x) s defned by μ c T(x) = μ T (x), x T 46 Example Le : Tμ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), ( 2, ) } T c μ = { (,08 ), ( 2, 0 ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, 08 ), ( 2, 08 ), ( 7, ), ( 2, 0 ), ( 2, 08 ) } Addonal operaons on fuzzy sacked se :

9 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN I has already been menoned ha mn and max are no he only operaors ha could have been chosen o model he nersecon or unon of fuzzy sacked ses respecvely The queson arses, why hose and no ohers! from a logcal pon of vew, nerpreng he nersecon as logcal and," he unon as logcal or," and he fuzzy sacked se Z as he saemen The elemen x belongs o se Z can be acceped as more or less rue l s very nsrucve o follow her lne of argumen, whch s an excellen example for an axomac jusfcaon of specfc mahemacal models We shall herefore skech her reasonng: Consder wo saemens, S and T, for whch he ruh values are μ S and μ T, respecvely, μ S, μ T [0, ] The ruh value of he and and "or" combnaon of hese saemens, μ(s and T) and μ(s or T), boh from he nerval [0, ] are nerpreed as he values of he membershp funcons of he nersecon and unon, respecvely, of S and T We are now lookng for wo real-valued funcons f and g such ha μ S and T = f(μ S, μ T ) μ S or T = g(μ S, μ T ) Tha he followng resrcons are reasonably mposed on f and g: - f and g are non decreasng and connuous n μ S and μ T - f and g are symmerc, ha s, f(μ S, μ T ) = f(μ T, μ S ) - f(μ S, μ S ) and g(μ S, μ S ) are srcly ncreasng n μ S v- f(μ S, μ T ) mn (μ S, μ T ) and g(μ S, μ T ) max(μ S, μ T ) Tha mples ha accepng he ruh of he saemens S and T" requres more, and accepng he ruh of he saemen S or T" less han accepng S or T alone as rue v- f(, )= and g(0, 0)= 0 v- Logcally equvalen saemens mus have equal ruh values and fuzzy ses wh he same conens mus have he same membershp funcons, ha s, S and ( S 2 or S ) s equvalen o ( S and S 2 ) or ( S and S ) and herefore mus be equally rue Usng he symbols for and (= nersecon) and for or (= unon), hs amouns o he followng 7 resrcons, o be mposed on he wo commuave (see ()) and assocave (see (v)) bnary composons and on he closed nerval [0, ] whch are muually dsrbuve (see (v)) wh respec o one anoher - μ S μ T = μ T μ S μ S μ T = μ T μ S 2- (μ S μ T ) μ U = μ S ( μ T μ U ) ( S μ T ) μ U = μ S ( μ T μ U ) - μ S ( μ T μ U ) = ( μ S μ T ) (μ S μ U ) μ S ( T μ U ) = ( μ S μ T ) (μ S μ U ) 4- μ S μ T and μ S μ T are connuous and non decreasng n each componen 5- μ S μ T and μ S μ T are srcly ncreasng n μ S (see ()) 6- μ S μ T mn(μ S, μ T ) μ S μ T max( μ S, μ T ) (see (v)) 7- = 0 0 = 0 (see (v)) μ S T = mn(μ S, μ T ) and μ S T = max(μ S, μ T ) For he complemen would be reasonable o assume ha f saemen S s rue, s complemen non S" s false, or f μ S = hen μ nons = 0 and vce versa The funcon h (as complemen n analogy o f and g for nersecon and unon) should also be connuous and monooncally decreasng and we would lke he complemen of he complemen o be he orgnal saemen (n order o be n lne wh radonal logc and se heory) 48 Defnon The Caresan produc of fuzzy sacked ses s defned f T μ, T μ2,, T μn be fuzzy sacked ses n T, T 2,, T n Then he Caresan produc of fuzzy sacked ses n he produc space T T 2 T n, and he membershp funcon f δ = T μ T μ2 T μn s : μ δ(x) = mn [μt μ ( x ) x = ( x, x 2,, x n ), x T ] 49 Example T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), (, ) } T μ2 = { (,0027 ), ( 2, ), ( 2, ), ( 4 2, 0 ), ( 5 22, ), ( 2, 0027 ), ( 7, ), ( 2, ), (, 0027 ) } T μ T μ2 = { [( ; ),0027 ], ([( ; 2 ), ], [( ;2 ), ], [( ;4 2 ), 0 ], [ ( ;5 22 ), ], [( ; 2 ), 0027 ], [( ;7 ), ],[ ( ; 2 ), ], [( ; ), 0027 ], [( 2 ; ),0027 ], ([( 2 ; 2 ), ], [( 2 ;2 ), ], [( 2 ;4 2 ), 0 ], [ ( 2 ;5 22 ), ], [( 2 ; 2 ), 0027 ], [( 2 ;7 ), ],[ ( 2 ; 2 ), ], [( 2 ; ), 0027 ], [(2 ; ),0027 ], ([(2 ; 2 ), ], [(2 ;2 ), ], [(2 ;4 2 ), 0 ], [ (2 ;5 22 ), ], [(2 ; 2 ), 0027 ], [(2 ;7 ),05 ],[ (2 ; 2 ), ], [(2 ; ), 0027 ], [(4 2 ; ),0027 ], ([(4 2 ; 2 ), ], [(4 2 ;2 ), ], [(4 2 ;4 2 ), 0 ], [ (4 2 ;5 22 ), ], [(4 2 ; 2 ), 0027 ], [(4 2 ;7 ),05 ],[ (4 2 ; 2 ), ], [(4 2 ; ), 0027 ], [(5 22 ; ),0027 ], ([(5 22 ; 2 ), ], [(5 22 ;2 ), ], [(5 22 ;4 2), 0 ], [ (5 22 ;5 22 ), ], [(5 22 ; 2 ), 0027 ], [(5 22 ;7 ),

10 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN ],[ (5 22 ; 2 ), ], [(5 22 ; ),0027 ], [( 2 ; ),0027 ], ([( 2 ; 2 ), ], [( 2 ;2 ), ], [( 2 ;4 2 ), 0 ], [ ( 2 ;5 22 ), ], [( 2 ; 2 ), 0027 ], [( 2 ;7 ), ],[ ( 2 ; 2 ), ], [( 2 ; ), 0027 ], [(7 ; ),0], ([(7 ; 2 ), 0], [(7 ;2 ), 0], [(7 ;4 2 ), 0 ], [ (7 ;5 22 ), 0 ], [(7 ; 2 ), 0], [(7 ;7 ),0 ],[ (7 ; 2 ), 0], [(7 ; ), 0], [( 2 ; ),0027 ], ([( 2 ; 2 ), ], [( 2 ;2 ), ], [( 2 ;4 2), 0 ], [ ( 2 ;5 22 ), ], [( 2 ; 2 ), 0027 ], [( 2 ;7 ), ],[ ( 2 ; 2 ), ], [( 2 ; ), 0027 ], [( ; ),0027 ], ([( ; 2 ), ], [( ;2 ), ], [( ;4 2 ), 0 ], [ ( ;5 22 ), ], [( ; 2 ), 0027 ], [( ;7 ), ],[ ( ; 2 ), ], [( ; ), 0027 ]} 40Defnon The nh power of a fuzzy sacked se T s a fuzzy sacked se wh he membershp funcon (Tμ) n (x) = [ μt (x) ] n, x T μ 4Example T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), (, ) } [T μ ] 2 = { (,004 ), ( 2, ), ( 2, 5 ), ( 4 2, 5 ), ( 5 22, 004 ), ( 2, 004 ), ( 7, 0 ), ( 2, ), (, 004 ) } 42Defnon The algebrac sum ( probablsc sum ) T μa + T μb s defned as : T μa + T μb = { ( x, μ (Tμa+Tμb) (x)) x T μ } Where : μ (Tμa+Tμb) (x)= μ (Tμa) (x) + μ (Tμb) (x) - μ (Tμa) (x) μ (Tμb) (x) 4Example T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), (, ) } T μ2 = { (,0027 ), ( 2, ), ( 2, ), ( 4 2, 0 ), ( 5 22, ), ( 2, 0027 ), ( 7, ), ( 2, ), (, 0027 ) } T μ + T μ2 = { (,26 ), ( 2, ), ( 2, 0592 ), ( 4 2, 055 ), ( 5 22, 06 ), ( 2, 26 ), ( 7, ), ( 2, ), (, 26 ) } 44Defnon The bounded sum T μa T μb s defned as T μa T μb = { ( x, μ (Tμa Tμb)(x)) x T μ } Where μ (Tμa Tμb)(x) = mn{, μ (Tμa) (x) + μ (Tμb) (x)} 45Example T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), (, ) } T μ2 = { (,0027 ), ( 2, ), ( 2, ), ( 4 2, 0 ), ( 5 22, ), ( 2, 0027 ), ( 7, ), ( 2, ), (, 0027 ) } T μ T μ2 = { (, 27), ( 2, ), ( 2, 05846), ( 4 2, 06 ), ( 5 22, 04 ), ( 2, 27), ( 7, ), ( 2, ), (, 27) } 46Defnon The bounded dfference T μa T μb s defned as T μa T μb = { ( x, μ (Tμa Tμb) (x)) x T μ } Where μ (Tμa Tμb) (x) = max{0, μ (Tμa) (x) + μ (Tμb) (x)-} 47Example T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), (, ) } T μ2 = { (,0027 ), ( 2, ), ( 2, ), ( 4 2, 0 ), ( 5 22, ), ( 2, 0027 ), ( 7, ), ( 2, ), (, 0027 ) } T μ T μ2 = { (, 0), ( 2, ), ( 2, 0 ), ( 4 2, 0 ), ( 5 22, 0 ), ( 2, 27), ( 7,0 ), ( 2, ), (, 0 ) } 48Defnon = { ( 2, ), ( 2, )} The algebrac produc of ow fuzzy sacked ses T μa T μb s defned as T μa T μb = { ( x, μ (Tμa) (x) μ (Tμb) (x)) x T μ } 49Example T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), (, ) } T μ2 = { (,0027 ), ( 2, ), ( 2, ), ( 4 2, 0 ), ( 5 22, ), ( 2, 0027 ), ( 7, ), ( 2, ), (, 0027 ) } T μ T μ2 = { (, 00054), ( 2, ), ( 2, 0092 ), ( 4 2, 005 ), ( 5 22, 004 ), ( 2, 00054), ( 7,0 ), ( 2, ), (, ) } V FUZZY STACKED SEMIGROUPS 5 Defnon Le T α,n be saked se,and (T α,n, τ) be a sacked semgroup, hen a map μ: T α,n [0, ] s called a fuzzy sacked semgroup f μ(x τ y) = mn {μ(x), μ(y)} for all x, y T α,n 52 Example

11 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December 204 ISSN From example above, suppose ha he dsrbuon ofproducs from alldsrbuon ceners,o hepon of salealesscosand akng no accounhahebggescosofdsrbuons7uns And : μ T (x) = ( + ( x 7 ) 2 ) - : 0 x 7 when T,2 = {, 2, 2, 4 2, 5 22, 2, 7, 2, 2 } and so : T μ (,2) [ a γ 0 ] = and so : Table or T,2 = lt μ (,2) [ a γ 0 ] = Table2 Table6 lt,2 ( ) =, lt,2 ( 2 ) = 6, lt,2 ( 4 2 ) = 5, lt,2 ( 5 22 ) = 8, lt,2 ( 7 ) = 9, lt,2 ( 2 ) = 7 lt,2 = Table and mn ( 4 2, 2 ) = 4 2, mn (, 7 ) =, mn ( 5 22, 2 ) = 5 22 from, μ T (x) = ( + ( x 7 ) 2 ) - : 0 x 7 T μ = { (,097 ), ( 2, 094 ), ( 2, 096 ), ( 4 2, 09 ), ( 5 22, 08 ), ( 2, 0 97 ), ( 7,0 ), ( 2, 094 ), ( 2, 0 97 ) } Or a γ = [ = Tμ= a a + a [ ] / a Table4 a a a a = [ ( 097 / ( )) + (097 / ( ))] ]/α lt μ (,2) [ a γ 0 ] = lt,2 When μ T ( 4 2 τ 2 ) = μ T [ mn (4 2, 2 )] = μ T (4 2 ) = 09 And, mn { μ T (4 2 ), μ T ( 2 )} = mn { 09, 094 } = 09 Then: μ T ( 4 2 τ 2 ) = mn { μ T (4 2 ), μ T ( 2 )} And x, y T,2, μ(x τ y) = mn {μ(x), μ(y)}, hen (T 2, τ ) s fuzzy sacked semgroup 5 Theorem Le ( T α,n, τ ) be a sacked semgroup, hen a map μ: T α,n [0, ] s called a fuzzy sacked semgroup f lt α = l(t μ) or lt = l(-t μ) Proof: Le a, b T α,n, where (T α,n, τ ) be a sacked semgroup, and μ: T α,n [0, ] Then mn (a, b ) = a f a < b or b f a < b So, mn (a, b ) corresponds mn[ a, b ] hen here s corresponds beween mn [μ T (a), μ T (b)] and mn[ a, b ] from: mn [a μ, b μ] = a μ f a μ < b μ or b μ f a μ < b μ When T α,n = T μ lt α,n = l(t μ) = r ( from he corresponds ) so lt α,n = {, 2, r } and lt μ = {, 2, r } lt α,n = l(t μ) Bu somemes μ T (x) s defne a max operaons so s corresponds l(-t μ) hen lt α,n = l(-t μ) or lt α,n = l(t μ) Le T α,n be a sacked semgroup A funcon f from T α,n o he un nerval [0, ] s a fuzzy sacked subse of T α A sacked semgroup T α self s a fuzzy sacked subse of T α such ha T α (x) = for all x T α denoed by T α Le μ and δ be any wo fuzzy sacked subses of T Then he ncluson relaon μ δ s defned by μ(x) δ(x) for all x T ( μ) s a fuzzy sacked subse of T α defned for all x T α 54 Defnon

12 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December ISSN ( μ)(x) = μ(x), μ δ and μ δ are fuzzy sacked subses of T defned by : (μ δ)(x) = mn {μ(x), δ(x)}, (μ δ)(x) = max {μ(x), δ(x)} for all x T α The produc μ δ s defned as follows: sup{mn[{ ( y), ( z)}]} x(μ yz δ)(x) = 0 f x s no expressble as x=yz ' ' s an assocave operaon 55 Example T μ = { (, ), ( 2, ), ( 2, 05 ), ( 4 2, 05 ), ( 5 22, ), ( 2, ), ( 7, 0 ), ( 2, ), (, ) } T μ2 = { (,0027 ), ( 2, ), ( 2, ), ( 4 2, 0 ), ( 5 22, ), ( 2, 0027 ), ( 7, ), ( 2, ), (, 0027 ) } 0 =[(/ (++2))+(/(+4+7))]/ 2 = = = = = = = = = T α = T α = Mn 0 [, T α ] =, so - =, 2 =, 2 =, 4 2 =, 2 =, 2 =, 5 22 =, 7 = =, 2 2 = 2, = 2, 2 2 = 2, 2 2 = 2, = 2, 2 7 = = 2, = 2, 2 2 = 2, 2 2 = 2, = 2, 2 7 = =4 2, =4 2, =4 2, =4 2, = = 2, 2 2 = 2, = 2, 2 7 = =2, =2, 2 7 = =5 22, = =7 sup{mn[{ ( y), 2( z)}]} x yz (μ μ 2 )(x) = 0 f x s no expressble as x=yz (μ μ 2 )( ) = sup { mn [ μ ( ), μ 2 ( )], mn [ μ ( ), μ 2 ( 2 )], mn [μ ( ), μ 2 ( 2 )], mn [μ ( ), μ 2 (4 2 )], mn [μ ( ), μ 2 ( 2 )], mn [μ ( ), μ 2 ( 2 )], mn [μ ( ), μ 2 (5 22 )], mn [μ ( ), μ 2 (7 )]} = sup{ 0027, 00588, 00846, 0, } = A fuzzy sacked subse f of T α s called a fuzzy sacked subsemgroup of T α f : f(xy) f(x) f(y), x,y T α 57 Defnon for all a, b T α, and s called a fuzzy sacked lef (rgh) deal of S f : f(ab) > f(b), (f(ab) > f(a)) 58 Defnon - Le T α be a sacked semgroup Le A and B be subses of T α Then mulplcaon of A and B s defned as follows: AB = { ab T α a A and b B} - A nonempy subse A of T α s called a sacked subsemgroup of T α f AA A - A nonempy sacked subse A of T α s called a lef (rgh) sacked deal of T α f T α A A (A T α A) Furher, A s called a wo-sded sacked deal of T α f s boh a lef and a rgh sacked deal of T α - A nonempy sacked subse A of T α s called an neror sacked deal of T α f T α A T α A, and a quas-sacked deal of T α f A T α T α A A A sacked subsemgroup A of T α s called a sacked b-deal of T α f A T α A A A nonempy subse A s called a generalzed sacked bdeal of T α f A T α A A - A semgroup T α s called regular f for each elemen a of T α, here exss an elemen x T α such ha a = axa 59 Defnon We denoe by L [a] (R [a], J [a], I [a], Q [a], B [a]) he prncpal ( lef, rgh, wo-sded, neror, quas- and b-) deal of a sacked semgroup S generaed by he elemen a T α, ha s, - L [a] = {a} T α a, 2- R [a] = {a} a T α, - J [a] = {a} T α a a T α T α a T α, 4- I [a] = {a} {a 2 } T α a T α, 5- Q [a] = {a} ( a T α T α a ), 6- B [a] = {a} {a 2 } a T α a 50 Defnon

13 Inernaonal Journal of Scenfc and Research Publcaons, Volume 4, Issue 2, December 204 ISSN A fuzzy sacked subse f of a sacked semgroup T α s called a fuzzy sacked b-deal of T α f : F(xyz) f(x) f(z), for all x, y, z of T α 5 Defnon A fuzzy sacked subse f of T α s called a fuzzy sacked neror deal of T α f f(xay) f (a) for all x, a and y of T α 52 Defnon A fuzzy sacked subse f of a sacked semgroup T α s called a fuzzy sacked quas-deal of T α f (f T α ) (T α f) f 5 Defnon A nonempy sacked subse A of a sacked semgroup T α s called a generalzed sacked b-deal of T α f A T α A A A fuzzy sacked subse f of T α s called a fuzzy sacked generalzed b-deal of T α f f(xyz) f(x) f(z) for all x, y and z of T α I s clear ha every fuzzy sacked b-deal of a sacked semgroup T α s a fuzzy sacked generalzed b-deal of T α, bu he converse of hs saemen does no hold n general REFERENCES [] Aymen A Ahmed Imam, M AlBshr, and Emad eldeen A A R, A se wh specal arrangemen and sem-group onanewsysemcalledhe sackedsysem, IJSRP, Volume 4, Issue 7, July 204 Edon [ISSN ], hp:///researchpaper-074php?rp=p288 [2] Aymen A Ahmed Imam, M AlBshr, and Emad eldeen A A R, The Sacked Sem-Groups and Fuzzy Sacked Sysems on Transporaon Models, IJSRP, Volume 4, Issue 8, Augus 204 Edon [ISSN ], hp:///research-paper- 084php?rp=P22990 [] H-J Zmmermann, Fuzzy Se Theory-and Is Applcaons, Second, Revsed Edon, Kuwer Academc Publshers Second Prnng 99 [4] J M Howe, An Inroducon o Semgroup Theory, Academc Press INC London (976), ISBN: [5] John M Howe, fundamenals of semgroup heory, Clarendon Press Oxford (995) [6] John N Mordeson, Fuzzy Semgroups, Davender S Malk, Nobuak Kurok, ISBN : , ISBN: (ebook), Copyrgh Sprnger-Verlag Berln Hedelberg 200 [7] K G Bnmore, Mahemacal Analyss: A Sraghforward Approach, Cambrdge Unversy press 977, 982, ISBN paperback [8] Kwang HLee, Frs Course on Fuzzy Theory and Applcaons, Sprnger-Verlag Berln Hedelberg ( 2005 ), ISBN [9] S Axler, KA Rbe, Perre Anone Grlle Absrac Algebra Second Edon, Sprnger Scence + Busness Meda, LLC (2007), ISBN-: [0] Zadeh, L A, Fuzzy Ses, Inform Conrol, 8 (965) 8-5 AUTHORS Frs Auhor Aymen Abdelmahmoud Ahmed Imam PhD sudena he Unversy ofsudanof Scence and Technology Frs Emal aymnaaa@gmalcom Second Emal : aymnaaa@homalcom Second Auhor Professor: Mohmed AlBashr,Drecor of he Academyof Engneerng Scences, Dean of hefaculy of Graduae Sudeshe Unversy ofneelanearler Supervsedmanydocoral dsseraonsandsuccessfulwhglobal resonance Thrd Auhor Dr: Emad eldeen AARahm, head ofdeparmen of Mahemacs, Faculy of Scence, Unversy of SudanWonseveralnernaonal awardsnhe feld of mahemacs