On Fuzzy Spheres in Fuzzy Lobachevsky Space and its Retractions

Transcription

1 Ida Joural of Scece ad Techology O Fuzzy Spheres Fuzzy Lobachevsky Space ad ts Retractos A E El-Ahady * ad K Al-Oe 3 Matheatcs Departet Faculty of Scece Tabah Uversty Madah Saud Araba; a_elahady@hotalco 3 Matheatcs Departet Faculty of Scece Tata Uversty Tata Egypt Abstract Ths paper attepts to troduce ad study ew coecto betwee fuzzy retractos fuzzy foldgs ad fuzzy deforato retracts of ope fuzzy spheres fuzzy Lobachevsky space ad fuzzy ope ball fuzzy Eucldea space Types of fuzzy foldgs ad fuzzy deforato retracts of fuzzy ope sphere are dscussed Types of al fuzzy retractos of fuzzy ope sphere are preseted The fuzzy foldgs of fuzzy ope sphere s deduced also the lt of ths foldg s obtaed The coecto betwee fuzzy foldgs ad fuzzy cetral projecto ap s acheved; also the coecto betwee fuzzy retractos ad fuzzy expoetal ap s dscussed Soe applcatos are preseted Keywords Fuzzy Retractos Fuzzy Foldgs Fuzzy Deforato Retracts Cetral Projecto Map Fuzzy Ope Sphere Fuzzy Lobachevsky Space Itroducto ad Backgroud Lobachevsky space represets oe of the ost trgug ad ebleatc dscoveres the hstory of geoetry Although f t were troduced for a purely geoetrcal purpose they cae to proece ay braches of atheatcs ad physcs Ths assocato wth appled scece ad geoetry geerated syergstc effect appled scece gave relevace to Lobachevsky space ad Lobachevsky space Allowed foralzg practcal probles [ ] At each pot p of a coplete Reaa afold M we defe a appg of the taget space T p (M) at p oto M the followg aer If X s a taget vector at p we draw a geodesc g(t) startg at p the drecto of X If X has legth a the we ap X to the pot g(α)of the geodesc We deote ths appg by exp p T p (M) Æ M the ap exp p s everywhere C ad a eghborhood of p M ad t s a dffeoorphs [ 9] As s well kow the theory of retractos s always oe of terestg topcs Euclda ad No- Euclda spaces ad t has bee vestgated fro the varous vewpots by ay braches of topology ad dfferetal geoetry [4 6 9] There are ay dverse applcatos of certa pheoea for whch t s possble to get relevat data It ay ot be possble to easure essetal paraeters of a process such as the teperature sde olte glass or the hoogeety of a xture sde soe taks The requred easureet scale ay ot exst at all such as the case of evaluato of offesve sells evaluatg the taste of foods or edcal dagoses by touchg [9 4 7] The a of the preset paper s to descrbe the above pheoea geoetrcally specfcally cocered wth the study of the ew types of fuzzy retractos fuzzy deforato retracts ad fuzzy foldg of fuzzy ope sphere S () g fuzzy Lobachevsky space as preseted by El- Ahady [ ] A fuzzy afold s afold whch has a physcal character Ths character s represeted by the desty fucto μ where μ Œ [] [7 8] * Correspodg author A E El-Ahady (a_elahady@hotalco)

2 444 O Fuzzy Spheres Fuzzy Lobachevsky Space ad ts Retractos A fuzzy subset ( A ) of a fuzzy afold ( M ) s called a fuzzy retracto of ( M ) f there exst a cotuous ap ~ r ( M ) ( A ) such that r a a a a ( )= ( ) a Œ A ~ Œ [ ] [ 5] A fuzzy subset ( M ) of a fuzzy afold ( ) M s called a fuzzy deforato retract f there exsts a fuzzy retracto r ( M ) ( M ) ad a fuzzy hootopy F( M ) I ( M ) [ 5 6] such that F (( x ) ) = ( x ) x M F (( x )) = rx ( ) ~ F(( a )) t = ( a ) ( a ) M t I [ ] Where r( x ) ~ s the retracto etoed above Topologcal foldg of fuzzy sphere fuzzy Lobachevsky space s reported [ 3 ] A ap I S S s sad to be a soetrc foldg of fuzzy sphere fuzzy Lobachevsky space to tself ff for ay pecewse fuzzy geodesc path g J S the duced path I g J S s a pecewse fuzzy geodesc ad of the sae legth as γ where J = [] If ~ does ot preserve legths the ~ s a topologcal foldg of fuzzy sphere fuzzy Lobachevsky space [8 93 9] The sofuzzy foldg of M S s a foldg I M M such that I ( M ) = M ad ay M ~ belog to the upper hyperafolds M j dow M ~ such that μ = μ j for every correspodg pots e μ(a ) = μ(a j ) [6] (Fgure ) Ma Results Theore Let S L + be a fuzzy sphere a fuzzy Lobachevsky space whch s hoeoorphc to D R I S S be a fuzzy foldg the there s a duced fuzzy foldg I D D such that the followg dagra s coutatve S S D - D - Proof Let I S S be a fuzzy foldg The uder the cetral projecto ap b bs D there s a duced fuzzy foldg I D D such that b I=I b Theore Let S L + be a fuzzy sphere hoeoorphc to a fuzzy ope ball D R The for ay fuzzy M a M j a j Fgure The sofuzzy foldg of fuzzy afold foldg I S S there are duced fuzzy foldgs I I D D D D Proof Cosder a fuzzy sphere S L + where the fuzzy pot of S have a dfferet ebershp degree Uder the cetral projecto ap ay fuzzy o S duced fuzzy syste whch wll be fuzzy ope balls D D Also f I ( S ) = S The the sofuzzy foldg I D D Corollary The relato betwee the fuzzy expoetal ap ad fuzzy foldg of a fuzzy sphere S L + dscussed fro the followg coutatve dagra Proof Sce exp - S T p( S ) be the fuzzy expoetal ap of S the fuzzy foldg of S T p( S ) are gve by I + + S L S L I T ( S ) T p p( S ) Hece the - - followg dagra s coutatve e exp I=I exp Theore 3 The fuzzy foldg of fuzzy sphere S of fuzzy Lobachevsky space L + s dffeoorphc to fuzzy foldg of fuzzy ope dsk D ( p r) R Proof Cosder a fuzzy sphere S a fuzzy Lobachevsky space L + wth fuzzfcato S S I S S Let be the sofuzzy foldg Now let us cosder the cetral projecto ap b S R The ap b ~ takes S dffeoorphcally to the fuzzy ope dsk D ( p r) R of ceter p ad radus r = Uder the dffeoorphs betwee L + ad R there exsts a fuzzy ope dsk D ( p r) R wth fuzzfcato D ( p r) D ( pr ) ad there s a duced sofuzzy foldg b I D ( ) p r D ( pr ) M wwwdjstorg Vol 6 (4) Aprl 3 Ida Joural of Scece ad Techology Prt ISSN Ole ISSN

3 A E El-Ahady ad K Al-Oe 445 Theore 4 Let S L + whch s hoeoorphc to D R ad r S S be a fuzzy retracto ap the there s duced fuzzy retracto r D D such that the followg dagra s coutatve Proof Let + + r S L S L r D R D R D D are opes ball - Eucldea space Usg the cetral projecto ap b b S L + D b R S L + D R the the followg dagra s coutatve Ths s the geeral coutatve dagra such that I + = I r r Now uder the fuzzy cetral projecto ap b ~ there exsts a fuzzy ope dsk D ( p r R ) wth retractos D p r D 3 pr D 3 p r D ( pr Dpr ) ( 3) ( 3) ( ) ( ) where r - < r - < r -3 < < r 3 < r < r Also we obta the followg cha e r b = b r Theore 5 If r S L T S L r S exp ( ) p ( ) + L + S L - adexp S T p ( S ) The exp- r = exp r Proof Cosder the fuzzy expoetal ap exp - S T p( S ) ad the fuzzy retractos be defed as r S S r S S the there are duced coutatve dagras gve by Such that exp r = r exp Theore 6 Let S L + be a fuzzy sphere a fuzzy Lobachevsky space L + ad D ( p r R ) be a fuzzy ope dsk - Eucldea space R The ( f r r f + = ) = = b ( I r =I r ) + 3 Proof Cosder the fuzzy retracto S S S S S a fuzzy sphere S a fuzzy Lobachevsky space + L ad I S S I S S I S S I S S be a fuzzy foldgs of S The we get the followg cha Ths s the geeral coutatve dagra such that I r = r I + Theore 7 If I + + S L S L S T ( S exp p ) I ad T p( S ) T p S S D also b The ( exp exp - I=I = b( exp I=I exp - ) Proof Cosder the fuzzy expoetal ap exp - S T p( S ) the the duced coutatve dagra s defed as wwwdjstorg Vol 6 (4) Aprl 3 Ida Joural of Scece ad Techology Prt ISSN Ole ISSN

4 446 O Fuzzy Spheres Fuzzy Lobachevsky Space ad ts Retractos Such that exp I=I exp Aga cosder the fuzzy cetral projecto ap b ~ such that + b S L D R ad D R I D R I T ( D q ) T ( D ) q the the duced coutatve dagra s gve by Such that exp I=I exp Theore 8 Let S S be two fuzzy spheres a fuzzy Lobachevsky space S S L + The ay fuzzy retractos of {S - p} duces fuzzy retractos of T p( S ) T p( S oto ) Also uder the fuzzy cetral projecto ap + b bs L D R ay fuzzy retractos of {D - ( ) q} duces fuzzy retractos of T q( D ) to T ( D ) The q ( exp r = r exp ) = b( exp r = R exp ) Proof Cosder two fuzzy spheres S S ersed wth a coo pot p The at p the fuzzy taget spaces wll be overlapped Now let r μ{ S - p} μ Ø {S- ( ) p} μ < μ be a fuzzy retractos of the fuzzy physcal character of { S - p} to { S p} The there s a duced fuzzy retractos r T - p (μ{ Ø S- -p}) T - p(μ Ø Ø {S- ( ) p}) ad exp - r = r exp - Now uder the fuzzy cetral projecto ap bb S L + D R let r {D - ( ) q} μ Ø {D - ( ) q} be a fuzzy retractos of the fuzzy physcal character of {D - ( ) q} to { D q} The there s a duced fuzzy retractos r T - p(μ{ D- Ø ( ) q}) D - ( ) q}) T - p(μ Ø Ø {D- ( ) q}) such that exp - r = r exp - Theore 9 Let S L + be a fuzzy sphere fuzzy Lobachevsky space whch s hoeoorphc to D R The the fuzzy deforato retract of the fuzzy sphere a fuzzy Lobachevsky space S S oto duces the fuzzy deforato retract of D D oto Proof Now we defe the fuzzy deforato retract of S as follows F S p I S {( ) } {( p) } such that ( F{( )} t ( ) t ) = (-t) + where F {( ) } = ad F ( ) {( )} = Hece we ca duce the fuzzy deforato retract of D as follows F D q I D {( ) } {( q) } such that ( x ) F{( x )} t = (-t)( x ) + t where F {( x ) } = ( x ) x ad F ( x ) {( x )} = x Corollary Ay fuzzy deforato retract of μ o {S - p} duces fuzzy deforato retract of μ o{d - ( ) p} Corollary 3 Ay fuzzy deforato retract of μ o {S - p} duces fuzzy deforato retract of T - p(μ) of Ø {S - p} Theore If + b b S L D R S D be a fuzzy cetral projecto ap ad the fuzzy re tractos of S are gve by r S S r S S The b r r + = b Proof Cosder the fuzzy cetral projecto ap b b b S L + D R S D S D b S D ad r S S r S S r 3 S S r S S The there are duced coutatve dagras Such that b r r b + j = j Theore Let S L + be a fuzzy deforato retract of S L S S + ad I be a fuzzy foldg ) wwwdjstorg Vol 6 (4) Aprl 3 Ida Joural of Scece ad Techology Prt ISSN Ole ISSN

5 A E El-Ahady ad K Al-Oe Theore If ( S ) { P} r L s S the fuzzy retracto of ( S ) { P} r L ( ) s S + ad the fuzzy foldg of S S { P} The there are coutatve dagra betwee fuzzy retracto ad fuzzy foldg such that ad the lt of fuzzy S L + Theore 3 The relato betwee the fuzzy retracto of ad the lt of fuzzy foldg dscussed fro the followg coutatve dagra Corollary 4 The fuzzy retracto whch preserves the deso of S L + s a type of the fuzzy foldg Corollary 5 The fuzzy retracto whch decreases the deso of S L + s a lt of the fuzzy foldg of S L + Theore 4 The fuzzy foldg of fuzzy sphere S L are dffeoorphc to fuzzy foldg of fuzzy ope ball D (-)- desoal Eucldea space Proof Cosder a fuzzy sphere S L Its fuzzfcato S S Let f S S be the sofuzzy foldg Uder the dffeoorphs γ betwee L ad R there exsts a fuzzy ope ball D a fuzzy (-) desoal Eucldea space R Its fuzzfcato D D ad there s a duced sofuzzsy foldg g f D D Corollary 6 Let S S D D be the upper ad lower fuzzfcato of fuzzy sphere S ad fuzzy ope ball D If g f D D j be a topologcal fuzzy foldg the there s duced topologcal fuzzy foldg g f D D j 3 Applcato -The strea fucto of the acoustc gravty trpolar vortces s geeralzed to pert a study of the Earth s atosphere uder coplex eteorologcal codtos characterzed by sheared horzotal flows ad parabolc desty ad pressure profles [7] (Fgure ) -Cosder the flow of the flud sde a tube [3] If we represet the velocty of the flud as a ebershp degree µœ []the µ = the d of the edu where the velocty of the flud takes a axu ad s syetrc roud ths le but at the edge of the tube the velocty of the flud vashes e µ = 3-The Rtz varatoal ethod [6] durg the calculato of the groud state eergy a fuzzy fraework Cosder a Halto H ad a arbtrary square tegrable fucto Ψ so that <Ψ/Ψ> = Cosderg Ψ as a fuzzy fucto ad the rakg syste as defed [] slar to [6] t ca be show that <Ψ/H/Ψ> s a fuzzy upper boud o E ( groud- stat eergy) Now <Ψ/H/Ψ> should be zg the dstace betwee E ad respect to a uber of paraeters (α α ) Ths ca be doe by zg dstace betwee E ad <Ψ/H/Ψ> The rest of the dscusso s the sae as that provded [6] 4 Cocluso I the preset paper we obta ad study soe types of fuzzy retractos of fuzzy ope sphere fuzzy Lobachevsky space Also we deduced soe types of fuzzy deforato retract of fuzzy ope sphere The relatos betwee the fuzzy Fgure Applcato of fuzzy afold wwwdjstorg Vol 6 (4) Aprl 3 Ida Joural of Scece ad Techology Prt ISSN Ole ISSN

6 448 O Fuzzy Spheres Fuzzy Lobachevsky Space ad ts Retractos foldg ad the fuzzy deforato retracts of fuzzy ope sphere are obtaed Soe applcatos are preseted 5 Ackowledgeets The author s deeply debted to the tea work at the deashp of the scetfc research Tabah Uversty for ther valuable help ad crtcal gudace ad for facltatg ay adstratve procedures Ths research work was faced supported by Grat o 366/434 fro the deashp of the scetfc research at Tabah Uversty Al- Madah Al- Muawwarah Saud Araba 6 Refereces El- Ahady A E (7) The varato of the desty o chaotc spheres chaotc space- lke Mkowsk space te Chaos Soltos & Fractals vol 3(5) 7 78 El- Ahady A E (7) Foldg of fuzzy hypertor ad ther retractos Proceedgs of the Matheatcal ad Physcal Socety of Egypt vol 85() 3 El- Ahady A E (6) Lts of fuzzy retractos of fuzzy hyperspheres ad ther foldgs Takag Joural of Matheatcs vol 37() El- Ahady A E (Accepted) Fuzzy elastc Kle bottle ad ts retractos Iteratoal Joural of Appled Matheatcs ad Statstcs 5 El- Ahady A E () Foldg ad ufoldg of chaotc spheres chaotc space- lke Mkowsk space- te The Scetfc Joural of Appled Research vol () El- Ahady A E ad El- Araby A () O fuzzy spheres fuzzy Mkowsk space Nuovo Ceto vol 5B() El- Ahady A E () Retracto of chaotc black hole The Joural of Fuzzy Matheatcs vol 9(4) El- Ahady A E () Retracto of ull helx Mkowsk 3-space The Scetfc Joural of Appled Research vol () El- Ahady A E (4) Fuzzy Lobachevska space ad ts foldg The Joural of Fuzzy Matheatcs vol () El- Ahady A E (4) Fuzzy foldg of fuzzy horocycle Crcolo Mateatco d Palero Sere II Too L III El- Ahady A E ad Shaara H M () Fuzzy deforato retracts of fuzzy horospheres Ida Joural of Pure ad Appled Matheatcs vol 3() 5 56 El- Ahady A E ad Al- Luhayb A S (3) Fuzzy retractos of fuzzy ope flat Robertso- Walker space Advaces Fuzzy Systes vol 3 3 El- Ahady A E ad Al- Oe K (3) O the foldg of Lobachevsky space Iteratoal Joural of Appled Matheatcs ad Statstcs vol 4() El- Ahady A E () The geodesc deforato retract of Kle bottle ad ts foldg The Iteratoal Joural of Nolear Scece vol 9(3) 8 5 El- Ahady A E (3) O the fudaetal group ad foldg of Kle bottle Iteratoal Joural of Appled Matheatcs ad Statstcs vol 37(6) El- Ahady A E (3) Foldg ad fudaetal groups of Buchdah space Ida Joural of Scece ad Techology vol 6() El- Ahady A E ad Al- Haz N (3) Foldgs ad deforato retractos of hypercylder Ida Joural of Scece ad Techology vol 6() El- Ahady A E ad Al- Luhayb A S (Accepted) O fuzzy retracts of fuzzy closed flat Robersto- Walker spaces Advaces fuzzy sets ad systes 9 El- Ahady A E ad Al- Luhayb A S (3) Retractos of fuzzy flat Robertso- Walker space Iteratoal Joural of Appled Matheatcs ad Statstcs vol 4() 6 9 Schekal B () Prordal Space The Metrc Case Chapter New York Naber G L () Topology Geoetry ad Gauge felds Chapter Sprger- Verlage New York Berl Hartle J B (3) Gravty A troducto to Este s geeral relatvty Chapter 3 Addso- Wesley New York 3 Grffths J B ad Podolsky J I (9) Exact Space- Tes Este s Geeral Relatvty Cabrdge Uversty Press Cabrdge New York 4 Zadeh L A (975) Fuzzy sets ad ther applcato to cogtve ad decso Chapter New York Acadec Press 5 Red M ad Szedro B (5) Topology ad Geoetry Chapter Cabrdge Uversty Press Cabrdge New York 6 Arkowtz M () Itroducto to hootopy theory Chapter Sprger- Verlage New York 7 Palaappa N () Fuzzy Topology Chapter Lodo New York 8 Straua N (4) Geeral relatvty wth applcato to astrophyscs Chapter 4 Sprger- Verlage New York Hedelberg Berl 9 Stro J () Moder classcal hootopy theory Chapter Aerca Matheatcal Socety 3 Kudryashov V V Kurochk Y A () Moto caused by agetc feld Lobachevsky space arxv 6 5vI [ath- ph] vol 5 8 wwwdjstorg Vol 6 (4) Aprl 3 Ida Joural of Scece ad Techology Prt ISSN Ole ISSN