Strong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity
|
|
- Jeffrey Newton
- 5 years ago
- Views:
Transcription
1 BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sc. Soc. () 7 (004), 5 35 Strog Covergece of Weghted Averaged Appromats of Asymptotcally Noepasve Mappgs Baach Spaces wthout Uform Covety D.R. SAHU, J.S. JUNG AND 3 R.K. VERMA Departmet of Appled Mathematcs, Shr Shakaracharya College of Egeerg ad Techology, Juwa, Dstt. Durg (C.G.) , Ida Departmet of Mathematcs, Dog A Uversty, Busa , Korea 3 Departmet of Mathematcs, Govt. Scece College, Pata, Pata Dstt. Durg (C.G.), Ida e-mal: sahudr@redffmal.com ad jugjs@mal.doga.ac.kr Abstract. Let C be a oempty closed cove subset of a refleve Baach space whose orm s uformly Gâteau dfferetable, T : C C a asymptotcally oepasve mappg ad P the suy oepasve retracto from C oto F (T ). I the paper, we troduce property (S) for mappg T as mmal codto for strog covergece to P of the sequece { } defed by ( ) T, N 0 +, where { } ad { } (,, ) are real sequeces satsfyg approprate codtos ad N 0 s suffcetly large atural umber. 000 Mathematcs Subject Classfcato: 47H09, 47H0. Itroducto Let C be a oempty subset of a real Baach space E ad let T : C C be a olear mappg. The mappg T s sad to be asymptotcally oepasve f for each, there ests a postve costat k wth lm k such that T T y k y for all, y C. T s oepasve f k for,,.
2 6 D.R. Sahu et al. Let C be a closed cove subset of E, T : C C be a oepasve mappg such that the set F (T ) of fed pots of T s oempty ad be a elemet of C. Browder [] proved that { t } defed by t ( t) T, 0 < < t + t t coverges strogly to a elemet of F (T ) whch s earest to F (T ) as t 0 case whe E s a Hlbert space. Rech [7] eteded Browder s result the framework of a uformly smooth Baach space. Lm ad Xu [6] partally eteded celebrated covergece theorem of Rech [7] for asymptotcally oepasve mappgs same framework. Recetly, usg a dea of Browder [], Shmzu ad Takahash [8] studed the covergece of the followg appromated sequece for a asymptotcally oepasve mappg the framework of a Hlbert space: a + T,,,, ( a) where { a } s a real sequece satsfyg 0 < a < ad a 0. Shoj ad Takahash [9] ad Jug et al. [5] eteded ther results uformly cove Baach spaces. I ths paper, we study estece ad strog covergece of the sequece { } defed by ( ) + a T,,,, () where T s a asymptotcally oepasve mappg o a oempty closed cove subset of a refleve Baach space whose orm s uformly Gâteau dfferetable. Our results mprove prevous kow results [5, 6, 7, 8, 9].. Prelmares ad Lemmas Let E be a Baach space ad E * be the dual space of E. The value of y E at E wll be deoted by, y. We also deote by J, the dualty mappg from E to E *, that s, { y E :, y, y } J ( ) * for each E.
3 Strog Covergece of Weghted Averaged Appromats 7 Recall that a Baach space E s sad to be smooth provded the lmt lm t 0 + ty t ests for each ad y S { E : }. I ths case, the orm of E s sad to be Gâteau dfferetable. It s sad to be uformly Gâteau dfferetable f for each y S, ths lmt s attaed uformly for S. The orm s sad to be Fréchet dfferetable f for each S, ths lmt s attaed uformly for y S. Fally, the orm s sad to be uformly Fréchet dfferetable f the lmt s attaed uformly for (, y) S S. I ths case E s sad to be uformly smooth. Sce the dual E * of E s uformly cove f ad oly f the orm of E s uformly Fréchet dfferetable, every Baach space wth a uformly cove dual s refleve ad has a uformly Gâteau dfferetable orm. Let C be a oempty closed cove subset of a Baach space E ad T : C C be a asymptotcally oepasve mappg. Wthout loss of geeralty, we may assume that k for all,,. Set A : a T, C,,,, where { a } (,, ) are sequeces of real umbers such that () a 0 for all,,, () a, () a k β for all,, ad lm β. Now, for a C ad a postve teger, cosder a mappg T o C defed by T u ( ) A u, u C, + () where { } s a sequece of real umbers such that β 0 <, lm 0, ad lm sup <.
4 8 D.R. Sahu et al. Lemma. Let C be a oempty closed cove subset of a Baach space E ad T : C C be a asymptotcally oepasve mappg. Let T be a mappg defed by (). The T has eactly oe fed pot C such that ( ) A + for all, (3) N 0 where N 0 s a suffcetly large atural umber. β < Proof. Sce lm sup, there ests a atural umber N 0 such that N, there ests the uque pot T defed by () satsfes u, v C ( ) β < for all N0. So, for each 0 C satsfyg + ( ) A, sce the mappg T u T v ( ) β u v for all. Lemma. Let C, T ad X be as Lemma. If F (T ) s oempty, the { } s bouded. Proof. Sce ( ) β < for all N0, for each ε > 0, there ests a atural umber 0 such that < ε for all ( ) 0. The for v F(T ), we have β mples for all 0. v + ( ) A v v ( ) β + v ( ) a T v ε v T v Let μ be a cotuous lear fuctoal o ad let a, a,, ). ( 0 a We wrte μ ( a ) stead of μ ( a0, a, a, ). We call μ a Baach lmt [] whe μ satsfes: μ μ ( ) ad μ a ) μ ( a ) ( + for all ( a 0, a, a, ). For a Baach lmt, μ we kow that lm f a μ ( a ) lm sup a for all ( a 0, a, a, ).
5 Strog Covergece of Weghted Averaged Appromats 9 Let { } be a bouded sequece E. The we ca defe a real-valued cotuous cove fucto o E by f ( z) μ z for all z E. The followg lemma s gve [3, 4, 0]. Lemma 3 [3, 4, 0]. Let C be a oempty closed cove subset of a Baach space whose orm s uformly Gâteau dfferetable. Let { } be a bouded sequece C, u C ad μ be a Baach lmt. The f ad oly f f ( u) m f ( z) z C μ z u, J( u) 0 for all z C. Let C be a cove subset of E, D a oempty subset of C ad P a retracto from C oto D, that s, P for all D. A retracto P s sad to be suy f P ( P + t( P) ) P for all C ad t 0. D s sad to be a suy oepasve retract of C f there ests a suy oepasve retracto of C oto D. For more detals, see [3]. The followg lemma s well kow (cf. [3]). Lemma 4 [3]. Let C be a cove subset of a smooth Baach space, D be a oempty subset of C ad P be a retracto form C oto D. The P s suy ad oepasve f P, J( z P) 0 for all C, z D. Lemma 5. Let C, T ad X be as Lemma. The β z for all z F(T ), where lm sup <. μ, J( z) μ Proof. Sce from (3) ( ) ( ) ( A ), we get for z F(T ) ad for each N0,
6 30 D.R. Sahu et al., J ( z) A, J ( z) A A z + z, J ( z) ( β ) z whch gves β z μ, J( z) μ z. Fally, we troduce the followg mmal property for covergece of sequece { } defed by (3): Defto. Let C be a oempty closed cove subset of a Baach space E ad T : C C be a mappg. The T s sad to satsfy the property (S) f the followg holds: for each bouded sequece } C, { lm T 0 mples M F(T ) φ. (S) where M { u C : f ( u) f f ( z) }. z C The eamples of such mappgs that satsfy the property (S) are oepasve mappgs uformly smooth Baach space (see [7]). 3. Ma results Theorem. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k whch satsfes the property (S), ad P be the suy oepasve retracto form C oto F (T ). Let { a } (,, } be real sequeces satsfyg: a 0, a ad ak β,,,.
7 Let { } be a real sequece such that Strog Covergece of Weghted Averaged Appromats 3 β 0, lm 0 ad lm sup <. The (a) for ay C, there s eactly oe C such that + ( ) a T for all, N 0 where N 0 s a suffcetly large atural umber. (b) If { } s a appromate fed pot sequece for T,.e., lm T 0 follows that { } coverges strogly to P. Proof. (a) The result follows from Lemma. (b) From Lemma, t follows that { } s bouded. Defe a real-valued fucto o E by, t f ( z) μ z for all z E. The, sce f s cotuous ad cove, f (z) as z ad E s refleve, f attas ts fmum over C. Let u C such that f ( u) f z C f ( z). The M { u C : f ( u) f z C f ( z)} s oempty because u M. Sce { } s bouded ad lm T 0, by property (S), T has a fed pot M. Deote such a pot y. It follows from Lemma 3 that μ y, J( y) 0. Ths equalty ad Lemma 5 yelds μ y μ y, that s, μ y 0.
8 3 D.R. Sahu et al. Therefore, there s a subsequece { } of } whch coverges strogly to y F(T ). The, by Lemma 5, we have j { y, J( y P) y P. Ths equalty ad Lemma 4 yeld y P y P. From <, we have y P. To complete the proof, let } be aother { k subsequece of { } whch coverges strogly to z. We shall show that y z. Sce y P, t follows from Lemma 4 ad 5 that P, J ( z P) 0 ad z, J ( z P) z P, ad hece we have y z z y, J( z P) z, J ( z P) + P, J ( z P) y z. Thus, { } coverges strogly to P. Recall that a oempty subset D of C s sad to satsfy the Property (P) (cf. [6]) f the followg holds: where ω ω () s the weak ω -lmt set of T, that s, the set D mples ω ( ) D, (P) { y C : y weak lm T for some }. The followg lemma s crucal to prove our et ma result. Lemma 6. Let E be a refleve Baach space whose orm s uformly Gâteau dfferetable, C be a oempty closed cove subset of E ad T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k. Let { } be a bouded sequece wth lm T 0. Assume that every oempty weakly ω
9 Strog Covergece of Weghted Averaged Appromats 33 compact cove subset of C satsfyg the property (P) has a fed pot for T. The T satsfes property (S). Proof. Note that M { u C : f ( u) f f ( z) } z C s oempty closed cove bouded subset of C. Although M s ot ecessarly varat uder T, t does have the j property (P). I fact, f u M ad y weak lm T u belogs to the weak j ω -lmt set ω ω (u) of T at u, the from weak lower semcotuty of f ad lm T 0, we have j f ( y) lm f f ( T u) lm sup f ( T u) j lm sup lm sup k μ ( μ T u ) lm sup ( T T m m μm m u ) m m u μ m m u f z C f ( z). Thus, y M ad hece M satsfes the property (P). It follows from assumpto that T has a fed pot M. Therefore, T has satsfes property (S). Theorem. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k ad P be the suy oepasve retracto from C oto F (T). Let a } (,, ) be real sequeces satsfyg: { 0, ad a a Let { } be a real sequece such that a k β,,,. β 0, 0 ad lm sup <, ad let { } be a sequece C defed by (3) such that lm T 0. Assume that every oempty weakly compact cove subset of C satsfyg the property (P) has a fed pot for T. The } coverges strogly to P. { Proof. The result follows from Lemma 6 ad Theorem.
10 34 D.R. Sahu et al. I the case whe a for,,, Theorem ad, we have the followg corollares. Corollary. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k whch satsfes the property (S), ad P be the suy oepasve retracto from C oto F (T). Let { a } be a real sequece such that β 0, lm 0 ad lm sup <, where β k. The (a) For ay C, there s eactly oe C such that + ( ) T for all N 0, (4) where N 0 s a suffcetly large atural umber. (b) If T 0 as, t follows that } coverges strogly to P. Corollary. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k ad P be the suy oepasve retracto from C oto F (T ). Let } be a real sequece such that { { β 0, 0 ad lm sup <, where β k. ad let { } be a sequece C defed by (4) such that lm T 0. Assume that every oempty weakly compact cove subset of C satsfyg the property (P) has a fed pot for T. The } coverges strogly to P. { I the case whe a for all ad a 0 otherwse, we have the followg: Corollary 3. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k whch satsfes the property (S), ad
11 Strog Covergece of Weghted Averaged Appromats 35 P be the suy oepasve retracto from C oto F (T ). Let { } be a real sequece ( 0, ) such that ad let { } be a sequece C defed by k lm 0 ad lm sup < + ( ) T, N. 0 Suppose addto that lm T 0. The } coverges strogly to P. Remark. () Corollary ad eted Theorem of [5] ad Theorem of [9] from uformly cove Baach space to refleve Baach space, respectvely. () Corollary 3 mproves Theorem of Lm ad Xu [6], where the space s assumed to be uformly smooth. Ackowledgemet. The frst author wshes to ackowledge the facal support of Departmet of Scece ad Techology, Ida, made the program year , Project No. SR/FTP/MS 5/000. { Refereces. S. Baach, Theore des Operatos Leares, Moograph Mat., PWN, Warsazawa, 93.. F.E. Browder, Covergece of appromats to fed pots of o-epasve olear mappgs Baach spaces, Arch. Rato. Mech. Aal. 4 (967), R.E. Bruck ad S. Rech, Accretve operators, Baach lmts, ad dual ergodc theorems, Bull. Acad. Polo. Sc. 9 (98), K.S. Ha ad J.S. Jug, Strog covergece theorems for accretve operators Baach spaces, J. Math. Aal. Appl. 47 (990), J.S. Jug, D.R. Sahu ad B.S. Thakur, Strog covergece theorems for asymptotcally oepasve mappgs Baach spaces, Comm. Appl. Nolear Aal. 5 (998), T.C. Lm ad H.K. Xu, Fed pot theorems for asymptotcally oepasve mappgs, Nolear Aal. (994), S. Rech, Strog covergece theorems for resolvets of accretve operators Baach spaces, J. Math. Aal. Appl. 75 (980), T. Shmzu ad W. Takahash, Strog covergece theorems for asymptotcally oepasve mappgs, Nolear Aal. 6 (996), N. Shoj ad W. Takahash, Strog covergece of averaged appromats for oepasve mappgs Baach spaces, J. Appro. Theory 97 (999), W. Takahash ad Y. Ueda, O Rech s strog covergece theorems for resolvets of accretve operators, J. Math. Aal. Appl. 04 (984), Keywords ad phrases: Fed pot, asymptotcally oepasve mappg, uformly Gâteau dfferetable orm.
Research Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings
Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Volume 009, Artcle ID 391839, 9 pages do:10.1155/009/391839 Research Artcle A New Iteratve Method for Commo Fxed Pots of a Fte
More informationComplete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables
Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl
More informationA Remark on the Uniform Convergence of Some Sequences of Functions
Advaces Pure Mathematcs 05 5 57-533 Publshed Ole July 05 ScRes. http://www.scrp.org/joural/apm http://dx.do.org/0.436/apm.05.59048 A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut
More informationAbstract. 1. Introduction
Joura of Mathematca Sceces: Advaces ad Appcatos Voume 4 umber 2 2 Pages 33-34 COVERGECE OF HE PROJECO YPE SHKAWA ERAO PROCESS WH ERRORS FOR A FE FAMY OF OSEF -ASYMPOCAY QUAS-OEXPASVE MAPPGS HUA QU ad S-SHEG
More informationQ-analogue of a Linear Transformation Preserving Log-concavity
Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, 87-94 Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua 362021, P. R. Cha ldzblue@163.com
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233-238 Etropy ISSN 1099-4300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationThe Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More information= lim. (x 1 x 2... x n ) 1 n. = log. x i. = M, n
.. Soluto of Problem. M s obvously cotuous o ], [ ad ], [. Observe that M x,..., x ) M x,..., x ) )..) We ext show that M s odecreasg o ], [. Of course.) mles that M s odecreasg o ], [ as well. To show
More informationExtend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables
ppled Mathematcal Sceces, Vol 4, 00, o 3, 637-64 xted the Borel-Catell Lemma to Sequeces of No-Idepedet Radom Varables olah Der Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira der53@gmalcom
More informationThe Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012
Sc. Math. Japocae Vol. 00, No. 0 0000, 000 000 1 The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally
More informationTHE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION
Joural of Scece ad Arts Year 12, No. 3(2), pp. 297-32, 212 ORIGINAL AER THE ROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION DOREL MIHET 1, CLAUDIA ZAHARIA 1 Mauscrpt receved: 3.6.212; Accepted
More informationInternational Journal of Mathematical Archive-5(8), 2014, Available online through ISSN
Iteratoal Joural of Mathematcal Archve-5(8) 204 25-29 Avalable ole through www.jma.fo ISSN 2229 5046 COMMON FIXED POINT OF GENERALIZED CONTRACTION MAPPING IN FUZZY METRIC SPACES Hamd Mottagh Golsha* ad
More informationOn Submanifolds of an Almost r-paracontact Riemannian Manifold Endowed with a Quarter Symmetric Metric Connection
Theoretcal Mathematcs & Applcatos vol. 4 o. 4 04-7 ISS: 79-9687 prt 79-9709 ole Scepress Ltd 04 O Submafolds of a Almost r-paracotact emaa Mafold Edowed wth a Quarter Symmetrc Metrc Coecto Mob Ahmad Abdullah.
More informationOn L- Fuzzy Sets. T. Rama Rao, Ch. Prabhakara Rao, Dawit Solomon And Derso Abeje.
Iteratoal Joural of Fuzzy Mathematcs ad Systems. ISSN 2248-9940 Volume 3, Number 5 (2013), pp. 375-379 Research Ida Publcatos http://www.rpublcato.com O L- Fuzzy Sets T. Rama Rao, Ch. Prabhakara Rao, Dawt
More informationUnique Common Fixed Point of Sequences of Mappings in G-Metric Space M. Akram *, Nosheen
Vol No : Joural of Facult of Egeerg & echolog JFE Pages 9- Uque Coo Fed Pot of Sequeces of Mags -Metrc Sace M. Ara * Noshee * Deartet of Matheatcs C Uverst Lahore Pasta. Eal: ara7@ahoo.co Deartet of Matheatcs
More informationSTRONG CONSISTENCY FOR SIMPLE LINEAR EV MODEL WITH v/ -MIXING
Joural of tatstcs: Advaces Theory ad Alcatos Volume 5, Number, 6, Pages 3- Avalable at htt://scetfcadvaces.co. DOI: htt://d.do.org/.864/jsata_7678 TRONG CONITENCY FOR IMPLE LINEAR EV MODEL WITH v/ -MIXING
More information. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1)
Chapter 7 Fuctos o Bouded Varato. Subject: Real Aalyss Level: M.Sc. Source: Syed Gul Shah (Charma, Departmet o Mathematcs, US Sargodha Collected & Composed by: Atq ur Rehma (atq@mathcty.org, http://www.mathcty.org
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationResearch Article Some Strong Limit Theorems for Weighted Product Sums of ρ-mixing Sequences of Random Variables
Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 2009, Artcle ID 174768, 10 pages do:10.1155/2009/174768 Research Artcle Some Strog Lmt Theorems for Weghted Product Sums of ρ-mxg Sequeces
More informationComplete Convergence for Weighted Sums of Arrays of Rowwise Asymptotically Almost Negative Associated Random Variables
A^VÇÚO 1 32 ò 1 5 Ï 2016 c 10 Chese Joural of Appled Probablty ad Statstcs Oct., 2016, Vol. 32, No. 5, pp. 489-498 do: 10.3969/j.ss.1001-4268.2016.05.005 Complete Covergece for Weghted Sums of Arrays of
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationMarcinkiewicz strong laws for linear statistics of ρ -mixing sequences of random variables
Aas da Academa Braslera de Cêcas 2006 784: 65-62 Aals of the Brazla Academy of Sceces ISSN 000-3765 www.scelo.br/aabc Marckewcz strog laws for lear statstcs of ρ -mxg sequeces of radom varables GUANG-HUI
More informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationThe Primitive Idempotents in
Iteratoal Joural of Algebra, Vol, 00, o 5, 3 - The Prmtve Idempotets FC - I Kulvr gh Departmet of Mathematcs, H College r Jwa Nagar (rsa)-5075, Ida kulvrsheora@yahoocom K Arora Departmet of Mathematcs,
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More informationA Study on Generalized Generalized Quasi hyperbolic Kac Moody algebra QHGGH of rank 10
Global Joural of Mathematcal Sceces: Theory ad Practcal. ISSN 974-3 Volume 9, Number 3 (7), pp. 43-4 Iteratoal Research Publcato House http://www.rphouse.com A Study o Geeralzed Geeralzed Quas (9) hyperbolc
More informationAlmost Sure Convergence of Pair-wise NQD Random Sequence
www.ccseet.org/mas Moder Appled Scece Vol. 4 o. ; December 00 Almost Sure Covergece of Par-wse QD Radom Sequece Yachu Wu College of Scece Gul Uversty of Techology Gul 54004 Cha Tel: 86-37-377-6466 E-mal:
More informationChapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
More informationAitken delta-squared generalized Juncgk-type iterative procedure
Atke delta-squared geeralzed Jucgk-type teratve procedure M. De la Se Isttute of Research ad Developmet of Processes. Uversty of Basque Coutry Campus of Leoa (Bzkaa) PO Box. 644- Blbao, 488- Blbao. SPAIN
More informationON THE LAWS OF LARGE NUMBERS FOR DEPENDENT RANDOM VARIABLES
Joural of Sees Islam Republ of Ira 4(3): 7-75 (003) Uversty of Tehra ISSN 06-04 ON THE LAWS OF LARGE NUMBERS FOR DEPENDENT RANDOM VARIABLES HR Nl Sa * ad A Bozorga Departmet of Mathemats Brjad Uversty
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationA tighter lower bound on the circuit size of the hardest Boolean functions
Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the
More informationAsymptotic Formulas Composite Numbers II
Iteratoal Matematcal Forum, Vol. 8, 203, o. 34, 65-662 HIKARI Ltd, www.m-kar.com ttp://d.do.org/0.2988/mf.203.3854 Asymptotc Formulas Composte Numbers II Rafael Jakmczuk Dvsó Matemátca, Uversdad Nacoal
More informationA Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming
Aerca Joural of Operatos Research, 4, 4, 33-339 Publshed Ole Noveber 4 ScRes http://wwwscrporg/oural/aor http://ddoorg/436/aor4463 A Pealty Fucto Algorth wth Obectve Paraeters ad Costrat Pealty Paraeter
More informationSUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE. Sayali S. Joshi
Faculty of Sceces ad Matheatcs, Uversty of Nš, Serba Avalable at: http://wwwpfacyu/float Float 3:3 (009), 303 309 DOI:098/FIL0903303J SUBCLASS OF ARMONIC UNIVALENT FUNCTIONS ASSOCIATED WIT SALAGEAN DERIVATIVE
More informationCS286.2 Lecture 4: Dinur s Proof of the PCP Theorem
CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat
More informationTHE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/2008, pp
THE PUBLISHIN HOUSE PROCEEDINS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/8, THE UNITS IN Stela Corelu ANDRONESCU Uversty of Pteşt, Deartmet of Mathematcs, Târgu Vale
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationBounds for the Connective Eccentric Index
It. J. Cotemp. Math. Sceces, Vol. 7, 0, o. 44, 6-66 Bouds for the Coectve Eccetrc Idex Nlaja De Departmet of Basc Scece, Humates ad Socal Scece (Mathematcs Calcutta Isttute of Egeerg ad Maagemet Kolkata,
More informationON THE LOGARITHMIC INTEGRAL
Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), 393 41 ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa Jolevska-Tueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x)
More informationLebesgue Measure of Generalized Cantor Set
Aals of Pure ad Appled Mathematcs Vol., No.,, -8 ISSN: -8X P), -888ole) Publshed o 8 May www.researchmathsc.org Aals of Lebesgue Measure of Geeralzed ator Set Md. Jahurul Islam ad Md. Shahdul Islam Departmet
More informationHájek-Rényi Type Inequalities and Strong Law of Large Numbers for NOD Sequences
Appl Math If Sc 7, No 6, 59-53 03 59 Appled Matheatcs & Iforato Sceces A Iteratoal Joural http://dxdoorg/0785/as/070647 Háje-Réy Type Iequaltes ad Strog Law of Large Nuers for NOD Sequeces Ma Sogl Departet
More informationA Strong Convergence Theorem for a Proximal-Type. Algorithm in Re exive Banach Spaces
A Strog Covergece Theorem for a Proxmal-Type Algorthm Re exve Baach Spaces Smeo Rech ad Shoham Sabach Abstract. We establsh a strog covergece theorem for a proxmal-type algorthm whch approxmates (commo)
More informationSTRONG CONSISTENCY OF LEAST SQUARES ESTIMATE IN MULTIPLE REGRESSION WHEN THE ERROR VARIANCE IS INFINITE
Statstca Sca 9(1999), 289-296 STRONG CONSISTENCY OF LEAST SQUARES ESTIMATE IN MULTIPLE REGRESSION WHEN THE ERROR VARIANCE IS INFINITE J Mgzhog ad Che Xru GuZhou Natoal College ad Graduate School, Chese
More informationUniform asymptotical stability of almost periodic solution of a discrete multispecies Lotka-Volterra competition system
Iteratoal Joural of Egeerg ad Advaced Research Techology (IJEART) ISSN: 2454-9290, Volume-2, Issue-1, Jauary 2016 Uform asymptotcal stablty of almost perodc soluto of a dscrete multspeces Lotka-Volterra
More informationExtreme Value Theory: An Introduction
(correcto d Extreme Value Theory: A Itroducto by Laures de Haa ad Aa Ferrera Wth ths webpage the authors ted to form the readers of errors or mstakes foud the book after publcato. We also gve extesos for
More informationOn the convergence of derivatives of Bernstein approximation
O the covergece of dervatves of Berste approxmato Mchael S. Floater Abstract: By dfferetatg a remader formula of Stacu, we derve both a error boud ad a asymptotc formula for the dervatves of Berste approxmato.
More information1. A real number x is represented approximately by , and we are told that the relative error is 0.1 %. What is x? Note: There are two answers.
PROBLEMS A real umber s represeted appromately by 63, ad we are told that the relatve error s % What s? Note: There are two aswers Ht : Recall that % relatve error s What s the relatve error volved roudg
More informationOn Eccentricity Sum Eigenvalue and Eccentricity Sum Energy of a Graph
Aals of Pure ad Appled Mathematcs Vol. 3, No., 7, -3 ISSN: 79-87X (P, 79-888(ole Publshed o 3 March 7 www.researchmathsc.org DOI: http://dx.do.org/.7/apam.3a Aals of O Eccetrcty Sum Egealue ad Eccetrcty
More information18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
More informationLecture 12 APPROXIMATION OF FIRST ORDER DERIVATIVES
FDM: Appromato of Frst Order Dervatves Lecture APPROXIMATION OF FIRST ORDER DERIVATIVES. INTRODUCTION Covectve term coservato equatos volve frst order dervatves. The smplest possble approach for dscretzato
More informationMATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More informationLarge and Moderate Deviation Principles for Kernel Distribution Estimator
Iteratoal Mathematcal Forum, Vol. 9, 2014, o. 18, 871-890 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/mf.2014.4488 Large ad Moderate Devato Prcples for Kerel Dstrbuto Estmator Yousr Slaou Uversté
More informationONE GENERALIZED INEQUALITY FOR CONVEX FUNCTIONS ON THE TRIANGLE
Joural of Pure ad Appled Mathematcs: Advaces ad Applcatos Volume 4 Number 205 Pages 77-87 Avalable at http://scetfcadvaces.co. DOI: http://.do.org/0.8642/jpamaa_7002534 ONE GENERALIZED INEQUALITY FOR CONVEX
More informationCSE 5526: Introduction to Neural Networks Linear Regression
CSE 556: Itroducto to Neural Netorks Lear Regresso Part II 1 Problem statemet Part II Problem statemet Part II 3 Lear regresso th oe varable Gve a set of N pars of data , appromate d by a lear fucto
More informationJournal of Mathematical Analysis and Applications
J. Math. Aal. Appl. 365 200) 358 362 Cotets lsts avalable at SceceDrect Joural of Mathematcal Aalyss ad Applcatos www.elsever.com/locate/maa Asymptotc behavor of termedate pots the dfferetal mea value
More informationSolution of General Dual Fuzzy Linear Systems. Using ABS Algorithm
Appled Mathematcal Sceces, Vol 6, 0, o 4, 63-7 Soluto of Geeral Dual Fuzzy Lear Systems Usg ABS Algorthm M A Farborz Aragh * ad M M ossezadeh Departmet of Mathematcs, Islamc Azad Uversty Cetral ehra Brach,
More informationLyapunov Stability. Aleksandr Mikhailovich Lyapunov [1] 1 Autonomous Systems. R into Nonlinear Systems in Mechanical Engineering Lesson 5
Joh vo Neuma Perre de Fermat Joseph Fourer etc 858 Nolear Systems Mechacal Egeerg Lesso 5 Aleksadr Mkhalovch Lyapuov [] Bor: 6 Jue 857 Yaroslavl Russa Ded: 3 November 98 Bega hs educato at home Graduated
More information1 0, x? x x. 1 Root finding. 1.1 Introduction. Solve[x^2-1 0,x] {{x -1},{x 1}} Plot[x^2-1,{x,-2,2}] 3
Adrew Powuk - http://www.powuk.com- Math 49 (Numercal Aalyss) Root fdg. Itroducto f ( ),?,? Solve[^-,] {{-},{}} Plot[^-,{,-,}] Cubc equato https://e.wkpeda.org/wk/cubc_fucto Quartc equato https://e.wkpeda.org/wk/quartc_fucto
More information( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model
Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch
More informationAbout k-perfect numbers
DOI: 0.47/auom-04-0005 A. Şt. Uv. Ovdus Costaţa Vol.,04, 45 50 About k-perfect umbers Mhály Becze Abstract ABSTRACT. I ths paper we preset some results about k-perfect umbers, ad geeralze two equaltes
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationPacking of graphs with small product of sizes
Joural of Combatoral Theory, Seres B 98 (008) 4 45 www.elsever.com/locate/jctb Note Packg of graphs wth small product of szes Alexadr V. Kostochka a,b,,gexyu c, a Departmet of Mathematcs, Uversty of Illos,
More informationCHAPTER 3 COMMON FIXED POINT THEOREMS IN GENERALIZED SPACES
CHAPTER 3 COMMON FIXED POINT THEOREMS IN GENERAIZED SPACES 3. INTRODUCTION After the celebrated Baach cotracto prcple (BCP) 9, there have bee umerous results the lterature dealg wth mappgs satsfyg the
More informationExchangeable Sequences, Laws of Large Numbers, and the Mortgage Crisis.
Exchageable Sequeces, Laws of Large Numbers, ad the Mortgage Crss. Myug Joo Sog Advsor: Prof. Ja Madel May 2009 Itroducto The law of large umbers for..d. sequece gves covergece of sample meas to a costat,.e.,
More informationMahmud Masri. When X is a Banach algebra we show that the multipliers M ( L (,
O Multlers of Orlcz Saces حول مضاعفات فضاءات ا ورلكس Mahmud Masr Mathematcs Deartmet,. A-Najah Natoal Uversty, Nablus, Paleste Receved: (9/10/000), Acceted: (7/5/001) Abstract Let (, M, ) be a fte ostve
More informationDimensionality Reduction and Learning
CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that
More informationarxiv: v1 [cs.lg] 22 Feb 2015
SDCA wthout Dualty Sha Shalev-Shwartz arxv:50.0677v cs.lg Feb 05 Abstract Stochastc Dual Coordate Ascet s a popular method for solvg regularzed loss mmzato for the case of covex losses. I ths paper we
More informationCHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
More information3. Basic Concepts: Consequences and Properties
: 3. Basc Cocepts: Cosequeces ad Propertes Markku Jutt Overvew More advaced cosequeces ad propertes of the basc cocepts troduced the prevous lecture are derved. Source The materal s maly based o Sectos.6.8
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationThe Lie Algebra of Smooth Sections of a T-bundle
IST Iteratoal Joural of Egeerg Scece, Vol 7, No3-4, 6, Page 8-85 The Le Algera of Smooth Sectos of a T-udle Nadafhah ad H R Salm oghaddam Astract: I ths artcle, we geeralze the cocept of the Le algera
More informationUnimodality Tests for Global Optimization of Single Variable Functions Using Statistical Methods
Malaysa Umodalty Joural Tests of Mathematcal for Global Optmzato Sceces (): of 05 Sgle - 5 Varable (007) Fuctos Usg Statstcal Methods Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal
More information9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d
9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,
More informationFibonacci Identities as Binomial Sums
It. J. Cotemp. Math. Sceces, Vol. 7, 1, o. 38, 1871-1876 Fboacc Idettes as Bomal Sums Mohammad K. Azara Departmet of Mathematcs, Uversty of Evasvlle 18 Lcol Aveue, Evasvlle, IN 477, USA E-mal: azara@evasvlle.edu
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationUnsupervised Learning and Other Neural Networks
CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all
More informationChapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements
Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK
Far East Joural of Appled Mathematcs Volume, Number, 2008, Pages Ths paper s avalable ole at http://www.pphm.com 2008 Pushpa Publshg House ANALYSIS ON THE NATURE OF THE ASI EQUATIONS IN SYNERGETI INTER-REPRESENTATION
More informationA New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming
ppled Matheatcal Sceces Vol 008 o 50 7-80 New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research
More informationLINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
More informationIntroduction to Probability
Itroducto to Probablty Nader H Bshouty Departmet of Computer Scece Techo 32000 Israel e-mal: bshouty@cstechoacl 1 Combatorcs 11 Smple Rules I Combatorcs The rule of sum says that the umber of ways to choose
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationMaximum Walk Entropy Implies Walk Regularity
Maxmum Walk Etropy Imples Walk Regularty Eresto Estraa, a José. e la Peña Departmet of Mathematcs a Statstcs, Uversty of Strathclye, Glasgow G XH, U.K., CIMT, Guaajuato, Mexco BSTRCT: The oto of walk etropy
More informationOn Some Covering Properties of B-open sets
O Some Coverg Propertes of B-ope sets Belal k Narat Appled Sceces Prvate verst Amma-Jorda Astract I ths paper we troduce ad stud the cocepts of -ope set, -cotuous fuctos, the we also stud the cocepts of
More information#A27 INTEGERS 13 (2013) SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES
#A27 INTEGERS 3 (203) SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES Emrah Kılıç Mathematcs Departmet, TOBB Uversty of Ecoomcs ad Techology, Akara, Turkey eklc@etu.edu.tr Neşe Ömür Mathematcs Departmet,
More informationA Family of Non-Self Maps Satisfying i -Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces *
Advaces Pure Matheatcs 0 80-84 htt://dxdoorg/0436/a04036 Publshed Ole July 0 (htt://wwwscrporg/oural/a) A Faly of No-Self Mas Satsfyg -Cotractve Codto ad Havg Uque Coo Fxed Pot Metrcally Covex Saces *
More informationDouble Dominating Energy of Some Graphs
Iter. J. Fuzzy Mathematcal Archve Vol. 4, No., 04, -7 ISSN: 30 34 (P), 30 350 (ole) Publshed o 5 March 04 www.researchmathsc.org Iteratoal Joural of V.Kaladev ad G.Sharmla Dev P.G & Research Departmet
More informationMaps on Triangular Matrix Algebras
Maps o ragular Matrx lgebras HMED RMZI SOUROUR Departmet of Mathematcs ad Statstcs Uversty of Vctora Vctora, BC V8W 3P4 CND sourour@mathuvcca bstract We surveys results about somorphsms, Jorda somorphsms,
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationCIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights
CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:
More informationSpecial Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
More informationECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity
ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data
More informationE be a set of parameters. A pair FE, is called a soft. A and GB, over X is the soft set HC,, and GB, over X is the soft set HC,, where.
The Exteso of Sgular Homology o the Category of Soft Topologcal Spaces Sad Bayramov Leoard Mdzarshvl Cgdem Guduz (Aras) Departmet of Mathematcs Kafkas Uversty Kars 3600-Turkey Departmet of Mathematcs Georga
More information